How important is knowledge in mathematics?

The importance of knowledge in any subject is well-known. Indeed, if you are yet to read Sarah’s  (@s_donarski) blog on questioning for knowledge, you can find it here. I was at the same conference as Sarah and Doug’s demonstration of the power of knowledge is something that will stick with me for a long time. Knowing more instantly allowed everyone in the room a lot more freedom to think and allowed us to think more deeply.

My question now is: as a maths teacher, how important is it that students know mathematical facts? How can I adjust and adapt my own practice in order to ensure that these facts are known and easy to access?

 

Knowing mathematical facts

I listened to Dani Quinn from Michaela (@danicquinn) talk over a year ago now and the massive takeaway I had was:

if we, as teachers, use a shortcut or “just remember” something, why do our students not?

At the time, we were using the example of multiplying by 0.1 and the educators in the room were “just dividing by 10”. A student does not necessarily do that. Why is it that we do? We have had hours and hours of practice across our mathematical life times. We need to get students to the same point as quickly as possible. Dani is a proponent of “drilling” whereby students are asked a lot of one-step questions in a short period of time. Things such as: “will this give a positive or negative answer?”.

I took this idea and ran with it. For example, I’m currently teaching trigonometry to a year 10 group and over the last 5-8 lessons, I think they have answered about 60 questions which are “which rule can we use?”. The students have picked up on the importance of speed in that decision. I’m not rushing them through the process of applying the sine or cosine rule but I am highlighting how experts would act with a year 10 trigonometry question.

Another part of knowing facts within maths is remembering formulae. The sine rule has to be readily accessible in order to answer a difficult, multi-part bearings question. The quadratic formula must be ready to come out at the drop of a hat at the end of a hard pair of simultaneous equations.

This is not a novel idea and comes from cognitive psychology. We can hold 7-9 things in working memory at any one time. We don’t want these things to be formulae. We want these to be relevant to that specific question (the arithmetic required for example).

Take the following GCSE exam question:

A sector with an angle of 144° is cut out of a circle of radius 10 cm. The remainder is turned into a cone by bringing two radii together.

If the volume of the cone is the same as a sphere, find the radius of the sphere, leaving your answer as a cube root.

This question is impossible if you do not have lots of things to hand in your long-term memory. See below.

My thinking:

  • Angle left in circle = 216 degrees
  • Length of arc = circumference of base = 12pi
  • 12pi = 2pi x radius therefore radius = 6 cm
  • vertical height, radius and slant height form right angled triangle so vertical = 8 cm
  • 4/3 pi r^3 = 1/3 pi (36) x 8
  • Solve so r is the cube root of 72

 

Look at how even the most competent of student will think about this problem:

  • Angle left in circle = 216 degrees
  • Length of arc will be the circumference of the base.
  • Length of arc formula is portion of circle x circumference
  • 2 pi r x 216/360
  • = 12 pi
  • The circumference of the cone base is 2 pi r and that is 12 pi so r = 6
  • How do I get the height from the radius?
  • The height is 10.
  • What’s the volume of a cone?
  • What’s the volume of a sphere?

 

You can see very quickly how a student has to hold and internalise a lot of thoughts. This comes after having to visualise and conceptualise this whole question – cognitively, this is exhausting! We must help them. These questions are hard even if you can remember all the relevant formulae but they become nigh on impossible if you have to struggle to remember them too.

Organise your students’ knowledge

Knowledge organisers are a hot topic at the moment and lots of people use them in a variety of ways. I must say that I am a complete novice in this area but I am a huge supporter of organising students’ knowledge for them. Every lesson, I will revisit my “things we have to remember” page. I make sure I am the one giving definitions and formulae and I ensure that all students have correctly copied this down.

We test every fortnight and these tests are designed in a way that students should be expecting to get everything correct if they have kept up in class. Part of all of these tests is a memory recall section. Recently, I have decided to be transparent with this, too. Students are given 5 minutes to answer 10-15 minutes of memory recall questions. We mark those ourselves and then they crack on with the “problem solving” questions – where they may be using some of the things in the memory recall section.

I have written tests before where I have asked “use the cosine rule to find the missing length” and I thought I had done enough to bullet-proof the question because I’d told them which rule to use. Unfortunately, I hadn’t foreseen people not remembering, misremembering etc. If I wanted to ask the same question now, I’d put a “write down the cosine rule” in the memory recall section and then the same question later in the problem solving question.

 

Testing for (and to deepen) knowledge

This really is the crux of the matter in mathematics. I’m not advocating doing a load of past exam papers and you’ll get better at maths. I am advocating short tests on lots of things. Each question should serve an obvious purpose and if a student can’t answer that question, there should only be one reason. Exam practice is a different animal and one for another time. For the purposes of this piece, we are thinking about deepening knowledge and memory – this comes from purposeful assessment and specific feedback.

Matthew Martin wrote about the forgetting curve in his blog and that’s perhaps more important in mathematics teaching. Students are very good at compartmentalising topics. This is a sequences lesson. This is geometry. This is calculus. When they are doing calculus, they are not thinking of geometry. Our tests should include things they have forgotten. Effortful retrieval has been shown to be a high-impact solution to deepening knowledge and strengthening memory (Make it Stick).

 

The key things to think about when thinking about memory in mathematics are:

  1. Think very carefully about what it is you want students to remember.

  2. Test very specifically whether or not they can remember these things.

  3. Test regularly. Explain why you are testing them.

  4. Let them mark these tests – you gain almost nothing from marking it yourself. They gain lots from knowing whether or not they actually remember it. The right/wrong feedback is enough for a memory recall question

  5. Don’t ask students to actively remember and apply within the same question until they are ready for it. This is the main reason past paper practice is a problem.

  6. Deliberate, specific practice of one or two step questions like Dani’s mentioned earlier alleviate this problem.

 

 

 

 

 

 

 

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How important is knowledge in mathematics?

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