The following is a list and description of some of the most common problems there are involved with choosing and completing projects, along with some ideas about how to overcome them. It is intended for primarily for teachers and as a guide that could be close to hand through out the process. No two schools, classes or students are the same and as such this list cant cover all, but could provide some useful warnings and will hopefully continue to evolve into a useful list.
The 'Project Planner' pages consist of tasks to help students overcome some of these particular difficulties.
The project curve
By way of a little mathematical modelling and conjecturing it is my view that the Maths Studies project follows an exponential growth curve like the one below! Greater time input is required for less progress in the beginning. Once the clarity and planning exists, students should just be executing tasks and as such making more progress. Obviously this argument hinges on the definition of the word progress!
Choosing a project
Probably the hardest part of the whole process and unfortunately the bit that has to be done first. It can be very hard for students to narrow down a focussed theme for a Mathematical investigation and a good chunk of time should be allowed to get this bit right. Here is a bullet point list of things that can be done to help with this process.
- Spend some time considering the very nature of 'Mathematical investigation', its different forms and purposes. Look at the page on 'Project Inspiration' for some help with this. It is important to consider what is meant by this concept and what possible outcomes can be
- Look at as many previous projects as possible. Previous projects can really help with the definition above and provoke ideas. Spend time discussing the themes or reading them if they are available and ask searching questions of them. There is a page on 'Possible Ideas' that could be useful here.
- Consider using the 'Choosing a theme' task or something similar to help narrow down some ideas based on students' areas of interest. Occasionally a spontaneous idea will occur and develop, but often a structured approach to choosing is more productive.
Clarity can be very difficult to achieve for any of us, but without it, work can be very frustrating and lack direction. As much as possible it is important for students to have a clear vision of what they are attempting to do with their projects and how they are going to do it. That is not to see they need to have a clear vision of what will happen during the process or what they will find out, but more that they know the questions they are trying to answer and the way in which they are going to try and answer them. If students proceed without this clarity then progress is often slow and aimless.
Combining a good idea with a clear vision is a great start. What must follow is a cast iron plan. Mismanagement of tasks and time (which often results from lack of the clarity mentioned above) is the single biggest handicap for students with these projects. The tasks are hard to quantify in terms of time, which does not make planning easy but it is more easily underestimated than anything else. As time runs out, bits of the investigation tend to get cast aside and the time for reflection, adjustment and accuracy disappears and the result is often 'nearly' a good project. Consider using the 'Planning and Scheduling' task for students.
Statistics or not?
It can be quite hard to distinguish between these two types of tasks especially when both of them will involve an 'Information collection' element. The collected information can almost always be analysed with statistics. The essential difference is that 'Statistical' projects generally refer to use of data that is not generated mathematically, although this could arguably be said of data collected for mathematical modelling that would come under the heading of 'Non-Statistical'. In a sense it is not really important to make the distinction, but more important to be aware that information can be gathered mathematically as well. Students should be aware of the different options here and that consequent difference in the type of analysis that can be done.
Turning a good idea into reality can be very challenging. Going from statements like 'I will analyse the data to look for a link....' to very specific statements about tasks, processes and data is often something students find very hard. This is often a consequence of the 'Grand Idea' where the overall aim of a project idea is so big that it is hard to breakdown into smaller parts. In the project planner, students are asked to consider identifying specific potential very early on in the piece and definitely before they have started the data collection.
It needs to be mentioned on this page that information collection is a major stumbling block that should be given very careful consideration before any work is done or decisions made. There is a big section on the site about this that explains some of the pros and cons of different ways of collecting information and again, a specific related task for students in the project planner.
This is often one of the first occasions that students realise that it can be at least as difficult if not more so to keep a word count down to a given number than it is to reach it. The guidelines suggest that 2000 words is a suitable length for a project and that requires students to be particularly succinct! This, of course, can be difficult for them when writing about maths. The standard student response to being asked to write about maths is often a lot of general or waffly remarks that are not helpful or necessary and have the overall effect of reducing the quality of communication. It is worth considering that this is not surprising given how rarely students are asked to write about mathematics. One response is to create more chances to do so throughout the course so that it is not so rare. Another is to ask students to read eac hothers work regularly to help them to see this for themselves.
Keeping a Theme
It can be difficult for students to ty their work all together under a particular theme. There is no particular requirement for them to do so but a project is almost always of better quality and is generally more coherent and interesting when it does. An initial idea can be narrowed down to some specific investigations and the challenge is for those to be in some way sequential or related. Students need to reflect on this regularly through out the project period. In the project planner there is a task that asks them to imagine writing a newspaper article about their project. What would the headline be? What would the conclusion be? This type of thinking can help students to 'keep a theme'.
The concept of Mathematical validity is definitely a very tricky one and as such hard for students to score on! For students benefit I have tried to describe the follow two categories for discussing mathematical validity.
- the nature of the information used questions the validity of the conclusions reached
- the process used or the way it was used or the place in which it was used was not valid
Students tend to find the former easier to do than the latter. It is a good idea to try and show students examples of this as early as possible to help them remember to consider it as they work.
Examples could include;
- Considering validity based on the expected frequencies in an independence test.
- reading from regression line outside the range of the data used
- plotting and/or using a line of best fit when there is no correlation to speak of.
What processes should be used at what times? This is a key difficulty that relates to a couple of key points;
- students attempting to use as many processes as they can will go looking for opportunities to use the process
- do students really have a good grasp of the statistical processes in question?
Whilst there is definitely merit in the former, the risk is a formulaic approach to projects that will result in a less interesting project. The ideal is that a project idea suggests that it would be interesting to test if two given data sets were independent from each other rather than looking for two different data sets that you could use for an independence test. Either way the concept of an independence test needs to be properly understood before it can be successfully applied and this begins address the second point. In the Statistics teaching ideas, there is an activity that asks students to reflect carefully and research the different processes they have learned so that they understand what they are for and when it is appropriate to use them. Work like this done during the course can be really helpful at these times.