'Too hard' Scottish Higher Maths exam
Sunday 9 August 2015
Although I live in Scotland, I'm not directly involved with the curriculum for Scottish secondary mathematics (worth mentioning that the education system in Scotland is separate from the rest of the UK - as are several other institutions). Nevertheless, I am familiar with mathematics teaching in Scottish secondary schools and often find mathematics education materials produced here very useful. So, when reports on issues related to mathematics exams in Scotland recently appeared in news publications here they caught my attention. And, as a maths teacher that has worked in international schools in a few countries, I'm just naturally curious about how aspects of mathematics teaching and assessment compare from one country, or system, to another.
The news stories were specifically about the perceived increased difficulty of new exams for Scottish Higher Mathematics (one of many exams operated by the Scottish Qualifications Authority, SQA). There is another secondary maths course in Scotland that is more advanced, with the appropriate course title of Advanced Higher Mathematics. The curriculum in Scottish secondary schools has been undergoing some changes the past few years - and one of the consequences of this was a 'new' exam for Higher Maths (the 'new' exam for Advanced Higher Maths first occurs in 2016). Not long after students sat the 'new' Higher Maths exam about 10 weeks ago, there were so may complaints about the exam being "too hard" that it became news, i.e. it had enough 'sensational' aspects so various media outlets deemed it worthy of reporting.
In my opinion (and some may differ with this), Scottish Higher Maths falls somewhere very roughly between IB Maths SL and IB Maths HL in terms of content and difficulty level. The Higher Maths exam has two papers - Paper 1 does not allow a calculator, and Paper 2 does. As a maths teacher, one of the aspects of the news reporting on the 'difficulty' of the Higher Maths exam that I found very intriguing was one particular question from the exam that different online news stories decided to highlight. One of the news stories even referred to the question in its title: New Higher Maths exam - why did the crocodile cross the stream? The question is shown below (it was #8 on Paper 2). By clicking on the question, you can obtain the entire 2015 Scottish Higher Maths exam.
To some this question may 'look' difficult - it has a 'messy' function. Remember, it is on Paper 2, so a student will have a graphing calculator. With that in mind, the question is really rather trivial. To answer part (a) (i) one simply needs to evaluate the time function when x=20 metres, that is - and since the units are tenths of a second this is approximately 10.4 seconds. And part (a) (ii) is answered by evaluating the time function when x=0 metres, that is which is equivalent to 11 seconds. To find the minimum possible time - and the value of x that produces this minimum - a student can simply graph on their GDC and use a standard built-in 'minimum' function to find the minimum point on the graph. The graph below on a TI-Nspire shows that the minimum time is 98 (this is 9.8 seconds) and that it occurs when x=8 metres.
If there is a 'difficult' aspect to this question, I think it is being very careful to correctly interpret the given information in the context of the time function, , that was given. For example, it was important to understand that the units of the function's output is tenths of a second, not seconds - and, also that 'not traveling on land' (part (a) (i)) means that x=20 metres; and, likewise, that 'swimming shortest distance' means that x=0 metres. And, perhaps most importantly, a student needs to resist being intimidated by the question because it might appear difficult given the amount of text and the 'not-so-simple-looking' time function.
It is my firm belief that one of the most important skills that a student in either Maths HL or Maths SL (more important in HL) needs to gain from the course (if they do not already have it) is to not be easily intimidated by questions that initially look more difficult than they actually are; not to become unsettled by questions that look unfamiliar. This is probably what the large number of Scottish students were actually complaining about. The recent 'new' Higher Maths exam had some questions which were unfamiliar to them - when, in fact, these questions (like the one above) were actually relatively straightforward. Too many students become easily intimidated by questions that look 'different'. As teachers, we need to continually present our students (again, especially HL students) with a wide variety of questions, and to design assignments and assessments so that students routinely confront unfamiliar questions. This makes them practice good fundamental problem solving skills and builds up their exam-taking confidence - and to successfully answer questions which some would claim they "have not seen before."