IB Maths Studies / Applications: Last blog entries
http://www.thinkib.net/mathstudies/blog
InThinking IB Maths Studies / Applications: www.thinkib.net/mathstudies2019 InThinking Educational Consultants. All rights reserved.Exam Season
https://www.thinkib.net/mathstudies/blog/29144/exam-season
Sun, 14 Apr 2019 00:00:00 +0000<img alt="" src="files/mathstudies/images/Exams/photo.JPG" style="float: left; width: 150px; height: 150px;" />I just wanted to write a quick blog post to help people be aware about resources that are available to support students through the exam season! I always have mixed feelings about this time of year. On the one hand, it is often the moment when the nature of actvities becomes more mechanical and the focus is a little more on cramming and exam technique than I would choose. On the other hand, it is also the moment where - hopefully - everything can come together. The last bits of hard work where all these ideas make sense simultaneously. Students are stressed, but also often they are switched on and ultimately receptive. I'll be glad for them when the exams are done. In the mean time, lets make the most of the remaining available time! Good luck to all tecahers and students out there for the coming season.<hr /><div class="greenBg"><h2>Whats on the site?</h2><p>Here is a list of pages that you will want to make sure you know about.</p><div class="index-list"><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/16571/exam-revision-packs'" style="background-image: url('files/mathstudies/images/revision/rev.jpg')" title="Exam Revision Packs"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/16571/exam-revision-packs">Exam Revision Packs</a></h4><p class="body">Helping students to prepare for the exams!NEW - The Calculus pack has landed - This page is the home of a series of revision packs designed for the final stages of the course. We all run our courses differently,...</p></div></div><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/18811/revision-calendar'" style="background-image: url('files/mathstudies/images/Exams/RC1.jpg')" title="Revision Calendar"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/18811/revision-calendar">Revision Calendar</a></h4><p class="body">In many schools, IB students will be granted study leave in the weeks leading up to exams. Sometimes this time coincides with holidays depending where you are. Either way, this period can be make or break...</p></div></div><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/23302/exam-advice'" style="background-image: url('files/mathstudies/images/revision/exam-advice.001.jpeg')" title="Exam Advice"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/23302/exam-advice">Exam Advice</a></h4><p class="body">This page is a simple list of things that you can do and prepare for relating to exam technique. Small things that can make a big difference. We should all know that there are more things to life than...</p></div></div><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/22912/studyibnet'" style="background-image: url('files/mathstudies/images/sib/sib.jpg')" title="StudyIB.net"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/22912/studyibnet">StudyIB.net</a></h4><p class="body">Visit studyIB.net/mathstudies to find out moreI would like you all to know about this resource I have been developing for students. It came from a genuine desire to be able to give my own students a really...</p></div></div><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/9335/lucky-dip'" style="background-image: url('files/mathstudies/images/Everything Else/lucky dip 1.jpg')" title="Lucky Dip"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/9335/lucky-dip">Lucky Dip</a></h4><p class="body">The lucky dip idea is really quite a simple one, its easy to explain and its a good one for last minute checking. In short, each of the syllabus items are printed, copied and cut out into small bits....</p></div></div><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/2375/notes-files'" style="background-image: url('files/mathstudies/images/Everything Else/Notes.jpg')" title="Notes Files"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/2375/notes-files">Notes Files</a></h4><p class="body">As mentioned before, these files have arisen from looking for a solution to note taking in class. The result has become a useful revision tool. I did not want to spend time creative comprehensive notes...</p></div></div><div class="item"><div class="bg-cover cover-96" onclick="window.location='/mathstudies/page/2108/common-errors'" style="background-image: url('files/mathstudies/images/Everything Else/No.png')" title="Common Errors"></div><div class="content"><h4 class="head"><a href="/mathstudies/page/2108/common-errors">Common Errors</a></h4><p class="body">Common errors are both surprising and then not! There are bound to be things that are that much more difficult or that much more likely to be left out, but at the same time, if we know about them then...</p></div></div></div></div><hr /><br /><h2></h2>https://www.thinkib.net/mathstudies/blog/29144/exam-season#1555200000Compound Interest
https://www.thinkib.net/mathstudies/blog/28203/compound-interest
Sun, 23 Dec 2018 00:00:00 +0000]]>Compound Interesthttps://www.thinkib.net/cache/blog-thumbs/21/28203-1544968178-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/28203/compound-interest
<p><img src="https://www.thinkib.net/cache/blog-thumbs/21/28203-1544968178-thinkib.jpg" alt="Compound Interest" /><br /><br />So! I have been meaning to post something about this for a while. Please let me start with a disclaimer - I am only passing on what I have heard at workshops and am open to being corrected in the comments! In the unit on compound interest, the IB syllabus requires that students adjust the interest rate based on the compounding period. If, for example, a bank offers 4% pa compounded quarterly, then students would be expected to divide the interest rate by 12 (so it is a monthly interest rate) and then multiply the number of years by twelve to get to the number of compounding periods. Logical enough right? Well, apprently, that depends on where you are from. A couple of workshop participants fro, Denmark (and possibly others from Norway or Sweden) have enlightened me to the fact that where they are from, if a bank quoates an interest rate of 4%pa, this already considers that interest is being calculated twelve times a year. As such, the monthly interest rate is what ever it needs to be so that the investment will be 4% bigger at the end of the year. Effectively 4% simple interest on the year, provided no funds are addedd or taken away in between......</p><p>As required by the IB syllabus....</p><p><span class="math-tex">\(100\times { \left( 1+\frac { \frac { 4 }{ 12 } }{ 100 } \right) }^{ 12 }=\quad 104.07\quad (2dp)\\ An\quad increase\quad of\quad just\quad under\quad 4.1%\\ \\ Solving\quad the\quad following\quad equation\\ 100\times { \left( 1+\frac { \frac { r }{ 12 } }{ 100 } \right) }^{ 12 }=104,\quad gives\\ r=3.93\quad (2dp)\)</span></p><p>Now consdiering the Danish model....</p><p><span class="math-tex">\(Solving\quad the\quad following\quad equation\\ 100\times { \left( 1+\frac { \frac { r }{ 12 } }{ 100 } \right) }^{ 12 }=104,\quad gives\\ r=3.93\quad (2dp)\)</span></p><p>Now I never knew that! Interesting no? Please correct me or comment,</p><p>Thanks, Jim</p>https://www.thinkib.net/mathstudies/blog/28203/compound-interest#1545523200This Years projects 2018
https://www.thinkib.net/mathstudies/blog/27380/this-years-projects-2018
Sat, 15 Sep 2018 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/IA/map.jpg" style="float: left; width: 150px; height: 113px;" />A run down on this year's ideas</h2><p>I did this last year because I think that the more we hear about and share the 'ideas' behind internal assessments, the better we get at understanding the scope of possibility. At workshops I often challenge teachers to work with any given 'area of interest' and turn it in to a potential project so we can practice that process to help our students complete work on their own areas of interest. </p><hr /><h3>Reminders</h3><p>Partly for my own benefit, I am trying to remind myself of some key parts of this process of taking ideas andmoulding them in to projects that meet the criteria. Here are a few....</p><div class="greenBg" contenteditable="false"><div contenteditable="true"><ul><li>I must show students lots of projects or at least talk them through the ideas. It is important for them to see a range, including some that are not heavily statistical. </li><li>Try to help match the project with the student. If the student is targetting a top grade then the project must have potential to reach all the criteria. If they are targetting a 3 or a 4 then it might be ok to compromise on the potential a bit. Better that a student pursues their own area of interest and gets 12/20 ( a mark that supports their target) than is shoe horned in to something more complicated that they dont do very convincingly - especially if the latter then results in the same mark. The ofrmer would be a more positive experience.</li><li>Try and help students to have 'themes' for their projects rather than base it on a single specific process. The former tends to offer more potential.</li><li>Remember that searching for specific secondary data can difficult and disappointing. Often a search for useful secondary data on a given theme can give rise to an idea and that some students might do this in this order.</li><li>Note that many students cite this as the most enjoyable aspect of the course because of the choices they were allowed to make about what to do.</li></ul></div></div><hr /><br /><h3>Some of this year's ideas</h3><p>Here are a few of the projects I am going to be seeing this year</p><p>100 books to read before you die - This was the inspration and the student is going to be analysing data about those books to try and see what it takes to get on the list. Everything from good reads ratings, to publication date and sales figures are up for grans here.</p><p>La Liga - a football fan has a wealth of stats about a single La Liga (Spanish football league) season and will be looking for the patterns within it.</p><p>Teenage Pregnancy - Where is it most and least prevalaent? What are the possible causes/reasons? these kind of social indicators and always worth exploring and students tend to find out a lot by doing it.</p><p>Carbin emmissions - Very topical and I have already enjoyed what I have learned already. Which countries emit the most and what might explain that? What patterns emerge when looking at carbon emissions over time and what might explain that?</p><p>...</p>https://www.thinkib.net/mathstudies/blog/27380/this-years-projects-2018#1536969600Final Report to teachers
https://www.thinkib.net/mathstudies/blog/26628/final-report-to-teachers
Mon, 18 Jun 2018 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/2019.jpeg" style="float: left; width: 200px; height: 115px;" />Curriculum Review for 2019</h2><p>This is just a quick post to try and spread the word about the latest curriculm review report which has been published on MyIB. You will need your MyIB log in details to access this, but here is a <a href="http://resources.ibo.org/dp/subject-group/Mathematical-studies-SL/resource/11162-47877/?c=4c8e6b37" target="_blank">link for you</a>. If you are not familiar with MyIB yet, you will need to go to the program resource center, then choose the DP section and then Maths Studies (or indeed ny of the mathematics courses) the you will find it very near the top. The report is labelled as the final report to schools May 2018 and has the most information yet about what is coming in the new syllabus.</p><hr /><h3>The New Structure</h3><p>Plenty is already known about the new structure and lots has been written here on the site in this <a href="https://www.thinkib.net/mathstudies/page/25551/curriculum-review-and-2019" title="Getting Started » Curriculum Review and 2019">Curriculum Review and 2019</a> section and these more detailed <a href="https://www.thinkib.net/mathstudies/page/23962/new-curriculum2019-applications" title="Getting Started » New Curriculum(2019): Applications">New Curriculum(2019): Applications</a> pages which were based on the last report. This new report says that it is a compliment to the April 2017 report and that the two should be read together. The April 2017 report offers much more in the way of rationale and thinking which is very useful. This new report goes in to a lot more detail about specifics of the syllabus.</p><hr /><h3>Syllabus Details</h3><p>The new report lists every syllabus item for both the Analysis and approaches and the Applications and interpretations courses. The references show clearly that which is SL and HL and also that which is common to both courses. The items are also explained in quite a lot of detail. As such this is going to be a very useful document for all of us. We also have some breakdowns of advised hours and so on. For us here, it means we can get very busy with designing an intial scheme of work to fit the new applications course and start some seruious structuring! You see previous posts about how we are planning to manage that.</p><h3></h3><hr /><br /><h3>Whats next?</h3><p>So we wont expect any further communication now until January 2019 when we can expect the publishing of the new guides, the support materials and specimen papers. This will be followed by the subject specific seminars for the launch. We'll be sure to share news about this as soon as we have it. Happy reading!</p>https://www.thinkib.net/mathstudies/blog/26628/final-report-to-teachers#1529280000Problems with surveys
https://www.thinkib.net/mathstudies/blog/26567/problems-with-surveys
Fri, 08 Jun 2018 00:00:00 +0000<img alt="" src="files/mathstudies/images/IA/QM.jpg" style="float: left; width: 100px; height: 100px;" />So one of the things I have always enjoyed about teaching this subject is that it is a continuous journey of discovery. There are always new things to explore and learn about mathematics and ways to communicate it. I dont suppose I have ever been the first to cross paths with one of those ideas, but that didn't make my discovery of them any less enjoyable. I often use the analogy of travel. Explorers discovered territories hundreds of years ago, but that doesn't mean I dont want to go there and discover them for my self. Anyway, all of this is my usual disclaimer for leading up to telling you about my issue of the moment that is ready for an outing. The point being, that I know I am only the latest in line to write about it, but that doesn't mean it isn't worth sharing what I am thinking. After all, I am bothered by an ongoing issue and so perhaps the more people share their issues the more likely general understanding is to improve. Maybe you wont share my views and maybe you can educate me further - please don't hesitate to do so in the comments. In any case, I think the following is ripe for discussion with maths studies and ToK students and should become even more so when the new syllabus comes in.<hr /><h3>Example 1 - The G2 form</h3><p>Like many of you I hope, I have completed the G2 form which is a review of the exam that students did this year in May. I think it is excellent to have the opportunity to do that and understand that the feeling is still of some disappointment that more teachers don't do it. In a relatively small organisation, our feedback will get read and this is terrific.</p><p>I would word the survey slightly differently myself, but then I imagine that most people would and so I am not suggesting that I would be right to do so. What prompted me to write this post was the question about whether or not the exam papers were accessible/appropriate to people with special educational needs, disability, and people of different genders, race and cultures. I actucally tried to avoid answering this question but then realised that it was compulsory.</p><p>Again, I am not complaining at the inclusion of an important question for our times which has come out of an important desire/need. My thought was simply that I don't feel qualified to answer the question properly. I think the related issues are a minefiled which leave people treading very carefully, but mostly because people are afraid of unwittingly writing questions that don't meet those accessibility requirements. It is the unwitting bit to which I am referring. Whilst I read and think about as much as I can, I can only answer that question to the best of my knowledge and I recognise that my knowledge on the topic is limited. Note - this point is less about the G2 survey and more about the general principle of asking compulsory questions to people who may not have the expertise they need to answer them fairly.</p><p>In this case, I answered as honestly as I could that I did not feel there were any issues that needed to be pointed out. With my untrained eye, I might have missed something important. Imagine, though, that a majority of people are like me (not saying they are, just imagining) then the majority view will be that there weren't issues and the more informed minority will be overuled. Again, I am not expresing any serious concern about this particular issue, because the survey gave the opportunity to elaborate with words if anyone felt there were issues and the organisation is small enough to pay attention to those remarks. That said, if the situation played out as imagnied then they would still be able to correctly state that a majority felt there were no issues.</p><hr /><h3>Example 2 - Percentage percpetion</h3><p>So, here I am drawn to think about this <a href="https://www.thinkib.net/mathstudies/page/18542/percentage-perception" title="Number & Algebra » N&A Teaching ideas » Percentage Perception">Percentage Perception</a> activity. The activity throws up lots of issues and you should follow the link to learn more if you have not already used it. The whole basis of that survey was about asking people 'What percentage of the population in your country do you think are muslim?' The exercise goes on to expose and consider misconception. The issue for me here though is that if I am aksed that question, unless I can draw on actual knowledge to answer it, I really ought to answer with something like 'I could't possible say'. I mean why do I think I might actually be able to answer that question? I wonder, if people were asked to give a confidence score to their response how that would change things. I wonder if people were given the option to answer, I genuinely wouldn't know? What if people were asked to give a maximum and/or a minimum? I think all of these approaches would have helped the surveyer get more interesting information.</p><p>I wonder, how many people answer survey questions from uninformed positions and about the significance of the results of those surveys being used to justify positions and decisions.</p><hr /><h3>Example 3 - End of year surveys</h3><p>So I am a teacher and a parent at my school. At the end of the year we always get a chance to feedback from both perspectives. Students are given the chance too. Again - this is great and how it should be. That said, I have an increasing problem with questions that ask us to put an answer on a scale. </p><p >Strongly disagree - Disagree - Agree - Strongly agree</p><p>First up is the discussion about the missing middle option. I have had lots of useful debates about this and know that there is much to read on the topic that centers on the merits of pushing people off the fence and the down side of leaving the sfatey of the fence for the weakminded. Well, I can see arguments here, but feel that the great danger of pushing people off the fence is an inaccurate inflation of peoples views which is potentially misleading and polarising. </p><p>Next up is the notion of how respondents are expected to synthesize all the factors that go in to answering the question into one of those 4 options. I wonder how much compromise is made there. I wonder how much useful truth comes out. This is related to the points above about how well a survey response actually reflects the truth.</p><p>Lastly - and on a slightly different track - I feel these kind of surveys would be worth a lot more if people were simply allowed to write sentences in answers to questions. Of course that is untidy for statisticians, but survey makers have to go back to the main goals of the survey in the first place. In the context of these surveys, I would suggest that I wanted to know what things respondents were happy and unhappy with (actual examples) and give them the chance to suggest solutions/alternatives.</p><p>Another such example is the end of year student survey about the IB programme. When the results are shown as pie charts that show the percentage of respondents in each category - I find myself wondering whether the exercise helps us converge on the truth or if the compounding compromises along the way mean that we diverge from information we can actually do something with.</p><p>In this sense - I would defer to the assertion that words are of more use than numbers and the notion that, not everything that counts be counted and not everything that can be counted counts.</p><hr /><br /><h3>In summary</h3><p>Perhaps, unhelpfully, I have thrown out more questions than answers here, but I feel they are important questions. With data being the new 'Oil' and playing such a leading role in the current studies course and the new applications course, they are the questions that have to be asked and dealt with. As surveys are designed, we have to keep referring back to what the point of the survey is and make sure that the way we ask questions allows respondents to help us converge on truth and not diverge from it.</p>https://www.thinkib.net/mathstudies/blog/26567/problems-with-surveys#15284160002019
https://www.thinkib.net/mathstudies/blog/26486/2019
Mon, 21 May 2018 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/2019.jpeg" style="float: left; width: 200px; height: 115px;" />Getting Closer</h2><p>At my school, we will have an evening for propsective IB students and their parents in late October or early November 2018, which means we need to have answers to some key questions before then. That seems quite near and so it is a topic we are talking about quite a lot at the moment! I have written about the questions that I think schools are facing <a href="mathstudies/page/25551/curriculum-review-and-2019" target="_blank">here</a> and we have given as detailed a breakdown as we can of the new courses here - <a href="https://www.thinkib.net/mathstudies/page/23962/new-curriculum2019-applications" title="Getting Started » New Curriculum(2019): Applications">New Curriculum(2019): Applications</a> if you haven't seen it already.</p><hr /><br /><h3>What are we doing?</h3><p>So this blog is really just about what we are planning on the site and what it is that teachers might be looking for. At this stage, this is the plan for development of the site......</p><div class="greenBg" contenteditable="false"><div contenteditable="true"><ul><li>It seems like the majority of resources on this site as it is will be easily transferable to the new course. As such, it will be reviewed and reorganised in to the new units by the spring of 2019 so that we are offering hige cross section of resources that you can use to teach the new course.</li><li>We will offer a section on making the transition, considering the changes and general management of bringing in the new courses.</li><li>We will offer a detailed scheme of work for the new Applications and Interpretations course for the 2 years starting in September 2019.</li><li>We will aim to have the first term fully resourced well in advance.</li><li>Then we will aim to keep at least few months in advance of the scheme </li><li>After that we will of course continue to add depth, questions and quality to the site so that we continue to offer you more.</li></ul></div></div><h3>What is on your mind?</h3><p>In the mean time we would be really keen to hear from you about what your main priorities are. What do you most want to know? What kind of resources would you most like? How would you like the site to develop? Please add suggestions to this <a href="http://docs.google.com/forms/d/e/1FAIpQLSeKmBR7Lw53SYVs4jvM75KfovRhk3-wDsLhFOCBFZYFjk5_jQ/viewform" target="_blank">google form</a> or comment on this blog post.</p>https://www.thinkib.net/mathstudies/blog/26486/2019#1526860800May TZ2 exams
https://www.thinkib.net/mathstudies/blog/26432/may-tz2-exams
Fri, 11 May 2018 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/Exams/photo.JPG" style="width: 150px; height: 150px; float: left;" />First Impressions</h2><p>I got my first look at the May 2018 TZ2 papers today and I thought I would write a quick reflection. Clealry it is important for us, as teachers, to have a good look through the papers so that we can prpeare ourselves for the <a href="http://ibo.keysurvey2.com/f/1208598/788f/">G2 forms</a> where we getr the chance to have our say and then so that we can think for ourselves about fine tuning we can do to prepare our suture stuents. It is a challenge for question writers to balance the need for an exam tha tests the syllabus and stays in the spiriti of the mathematical studies course, but also one that is not too predictable. I take my hat off to paper editors as it strikes my as a very difficult and probably thankless task! It is a really salient task to go through an exam ppaer like this and try to imagine how your students might have paporached the questions and relate it to the experiences they have had in the classroom. There is always room for a bit of reflection on how we might do this better. Anyway, here goes, here are my intial thoughts on this paper. (Note - obviously I can't publish the paper itself here is it is IB copyright - check with your exams officer to see a copy) </p><hr /><h2>Paper 1</h2><p>On the whole it seemed lile a fair paper. Perhaps there a more little twists than there are at other times which cvan be off putting......</p><p><em><strong>Question 1 - Lines of best fit</strong></em></p><p>A fairly gentle start and no complaints for me. that said I am a little disappointed that linear rgression did not appear anywhere else. I spend a lot of time helping students understand how to use their GDCs for 1 and 2 variable data and this question does just give them the answer. It also seems a bit odd to focus on lines of best fit by eye when students know how get their GDCs to find the accurate answers. </p><p>NOTE - Question emphaisises the importance of students knowing that a line of best fit must go through the mean point.</p><p><em><strong>Question 2 - Truth tables</strong></em></p><p>Another fairly gentle starter question. The catch is that, as the table is drawn the two columns being compared for implication are in the wrong order. I generally encourage my students to rewrite some columns so that the implication can be in the right order for easy analysis. perhaps the question could have helped there for question 2. i'll bet this caught a few out.</p><p>NOTE - Emphasis on managing implication where columns are in 'the wrong' order.</p><p><em><strong>Question 3 - Standard form</strong></em></p><p>I have seen esier standard form questions at the start of an exam. The context takes a bit of thinking about but is otherwise a fairly standard kind of question</p><p>NOTE - Are students exposed to enough variety of contexts for questions.</p><p><em><strong>Question 4 - Number sets</strong></em></p><p>No complaints and I like that the second part asks students to reflect on the nature of sets and subsets a bit more than these questions usually do. That said, I hoe that the markscheme alows for students to write the same number in each of the boxes for part A. I bet not many were brave enough.</p><p>NOTE - Show my IB1 students some different answers to this question.</p><p><em><strong>Question 5 - Compound interest and currency exchange</strong></em></p><p>Part a) was fairly standard and a bit of thinking required for part b). I find myself wondering how many students would have set this up as an equation. - and whether or not they needed to...</p><p>NOTE - to think about approaches I might expect students to take to this kind of question.</p><p><em><strong>Question 6 - Linear functions</strong></em></p><p>Good fair question - Once again, I wonder about the approaches my students might have taken to part b). I wonder how many of them used the simulatenous equetion solver, how many plotted the functions and found the intersection, how many did it by eye.</p><p>NOTE - Thinking about the merits of showing muliplte approaches to questions. How often do I let students show each other the multiple approaches they actually use?</p><p><em><strong>Question 7 - Probability from the table</strong></em></p><p>A good start with 3 marks fairly esily available. Part c) is a bit of 'without replacement'. I wonder how many students visualise a tree diagram here. For how many is there in instinct to think of combined events and multiplication. Sometimes I think the tables are easier, but for combined events I think the tree diagram leads students more successfully.</p><p>NOTE - To make more explicit links between different representations for probability.</p><p><em><strong>Question 8 - Non right angmed trig</strong></em></p><p>Not much to say about this one. Fairly straightforward. I would be curious to know how many candidates labelled the trinagle with As Bs and Cs. I think I have a helpful instinct to see D2 as an intersection and D3 as a union straught away. Increasingly find this is a helpful approach.</p><p>NOTE - Nothing coming to mind just yet for this one.......</p><p><em><strong>Question 9 - Shading Venn Diagrams</strong></em></p><p>Ooh, I think students may have struggled with Diagram 2 and 3. I dont object to the question though. I suspect that the subset structure in Diagram 2 is seen less often. Of course there will be more ways to say Diagram 3 - I am sure the markscheme allows for it!</p><p>NOTE - to think about activity that helps students decide if they are looking at an Intersection or a union.</p><p><em><strong>Question 10 - Exponential models</strong></em></p><p>Good question and I like part b) - I think we can expect to see more of this in the new applications and Interpretations course. For part c, I am guessing (hoping?) students multiplied their value of A by 40 and looked at the table function. Is that more instincitve than solving 2 to the t = 40? The later seems quicker....</p><p>NOTE - I want to think about asking more 'what does this represent?' type questions</p><p><em><strong>Question 11 - Rational functio and asymptote</strong></em></p><p>Hmmm - again, I hope that students who needed it plotted the function on their GDC. Its another case of looking at multiple methods though. STudents could look at the table for the gaps in the x and y column. I like them to think about what f(x) can never be. I wonder how many went from the condition that x cant be zero to the vertical asymptote..... Then i wonder how many used equation solver for part c.</p><p>NOTE - I am reflecting on the formal treatment of rational functions. Again, perhaps this will be more obvuious in the new course when students will be expected to reflect on the significance of the asymptotes.</p><p><em><strong>Question 12 - Histograms and estimate of the mean.</strong></em></p><p>Estimates of the mean are notorious stumbling blocks. Possibly even this question with a frequency missing might even encourgae some better thinking. Part a) does lead students a little in the right direction....</p><p>NOTE - ....</p><p><em><strong>Question 13 - Quadratic models</strong></em></p><p>This seems doable for a Q13. I think it can all be done on the GDC right?</p><p>NOTE - Alternative forms for quadratic models - how comfortable are students in dealing with these?</p><p><em><strong>Question 14 - Calculus</strong></em></p><p>It is nice that parts a and b are aproachable. Clearly at the business end, I can see that part c) will trip up a lot of students., although it is not really that hard once you have worked the question can be seen as finding the x value where the gradient is 8.</p><p>NOTE - Again - thinking about exposure to this kind of twist on the key ideas is difficult. I think this is where we as teachers need to try and make sure we are encouraging students to speculate with information. For example - ideally, a student would say something like 'So, if I know that the graident of the normal is this..... then what else can I find out?' and then the question could iopen up nicely. This is tough but important.</p><p><em><strong>Question 15 - 3D Geometry</strong></em></p><p>This is not too bad for a Q15. I think there is plenty of potential for students to get method marks for beginning to solve this by writing expressions for surface area. Sure, they will need to be fairly precise to solve the whole question, but I would hope that most students could get at least 2 or 3 marks on this question.</p><p>NOTE - I am reminded of what I often refer to as 'scrapping around for marks' and how pertinent that could be here. Also, I wonder how many of them didn't get to this question and have that chance. Exam technique!!!</p><hr /><br /><h2>Paper 2</h2><p>Dangerous though it is to say so out loud, paper 2 did seem very approachable and did not contain many (if any) of the twists that we sometimes see...... but we will have to wait and see how students got on with it!</p><p>Question 1 - Sets and Probability</p><p>This looks like a nice question to start woth. There are plenty of approachable marks and probably the most challenging parts were not in a position where they could put you off or hinder progress on the rest of the question.</p><p>Question 2 - Cumulative Frequency</p><p>Again - this seems like an easy question without any twists or challenges really. I would hope students could cash in here. It does seem as hame though that there is a 15 mark paper 2 question on statistics that doesn't involve any interpretation at all. Clearly, those questions are harder to ask and assess (and maybe even harder to answer) but it is really the point of statistics. Thik there is room for more of this and I expect to see more of it in the new course.</p><p>Question 3 - Normal Distribution and Chi Squared test.</p><p>Another fairly straightforward question I think. perhaps in part b) students might be held up dealing with 1.5 standard deviations instead of a given maximum and minimum, but b)ii) asks them to sketch the curve which, in turn, invites them to work out the uppoer and lower limits. Here is another stats question without any significant interpretation though. </p><p>Question 4 - Sequences</p><p>The context can often make it a little harder to get in to, but I would expect students to spot this as a fairly standard sequences question - again without any real twists.</p><p>Question 5 - Trigonometry</p><p>Part a) really need students to have that 'speculative' approach I mentioned earlier. 'Given what I know - what else can I work out?'. otherwise, since it is a show that question, they could move on using 85° for the rest. There is a a bit of problem solving to do here, but nothing particularly tricky.</p><p>Question 6 - Calculus</p><p>And once more - compared to some optimisation questions that come up on paper 2, I think this is very approachable. I raised an eyebrow at the use of the term stationary points and certainly part f) will leave out a lot of people.</p>https://www.thinkib.net/mathstudies/blog/26432/may-tz2-exams#1525996800Subject report and IA
https://www.thinkib.net/mathstudies/blog/25856/subject-report-and-ia
Thu, 01 Mar 2018 00:00:00 +0000]]>Subject report and IAhttps://www.thinkib.net/cache/blog-thumbs/21/25856-1519901175-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/25856/subject-report-and-ia
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/25856-1519901175-thinkib.jpg" alt="Subject report and IA" /><br /><br />https://www.thinkib.net/mathstudies/blog/25856/subject-report-and-ia#1519862400New Curriculum
https://www.thinkib.net/mathstudies/blog/25772/new-curriculum
Sat, 10 Feb 2018 00:00:00 +0000]]>New Curriculumhttps://www.thinkib.net/cache/blog-thumbs/21/25772-1518274221-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/25772/new-curriculum
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/25772-1518274221-thinkib.jpg" alt="New Curriculum" /><br /><br /></p><p>The above is a bit crude and full details are yet to be made public but it is a good guide for now. So in tems of subscribing to ThinkIB.net sites the following is the planned model</p><p><strong>This MATHSTUDIES site will become the site for the Mathematics Applications and Interpretations and cater for both courses during the transition.</strong></p><p><strong>The HL/SL site will become the site for Mathematics Analysis and Approaches and cater for both courses during the transition.</strong></p><p>So, stay tuned for news of how the restructuring will take place!</p>https://www.thinkib.net/mathstudies/blog/25772/new-curriculum#1518220800Re-organisation
https://www.thinkib.net/mathstudies/blog/23476/re-organisation
Mon, 25 Sep 2017 00:00:00 +0000]]>Re-organisationhttps://www.thinkib.net/cache/blog-thumbs/21/23476-1496056870-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/23476/re-organisation
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/23476-1496056870-thinkib.jpg" alt="Re-organisation" /><br /><br />The IA section</h2> <p>This is just a quick blog post to explain some reorganisation that is coming up for this section of the site so as to explain any gaps or unfinished looking parts. Currently under assessment you have a choice between 'Internal assessment' and 'Exams and Revision' - During reorganisation, this will stay exactly as it is so that you can find things in the same place. During re organisation and building, there will be 2 more pages appearing under that list. The first is 'Internal Assessment teacher management' and the second is 'Internal assessment - planning and writing'. Gradually, the two sections will subsume all the old material. In the mean time, I will leave the old structure in place for reference.....</p> https://www.thinkib.net/mathstudies/blog/23476/re-organisation#1506297600First Draft Feedback
https://www.thinkib.net/mathstudies/blog/23656/first-draft-feedback
Sun, 02 Jul 2017 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/IA/fdfeedback.jpg" style="float: left; width: 150px; height: 142px;" />What am I saying the most?</h2> <p>This post will eventually be a full page in the new structure of the IA section. For now I thought it might be interesting to share as a blog post. This year I have 17 students in my IB1 class and we have been determined to complete the IA before the summer break. This prompted much of the thinking about the restructuring I have planned. Part of that thinking has been about providing students with the types of resources that can make them more self sufficient. I am sure we have all had the experience of repeating ourselves when talking to indivudal students. Sometimes that is just the way it is, at other times I think it is just inefficient. This year I lead students through a more rigorous process of going from idea to information to plan and insisted on all of those things being in place before students actually starte writing. I set a clear set of dates for each stage and to their credit, my students have done very well in sticking to them. I have spent the last week pouring through their first drafts. As I did, I copied all the annotations that I made so that I could go through them and loo, for patterns! This post is about that, my findings and consequent plans for next time.</p> <hr /> <div class="greenBg"> <h3>The Process</h3> <p><em><strong>Stage 1</strong></em> - Students made me 3 project idea maps that klaid out the ideas, questions, hypotheses and potential of the the three ideas. I then advised them which of the three I felt would be most effective.</p> <p ><img alt="" src="files/mathstudies/images/IA/ir.jpg" style="width: 600px; height: 444px;" /></p> <p><em><strong>Stage 2</strong></em> - Once settled on an idea, students were required to set out clearly exactly the information that they thought they would collect. In doing so, they were required to meet certain criteria I presented.</p> <p ><img alt="" src="files/mathstudies/images/IA/ir-info.jpg" style="width: 600px; height: 426px;" /></p> <p><em><strong>Stage 3</strong></em> - Planning - This was about asking students to make explicit statements about exactly what processes they would use, which data they would use and what the purpose of the process was (making sure that this was explicitly linked with the statement of task). Students were asked to have this approved before they took the idea any further.</p> <p ><img alt="" src="files/mathstudies/images/IA/ir-plan.jpg" style="width: 600px; height: 430px;" /></p> <p><em><strong>Stage 4</strong></em> - The first draft - At this stage, students were given the detailed <a href="mathstudies/page/19021/student-writing-guide" target="_blank">'Writing Guide' </a>and set off to write their first draft. In addition to the writing guide, I created this page on <a href="http://studyib.net/mathstudies/page/568/using-technology">Using technology in projects</a> for students on the 'StudyIB.net' sister site.</p> </div> <h2>What did I get?</h2> <p>Well, we all know that there is a difference between giving students things to read and watch and them actually doing so. It was a very pleasant surprise to find that majority did! Because so many of the ideas I am usually having to comment on did not come up, I thought it woryn paying attention to what did so I know what to work on for next year!</p> <div class="pinkBg"> <h3><img alt="" src="files/mathstudies/images/IA/fdfeedback.jpg" style="float: left; width: 300px; height: 284px;" /><br />A note on annotations</h3> <p>For anyone out there using managebac, I have found this a really useful way of providing feedback via annotations. If students upload their work to an assignement, then you can annotate the file where it is. When students gon to the assignment and choose 'Annotate' themselves, then they can see your annotations and add their own. This is appealing for 2 key reasons...</p> <ol> <li>There is no subsititute for adding annotations directly on the work to which you are referring. There are a few tools to help and students have a better chance of understanding exactly what you are saying.</li> <li>There is only one copy of the cdocument and its annotations, rather than multiple copies flying around via e-mail etc.</li> </ol> <p>Other tools like, turnit in offer this facility. I don't know enough to say which tool does it better, but I know that many of us are already using managebac so, for that reason, it seemed the most sensible choice.</p> </div> <h2>What did I say?</h2> <p>For interest, I have copied the comments directly below, but thought it would be useful to summarise them in to categories. I was really surpirsed to see that most of what remained for students to do centered around communcation. this was reassuring in some ways, because it meant that my resources had helped them really understand the IA process, the mathematics involved and the ideas of simple relevant investigation. Then I had some refelction to do about why these communication issues are what they are. My conclusion is that actually, theyt are rzquired to do this so rarely that it is to be possibly expected. Interesting that skills they learn for other subjects that require more written communication have not necessarily been translated to maths. These are the key points that I will reflect on in future. </p> <div class="blueBg"> <ol> <li>How do build in more opportunity to practice this kind of communication?</li> <li>How do I help students transfer communication skills from other subjets to this one?</li> </ol> </div> <p>Here is a summary of the categories.....</p> <div class="greyBg"> <h3>Introductions and structure</h3> <p>Much of the advice was about having an explicit plan that was then followed. Even if I am only giving 2/3 here, I want see to this done better. Either separate paragrpahs or ullet points that show clearly how the project will be chopped up in to 3, 4 or 5 pieces that follow a logical order. As the reader, this sets the tone and I know what toe xpect. If the sections that follow mirror that then this does a lot for the communication</p> <h3>Sign posting</h3> <p>Lots of the projects had new sections that started without a heading or an opening paragraph (sentence or two) to help the reader know where they are and what was coming. A seemingly random graph would appear... The result was lost of comments about 'Signposting'. Where are we? Where are we going next?. Use of a headings structure, titles and labels. Next year I think I might introduce another stage where students are required to show me a document outline using the heading structure before they start writing. I was surprised to learn how many did not know about the navigation pane in word.</p> </div> <h3>My Comments</h3> <p>The following are a selection of the comments I found my self writing...</p> <p>Make sure you have an explicit list of the things you are going to do that you then follow exactly. Each process must be justified by a purpose. This was the planning phase. At the moment this is a bit grey and not easy to tell.</p> <p>Consider using a headings structure where the headings match up with the sections you set out in your introduction match up.</p> <p>Again, this just appears without a heading or any explanation. Give it a subheading and then a sentence or two to explain what is about to happen....</p> <p>I think it would be useful to show an example of this in action.... Just show the process for calculating one of the means.</p> <p>It would be good to get a PMCC using tech here just to back up what you are saying....</p> <p>I really have no idea what this table is showing.....</p> <p>I am worried about this. It seems like it could be really interesting, but I still don't get it. I am being a bit devils advocate about this just to point out that to a neutral reader this might get the same reaction</p> <p>I don't think we do see that actually..... I also think this should be a scattergraph..... not a bar chart</p> <p>I have you somewhere between 2 and 3 here. I think it would be useful to hear more of an explanation about your numbers. For example – how is it possible get the number of species in a given country. There must be some problems with this?</p> <p>I am not seeing anything explicit enough here and think you have plenty to talk about…… I think the validity of your actual data is questionable. Not your fault, but you could talk about it. Also You might like to discuss how likely the link with GDP is – if there is a link what do we think about causation?</p> <p>I think you have some work to do here. Pay careful attention to the advice given in the writing guide and in my annotations. Much more signposting is needed. The reader needs to know where they are at and what is about to happen at all points. This does not need to be long winded and can be solved with headings and short explicit sentences.</p> <p>½ at the moment, there is not a lot of maths here to use as evidence….. Also I, like I said, one of your graphs should be a scattergraph and not a bar chart and so this might even knock you back to 0 here…..</p> <p>You need to get all ToK here Freddie. The list was made by an Anglophone magazine based in the USA... Is the outcome really shocking?</p> <p>5/5 This is good stuff Freddie, BUT, I am hesitating a bit about the fact that you calculated a PMCC when it was pretty obvious that there was not going to be any correlation…..</p> <p>What there is is evidence that most of the songs are older. There for there is a preference for older songs. I wonder, can you demonstrate this for us somehow</p> <p>What about…. Grouping your songs in to decades</p> <p>Plot a time series of Decade on the x-axis and number of songs included in the top 500. This might show an interesting pattern – You could then do a PMCC on this….</p> <p>Then you could do it for the top 100. This might be even better because it looks a bit different to the rest….</p> <p>I just think you need to bring this pattern out somehow… Then I will be happier to justify the mark.</p> <p>Have you got the whole table Freddie? Might be worth puttin gin an appendix</p> <p>You are making a good point, but you might need to emphasise that there is difference between judging by eye and calculating. It might also be worth searching for a 'perfect' lack of correlation</p> <p>So at the moment, your introduction does not contain a plan. this is key, please read this section and add.</p> <p>This is a good start at explaining your data. Can I recommend that you have 2 separate sections. 1 for an introduction and another for explaining everything we need to know about the info you collected. Perhaps some annotations of your shown sample would be useful</p> <p>I think you need to tell the reader a bit more. even though I know what your project is a bout, I am still asking 'the mean what?'</p> <p>What is 100 more? 100 what?</p> <p>Explicit labelling needed to stop your reader getting lost</p> <p>These labels are so small as to be nearly illegible</p> <p>use an equation and get a fraction 'Insert Equation'</p> <p>What formula did you use for covariance? Where is it? Where can I see you substituting the numbers in?</p> <p>This doesn't mean anything... A regression line should be in the form of a linear equation (y=mx+c for example)</p> <p>This is not a good scale to use - you need to focus on a section of the scale so we can see differences better.</p> <p>Why? Each section should start with a brief outline of what the reader can expect. This can be a copy of what you put in the plan. this helps the flow a lot.</p> <p>You might ned to give the uninitiated some background on 'Dr Who' Something about its cult following will help your project to make sense</p> <p>I am considering three here because I think it is good data and you worked hard to put it together….. I’ll think about it. Perhaps if, in your section on data, you go through each data heading and tell me why you collected it what patterns you think it might show.</p> <p>This could be clearer. if you wrap the text in the heading box and then narrow the columns, you'll get a better image with bigger text and less empty space....</p> <p>These long left tails are interesting too... why might they be there?</p> <p>And what might that tell us about that population?</p> <p>You might like to try and say more about why you think some results turned out the way they did</p> <p>I think it would also be good to explain a bit more about how you decided on your categories for the test….</p> <p>In general, I think it needs a little more sign posting. The reader should know where they are at any time and what is happening. A sentence or two at the start of each section, clear labels and titles and annotations. Don’t forget, the reader has not been involved with your project and needs everything to be made clear.</p> <p>Each section should begin with sentence or 2 about what is about to come. DOn't leave the reader guessing</p> <p>Make a reference to the whole table appearing at the end. Say that you didn't want to interrupt the flow with the whole thing here</p> <p>You have got two sections mixed up here. You should have an intro that has a very clear plan with the purpose of each thing in it.... THEN you should describe your data collection process. The stroy about the politics is a good one and you should tell it...</p> <p>Does this mean average? If so then say so....</p> <p>All these graphs should have their interpretations nearby... too much work for me to figure which bit of text goes with which graph</p> <p>NO - IT DOESN'T SHOW ANY CORRELATION AT ALL</p> <p>What about a sentence or two that outlines why you are interested in cyber attacks and why you think this might be a worthwhile exercise</p> <p>This whole section is incomprehensible because nothing has labels or titles. The reader has no idea what is happening</p> <p>DC? I am probably not alone in not knowing what you mean.....</p> <p>So we need a clearer outline of your plan. I recommend either separate paragraphs or bullet points for each process....</p> <p>This is a bit blurry and hard to read. Also lots of space - can you format your table so there is less space and the numbers are bigger?</p> <p>Now here I need a sub heading. Clearly something new is about to happen. Sign post it with a heading, then a couple of sentences to explain what you are going to do and why. This might be a repeat of your plan.</p> <p>Again, this is missing sign posts - the reader will be lost. Also the box plots need explicit labelling. I am left asking what variable you have used - guessing... searching... Frustrating..... This is all key to good communication.</p> <p>Is this Marvel and DC combined? (still don't know what DC is)</p> <p>Where are these numbers coming from? How can I check them?</p> <p>Anyway you could format the table so that the numbers are clearer and there is less space?</p> <p>Why might there be?</p> <p>Observations and interpretations should really be put near or on the diagram so we can easily associate the two. here I am having to scroll too far down to see what you are going to say...</p> <p>Wow - I am really impressed with this, although there are lots of problems. In short though, you have too many categories and not enough data, so too many of you expected frequencies are below 5. We need to rethink how you could combine some of your categories.</p> <p>Perhaps you can tell us more about why it is significant that you have used ‘rate’ and ‘per capita’. It is significant!</p> <p>You might need to elaborate here for the neutral reader....</p> <p>It might help to see this as separate paragraphs/bullet points so that the plan leaves a mark on the reader....</p> <p>so what rate - per 1000?</p> <p>Heading needed here.... signpost - what is coming next, then a sentence or two to set the tone....</p> <p>Don't think we need all of these decimal places.... round</p> <p>Distribution? I don't know what is on the Y axis here...</p> <p>OK - we need a more descriptive introduction. What are you going to do and why? What do you think might happen.. Specifically what processes will you go through and why?</p> <p>Would be good to see an example how you calculated one of these....</p> <p>Some work to do here on wording.... correlation goes with scatter graph Not sure what you mean by mean predominant religion.... think is just a case of wording...</p> <p>.....</p> https://www.thinkib.net/mathstudies/blog/23656/first-draft-feedback#1498953600Venns in the news
https://www.thinkib.net/mathstudies/blog/23453/venns-in-the-news
Wed, 24 May 2017 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/wiki-venn.jpg" style="float: left; width: 200px; height: 167px;" />Venn diagrams everywhere</h2> <p>I love how often we see venn diagrams in a variety of different contexts. I think they are seen as particularly powerful visualisations and rightly so. Like most mathematics in the media though, I wonder how often people stop to ask questions of them. Sometimes the questions I might ask might be seen as trivial - asking them just because I can, otherwise though I think the lack of questions might reflect the lack of understanding in the visualisation. On the one hand - misuse or or lack of questioning of this is a concern, on the other, as a mathematics teacher, it is a gift for us to bring that kind of relevance to our classrooms. I am just going to show a couple here to deomonstrate my points.</p> <hr /> <h3>The Wiki Tribune</h3> <p>This is a really exciting project and I am going to support it. If you don't know about it then read here - <a href="http://www.wikitribune.com/">The WikiTribune</a>. Of course you will have seen the Venn diagram in the article. I hope they dont mind me using it. Clearly it is theirs and I am referencing it...... What are the questions we want to ask about this diagram.....</p> <hr class="hidden" /> <p><img alt="" src="files/mathstudies/images/blog/wiki-venn.jpg" style="width: 390px; height: 325px; float: left;" /></p> <p>Are all the possible intersections shown?</p> <p>What is left out?</p> <p>Does it matter?</p> <p>What do the blue areas represent?</p> <p>What other sets might we include here?</p> <p>Can the wikitribune reallyy rest entirely in the set of 'facts'</p> <p>.....</p> <p>What other questions does this venn diagram provoke? Doubtless we would come up with different answers to the one I have asked.....</p> <hr class="hidden" /> <h3>The great education debate</h3> <p>I spend quite a bit of time on twitter, or particularly those parts related to education. There is a debate raging about differences between 'Progressive' and 'Traditional teachers'. I am certainly not going to offer my views on that in this blog post - I find it interesting on some levels and frustrating on others. A number of people are suggesting that it could be seen as a false dichotomy. In any case, it prompted one such teacher to produce this Venn diagram....</p> <p ><img alt="" src="files/mathstudies/images/blog/teacher-venn.jpg" style="width: 518px; height: 359px;" /></p> <p>My instant thought was about the lack of a defined Universal set of 'Things that teachers do'........... Again, I am not complaining, just asking questions. If our students saw it, they would say (hopefully) that both not P and not T should include the area outside the two sets offerred. Again, I find myself asking if this matters, but in any case it prompted me to ask questions!</p> <hr /> <h3>A thought</h3> <p>And finally, I will share with you a diagram I produced to give a presentation about what we do in our maths department. Even my own diagram made me ask lots of questions and in many ways, I made use of general understanding of Venn diagrams to visualise a point I wanted to make and maybe didn't stick to all the protocols I should have..... </p> <p ><img alt="" src="/files/mathstudies/images/blog/goals-venn.jpg" style="width: 600px; height: 337px;" /><br /></p> <p>In conclusion, I ask to what exetnet are we allowed to be flexible with the 'protocolas' of mathematical diagrams in order to help communication. In doing so there is an implication that perhaps mathematical diagrams in their purest form don't communicate as well as they might? All just questions and certainly no criticism of the Venn diagrmas used is intended or implied. We teach Venn diagrams, the mdeia uses them. That is a good combination as far as I am concerned.</p> <p>Now I am looking for the ToK angle,</p> <p>cheers!</p> https://www.thinkib.net/mathstudies/blog/23453/venns-in-the-news#1495584000This year's projects
https://www.thinkib.net/mathstudies/blog/23433/this-years-projects
Fri, 19 May 2017 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/ir.jpg" style="float: left; width: 200px; height: 140px;" />Enthused students!</h2> <p>I thought I would write this post about the topics my students have chosen for their IAs this year. I am busy re writing the IA section of the website (Big job) which will include some new resources about how to help students get solid, focussed ideas for their projects. I have been testing the resources out and am quite pleased with the results so far! Here is a run down of some of the ideas they have come up with.</p> <hr /> <h2>No Apologies</h2> <p>I have read the examiners reports about Non Stats projects and understood them. I do offer and encourage students the opportunity to branch out and recognise how much potential there is. The fat remains though, that, from my experience, students find statsitcis comfortably the most relevantly applicable aspect of the course. They find that statistics offer them an opportunity to explore areas of interest to them and get enthused by the idea. We may like to say more variety and we may have seen a hundred projects about GDP and so on, but for our students it is the first time and they are having exactly the experience the program sets out to offer. So I make no apology that the ideas presented here are all stats based. the new resources will , however, try to offer more help and encouragement to students to explore different avenues so that students can see all the potential.</p> <p>I am also determined to make sure that all ideas come from students themselves so that they are motivated to work on a topic that interests them.</p> <hr /><br /> <h2>Projects</h2> <p><strong>Cyber attacks</strong> - What is the damaged caused, the danger associated. Who gets targetted and when? This promises to be really interesting and the data has all the potential it needs.</p> <p><strong>How effective is the TB vaccination</strong> - Never seen this one before. Comparing data about prevalanece of the vaccination and the disease along with other factors that might affect both of those.</p> <p><strong>Dr Who Episodes</strong> - Which ones are most popular and why? Some good data here could give the show producers some useful tips on the formula for success!</p> <p><strong>The Happy Planet Index</strong> - This is a really interesting organisation that I have been aware of for a while, but I have never seen the data that this student uncovered before and look forwrad to some of the outcomes.</p> <p><strong>Homework</strong> - although this is a stayed theme, this student looks determined to get some really rigours data on what is going on here. I for one am keen to see the results, because I think perception might be different from reality. This data collection is being well planned so that students in our school will log the time they spend over the period of a week. I have a feeling I might be taking this oroject to a staff meeting!</p> <p><strong>Flexibility</strong> - Nice! This will be a primary data collection exercise exploring how flexible we are and what the factors that cause that might be. </p> <p><strong>US Election results</strong> - Wow - some fascinating figures coming out of this one so far on the 2016 Presidential election. This is the kind of work journalists are doing everywhere (and often badly). it couldn't be more relevant.</p> <p><strong>University enrollment</strong> - What factors affect this in different parts of the world.</p> <p><strong>Cancer type prevalance</strong> - This promises to be fairly sobering reading so far. Personally I wonder what differences there are between reality and public perception on this topic.</p> <p><strong>Murder rate</strong> - I have seen this as a bsis before, but this one will focus on links with urbanisation which could be intriguing</p> <p><strong>The Rolling Stone top 100 songs</strong> - A music fan will be breaking these down to figure out what they have in common - or not as the case may be. </p> <p>and seven more.....</p> <p>I know it will be hard work and won't all go well, but I like this stage when students seem genuinely interested in the idea of a mathematical investigation.....</p> <p>Work on the re write of the IA section continues.......</p> https://www.thinkib.net/mathstudies/blog/23433/this-years-projects#1495152000Project Planning
https://www.thinkib.net/mathstudies/blog/23405/project-planning
Fri, 05 May 2017 00:00:00 +0000]]>Project Planninghttps://www.thinkib.net/cache/blog-thumbs/21/23405-1493992727-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/23405/project-planning
<h2>Project Maps</h2> <p>Just a quick post! As one class gets through exams, the other is starting to think about projects. Here is a new image I made to help them brainstorm ideas! More to come soon in the IA section. I spent far too long on it, but I was happy with it....</p> <p ><img src="https://www.thinkib.net/cache/blog-thumbs/21/23405-1493992727-thinkib.jpg" alt="Project Planning" /><br /><br /></p> <p ></p> https://www.thinkib.net/mathstudies/blog/23405/project-planning#1493942400StudyIB.net
https://www.thinkib.net/mathstudies/blog/22937/studyibnet
Sat, 11 Mar 2017 00:00:00 +0000]]>StudyIB.nethttps://www.thinkib.net/cache/blog-thumbs/21/22937-1489226175-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/22937/studyibnet
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/22937-1489226175-thinkib.jpg" alt="StudyIB.net" /><br /><br /></p> <p>The StudyIB Maths Studies site is a web based learning resource for students, packed with resources to help you to improve your understanding and performance.</p> <p>The site is written by Jim Noble, a hugely experienced Maths Studies teacher, and it has been designed to support student learning from the beginning of the course until the final exams. It is clearly organised by s yllabus topic, simple, attractive layout, and is optimised for mobile devices.</p> <ul> <li>The site includes:</li> <li>100+ Teaching videos covering the key concepts on the syllabus.</li> <li>150+ Slides with visual explanations and examples.</li> <li>200+ Onscreen practice questions with feedback.</li> <li>50+ Revision Flashcards.</li> <li>50+ original exam style questions with video solutions.</li> <li>Regular updates with more videos and questions going up every month.</li> </ul> <p><strong>Teaching videos</strong> allow you to review key concepts you have learned in class.These thoughtful, well-paced, demonstrations and explanations, which you can watch again and again, are designed to make you think and improve your understanding.</p> <p><strong>Visual slides</strong> back up the videos with you can see visual notes that describe and summarise the key ideas.</p> <p><strong>On-screen quizzes</strong> enable you to check your understanding of ideas by giving you instant feedback and advice. Return to them as often as you like.</p> <p><strong>Revision Flashcards</strong> boil the syllabus down to key points which help you to review as exams approach.</p> <p><strong>Exam Style questions</strong> which you can try out on paper yourself and then check your answers against the video solutions.</p> </div> <p>....</p> https://www.thinkib.net/mathstudies/blog/22937/studyibnet#1489190400Percentage perception
https://www.thinkib.net/mathstudies/blog/22414/percentage-perception
Wed, 14 Dec 2016 00:00:00 +0000]]>Percentage perceptionhttps://www.thinkib.net/cache/blog-thumbs/21/22414-1481718583-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/22414/percentage-perception
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/22414-1481718583-thinkib.jpg" alt="Percentage perception" /><br /><br /></p> </div> <p>Even better, the new data comes with a new headline for us to explore. There is a new graphic and a scetion on predictions for 2020 as well! So much to talk about! I am looking forwrad to getting in to the data and updating the activity!</p> https://www.thinkib.net/mathstudies/blog/22414/percentage-perception#1481673600Picture this
https://www.thinkib.net/mathstudies/blog/21597/picture-this
Thu, 01 Sep 2016 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/picturethis1.jpg" style="float: left; width: 150px; height: 150px;" />Paying attention to detail</h2> <p>Just sharing what I did today with one of my Mathematical Studies classes as a warm up exercise for the new term. We played a few rounds of this <a href="http://www.amazon.co.uk/Paladone-PP2804-Picture-This-Game/dp/B010T8PCXM" target="_blank">'picture this game'</a> with a view to reflecting on how students look at mathematics problems. The idea of the game is very simple. There are two large packs of cards and each card has on it an image which is a close up photograph of a well known object. The photos are so close up that it is not at all obvious what the 'object' is. The goal of the game is obviously to try and figure it out. It is both incredibly satisfying and frustrating and, I think, a lovely analogy for reactions we can all have to mathematics problems!</p> <hr class="hidden" /> <p><strong>Frustratingly difficult</strong> - depending on which card you get first, it can be really tough because there is nothing that gives it away. Often it takes a few goes to get used to it..... try these two shown below........</p> <p ><img alt="" src="files/mathstudies/images/blog/picturethis.jpg" style="width: 500px; height: 400px;" /></p> <p><strong>Blindingly obvious</strong> - Then, once you know what it is, you often can't believe that you couldn't see it before!</p> <p><strong>Satisfying</strong> - and eventually you begin to get the hang of it and an eye for the key features of the images that give it away and experience a wonderfully satisfying certainty that you know what it is. It is a similar feeling to the one you get when you first <em>see</em> the <a href="http://www.magiceye.com/" target="_blank">'magic eye'</a> pictures.</p> <p>Click below to see the answers......</p> <section class="tib-hiddenbox"> <p>The two images shown are a) a tape measure and b) a box of toothpicks</p> </section> <hr class="hidden" /><br /> <h2>Why valuable?</h2> <p>Well, I think what happens to us when we look at these pictures is fascinating, particularly when you have an instinctive reaction to what you see. For example, maybe, like me, you first thought the toothpicks were pencils. Once you have an idea in your head it is very difficult to shift that mindset and imagine it being something completely different. Because we are not seeing 'the big picture' we are prone to making snap, unjustified judgements about what it is. Unless we check ourselves and try to look from a different angle, we are at the mercy of these judgements. </p> <p>I think this happens to students when they tackle exam questions and they go on to make related mistakes. Often when you give the paper back and point out the important feature of the diagram/question that they missed, they experience clarity and say 'I can't believe I didn't see/do that the first time' - much like you can't believe you didn't see the toothpicks in the picture above (perhaps you did, but believe me there are others there you wont get first time!). Bridging the gap between the first 'instinctive response' to the information shown and the 'clarity' becomes a significant goal. Paying attention to key details that give the bigger picture away is crucial and often the reason students do well (or not as the case may be).</p> <p>I found myself recounting to a class today, the lovely picture that Andrew Wiles paints of his exepriences with mathematics that goes something like this.....</p> <div class="greenBg"> <p><em>When you first confront a new bit of mathematics, it is like going in to an unknown room of a house that is pitch black. To start with, you have to move carefully around the room and build a map of the furniture in it - as you do so, you effectively turn the light on in the room and can see it clearly and the door that leads out the other side so you can repeat the process in the next room until eventually, you have illuminated the whole house perfectly.</em></p> </div> <p>WIles was more eloquent and emotionally charged given that his house was 'Fermat's last theorem', but the analogy holds with the card game and classroom mathematics. Those moments of illumination and clarity are priceless right!</p> <p>So I think the exercise is very valuable and I will get the odd card out for a while and warm students up with this idea. Lets illuminate the toothpicks as much as we can this year I say!</p> <p>.....</p> https://www.thinkib.net/mathstudies/blog/21597/picture-this#1472688000IB Americas
https://www.thinkib.net/mathstudies/blog/21359/ib-americas
Sun, 31 Jul 2016 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/img_6369.jpg" style="float: left; width: 200px; height: 150px;" />A trip across the pond!</h2> <p>So I was lucky enough to be taken to the IB Americas conference with the 'inthinking' team a couple of weeks ago. We went to man an exhibition stand aimed at spreading the gospel of the inThinking subject sites on that side of the pond. Of course, that comes with being able to attend one of these events and meet a whole host of educators from different backgrounds and this is always a fascinating experience and broadens the horizons. As an added bonus for me, <img alt="" src="files/mathstudies/images/blog/cnt.jpg" style="float: right; width: 100px; height: 100px;" />the conference was in Toronto, Canada, where I lived for a while as a boy. We returned home as a family in 1983 and this was my first visit back since! I was given a warm welcome by old family friends but also found the city buzzing with a genuine firendliness that was very welcoming. That was then extended into the conference where I found educators, mostly from the americas region, to be very warm, friendly, curious and interesting. It was great to chat to teachers hopefully many of them will become subscribers soon! </p> <hr /> <h3>What was my sales pitch?</h3> <p>There was a constant flow of educators through the exhibition hall and at breaks and lunches they were lured in with coffee, snacks and treats which menat that we were very busy indeed. As such, it became quite important to try and be as succinct as possible about what it is we do with the thinkIB.net sites so here is what I generally said.... It was nice to be constantly reminding myself of what I am setting out to do! </p> <div class="greenBg"> <p>Our websites are all about helping teachers to make their classrooms rich, engaging and effective</p> <p>The DP teacher has a lot on their plate at the moment - the ATTL initiative is a great summary of all the apsects of the IBs vision for great education and reading it can be daunting! Our websites offer support for teachers to deliver that philosophy including,</p> <ul> <li>Great ideas for rich, thought provoking and inquiry based classroom activity</li> <li>Ready to go resources to support those ideas</li> <li>Huge amounts of support resources for essential practice</li> <li>All the advice a teacher could need for mananging the course and its assessment (both internal and external)</li> </ul> <p>Would you like to have a look?</p> </div> <hr /> <h3>The Americas</h3> <p>It was also reallly good to talk about some of the factors of teaching IB that are particular in this region. I was struck reading statistics about the number if IB schools in Canada that are bilingual schools (French and English) and the issues this presents. On the same theme many of the educators at the conference are offering the IB in Spanish - whilst I could do the French I had to refer to my colleague to handle the spanish requests.</p> <p>I have long been aware, but became more so by talking to teachers, about the number of public schools that are offering IB alongside the American curriculum in US schools. This is a real challenge for those teachers and hopefully this site does make life a little easier.</p> <p>On all of these issues, I'd be interested to know if there is anything we could add to help. I'd love to make all of this available in different languages, but this would be a huge challenge and is not on the immediate horizon.</p> <hr /> <h3>Margaret Atwood<img alt="" src="files/mathstudies/images/blog/img_6438.jpg" style="float: right; width: 200px; height: 150px;" /></h3> <p>It is always interesting to listen to keynote speakers at these events and often a great opportunity to hear about interesting lives and experiences. The art of good public speaking really interests me. At conferences, that art needs to be carefully balanced with offering eductaors more than just entertainment but some genuine food for thought. Of course there is a big role for the listener there as well! The closing plenary was delivered by author Margaret Atwood and the main theme of her discourse was to defend the role of the arts in schools. Clearly a popular topic and I have not met a teacher in my career who is not on board with the arts. A few parents and often their children, but mostly it seems that governments and politicians are the biggest threat to the arts and I know there is good reason for some concern. This talk offered an opportunity to reflect on this and mostly contributed to my current wish to see more integration of subjects and less dsicrete separation between the Arts and the sciences. I think this has mostly developed in me as a ToK teacher where I have had more opportunities to reflect on the similarities as well as the differences between the Arts and the Sciences. Its just food for thought for now becuas the structure of the DP makes integration quite a challenge. It is always worth thinking about though.</p> <p>My abiding memory form this talk will be her opening offering which went something like this...</p> <div class="pinkBg"> <p ><em>'We are always hearing that students should be pushed more towards Mathematics and the Sciences because this will make them more employable - well if employability is our main concern then we should seriously consider pushing students in to the porn industry - there is lots of money to be made there'</em></p> </div> <p>Sometimes we need to hear policies taken to extremes to think about their wisdom - and again it makes me reflect on this choice that we have about what we put at the top of our priority list - Economics first and then social progress or the other way round. We can argue that either way, the former will take care of the latter, but lately I am not so sure....</p> <hr /> <h3>Breakout sessions</h3> <p>I didn't get to go to as many of these as I would have done as a delegate since I had exhibition hall duties, but there were some great sessions on offer. I did try to get to the session where there would be information on the changes to the mathematics courses to get a sense of peoples reactions. I made it at the end where I learned that it would be offered again because of the number of people who had not been able to get in! It is not surprising that there is interest in these changes since they are going to take some adjustment. I made the second session and was asked to say a few words about the curriculum review process to reassure teachers that we are fellow classroom practioners working hard to try and do the right thing and make the transition as smooth as posisble. If you haven't already read the latest reports on the OCC then I recommend that you do! You can start by reading the last blog post I wrote on this subject - <a href="http://www.thinkib.net/mathstudies/blog/19691/curriculum-review" target="_blank">Curriculum review</a>. There should be another one soon.</p> <hr /> <h3>Inthinking brochure</h3> <p>It was great to see and hand out the new brochure we have for subject sites. If you are ever teling a colleague about the sites and want to show them the brochure then hit the 'Download our brochure' button at the top of <a href="http://www.thinkib.net/sites-overview" target="_blank">this page</a>.</p> <hr /> <h3>Photos</h3> <p>As ever, like most of us these days, I am always on the look out for bits of mathematics to photograph. We came across this fabulous inflatable sculpture by the harbour which really caught my eye. With a distinctly parabolic appearance and lots of all kinds of curves, I thought it was another nice example of some mathematical art.</p> <p ><img alt="" src="files/mathstudies/images/blog/img_6197.jpg" style="width: 250px; height: 188px;" /><img alt="" src="files/mathstudies/images/blog/img_6201.jpg" style="width: 250px; height: 188px;" /></p> <p >In search of symmetry - I have an underused instagram account where I post pictures of (often) lovely bits of symmetry that I see. Here are a few that I caught on my trip to Canada, including the fabulous, if a little brown, carpet designs on the exhibition room floor!</p> <p ><img alt="" src="files/mathstudies/images/blog/img_6391.jpg" style="width: 150px; height: 150px;" /><img alt="" src="files/mathstudies/images/blog/img_6397.jpg" style="width: 150px; height: 150px;" /><img alt="" src="files/mathstudies/images/blog/img_6258.jpg" style="width: 150px; height: 150px;" /><img alt="" src="files/mathstudies/images/blog/lake.jpg" style="width: 150px; height: 150px;" /><br /></p> <p >Thanks Toronto - it was nice to see you again</p> https://www.thinkib.net/mathstudies/blog/21359/ib-americas#1469923200Results!
https://www.thinkib.net/mathstudies/blog/21262/results
Thu, 07 Jul 2016 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/results.jpg" style="float: left; width: 100px; height: 100px;" />What is the bottom line?</h2> <p>As my school enters its summer holidays and a new set of exam results are due, I wanted to share some related reflections I have had. The truth is, this time of year terrifies me and excites me all the same. The prospect of a long summer holiday is great and usually comes with a flurry of ideas, plans and resolutions for the future. Then, before the holidays have even really kicked in to gear, the IB results come out and I get as nervous about them as I did when they were mine! I have always subscribed to the philosophy that exam results should be a symptom of a good education rather than the goal itself, but I know how important the right numbers are to a) students who need them to pass on to their next stage and b) managers and directors who care for the reputation of the school. I would be in denial if I said they weren't important to me too. Obviously they are for the two reasons listed above, but also because you need them to justify the afore mentioned philosophy. I was recently asked (in an environment where I needed to answer carefully) what I was expecting from exam results this year by some one who is a former teacher. In typical fashion my focus is on what I wish I had said rather than what I did. Here is what I wish I had said....</p> <div class="greenBg"> <p><em>Well, as you might remember from your classroom days, you have tried your best to offer the rounded, engaging, focused, sound, rigorous and worthwhile mathematics education that you set out to do. You have designed tasks, given feedback, had meetings, given pep talks, run extra sessions, set personal targets and had a few sleepless nights worrying if you have done everything you can for the students. Despite all that, there is no real way of predicting what will happen because, at the end of the day, our business is people not products.</em></p> <p><em>I expect some students to be elated and proud of how their efforts paid off as they achieved their goals or higher. I expect some to be disappointed and some to wonder why they bothered when all the system has done is confirm their lack of worth. I expect others to have a cursory glance and then go back to their xbox. I expect some to be grateful and others to disappear in to the ether. I expect to feel a wave of personal pride at having made a difference to some peoples lives, and then to put my head in my hands and wonder what else I could have done to make a bigger difference even after 20 years in the classroom.</em></p> <p><em>I expect that all the attention will go on the number of top grades and the number of those that missed a passing grade. I expect that no one will reflect on the 5 that should have been a 6 or vice-versa.</em></p> <p><em>Above all that, I expect students to know how to quote and question statistics they read in the news. I expect them to know how mathematics is a powerful tool that helps us describe the world around us, from its diversity, vast inequalities, misconceptions and models for the future. I expect them to have a sense of how mathematics underpins so many other areas of study and how, in itself, it can be a source of wonder and beauty. I expect them to be able to reflect on activities, experiences and discussions that have helped them understand all the things mathematics can be and how that understanding, combined with all their other experiences, helps them to become critical thinking, global citizens for the future. I hope and expect that my students have become more human as a result of their experiences as our charges.</em></p> <p><em>All the while, I hope that students get what they need for the next stage of their lives. I hope that the percentage breakdowns fit nicely on a graph that can be shown and shared to make sure that my focus can remain on the above and the philosophy that exam results are a desirable symptom of a good education rather than the principle goal.</em></p> <p><em>Sure – I understand about bottom lines, believe me, they are never far from my thoughts. I am actually really proud of a host of wonderful results throughout the history of our department, but nothing like as proud as I am of what we offer students by way of an education and of the students who grab those opportunities at what ever level they are operating.</em></p> <p><em>That’s what kind of results I expect this year!</em></p> </div> <p>As term wound down and I had a couple of odd lessons left over, I showed my Maths Studies class an episode of 'Numbers' - the FBI show where they employ a consultant mathematician to help them solve crimes. The episode focussed on a professor who was working on a system of 'sabermetrics' that was being apllied to people in general. An analysis of statistics that determine which people, which neighbourhoods etc that would benefit most from investment so that money wasn't wasted on the others. A really scary, but beleivable idea for all the holywood licence that they use. The end of the episode focusses on the differences between developing mathematics and using mathematics and about recognising how it can be used as a force for good and evil. The episode prompted me to to remember this anonymous quote...</p> <div class="yellowBg"> <p><em>Dear Teacher,<br /> <br /> I am a survivor of a concentration camp. My eyes saw what no man should witness:<br /> <br /> Gas chambers built by learned engineers.<br /> <br /> Children poisoned by educated physicians.<br /> <br /> Infants killed by trained nurses.<br /> <br /> Women and babies shot and burned by high school and college graduates.<br /> <br /> So I am suspicious of education. My request is: Help your students become human. Your efforts must never produce learned monsters, skilled psychopaths, educated Eichmanns.<br /> <br /><br /> Reading, writing, arithmetic are important only if they serve to make our children more human.</em></p> </div> <p>So - I suppose I have just been reminding myself that YES, we and our students will be judged by numbers, grades and statistics and YES that is important and OK, BUT NO it is not the main thing by which we should judge our success as teachers nor that by which students should judge their own. The IB philosophy, learner profile and ATTL demands a good deal more than that which will ultimately not be tested by exams and there is always more work to be done on that!</p> <p>Have a great summer Northern Hemisphere teachers and a great weekend everybody else!</p> <p>Jim</p> <p>....</p> https://www.thinkib.net/mathstudies/blog/21262/results#1467849600Further Processes
https://www.thinkib.net/mathstudies/blog/20388/further-processes
Thu, 11 Feb 2016 00:00:00 +0000]]>Further Processeshttps://www.thinkib.net/cache/blog-thumbs/21/20388-1455222238-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/20388/further-processes
<h2><span ><img src="https://www.thinkib.net/cache/blog-thumbs/21/20388-1455222238-thinkib.jpg" alt="Further Processes" /><br /><br /></span>What counts and what doesn't?</h2> <p>The following is a reply I have found myself giving often regarding the use of 'Further processes' in Internal assessment. the question is <em>'Is there a comprehensive list of processes that can be considered as 'Further processes' in criterion C of the Internal assessment?' </em>It is a great question and it makes perfect sense to ask it. Unfortunatelty, the short answer is no, but there are some related issues worth considering. Here is my reply.....</p> <p><em>'I understand the frustration, but there is not a comprehensive list. In defence of their (IB) position, there is a desire to leave some options open and if there was a comprehensive list then this would rule out anything that wasn't on it. In criterion 4 of the IA markscheme it says 'Examples of further processes are differential calculus, mathematical modelling, optimisation, analysis of exponential functions, statistical tests and distributions, compound probability' Of course, this doesn't tell us what exactly we would expect to see. eg what would we expect to see for compound probability? This is part of the reason so many opt for the statistical test, because it is easy to know what is is expected. Of course, many studies students find stats the more interesting and relevant part of the course for them too. The next level issue is that given the greyness surrounding what we might expect to see for - say modelling - it is open to interpretation and so whilst you may argue well that it should be considered further, a moderator may disagree... BUT, whilst this may all sound a bit inconclusive, <strong>I honestly believe that the best approach is to help students follow what they are interested in and do some meaningful mathematics, then try your best to justify the marks you give them.</strong> If you think the modelling is good enough (ie not technology only) then go ahead and mark it as further. Happy to hear anyone elses views on this post and my conclusions and hope I have helped a little..... Jim' </em></p> <p>So you can see the problem! What I would like to see in future guidance though is some specific examples of what a moderator will be looking for if students are to get credit for processes other than statistical tests as further processes. I have a few suggestions on that based on the examples on the TSM and years of podering that question! DISCLAIMER - The following are just suggestions based on my opinion. I would give the following credit as further processes if done correctly and relevantly.</p> <div class="blueBg"> <h3>Optimisation</h3> <p>The general model for this is to set up a problem like, how to package a given volume with minimum surface area of material and given constraints. Then students collect data by examining particular cases and build towards deducing a model for the surface area and optimising with calculus.</p> </div> <div class="greenBg"> <h3>Modelling</h3> <p>This is essentially about collecting information/measurements and trying to find a function that fits the data so that it can be used for forecasting. Of course, like with statistical tests, this can be done with a variety of tech so we have think hard about what manual mathematics we might expect to see....</p> <p>Example - Say a students is trying to fit an exponential curve to some growth (population?) data. I would expect students to sketch a curve through the points and from tjat curve and the data make estimates or duduce the value of both the y - intercept and the equation of the assymptote. Since...</p> <p ><span class="math-tex">\(f(x)=k\times { a }^{ x }+c \)</span></p> <p >and the y-intercept is at (0, k+c) and the assyptotoe is at y = c</p> <p>we could reasonably expect that students then deduce the values of k and c. They might then create a dynamic function that allows them to find the best value of a to make that fit. In this sense the student has done better than a tech only solution and better than a random sliding of variables to make it fit. they have understood the key properties of the data and used them to deduce a possible model.</p> </div> <div class="pinkBg"> <h3>Compound Probability</h3> <p>I have always thought that this should be a problem that involves more than 2 events and conditional probability with probabilities of various combinations being calculated by hand. I am just not confident that a problem with 2 idependent events would really qualify.</p> <p>Example - If you play with the ideas on the <a href="mathstudies/page/2563/fairground-games" title="Logic, Sets & Probability » L,S&P Teaching Ideas » Fairground Games">Fairground Games</a> activity, there are a number of possible avenues. For example, if you get three throws to get the paper ball in the bin, are you more likely to get the second or thrid throw in? A student might determione a relative frequency for the success with the experiment and compare it to more data where people get three throws and so on. thin about it for a while. I think there is lots of possibility.</p> </div> <div class="yellowBg"> <h3>3D geometry</h3> <p>I think this one is definitely harder, but it often goes hand in hand with optimisation. It is hard to think of examples where this is would be used to solve a problem.</p> <p>Example - With the <a href="mathstudies/page/19544/cuboid-challenge" title="Geometry & Trig » G & T Teaching Ideas » Cuboid Challenge">Cuboid Challenge</a> it is possible to make the cuboid from three different pyramids which is nice, for lots of reasons, but mostly because it means they must all have an equal volume because they are all a third of the cuboid they fit in. I think this has potential to be explored in different ways and for some proof, but it would get pretty tricky.</p> <p>Example - When doing this <a href="http://www.teachmathematics.net/page/3096/the-rice-show" target="_blank">rice show activity</a>, the rice was falling in some wonderful shapes. I wonder, for example, how many cones of which dimensions could be made with a kilo of rice..... Again, there is thinking to do, but there is potential.</p> </div> <p>As I said, these are just some suggestions to provoke a little thought on this topic. To finish, I'll re iterate something from my response above. </p> <p><em><strong>'I honestly believe that the best approach is to help students follow what they are interested in and do some meaningful mathematics, then try your best to justify the marks you give them.</strong> '</em></p> <p>Happy marking!</p> https://www.thinkib.net/mathstudies/blog/20388/further-processes#1455148800Exponential (and tooth) decay
https://www.thinkib.net/mathstudies/blog/20298/exponential-and-tooth-decay
Sun, 17 Jan 2016 00:00:00 +0000]]>Exponential (and tooth) decayhttps://www.thinkib.net/cache/blog-thumbs/21/20298-1453029601-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/20298/exponential-and-tooth-decay
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/20298-1453029601-thinkib.jpg" alt="Exponential (and tooth) decay" /><br /><br /></p> <p>Once the technique has been practiced it can be used on other, more complex exaples such as those here in the <a href="mathstudies/page/15507/modelling-world-population-growth" title="Mathematical Models » Functions Teaching ideas » Modelling World Population Growth">Modelling World Population Growth</a> activity.</p> <p>Short, sweet (literally) and powerful!</p> https://www.thinkib.net/mathstudies/blog/20298/exponential-and-tooth-decay#1452988800Approaches to teaching and Learning
https://www.thinkib.net/mathstudies/blog/19754/approaches-to-teaching-and-learning
Wed, 07 Oct 2015 00:00:00 +0000<p ><img alt="" src="files/mathstudies/images/IBcore/ATL.jpg" style="width: 700px; height: 240px;" /><br /></p> <h2>Collecting the evidence</h2> <p>This post comes from my thoughts and experiements related to the unit planners that are likely to be a part of the DP teacher's life very shortly as a result of the <a href="mathstudies/page/19408/attl-and-planning" title="IB Core » ATTL and planning">ATTL and planning</a> initiative. That link outlines a lot of the key issues facing teachers. This post is about the process of writing about teaching and gathering evidence of all the things that are happening. As I have already said, I think it is an admirable aim to ask schools and teachers to reflect on the aspects of the practice and try to make sure they are delivering the aims, objectives and philosophy of the course. They key question still remains - 'how?'</p> <p>I have recently been expriementing with some new pages on the site to try and tackle this. The 'Focus on' pages are designed to allow teachers and students to work more easily by topic. See this example - <a href="mathstudies/page/19749/focus-chi-squared" title="Statistics » Focus - Chi Squared">Focus - Chi Squared</a> - that I published today and am still working on. The idea is kind of like a mini 'unit planner' for teaching this topic. The teaching activities are listed, there are some teaching slides, practice activites and then references to the opportunities for Tok. After that it gets difficult. With unit planners, we are supposed to be showing when we provide opportunities for students to develop the attributes of the learner profile and the key teaching and learning objectives outlined in the ATTL document. </p> <div class="greenBg"> <p><strong>Teaching</strong> - Inquiry based, conceptual understanding, collaboration in local and global contexts, assessment and differentiation</p> <p><strong>Learning</strong> - Thinking skills, research skills, social skills, communication skills and self-management skills.</p> </div> <p>These, of course, blend nicely with the attributes of the IB learner profile and the ways of knowing. These aims shouldn't really be seen as mutually exclusive and so it gets tricky when you start to try and list evidence that you have provided the opportunities. Many of the goals are generic teaching goals that infuse practice in different ways. For that reason, it can seem a little contrived to start listing specific instances when these opportunities occur. As teachers who talk to each other, we know when they crop up, we know they will and we know it wont always be as we planned. Still, the challenge is out there to find a solution to this need for providing evidence of planning without making tenuous lists.</p> <p>In response I have bundled the teaching and learning points together with the learner profile attributes - see the diagram above. Becoming a good commuincator and developing communication skills clearly go hand in hand. Developing thinking skills whilst aspiring to be a thinker and engaging in inquiry based tasks again go well together.</p> <p>As such, I am going to write a little about the teaching of each subtopic on the 'focus on' pages and highlight the use of any of these key words as I talk about the teaching and learning of the topic. Check out my latest effort here - <a href="http://www.thinkib.net/mathstudies/page/19756/focus-arithmetic-sequences" title="Number & Algebra » Focus - Arithmetic Sequences">Focus - Arithmetic Sequences</a></p> <p>Let me know what you think!</p> <p>....</p> https://www.thinkib.net/mathstudies/blog/19754/approaches-to-teaching-and-learning#1444176000Curriculum Review
https://www.thinkib.net/mathstudies/blog/19691/curriculum-review
Thu, 01 Oct 2015 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/CR.jpg" style="width: 150px; height: 144px; float: left;" />Cat amongst the pigeons!</h2> <p>One of the unquestionable strengths of the IBDP is the committment to curriculum review. Each course offered is reviewed in seven year cycles to ensure that the curricula are current, coherent and relevant. In many cases, changes are small, fine tuning, but occasionally some more significant changes are required. For the DP mathematical Studies course, the last review was fine tuning, but the next one is going to be a bit bigger! There are now two documents on the OCC from last June and December respectively that outline the current state of the proposed changes for the next change which is due for first teaching in 2019. This post is a discussion of what we have seen so far!</p> <hr /><br /> <h3>Disclaimer!</h3> <p>The only information published about this is in the two review reports on the OCC. I am not the IB and everything I write here is based on having read and thought about those two documents. As such, the facts are the same as the facts given in the reports, the rest is just discussion of the significance and speculation! Its early days, there is much yet to be decided.</p> <h3>Summary</h3> <p>The following is a summary of the key changes being proposed</p> <div class="greenBg"> <ul> <li>Choice of two courses in group 5 for mathematics - as yet without names</li> <li>Both courses can be done at HL and SL.</li> <li>1st course, much like exisiting SL/HL course only with reduced content to allow for more time developing mathematical thinking skills.</li> <li>2nd course based more on wider 'applications' of mathematics. Possibly like the current mathematical studies course but with a new HL element.</li> <li>A common element to all courses (60hrs)</li> <li>Both SL parts will be subsets of the respective HL courses</li> <li>There will be some intersection between the elements.</li> <li>A combined 'Mathematical Skills and Concepts' element that will encompass the IA.</li> </ul> </div> <h3>The Distinction</h3> <p>This is the key point and we await further clarity on the key difference between the courses. That said, there is plenty of evidence within the reports that points to the following. Again, I must stress that following is speculation based on the reports.</p> <div class="blueBg"> <p>One of the courses will be a development of the current HL/SL arrangement. One imagines that this is still the route we would advise for those with ambitions in the field of mathematics, engineering and other areas that require a significant degree of mathematical competency. This will be more like mathematics as we know it.</p> </div> <div class="yellowBg"> <p>The other course will be a development of the principles behind the current mathematical studies course. This is based on the idea that understanding of mathematical concepts and knowledge is still a fundamental part of any knowledge base AND that there are a broad range of 'applications of mathematics' that apply across a wider range of career paths. </p> <p>The <strong>big change here</strong> is the introduction of an HL element. This is accredited to recognition that fluency in the application and the significance of a wide range of mathematical ideas is increasingly relevant and desirable. The focus here will be less on the mathemmatical structure and more on the mathematical significance. </p> </div> <h3>The Rationale</h3> <p>If anyone has been following <a href="http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en" target="_blank">Conrad Wolfram's computer based mathematics movement</a>, then you will have already have considered his key contention that there is now less need for 'calculation' and more need for 'application' and that this a natural result of significant technological developments (see <a href="http://www.wolframalpha.com">Wolfram Alpha</a> as an example). Watch the talk to get a fuller idea, but the main thrust is that mainstream compulsory mathematics education should move towards teaching the significance and application of techniques and let computers do the calculation. See the example below..</p> <div class="pinkBg"> <p>Consider the Pearson's product moment correlation coefficient. The mathematical structure behind that calculation is a wonderful bit of mathematics, but I would suggest that few truly understand the structure, whilst many understand the principle of the test and significance of the result.</p> <p>The contention is that, for most, only the latter is necessary, BUT that being an expert in the latter is something we should value highly (hence the HL extension)</p> </div> <p>It is important to note that no one in the field is arguing that mathematics as we know it, no longer has a place, but simply that what we consider what is best for compulsory mathematics education to age 18.</p> <p>Of course, the key knowledge question here (sorry, couldn't resist the ToK reference) is the one about the age/stage at which we might start diversifying. If we still need mathematicians then we mustn't close that route to too many too soon. In the case of the IBDP, this is easily handled. At 16 it is fine to ask students to make this choice.</p> <h3>Avoiding Confusion</h3> <p>Some conversations I have had already with teachers suggests that there will be work to do to avoid confusion. It is currently possible to read the documents and conclude that there will be two new equivalent courses called (for now) 'Pure maths' and 'Applied maths'. I think this is potentially confusing since there is no obvious differentiation in the level of mathematics to be seen there. Clearly, closer reading will help, but there will be work to do to help people understand.</p> <p>Let us not beat around the bush, just as everyone knows that the current Studies course is easier (mathematically) than the SL course despite their equal weighting, surely the 'Applications' course will be easier than the 'Pure' course. As with the current structure, the courses serve different purposes. The 'Pure' course will be for mathematicians as we know them whilst the 'applied course' will be for a new kind of mathematician.</p> <p>It is my belief that the net result will be more 'Mathematicians', and more engagement and understanding of mathematical ideas, concepts and applications and that can only be a good thing.</p> <h3>And finally</h3> <p>At presentations and as part of our philosophy for teaching and learning mathematics where I work, we often describe the following three facets of mathematics. (not always using the same words I might add!)</p> <div class="greenBg"> <p >A unique area of knowledge that is built on logical foundations. Its beauty and elegance is to be appreciated and explored both for pleasure and to develop critical thinking skills.</p> <p >A functional tool for survival. These skills are to be thoughtfully explored, acquired and applied.</p> <p >A crucially important way to understand the world that we live in. Mathematical literacy is essential to understanding, inquiring and pursuing the truth about the world around us.</p> </div> <p>As we try our best to make mathematics education about all of those things, it is poossible to conceive that we can have two IBDP maths courses (HL and SL) that are different blends of those ingredients suited to different students for different purposes. I think so!</p> <p>We will watch with interest!</p> <p>.....</p> https://www.thinkib.net/mathstudies/blog/19691/curriculum-review#1443657600To Chi or not to Chi?
https://www.thinkib.net/mathstudies/blog/19648/to-chi-or-not-to-chi
Mon, 21 Sep 2015 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/Stats/ID.jpg" style="width: 150px; height: 150px; float: left;" />When and how can students consider using the Chi squared test of independence?</h2> <p>This blog post is a result of having received lots of related questions on this topic either online or at workshops. I thought I might just write some thoughts down ina blog post before perhaps making a more formal entry about this on the website. So, the following is just a list of key points for students and teachers to consider about this test.....</p> <hr /> <h3>In exams</h3> <p>It is probably worth noting here that students will need to know how to conclude these tests using both of the following methods.</p> <div class="greenBg"> <p>Using the <strong>p-number</strong> - as a measure of the probability of independence. If it is <strong>lower</strong> than significance level (eg 10%, 0.1) then that means that there is a less than 10% chance that the variables are independent so you can conclude that they are not.</p> <p ><em><strong>or</strong></em></p> <p>Using the <strong>calculated chi squared statistic</strong> - as a measure of the error between the observed and expected values. If the error is <strong>bigger</strong> than the critical value (which is related to both the significance level and the degrees of freedom) then we reject the hypothesis that the variables are independent. In exams, the critical value will be given</p> </div> <p>I have heard some brilliant acronyms for helping students to remember this and I am sure they work well. My preference though is to help students really understand what these statitsics mean. P is a measure of probabilityb of independence, the calculated statistic is a measure of error.</p> <div class="greenBg"> <p>Students may also be expected to demonstrate that they know how to <strong>calculate an expected frequency</strong>.</p> </div> <h3>In projects</h3> <p>Students are expected to conclude their tests using the calculated statistic and comparing it to the critical value. (The second of the methods used above). This is presumably because this method allows students to demonstrate the stages of the calculation where the p-number would not.</p> <h4>When is a test appropriate?</h4> <p>Although not an official rule, it is good guidance for our students that .....</p> <div class="pinkBg"> <ol> <li>IF they are looking for a relationship between two numerical data fields they should use a scattergraph.</li> <li>IF one or both of the data fields are categorical, they should use a Chi squared test of independence..</li> </ol> </div> <h4>What if one of the data fields is numerical?</h4> <p>In this case, students need to put this data in to categories. For example, a student might test to see if GDP is dependent on the hemisphere in which the country is in. In this case students would need to decide on categories for GDP. These might be Low, medium and high. In many cases, this is an arbitrary decision and students should try to justify it. One approach in this case might be to use the first quartile of the results as the 'low' category, the interquartile range as the 'medium' and the upper quartile as the 'high'.</p> <h4>Contingency table</h4> <p>Students should then tally all of their data in to one of these tables. For the example gievn below, the table might look like this.....</p> <p ><img alt="" src="files/mathstudies/images/blog/chis.jpg" style="width: 600px; height: 151px;" /></p> <p>This would be a 2 x 3 table ans students would take each country in the survey and put a tally in the appropriate cell. Perhpas the country is in the Northern hemisphere but has a low GDP and so you would put a tally in the top left column. At this stage, it is good to have data on lots of countries so there is a decent total. Otherwise there is a risk that the low frequencies will undrmine the test (see below).</p> <h4>Beware</h4> <p>Once students have put the observed frequencies in their contingency table, then they should do a quick test - probably with the GDC - to check the expected frequencies. They must stick to the following rule.</p> <div class="pinkBg"> <p ><strong>NO EXPECTED FREQUENCIES MUST BE LESS THAN 5</strong></p> </div> <p>This is said to undermione the test since at some point in the calculation we divide by the expected frequency. If that number is too small then the effect of small variations is disproportionately large.</p> <h4>Calculations</h4> <p>There after, students are expected to show how they calculate the error for each of the six (in this example) cases. See the example below. It is not a complete example, but just shows the highlights. This test was done to test for a relationship between the gender of a youtuber and the number of subscriptions to their channel.</p> <p ><img alt="" src="files/mathstudies/images/blog/LF1.jpg" style="width: 500px; height: 138px;" /></p> <p ><img alt="" src="files/mathstudies/images/blog/LF2.jpg" style="width: 500px; height: 200px;" /></p> <p ><img alt="" src="files/mathstudies/images/blog/LF3.jpg" style="width: 500px; height: 156px;" /></p> <p ><img alt="" src="files/mathstudies/images/blog/LF4.jpg" style="width: 500px; height: 376px;" /><br /></p> <h4>Yates's Continuity Correction</h4> <p>If students end up with a 2 x 2 contingency table then they are required to use this correction approach. Again, this is related to disproportionate effe t of small variations on smaler numbers (degree freedom just 1).</p> <p>This is easily done - When we do Observed frequency subtract expected frequency, we then subtract a further 0.5 before squaring. Students are only required to recognise the need and use the correction rather than explain why it is needed.</p> https://www.thinkib.net/mathstudies/blog/19648/to-chi-or-not-to-chi#1442793600Life's Complexities
https://www.thinkib.net/mathstudies/blog/19589/lifes-complexities
Thu, 10 Sep 2015 00:00:00 +0000]]>Life's Complexitieshttps://www.thinkib.net/cache/blog-thumbs/21/19589-1441876258-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/19589/lifes-complexities
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/19589-1441876258-thinkib.jpg" alt="Life's Complexities" /><br /><br />Hope that we can understand</h2> <p>This is a great <a href="http://www.ted.com/talks/hannah_fry_is_life_really_that_complex" target="_self">10 minute video</a> from <a href="http://www.hannahfry.co.uk/" target="_blank">Hannah Fry</a> based on how mathematics might be able to help us answer the question above. She starts by giving us examples of why things might seem so hopelessly complex that we are never likely to get any understanding of them and then cleverly shows us some other examples of how maybe we might. There is a nice useable example in there about the last London riots and the distances the rioters lived from the places they rioted. Equally there is some more complex mathematics about how 'Burglary hotspots' resemble the patterns of leopard spots - and an understandable explanation for why!</p> <p ><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/LnQYJa9-aR0?rel=0" width="560"></iframe></p> <p>I think this cane be used in two ways.......</p> <p>1. The very real example of the correlation between distance rioters live from the riot sites and the number riots is a good contextual example to use when looking at this bit of statistics. </p> <p>2. That combined with the other example is great for 'Project Inspiration' it is a good length and I think students will find it interesting.</p> https://www.thinkib.net/mathstudies/blog/19589/lifes-complexities#1441843200Sabbatical
https://www.thinkib.net/mathstudies/blog/19267/sabbatical
Mon, 13 Jul 2015 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/blog/Richelonhue13smal.jpg" style="width: 120px; height: 120px; float: left;" /><br />Something different!</h2> <p>Hi all, this is just a quick post to share my plans for the near future with you. Term has just ended for us here in Toulouse and the long summer holidays are just beginning. This year felt a bit odd for me though since I am not planning to go back to work there until January 2016. I have been negotiating with my school for ways to change things a little so that I can develop some more ideas I have to help teachers and students of mathematics and, in particular, those of the IB Mathematical Studies courses. So, all that means that I have the next 6 months set a side to really take this website to the next level amongst other things.</p> <p>I must confess that I am quite looking forward to a little break from the madness that is school life. I guess most of us feel like that as holidays approach. I like the idea of being able to manage my own time and am about to find out if I have the discipline. I also think that I will be really keen to get back to school in January. That will be swiftly followed by being totally exhausted 2 weeks later no doubt.</p> <h3>Priorities</h3> <p>Anyway, as I have already said, taking this site to the next level is high up my priority list. I will also be spending some time on <a href="http://www.teachmathematics.net/">teachMathematics</a> posting lots of new ideas. Along with that, I am working on some student support resources for Maths studies - more to follow. With reference to this site, here are some of my immediate priorities.....</p> <p><em><strong>More great ideas and resources for teaching </strong></em>- I still think this is the most important part of our jobs and I have lots of ideas in the making that need polishing and posting.</p> <p><em><strong>More IB style questions</strong></em> - A number of you have made this request and so this is a major area that I hope will blossom in this time period.</p> <p><em><strong>More on the IB core</strong></em> - with the new requirements for evidence of unit planning, I will help teachers to document and take advantage of the opportunities there are to help students develop learner profile attributes and make the important links with ToK and so on.</p> <p><em><strong>Correction of errors</strong></em> - I know that these errors are littered around the site and I do try my best to track them down and correct.</p> <h3>Feedback</h3> <p>I'd love to hear from people about what they think of the site and what areas they would like to see developed. Let me steer you to this page again so you can <a href="mathstudies/page/18637/tell-us-what-you-think">'Tell us what you think!'</a></p> <h3>Workshops</h3> <p>I will also be doing some face to face and online workshops in that time. Please think about signing up if you can and recommend to your colleagues and friends in other schools</p> <p><em><strong>Inthinking Workshops</strong></em></p> <p><a href="http://www.inthinking.co.uk/site/workshop/2015-09-25-bcn.htm" target="_blank">Category 1 - New teachers (new and new to IB) Barcelona, 25th to 27th September 2015</a></p> <p><a href="http://www.inthinking.co.uk/site/workshop/2016-02-05-bcn.htm" target="_blank">Category 2 - Experienced teachers Barcelona, 5th to 7th February 2016</a></p> <p><em><strong>Philpot Education</strong></em></p> <p><a href="http://philpot.nl/maths-studies">Category 2 - Experienced teachers, Amsterdam, 17th - 19th September</a></p> <p><em><strong>IB Online workshops</strong></em></p> <p><a href="http://www.ibo.org/en/event/279678" target="_blank">Category</a><a href="http://www.ibo.org/en/event/279678"> </a><a href="http://www.ibo.org/en/event/279678" target="_blank">3 </a><a href="http://www.ibo.org/en/event/279678">- </a><a href="http://www.ibo.org/en/event/279678" target="_blank">IBDP</a><a href="http://www.ibo.org/en/event/279678"> </a><a href="http://www.ibo.org/en/event/279678" target="_blank">Mathematics</a><a href="http://www.ibo.org/en/event/279678"> </a><a href="http://www.ibo.org/en/event/279678" target="_blank">and</a><a href="http://www.ibo.org/en/event/279678"> </a><a href="http://www.ibo.org/en/event/279678" target="_blank">ICT</a><a href="http://www.ibo.org/en/event/279678">, </a><a href="http://www.ibo.org/en/event/279678" target="_blank">19th August</a><a href="http://www.ibo.org/en/event/279678"> - </a><a href="http://www.ibo.org/en/event/279678" target="_blank">16th</a><a href="http://www.ibo.org/en/event/279678"> </a><a href="http://www.ibo.org/en/event/279678" target="_blank">September, </a>2015</p> <p><a href="http://www.ibo.org/en/event/279702">Category 3 - IBDP Mathematics and ICT, 25th November - 23rd December, 2015</a></p> https://www.thinkib.net/mathstudies/blog/19267/sabbatical#1436745600Chi Squared, regression and causation
https://www.thinkib.net/mathstudies/blog/18868/chi-squared-regression-and-causation
Tue, 14 Apr 2015 00:00:00 +0000<h2><img alt="" src="files/mathstudies/images/Stats/ID.jpg" style="width: 150px; height: 150px; float: left;" />Both tests on the same data?</h2> <p>This is a quick blog post about a couple of discussions I have had recently about students using both Linear regression and independence tests with the <em>same</em> data! I wrote this in a correspondence recently and thought it worth sharing. Comments welcomed!</p> <div> <div> <div> <div>'There is not an official line that you cannot use both tests on the same data. My hypothesis is that it is irrelevant to use the two tests on the same data. As I rule I think is good to encourage students to use scatter graphs with numerical data and Chi<sup>2</sup> if one or both of the variables are categorical. Clearly correlation does not imply causation, but a chi² test does not imply causation either. The two tests both look for a relationship between 2 variables. If both variables are numerical then a scatter is appropriate, if either or both are categorical then chi² is appropriate. Clearly a chi² test can be done with 2 numerical variables by categorising and it would not be wrong but I can't see a need for it? The outcome of the chi² test would be entirely dependent on the chosen class intervals used to create the categories which could maybe be adjusted to suit an outcome.</div> I am happy to be corrected on this and usually raise this at workshops because I find it interesting. My challenge to teachers is to produce example data where one test genuinely offers something the other doesn't (ie no correlation, but dependence). I am not saying it doesn't exist, but given the somewhat arbitrary nature of choosing class intervals I suspect that it doesn't..... but I haven't ruled it out. Pragmatically, for mathematical studies I think it highly unlikely that students will successfully differentiate between the value of the different tests done on the same data.<br /> <br /><br /> HOWEVER - this does not mean students cant use both.... eg a project might be investigating literacy rates and involve a scatter graph of literacy against GDP, and then a chi² to see if literacy rate is dependent on continent for example. This is great. In this case the student has used both tests to investigate a theme involving literacy rates but not on exactly the same data.</div> </div> IF students have done both tests on the same data then I think it is difficult to mark because the student would need to justify what one offers that the other doesn't in order for it to be considered relevant.</div> <p><span >I summary, my advice is that students avoid doing both tests on the same data, but there is no official line that it cant be done. If you want to give students full marks then I would advise noting to the moderator where you see the difference. Otherwise I think the relevance can be questioned.'</span></p> <p><span >......... to be continued</span></p> https://www.thinkib.net/mathstudies/blog/18868/chi-squared-regression-and-causation#1428969600This years projects
https://www.thinkib.net/mathstudies/blog/18737/this-years-projects
Sun, 22 Mar 2015 00:00:00 +0000]]>This years projectshttps://www.thinkib.net/cache/blog-thumbs/21/18737-1427053909-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/18737/this-years-projects
<img src="https://www.thinkib.net/cache/blog-thumbs/21/18737-1427053909-thinkib.jpg" alt="This years projects" /><br /><br /> <h2>Some Good ideas</h2> <p>As many of you probably are, I am currently putting the finishing touches on this years batch of Internal Assessment projects. It always seems like a monster exercise, but one that is worth it in the end. This year, I am particularly pleased and the main reason is that I feel that all the students in my class got lots out of their projects. The understood the goals and the exercise really felt like it is supposed to philosophically. It isn't that it normally doesn't just particularly so this year. Marking the projects was a real pleasure because students had done some really interesting work and because they have paid attention to my advice! I thought I would quickly share some of the topics they chose.</p> <p><em><strong>Youtube</strong></em> - inspired by the picture above, this was a great project about you tube channels. Who runs them? What are they about? How many subscribers - and so on. There is so much data there and youtube really plays a part in peoples lives these days one way or another. </p> <p><em><strong>Flickr</strong></em> - What makes a photograph popular on Flickr - again, these websites collect so much data and we can be sure that they are analysing it so it seems ripe for students to do so.</p> <p><em><strong>Dance</strong></em> - This seemed like a challenge at first, but this student worked really hard to collect some useful primary data on flexibility and then looked to see what types of people (ie dancers) might be more flexible.</p> <p><strong>GDP and CO2 emissions</strong> - is there a link?</p> <p><strong>Book publishing </strong>- These were some interesting stats I had never seen before. Data on how many books are published in a given year and the literacy rates for different countries.</p> <p><strong>Spotify</strong> - More online data collection - what makes a song popular.</p> <p><strong>Broadway</strong> - what makes more money and how, musicals or plays?</p> <p><strong>Films, cars, smoking and divorce rates</strong> made up the rest and all round I really enjoyed the diversity! No non stats projects this year, but students chose their own areas of interest and I, for one, was pleased with the results!</p> <h2></h2> https://www.thinkib.net/mathstudies/blog/18737/this-years-projects#1426982400Project Ideas
https://www.thinkib.net/mathstudies/blog/18680/project-ideas
Sat, 14 Mar 2015 00:00:00 +0000]]>Project Ideashttps://www.thinkib.net/cache/blog-thumbs/21/18680-1426321139-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/18680/project-ideas
<h2>Mind maps</h2> <p><!--cke_bookmark_77S--><!--cke_bookmark_77E-->Projects are on our minds at the moment! I am just final checking my IB2 projects before submitting marks and just starting my projects with IB1s. this year I have been really happy with the build up to that and really feel like students have a good handle on how to choose good themes. I have more to say about this, which I will put on the <a href="http://www.thinkib.net/mathstudies/page/2075/internal-assessment" title="Assessment » Internal Assessment">Internal Assessment</a> section! Essentially though, the build up results in the production of three 'Mind maps' about project ideas that show the theme, the information and the potential for different analysis. It seems simple, but it has been really effective. Students are asked to produce three of them and then I will help them choose the one with the most potential. Like I said, there is much more to say about this, but for now - which of these ideas would you encourage your students to pursue?</p> <p ><br /> <img src="https://www.thinkib.net/cache/blog-thumbs/21/18680-1426321139-thinkib.jpg" alt="Project Ideas" /><br /><br /></p> https://www.thinkib.net/mathstudies/blog/18680/project-ideas#1426291200Perceived Corruption
https://www.thinkib.net/mathstudies/blog/18049/perceived-corruption
Thu, 04 Dec 2014 03:30:00 +0000]]>Perceived Corruptionhttps://www.thinkib.net/cache/blog-thumbs/21/18049-1417688584-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/18049/perceived-corruption
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/18049-1417688584-thinkib.jpg" alt="Perceived Corruption" /><br /><br /> <div><iframe frameborder="0" height="520" src="http://media.transparency.org/maps/cpi2014-640.html" width="640"></iframe></div> <div>There is so much potential here for statistical analysis and discussion and probably a whole project. Here are just a few ideas...</div> <div></div> <ul> <li>Make hypotheses about the table before the results are shown. Maybe even survey students based on these hypotheses.</li> <li>Use measures of central tendency to compare different geographical areas.</li> <li>Plot box and whisker diagrams to show spread in those areas.</li> <li>Plot scattergraphs of the results of two versions of the survey and do some linear regression.</li> <li>Do the same for geographical areas - which areas show the most correlation? - why might some show more than others?</li> <li>Ask questions about the data collection methods and samples. Look through the website for answers.</li> </ul> <p>I am sure there is so much more, but I am just getting the ball rolling. Thank you Internet for continuing to supply such rich and relevant data for us to work with.</p> <div></div> https://www.thinkib.net/mathstudies/blog/18049/perceived-corruption#1417663800Theory of Knowledge
https://www.thinkib.net/mathstudies/blog/17966/theory-of-knowledge
Sun, 16 Nov 2014 03:30:00 +0000]]>Theory of Knowledgehttps://www.thinkib.net/cache/blog-thumbs/21/17966-1416145522-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/17966/theory-of-knowledge
<h3><img src="https://www.thinkib.net/cache/blog-thumbs/21/17966-1416145522-thinkib.jpg" alt="Theory of Knowledge" /><br /><br />A personal perspective</h3> <p>I want to use this opportunity to explain how my own personal approach to ToK has developed. Like many teachers who start the IB, the whole idea was very new to me and seemed very much like a distant 'bolt on that other teachers looked after and didn't need to affect the way I taught. After 2 years of teaching IB, I took over the department in my school and the IB coordinator approached me and asked if I would prepare some subject specialist sessions on ToK and Mathematics for the IB1 students as my predecessor had done! The only answer was yes of course, but I found myself quickly in unchartered water! Eight years later I argue that the ToK course has had a bigger impact on my teaching in general, and in particular Mathematics, than anything else. When teachers feel the time is right, I would recommend and encourage as much understanding of the ToK course as possible. Equally, it is very effective to help students make links between mathematics and ToK from an early stage. Understanding the differences in the way bodies of knowledge grow, develop and change in different subject areas is bot fascinating and lends a sharp perspective to what we expect our students to do day on day in our schools.</p> <p>I am now a teacher of the whole ToK course which has brought new riches in terms of the time I get to spend talking to, watching and learning from my colleagues about their subjects and the rich and fascinating links between them and the way we all accumulate knowledge.</p> <p>I am sure many readers are experienced with ToK and equally sure that there are many who haven't traveled there yet. All I can say is that I recommend that you take the chance as and when it comes along.</p> <h3>ToK, Maths Studies and this site</h3> <p>Many of the resources on this site are already charged with ToK issues as I am sure some of you have noticed. I am now developing a section on <a href="http://www.thinkib.net/mathstudies/page/17595/theory-of-knowledge" title="IB Core » Theory of Knowledge">Theory of Knowledge</a> which aims to be more explicit about how and when ypu might bring ToK discussions in to the Maths Studies classroom.</p> https://www.thinkib.net/mathstudies/blog/17966/theory-of-knowledge#1416108600Logic Lab
https://www.thinkib.net/mathstudies/blog/17708/logic-lab
Mon, 13 Oct 2014 03:30:00 +0000]]>Logic Labhttps://www.thinkib.net/cache/blog-thumbs/21/17708-1413206544-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/17708/logic-lab
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/17708-1413206544-thinkib.jpg" alt="Logic Lab" /><br /><br /></p> <p>Here is a list of things I have been thinking about and experimenting with!</p> <ul> <li>Human logic gates - drawing some big ones on the ground and getting the students to be the inputs being either on or off. This could produce some nice pictures and video.</li> <li>Modeling compound logic statements with logic circuits.</li> <li>Getting students to come up with the idea of truth tables themselves as a way of recording all the possible outcomes.</li> <li>Using logic circuits and their outcomes to generate and demonstrate the idea of 'Logical Equivalence'</li> <li>Using logic circuits to demonstrate tautolgies and contradictions.</li> <li>Using logic circuits to model, inverse, converse and contrapositive.</li> </ul> <p>All if this is likely to appear as an activity or series of activities on the <a href="http://www.thinkib.net/mathstudies/page/2233/ls-p-teaching-ideas">Logic, Sets and Probability activity page</a> eventually. In the mean time I can highly recommend that you have a play!</p> https://www.thinkib.net/mathstudies/blog/17708/logic-lab#1413171000May 2014 Results!
https://www.thinkib.net/mathstudies/blog/17283/may-2014-results
Thu, 14 Aug 2014 03:30:00 +0000<h2>Reflections and Resolutions</h2> <p>Each year I am required to write a report on the IB Mathematics results and, although I am still in holiday mode, I am beginning to think about what I will write this year. Obviously I wont publish confidential results here on this blog, but rather some related reflections. The end of one school year and the start of a new year offer great opportunities for reflection, resolution and new starts!the following are just some thoughts related to the teaching of IB Maths Studies that I have had following the results of the last class.</p> <h3>IA marks</h3> <p>I am a great advocate of not worrying too much about what moderations have been made to your IA marks. The system is such that this is beyond our control. If we have done our best to think about and interpret the criteria as fairly as possible and then justify why we did so then that is the best we can do. Sometimes the sample will be simple and representative and sometimes it wont. Sometimes our moderator will agree with us but theirs wont and so on. That said, it is reassuring when your grades are not changed at all which is what happened this year! I have marked up, marked down and both in the same year, but only twice have all of my marks been unchanged. It seems funny to me that the first time was my very first year teaching IB Maths Studies and the second the first run through of a new syllabus! Anyway, I would be citing the system if my marks had been moved and so I will put this down to laws of probability as much as my efforts. I did write these two posts about IA whilst marking this year if that is of interest to anyone....</p> <div><img src="img/user/generic/48-generic-editor.png" style="width: 48px; height: 48px; vertical-align: middle; margin: 4px 12px 4px 0" /> <a href="mathstudies/blog/15939/ia-issues" target="_blank"><span>IA Issues</span></a></div> <div><img src="img/user/generic/48-generic-editor.png" style="width: 48px; height: 48px; vertical-align: middle; margin: 4px 12px 4px 0" /> <a href="http://www.thinkib.net/mathstudies/blog/16828/ia-moderation" target="_blank"><span>IA Moderating</span></a></div> <div><img src="img/user/generic/48-generic-editor.png" style="width: 48px; height: 48px; vertical-align: middle; margin: 4px 12px 4px 0" /><br /><a href="http://www.thinkib.net/mathstudies/blog/16777/moderator-notes" target="_blank">Moderator notes</a></div> <div></div> <div>Whatever the outcome use it to inform what you do in the future and don't worry too much about it. All we can do is our best.</div> <h3>Exam Marks</h3> <p>I think one of the most important indicators for results is how close your results were to the predicted grades we gave to IB earlier in the year. Sure, there are lots of 50/50 students and some of us go for optimism and some are more cautious and you must bear this in mind as we interpret the correlation. This year I over predicted by an average of half a grade. Although I am self confessed optimist this is unusually high for me and so it has been quite a point for reflection for me. Without going in to details about the class in question I have been asking myself the following questions.....</p> <ol> <li>Did my class, in general, react in the same way as other classes to the mounting pressure of impending exam?</li> <li>Did I do anything particularly differently from previous years throughout the course?</li> <li>What impact has the new syllabus possibly had on my ability to predict?</li> </ol> <p>I have answered the first two questions for myself and drawn some useful conclusions. The third is, however, harder to answer and one I will be thinking about a lot this year as I prepare the next class for exams. As author of this site and a workshop leader for Maths Studies I have spent a lot of time reflecting on the subtle changes to the syllabus, trying hard to make sure I covered them all. After the exam and before completing the G2 form in May I did do the papers myself and have published the <a href="http://www.thinkib.net/mathstudies/page/16886/past-paper-review" target="_blank">written solutions here</a>. As I did the paper I did find myself identifying a number of questions I thought my students would find difficult, but did not conclude that they were particularly difficult in comparison to previous years, although I have made a note to check again as term starts. What I did conclude was that still, students are required to take their understanding of mathematical ideas and apply it to less familiar contexts, even on this course. As produce more <a href="http://www.thinkib.net/mathstudies/page/15871/exam-style-questions" target="_blank">'exam style questions'</a> I will bear this in mind.</p> <h3>Implications for teaching</h3> <p>This is the most important part - what might I do differently as a result?</p> <p>I took the decision to get my IB1 class through their IA already while I was thinking about so much during moderation. I think I will repeat this for the new IB1 class. This will have the added benefit of allowing a bit more time to do some more exam preparation with classes in the IB2 year.</p> <p>Pedagogy Vs passing exams - this is the raging debate in education and fascinating one. As day to day teachers we are the ones that have to walk this line. I have always justified an emphasis on 'thinking skills, discovery and exploration' in mathematics teaching by saying that exam results should be a symptom of what we do not the goal. I stand by this and the philosophy that education and in particular maths education, is about so much more than passing exams. It is always worth stopping to check though - if the symptoms stop appearing then there is a problem. I remain certain that the emphasis must remain on sound educational experiences, but having over predicted this year, I am checking myself by thinking about some ways to help students be better prepared for exams.</p> <p>Anyway I really enjoy the time and space to have these reflections and am looking forward to a few more in the weeks before the new term starts for those of us in the Northern Hemisphere.</p> <p>Thanks,</p> <p>Jim</p> https://www.thinkib.net/mathstudies/blog/17283/may-2014-results#1407987000The Intuition Line
https://www.thinkib.net/mathstudies/blog/12473/the-intuition-line
Wed, 02 Jul 2014 03:30:00 +0000]]>The Intuition Linehttps://www.thinkib.net/cache/blog-thumbs/21/12473-1404321344-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/12473/the-intuition-line
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/12473-1404321344-thinkib.jpg" alt="The Intuition Line" /><br /><br /></p> <h3> When do we trust out guts?</h3> <p> This blog post is about a concept I am defining as 'The Intuition Line'. I am sure I am not the first to do so, but it is currently present in my thoughts so I wanted to write it down! In short, I am defining the intuition line as the point at which intuition stops working for students and needs to be replaced with logical analysis. This comes to mind primarily because of recent work my students and I have done on Probability. As well as this I am struck by the number of questions I see that appear to fall one side or the other of this line. It is perhaps best to start by stating an example of what I am talking about.</p> <p> The question, 'what is the probability of throwing a 6 on an ordinary dice?' seems to draw an intuitive correct response - If the question goes on to ask, 'what is the probability of throwing two 6s?', this tends to draw an incorrect intuitive response.</p> <p> All I wish to do really is to ask people to think about when and where this intuition line occurs. How different is it for different people? How is the position of the line affected by teaching? How much of any given exam can a student answer with intuition?</p> <p> I dont think I can answer these questions yet, but I really think they need answering! I'd welcome any thoughts........</p> <p> <iframe id="buffer_tpc_check" src="https://d3ijcis4e2ziok.cloudfront.net/tpc-check.html" style="display: none;"></iframe></p> https://www.thinkib.net/mathstudies/blog/12473/the-intuition-line#1404271800The Problem with Monty Hall
https://www.thinkib.net/mathstudies/blog/16848/the-problem-with-monty-hall
Sun, 11 May 2014 03:30:00 +0000]]>The Problem with Monty Hallhttps://www.thinkib.net/cache/blog-thumbs/21/16848-1399831127-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/16848/the-problem-with-monty-hall
<h2><img src="https://www.thinkib.net/cache/blog-thumbs/21/16848-1399831127-thinkib.jpg" alt="The Problem with Monty Hall" /><br /><br />The 'Google' generation</h2> <p>Those of you familiar with the Monty Hall problem will know what a wonderfully rich bit of mathematics it is. Those of you who are not are faced with a problem the Internet has brought us. I would suggest that you just google it and get stuck in, but the problem is that the Internet ,as it is today, offers many more answers than questions and it is difficult to find things that dont give the answers away before anyone has had time to experience the richness of the problem.</p> <p>This post was prompted by a post I saw on facebook yesterday from the Khan Academy about the Monty Hall problem. Infact, it was a video explanation of the solution to the problem. I am not going to turn this post into a KA bashing session and there are some wonderful explanations of the Monty Hall problem out there. The one I enjoyed most is the simple one printed in the novel 'The curious Incident of the Dog in the Nightime'. In fact the novel offers a lovely summary of the whole story. I particularly enjoyed the quotes from famous mathematicians who were arguing that Marilyn Vos Savant was wrong. This is a lovely story for ToK lessons. Again, though, the problem is that this is only a wonderfully rich problem if people experience engagement with it themselves. Going straight to the solution is like watching the end of the 'Sixth Sense' first. Actually, I like that analogy, because that film is really good the second time too, because you look at everything differently, but the first time is the real experience.</p> <p>The google generation need to be wary that they have so many 'soultions' on offer to them that they are in danger of not experiencing problems. Knowledge will become a collection of solutions, rather than experience of problems. Teachers have the challenge of presenting problems in a way that lets students experience them. This is a must for Monty Hall. I would love to see a web page on the Monty Hall problem, that did not offer a solution anywhere! In fact, In think I'll put it on my to do list!</p> https://www.thinkib.net/mathstudies/blog/16848/the-problem-with-monty-hall#1399779000IA Moderation
https://www.thinkib.net/mathstudies/blog/16828/ia-moderation
Sun, 04 May 2014 03:30:00 +0000<h2><img alt="" src="files/mathstudies/images/IA/IATitle.jpg" style="float:left; height:124px; width:150px" /><br />Moderating</h2> <p>I have just completeed another set of moderating for the Internal Assessment element of the course. It is really important to say that I am only a moderator and that I too am subjecto to moderation and my opinion on the interpretation of marking criteria and the natuire of work is just that, only my opinion. I thought it was worth sharing some thoughts though, just the same</p> <p>In my view it would be more helpful if there was an obligation for teachers to make specific references to the work where they have justified marks. For example. For example, where you might write under criterion C '2 simple relevant correct processes and further process were used' where it would be more helpful to write 'The two simple processes were ..... and ..... which can be seen on page .... and so on.' This is not just born of a desire to make the moderators life easy, but the process would help teachers to check thoroughly for themselves if the candidate has actually completed correct relevant processes. This applies to all of the criteria.</p> <p>Out of 88 projects seen, I only had one 'Non stats' project. I think the new criteria lend themselves more to non stats work than the older criteria did, but they still lend themselves much more to statistics. I don't have a problem with this. I think there are many possibilities to do useful project work with statistics and I saw a number of very interesting project topics. Statistics is a very applicable tool, especially for maths studies students whose interest probably lie outside of mathematics. In general, there was much less of a formulaic approach taken to project work than in previous years. I had a strong sense that students were allowed to choose and pursue their own areas of interest. The downside is that many students chose some very poor limited topics/ideas. The overwhelming majority of projects were base don primary data. Whilst this is fine, it seems a shame given the ever increasing wealth of availability of real relevant data on the internet. Data is the new oil and working with these large quantities of it is a real pursuit these days and so I would like to see more students doing this.</p> <p>The new criterion A and B posed a few issues for me as a moderator. The fine details of both are to be applauded. Students need to explain the purpose of their planned approach and be concise and consistent in criterion A and in B they need to describe their data collection process thoroughly to get full marks. I found that very students had done this. As such the vast majority were awarded 2 in both of these criteria. This has an implication for future teaching. Again, I think these are great additions to criteria and will certainly be placing more emphasis on them in my own teaching.</p> <p>The increased emphasis on 'relevance' and 'conciseness' is excellent, but definitely a challenge for teachers and students. I think it is important that students are exposed to numerous examples of simple, relevant and irrelevant applications of simple statistics and that there is a corresponding emphasis placed on this understanding in classrooms. I think this is a positive change.</p> <p>I have enjoyed working with the new criteria, although it has been a challenge to make the subtle shifts from the old ones. I think the changes are positive and that the project work has become more purposeful as a result.</p> <iframe id="buffer_tpc_check" src="https://d3ijcis4e2ziok.cloudfront.net/tpc-check.html" style="display: none;"></iframe>https://www.thinkib.net/mathstudies/blog/16828/ia-moderation#1399174200Moderator Notes
https://www.thinkib.net/mathstudies/blog/16777/moderator-notes
Mon, 21 Apr 2014 03:30:00 +0000]]>Moderator Noteshttps://www.thinkib.net/cache/blog-thumbs/21/16777-1398088718-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/16777/moderator-notes
<p><img src="https://www.thinkib.net/cache/blog-thumbs/21/16777-1398088718-thinkib.jpg" alt="Moderator Notes" /><br /><br /></p> <p>There are more examples of this in the <a href="http://www.thinkib.net/mathstudies/page/2095/marking-moderation" target="_blank">IA section</a> of the website. You might also find this blog post on<a href="http://www.thinkib.net/mathstudies/blog/15939/ia-issues" target="_blank"> IA issues </a>helpful.</p> https://www.thinkib.net/mathstudies/blog/16777/moderator-notes#1398051000Revision Resources
https://www.thinkib.net/mathstudies/blog/16218/revision-resources
Mon, 27 Jan 2014 08:45:00 +0000]]>Revision Resourceshttps://www.thinkib.net/cache/blog-thumbs/21/16218-1390229568-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/16218/revision-resources
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/16218-1390229568-thinkib.jpg" alt="Revision Resources" /><br /><br />So I am writing this blog post as a statement of intent about revision resources that I need to make for my IB2 class sitting their exam in May. This is the first time through the new syllabus and so I have some replacement of old resources to do. For years I have used some revision packs that I made at the start of the last new syllabus. Most of this is not available through this site because it uses lots of IB copyrighted questions. This year I am going to be revising all of these and making some new original IB style questions to go with them. The following outlines my plans for creating revision packs.</p> <p> I am going to create 6 revision packs, one for each of the main syllabus areas,</p> <div class="greenBg"> <ol> <li> Number and Algebra</li> <li> Statistics (Both sections together)</li> <li> Logic, Sets and Probability</li> <li> Geometry and Trigonometry</li> <li> Mathematical Models (Functions)</li> <li> Calculus</li> </ol> </div> <p> Each pack will consist of the following structure,</p> <div class="pinkBg"> <p> <em><strong>A comprehension task</strong></em> - This is a series of questions designed to prompt students to think about their knowledge. As such they are different from exam style practice questions. Students might typically search their notes or texts in order to answer these questions.</p> <p> <em><strong>Creating quick reference notes</strong></em> - This is an exercise in studnets making the note cards they can use for last minute reviews. making these 'cards' is an excellent exercise and they can be really useful. We need to be realistic about what can be acheived and so I am suggetsing a handful of note cards for each main topic.</p> <p> <em><strong>Practice IB Style Questions</strong></em> - Once students have reviewed ideas they need to try and put them i to practice with some new questions. To do this there should be 10 - 15 paper 1 and 2 style questions that students have never seen before and look like those they might get in the exam.</p> </div> <p> So obviously there is a good deal of work involved in creating these resources, although I will be reworking some old stuff too. the aim is to produce these one at a time based on the following timetable,</p> <div class="blueBg"> <ol> <li> Number and Algebra - Thursday 26th February</li> <li> Statistics (Both sections together) - Thursday 26th February</li> <li> Logic, Sets and Probability - Thursday 5th March</li> <li> Geometry and Trigonometry - Thursday 5th March</li> <li> Mathematical Models (Functions) - Thursday 12th March</li> <li> Calculus - Thursday 12th March</li> </ol> </div>https://www.thinkib.net/mathstudies/blog/16218/revision-resources#1390812300La Face
https://www.thinkib.net/mathstudies/blog/16160/la-face
Mon, 30 Dec 2013 17:04:00 +0000]]>La Facehttps://www.thinkib.net/cache/blog-thumbs/21/16160-1388437483-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/16160/la-face
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/16160-1388437483-thinkib.jpg" alt="La Face" /><br /><br /></p> <h3> Challenges for the new year!</h3> <p> So I am lucky enough to be spending the New Year in the French Alps and enjoying some great wintery weather, lovely snow and some great skiing with my family. The photo on the left is a picture of a black run in Val D'Isere called La Face which bears down on the village offering us all the challenge it presents. It got me thinking about what challenges I will set myself for teaching the maths studies course this year (I may yet chicken out of La Face and satisfy myself with some good teaching challenges instead!) I have also chosen 'Fermats Last Theorem' by Simon Singh as my holiday read am I am really enjoying the ease with which Simon Singh tells the story and makes the mathematics accessible. I love how holidays and all that give you a proper chance to reflect and think about things.</p> <p> Anyway, I have two key drivers for the new year...</p> <p> 1. Applications of mathematics. I want to find more, real applications of the mathematics on the studies course to bring in to the classroom.</p> <p> 2. Bring some mathematical stories to life in the classroom. I want more activities that help students understand what mathematics really is and how it has developed.....</p> <p> watch this space.....</p> https://www.thinkib.net/mathstudies/blog/16160/la-face#1388423040Making Calculators
https://www.thinkib.net/mathstudies/blog/16089/making-calculators
Fri, 22 Nov 2013 07:30:00 +0000]]>Making Calculatorshttps://www.thinkib.net/cache/blog-thumbs/21/16089-1385124034-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/16089/making-calculators
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/16089-1385124034-thinkib.jpg" alt="Making Calculators" /><br /><br /> trig calculator</a> to see how dynamic geometry can be used to create a trig ratio calculator. the main point is that the programming of these calculators is a rich tasks that requires students to really explore with the mathematics and reason with each other to make them behave properly! I am planning to develop these in to classroom activities and will post links here when I have!</p>https://www.thinkib.net/mathstudies/blog/16089/making-calculators#1385105400IA Issues
https://www.thinkib.net/mathstudies/blog/15939/ia-issues
Thu, 10 Oct 2013 07:50:00 +0000]]>IA Issueshttps://www.thinkib.net/cache/blog-thumbs/21/15939-1381427692-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15939/ia-issues
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/15939-1381427692-thinkib.jpg" alt="IA Issues" /><br /><br /> experienced workshop leader</a> for IB Maths Studies and a moderator for the Internal Assessment. Perhaps more importantly than that, I am a full time teacher with Maths Studies classes.</p> <p> Anyway, I hope this was useful - writing it certainly was!</p> <p> *Update on chi<sup>2</sup> tests - Following an exchange with one of the users of this site, I can see that I haven't been quite clear about this regarding how it might be tested in exams. It is quite possible that students will be asked to calculate the chi<sup>2 </sup>statistic and compare it to a critical value in and exam BUT if this happens then the critical value will be given. It is also possible and I think, more likely, that students will be asked to conclude on chi<sup>2 </sup>tests using the p-number from their calculator.</p> https://www.thinkib.net/mathstudies/blog/15939/ia-issues#1381391400IA Surveys
https://www.thinkib.net/mathstudies/blog/15911/ia-surveys
Sun, 22 Sep 2013 13:32:00 +0000]]>IA Surveyshttps://www.thinkib.net/cache/blog-thumbs/21/15911-1379879439-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15911/ia-surveys
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/15911-1379879439-thinkib.jpg" alt="IA Surveys" /><br /><br /> Comparing data distributions</a> activity that both help students to think about good project ideas.</p> https://www.thinkib.net/mathstudies/blog/15911/ia-surveys#1379856720Communicating Mathematically
https://www.thinkib.net/mathstudies/blog/15889/communicating-mathematically
Sun, 15 Sep 2013 11:19:00 +0000]]>Communicating Mathematicallyhttps://www.thinkib.net/cache/blog-thumbs/21/15889-1379266793-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15889/communicating-mathematically
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/15889-1379266793-thinkib.jpg" alt="Communicating Mathematically" /><br /><br /> 3D uncovered activity</a> and came up against some similar issues. The whole point of this activity is about students extracting 2D planes from 3D constructions and then applying trigonometry. In most cases, it was near impossible to trace a students conclusion back to the question because thay had written little or nothing along the way to help.</p> <p> I tried to make my point with students with two examples related to writing,</p> <ol> <li> If asked to do a piece of writing, would students write the words in random unsequential places on the page and expect the teacher to just 'know' what order they are supposed to come in?</li> <li> What use would a book be if you were only offered the first and last chapters?</li> </ol> <p> I know analogies can be dangerous and distracting, but instinctively I suppose I was just observing how there is so much to do to teach people to communicate properly and how that is actually an integral part of strengthening understanding.</p> <p> For maths studies students this a really important point related to project work.</p> <p> Next task - how to place more emphasis on this.</p> https://www.thinkib.net/mathstudies/blog/15889/communicating-mathematically#1379243940Rubik's Race
https://www.thinkib.net/mathstudies/blog/15786/rubiks-race
Thu, 29 Aug 2013 07:01:00 +0000]]>Rubik's Racehttps://www.thinkib.net/cache/blog-thumbs/21/15786-1377694968-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15786/rubiks-race
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/15786-1377694968-thinkib.jpg" alt="Rubik's Race" /><br /><br />this isn't possible because it shows 5 yellows!" I stop and think for a while and then realise she is right and that since each cube in the shaker has one of each of six colours on it that it is technically possible to get them all the same colour. Rather than concluding that the game designers have missed an obvious trick I guess that they have concluded that the odds of the shaker throwing up an impossible combination are so small that it isn't worth worrying about. At this point you wonder whether someone has actually worked it out? made some assumptions? tested it out? What ever the outcome you realise that probability has played an important part in the design of this game as it has with so many others. Quickly thinking that I might blog about this, I went to take a picture of the impossible combination, only to notice that my daughter has already shaken again! I said, "do it again until you get another 5 yellows" She dilligently sets out on this mission and quickly realises that it doesn't happen very often. She then asks "can it be 5 of any colour?" and you see a lovely example of how intuitive <em>some</em> probability can be! So now we are left with the question 'What is the probability that the shaker will show an impossible combination?'</p> <p> Happy summing!</p> <p> .....</p> <p> *** For teaching, I am thinking that whilst the theortical probability might get out of hand here, it could be a good example for some experimental probability. How about keeping records of how many times 2 of the same colour turn up and so on....</p> <p><strong>Tags:</strong> <em>probability</em></p>https://www.thinkib.net/mathstudies/blog/15786/rubiks-race#1377759660Planning Diary
https://www.thinkib.net/mathstudies/blog/15670/planning-diary
Tue, 13 Aug 2013 13:50:00 +0000]]>Planning Diaryhttps://www.thinkib.net/cache/blog-thumbs/21/15670-1376424479-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15670/planning-diary
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/15670-1376424479-thinkib.jpg" alt="Planning Diary" /><br /><br /> 3D uncovered.</a></p> <p> Anyway, I will give some more thought about how best to share my planning through this blog!</p> <p> thanks!</p> https://www.thinkib.net/mathstudies/blog/15670/planning-diary#1376401800How many ways to use a Venn diagram?
https://www.thinkib.net/mathstudies/blog/15498/how-many-ways-to-use-a-venn-diagram
Sat, 29 Jun 2013 11:20:00 +0000]]>How many ways to use a Venn diagram?https://www.thinkib.net/cache/blog-thumbs/21/15498-1372525840-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15498/how-many-ways-to-use-a-venn-diagram
<h3> A functions Venn diagram</h3> <p> Whilst twitter and the like can be a huge distraction, I persist with it because of the frequency with which it offers me a new idea to work with. Yesterday I had a quick look and saw a post from who <a href="http://twitter.com/tesMaths" target="_blank">Craig Barton </a>who was obviously attending a session at the UK based <a href="http://www.conference.mei.org.uk/" target="_blank">MEI Conference</a>. It was simply a photograph, which I have included below....</p> <p > <img src="https://www.thinkib.net/cache/blog-thumbs/21/15498-1372525840-thinkib.jpg" alt="How many ways to use a Venn diagram?" /><br /><br /></p> <p> I retweeted and later in the evening it turned up again as posted by Andy Donahue in our <a href="http://www.facebook.com/groups/149577565103232/" target="_self">IB Maths Studies Teachers facebook group!</a> Anyway, I liked it because it made me think about using venn diagrams as a teaching tool probably for the first time, which of course has double benefits! What else could the three sets be? What else could we isolate with the intersections? I just wanted to share this quickly for now, but will certainly come back to it at a later date.....</p> <p><strong>Tags:</strong> <em>slp,venn,idea,functions</em></p>https://www.thinkib.net/mathstudies/blog/15498/how-many-ways-to-use-a-venn-diagram#1372504800The Bat Cave
https://www.thinkib.net/mathstudies/blog/15440/the-bat-cave
Fri, 21 Jun 2013 05:43:00 +0000]]>The Bat Cavehttps://www.thinkib.net/cache/blog-thumbs/21/15440-1371816088-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/15440/the-bat-cave
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/21/15440-1371816088-thinkib.jpg" alt="The Bat Cave" /><br /><br /> </iframe></p><p><strong>Tags:</strong> <em>modelling,functions</em></p>https://www.thinkib.net/mathstudies/blog/15440/the-bat-cave#1371793380Facebook for Schools
https://www.thinkib.net/mathstudies/blog/11987/facebook-for-schools
Sun, 18 Mar 2012 07:04:10 +0000]]>Facebook for Schoolshttps://www.thinkib.net/cache/blog-thumbs/21/11987-1332065050-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/11987/facebook-for-schools
<p> <img alt="" class="left" height="230" src="/files/mathstudies/images/blog/10fbreasons.jpeg" width="300"></p> <h3> 10 reasons why facebook should be allowed in schools</h3> <p> I am a vociferous opponent of one sided arguments, but recognise their value in helping people to see the issues. I am not unaware of the risks of embracing facebook in schools and certain that one bad experience could change my views. I am even naturally cautious about these things, but do find myself increasingly incredulous that there is such opposition to it. The following is just what the title says it is and not intended as a fullproof argument. The aim is to get people thinking.</p> <h4> More sophisticated communication tool</h4> <p> This is often overlooked. Facebook is - hands down - a considerable evolution of a communication tool. It offers everything that e-mail does on the side, whilst facilitating and encouraging a whole new level of collaboration and community. No doubt that communication in many industries could benefit hugely from such a tool. Why then will it take so many so long to work it out?</p> <h4> Groups not friends</h4> <p> The whole ability to interact with different groups of people without being 'friends' in a facebook sense has really opened up the options for schools to make use of this tool. Whilst it will always be true that all users must think very carefully about what they post, and that privacy and social networking are rarely synonymous, this is an important barrier.</p> <h4> Sharing links</h4> <p> The other day, I wanted to spontaneously share a link with my class. They all have computers with them so I copied the link and pasted it in to a word document which I then saved on a shared network drive with a memorable name. I shouted out the memorable document name to the class who, one by one, navigated to it to click on the link. This was the most efficient method for this spontaneous sharing. I rest my case.</p> <h4> It is real</h4> <p> We can pretend it is not, but it is. It is a phenomenon and it is happening. Making school a place where it does not happen is just helping detach school from reality.</p> <h4> Preferred choice</h4> <p> In my experience, and it wont last forever, facebook is the communication tool of choice amongst students (and an increasing number of adults). When such a high premium is placed on effective communication why would not use the tool of choice, especially when it is more sophisticated than the alternatives? Stood perfectly still in a traffic jam on the way to school I realised I would be late. I posted instructions to the facebook group for the class concerned and when I got to work 20 minutes late students all knew what they were supposed to be doing and were. The e-mail might have been picked later that evening.</p> <h4> Distractions</h4> <p> It is often argued that students having access to facebook in schools would just add another distraction for them and stop them focussing on the task. Well if that is the case then I think we should ban day dreaming as well. We should all ask ourselves what distracts us and why? When are we truly engaged in an activity organised by someone else? What is it about those moments that stops us following the numerous distractions on offer? If a student is not in to the activities in my lesson then there are already a million other things they could be doing as a distraction. There are hundreds of other apps on their computers, not even mentioning the internet, they could be doing homework for another subject, chatting to their neighbours, playing with their webcams or just plain day dreaming. Yes it is another possible distraction but what it adds to the existing choice is minimal. It is not a problem.</p> <h4> When did banning work?</h4> <p> OK, so there must be some answers to this question, but in the case of facebook, well, we can ban it from networks and computers and so they all get out their phones! Not all of course, but I am just trying to point out how difficult it is to stop a phenomenon like this.</p> <h4> Make it work for you not against you!</h4> <p> No explanation necessary</p> <h4> Get staff collaborating</h4> <p> Rightly or wrongly and even though they work ion the same building, staff do not get much time to talk to each other in school. A facebook group for teachers is a fabulous way for staff to share what they are doing, swap opinions and share interesting and relevant links with each other. It is really great.</p> <h4> Get students collaborating</h4> <p> Again, no explanation necessary</p> <h3> How can this work for teachers of Maths Studies?</h3> <p> Here are three ways to get in to this straight away.</p> <h4> Facebook groups for classes</h4> <p> I can recommend highly the use of these groups for effective communication between students and teachers. I have one group that all students doing maths studies in the school belong to All the maths teachers in the school also belong and students stay in the group even after they have left the school which adds a nice dimension as well.</p> <h4> Facebook groups for teachers</h4> <p> Here is a group for IM Maths studies teachers all over the world. <a href="http://www.facebook.com/groups/149577565103232/" target="_blank">IB Maths Studies Teachers</a> - it is a fantastic opportunity to be able to collaborate and share with teachers with the same issues.</p> <h4> Worldwide IB Maths Studies classroom</h4> <p> This <a href="http://www.facebook.com/groups/231815136900310/" target="_blank">group for students of maths studies students all over the world</a> is just that. I am hoping that we could get students and classes all over the world collaborating in this way.</p> https://www.thinkib.net/mathstudies/blog/11987/facebook-for-schools#1332054250Enquiry based activities
https://www.thinkib.net/mathstudies/blog/11818/enquiry-based-activities
Mon, 20 Feb 2012 07:52:51 +0000]]>Enquiry based activitieshttps://www.thinkib.net/cache/blog-thumbs/21/11818-1329735171-thinkib.jpghttps://www.thinkib.net/mathstudies/blog/11818/enquiry-based-activities
<p> <span ><img alt="" class="left" height="150" src="/files/mathstudies/images/GandT/WR.jpg" width="150">The blog post is just to give an idea of some of the ideas on this website. The theme here is 'Enquiry based learning' and by that I am talking about the sorts of activities that provide opportunities for students to make discoveries on their own through engagement induced enquiry! At least that is the aim! Let explain the point with the following examples The title each links to a fuller, resourced, outline of the activity.</span></p> <p> <span ><a href="http://www.https:thinkib.net/teachmaths/activities/probability-trees.htm" target="_blank">Probability trees</a> - <span >This activity is about bridging the gap between the intuition of sample space diagrams and the efficiency of tree diagrams. Students will look at a problem from the two points of view, play with multiplying and adding fractions and hopefully see how tree diagrams are a more efficient way of doing the same thing! The activity is good for group work and physical manipulation, although it could be </span><span >completed on computers by individuals if required. It may well take 2 to 3 hours to complete all of the tasks, but at the end, the hope is that students have a strong understanding of how tree diagrams work that they can apply to different problems.</span></span></p> <div > <span ><a href="http://www.https:thinkib.net/teachmaths/activities/scattertastic.htm" target="_blank">Scattertastic</a> - <span >This activity makes use of two excellent virtual manipulative that are freely available on the web. The activity helps students to begin understanding the concepts of correlation, lines of best fit and degrees of correlation through the use of these manipulative.</span></span></div> <div > </div> <p> <span ><a href="http://www.https:thinkib.net/teachmaths/activities/meeting-functions.htm" target="_blank">Meeting Functions</a> - <span >Challenge students to really understand the concept of a function. Match a set of input values with a function and a corresponding set of output values. There are eight sets of three to make and only one correct solution. This activity is 'old meets new'. Students work with cut out bits of paper but can use calculators/computers to help them solve the puzzle!</span></span><br> <br> <span ><a href="http://www.https:thinkib.net/teachmaths/activities/quadratic-links.htm" target="_blank">Quadratic Links</a> - <span >This activity is about linking the graphing of quadratics with the equations themselves by looking at their key features. Students match pieces of information with different graphs using logical deduction. This practical group activity leads to being able to sketch graphs from their equations.</span></span><br> <br> <span ><a href="http://www.teachmaths-inthinking.co.uk/activities/making-cones.htm=cl" target="_blank">Making Cones</a> - Ex<span >plore cones by making one! This activity helps students understand where the formula for the surface area of a cone comes from and play with the associated mathematics. A great practical task that seems easy and works out to be more of a challenge. In making the cone students will confront some great mathematical reasoning and maybe even some algebraic proof!</span><span > </span></span></p> <p> <br> <span ><a href="http://www.https:thinkib.net/teachmaths/activities/which-rule.htm" target="_blank">Which Rule</a> - <span >This activity is designed to help students solve trigonometry problems by encouraging them to 'Speculate' about what might be possible. Students are asked to state different truths or complete different equations for a given diagram without being told what to solve for. Having completed the equations they are asked to think about which of them is most useful for solving for a particular variable. So often </span><span >students feel that they must know 'the right thing to do' before they proceed and are afraid to try things out to see what happens. Yes, it is possible to learn how to recognise certain types of problems but it is equally important to learn that problems can be solved by trying to use the different pieces of knowledge you have to make new ones. The sine rule and cosine rule can both be applied to any given triangle it is just that often only one of them generates an equation that can be solved. We can either learn to spot types of problems or to speculate with both. In practice, one often leads to the other and then we are better equipped to solve more problems. </span></span></p>https://www.thinkib.net/mathstudies/blog/11818/enquiry-based-activities#1329724371Natural Medium
https://www.thinkib.net/mathstudies/blog/11814/natural-medium
Sun, 19 Feb 2012 18:00:15 +0000 <h3> Are computers a natural medium for mathematics?</h3> <p> One of the reasons I both love and hate twitter! I am casually flicking through some pages over breakfast and I happen on this <a href="http://blog.mrmeyer.com/?p=12782&utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+dydan1+%28dy%2Fdan+posts+%2B+lessons%29" target="_blank">blog post from Dan Meyer</a>. In fairness the blog post seemed mostly to point out how helping mathematics education has not ever risen to the top of silicon valley's priority list. Whilst this is an interesting question, it was the question phrased in the title above that caught my attention. I love this because it is great when some one else's writing makes you stop and think - I hate it when the question pre-occupies your mind when you are trying to do other things. The result is that I am writing this long after I should be asleep, getting ready for tomorrow. Anyway, I think the below can stand alone, but can be put in to context by reading the blog post linked above. This was the response I left on the blog post.</p> <p> As #57 says, who is still reading! I find though that putting these thoughts and reactions in writing is mostly only for my own benefit! In this case, it is beacuse, whilst I understand and sympathise with the general view being expressed, I think I actually disagree with the statement about 'natural medium'! I read most of the responses and scanned the rest but the response from David Wees came closest to my reaction when he said '</p> <p > <em>'There are some tasks for which computers are perfectly suited in terms of mathematics'</em></p> <p > and</p> <p > <em>'What you have suggested is that they are less than ideal for the quick communication of mathematics, and for deeper assessment of what mathematics students understand.'</em></p> <p> Regarding the first point....</p> <p> My relatively short teaching career (13 years) has spanned 'almost no access to computers' to 'working with a one to one program at my current school'. There is no doubt in my mind that computers have had a hugely significant effect on the way mathematics can be taught and, more importantly, discovered,<em>beacuse</em> they provide a considerably <em>more</em> natural, able and versatile medium. A lengthy description of cases could follow, but I will limt myself to just a few...</p> <p> Dynamic geometry, as has been mentioned by some already. This tool has done amazing things for helping teachers to create opportunitites for students to make discoveries on their own and thus enage with mathematics. It can go beyond the teaching of geometry as well.</p> <p> Graphing software - largely by labour saving, but also through dynamic functionality - these tools as well have created new opportunities for exploring relationships.</p> <p> Data Handing - This has come to life through computers with access to real, live data, the functionality to collect it and the ability to process it. All this means that the nature of data handling tasks can now vary in new ways. (I will not say 'more mathematical ways' although that it is what <em>I</em> think.)</p> <p> As suggested, I could go on and will in my head!</p> <p> Regarding the second point....</p> <p> Yes I agree that progress is slow on more able and intuitive user interfaces for communicating mathematics. I think that this has worked in our favour as teachers though. For example, taking the fractions, modern calculators now make it much easier to input and work with fractions than it used to be and this may have resulted in a poorer understanding of what fractions actually mean. The fact that computers dont find it easy to accept fractions means that users have to think about what the fraction actually means in order to input it. A fraction is easily written on a piece of paper with no understanding of its meaning.</p> <p> Likewise, when programming with dynamic geometry (and I do consider constructions a type of programming), there is no 'rectangle tool', in order to construct one you have to know that a rectangle is made by two pairs of parallel sides intersecting at right angles. When you program it correctly it will always be a rectangle regardless of which points are moved. The process of drawing a rectangle on a piece of paper is not at all the same.</p> <p> In summary, dont get me wrong, I estimate that computers are used for about 50% of our lesson time and I am a committed believer in variety of tasks that range from the pencil and paper, to the practical, to the virtual. That said, I am a passionate supporter of what computers have done for mathematics education. I am also a relatively new blogger and always have a sense of fear when 'submitting' such responses. I think most bloggers understand that expressing your views and reactions is the best way to develop them, so thanks Dan for making me think! Apologies if I have missed the point somewhere along the line, I feel better for writing this down either way.</p>https://www.thinkib.net/mathstudies/blog/11814/natural-medium#1329674415