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The Future of Maths Exams

June 10, 2013

Maths exams as they stand are poor. I particularly dislike any question that basically asks you to recall something or do some maths skill (add fractions say) with no context at all. Although I appreciate the need or even desire to show that solving maths questions is an enjoyable thing to do in its own right. I’m quite happy with (or even in favour of) a question that needs multiple steps and doesn’t guide you through. That’s fine. I don’t feel the need to make questions contextual or ‘relevant’ unless they genuinely are relevant questions. I can not stand and question that starts “Sarah is three times the age her dad was when he …” This is the way of pseudo context as Dan Meyer and Jo Boaler point out.

So, my HoD and I are considering a test for students to take at the end of a year after they’ve spent time doing problem solving and investigations. In this test, I want there to be no marks for getting the right answer and all the marks for showing mathematical thinking. I believe, perhaps in an oversimplifying way, that there are two main things that should be held up as ‘doing maths’:

  • Trying to prove or disprove something already thought to be true. The most famous bits of maths recently are to do with finally proving some old conjectures Eg Fermat’s last theorem. The way this sort of maths works is that we have an idea such as Goldbach’s conjecture which could be thought of as ‘the answer’ and the joy is in trying to show that it’s true. It is the method that gets you to that answer  which everybody wants to find. I forget the person that said this but it’s easy to come up with conjectures, proving them is where the real skill lies,
  • Trying to model a situation in a suitably accurate way. Trying to think what matters in a question and how the parts that are important tie together to give you an answer. In this type of maths, we know the answer isn’t likely to be ‘right’ or perfect but we want to know if it’s going to be close enough to be helpful. As Box would say, “All models are wrong, but some are useful”. [Good name for a podcast.]

I would envisage having around eight questions and the student picks three to answer.

Questions could be something like:

How many pencil crayons do you think there are in Birmingham? (Modelling)

How much pocket money do students at this school get given over a year? (Modelling)

Can you show why four digit, palindromic numbers are divisible by 11? (Proof) [Thanks to @stecks and @aperiodical]

I think the mark scheme for each question would include something like up to 10 marks for being systematic, 10 for being organised (ie how easy is it to follow your thinking) and some marks for make conjectures/suitable statements. Clearly, we’d need to think about this more. I’m aware that this is similar to GCSE coursework but I liked that and in an exam situation, teachers wouldn’t be able to help.

Obviously it’s early days yet but I’d be interested in your thoughts.

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  1. Sounds like GCSE Maths investigations! They had a three-aspect 10-mark scale. SO subjective, took AGES to mark, and of course teachers would “help” students. Would this be in an exam hall and externally marked?

  2. Ben permalink

    I really like this idea – far too often we are forced to focus on procedural elements of maths rather than the reasoning skills this would promote.

    Maths assessments as they are force maths teaching to rely on receptive models of learning – focussing the assessment on skills allows the teaching to focus on skills too.

  3. Racheal Jukes permalink

    This is a great idea! We have started to use what we are calling ‘learning portfolio tasks’ which are open ended tasks which demonstrate understanding of the topics we have taught. They are also marked against a similar criteria and firm part of each childs maths portfolio which they keep throught their 5 years at the school. For example the mystic rosé investigation for our sequences topic and we are currently in the process of writing a task based on designing a sports bag to cover surface area and volume. I would be interested in seeing how you structure and support pupils with these types of tasks.

  4. I agree – far too many exams just have recall type questions and as you say with no context at all. Your example questions there are one giant leap though! Would you consider shorter ‘rich’ questions? I have used some problems from the excellent AQA resources ‘Problem Solving Questions’ I have linked to below as exam questions, they are certainly significantly more challenging than standard questions.

  5. Sarah permalink

    I have been thinking about this too but haven’t actually investigated it yet. I am doing a masters in maths Ed (part time) and have been thinking of looking into it for my dissertation. I.e. how do we teach skills/ what should maths exams involve/ how do we assess higher order thinking skills? Marker subjectivity seems like an issue but surely the same is true of English exams? I’d be keen to know how you progress.

  6. I love those questions! I think this idea is superb, but i feel you are only shaving the tip of the iceberg as far as problems with maths exams (specifically the GCSE variety) go. I do like abstract questions, but agree that adding fractions out of context seems a little daft, (unless the fractions are algebraic, of course!)

    The content of the maths GCSE needs looking at as a whole, there are things that appear on there that should be cvered in KS1/2/3 and there is no rigour or even stretch for the birghtest pupils. I think that set and group theory should be on the syllabus, along with a basic introduction to calculus, and thats just for starters!

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