Since my last post, the White Rose Maths Hub was renamed West Yorkshire Maths Hub. There are many reasons behind the change but it doesn’t affect my position, thankfully!

This week sees the first residential of the Maths Mastery course in Leeds on Thursday and Friday. I had a list of readings to go through before the event. Here are my thoughts from those texts.

  • Lai, M. Y., & Murray, S. (2012). Teaching with procedural variation: A Chinese way of promoting deep understanding of mathematics. International Journal of Mathematics Teaching and Learning.

The examples in this article are the main examples from Secondary Mathematics; however, the introduction is helpful in exploring the idea of “variation” a key strategy that supports deep conceptual learning and mastery of mathematics in East Asian countries.

I ended up reading the whole article – and the examples within were moderately useful in helping to understand the three main ways of procedural variation. I say moderately because the secondary maths ideas don’t apply at all to the curriculum I teach from. What it felt like is almost comparable to conducting a series of fair tests in Science; for each fair test you would alter one variable and see how that affected the outcome. Lai and Murray recommend that variation should be ‘controlled and systematic’ in such a way. They go on to indicate that a teacher guiding pupils in their thinking by asking questions like, “what changes and what stays the same,” it offers a way in to this systematic nature they are trying to attain and opens up a deeper way of thinking.

From this article, the three forms of problem-solving that promote procedural variation are:

  1. varying a problem: extending the original problem by varying the conditions, changing the results and generalisation – this is the fair test method mentioned above.
  2. multiple methods of solving a problem: varying the different process of solving a problem and associating different methods of solving a problem – using different given methods for the same problem
  3. multiple applications of a method: applying the same method to a group of similar problems – this was a little confusing from the paper, but I think this is similar to the way a small step curriculum works in my head currently.
  • Sun, X. H. (2013, April). The structures, goals and pedagogies of” variation problems” in the topic of addition and subtraction of 0-9 in Chinese textbooks and reference books. In Eighth Congress of European Research in Mathematics Education (CERME 8), Apr. 2013.

This compared a common Chinese textbook with a common American textbook. The examples given from the Chinese book display the cardinality of 4, 6 and 7. I haven’t got much experience of teaching maths at this level and so don’t have a huge amount to compare it with. One thing that the examples clearly show is that the number of different ways to represent numbers is striking. There are children with apples, sunflowers (?), part-part-whole diagrams (something which hasn’t been used much in teaching in England at all up to now), thought bubbles of children and hands holding objects leading to further part-part-whole models. And that’s just on one page! 6 and 7 get a further representation of two towers of cubes showing 7s decomposition into 6 & 1, 5 & 2 and 4 & 3.

It goes on to discuss concept connections in similar ways. Showing the same examples in as many ways as possible. Out of everything, these remind me certainly of the current Maths No Problem! books. It criticises the approach of stating facts without showing them in differing ways – clearly preferring the concrete, pictorial, abstract approach to a solely abstract one. It also seems to me that the accompanying book for the teacher goes into a lot of pedagogical depth, something which just isn’t seen in this country too often. Overall the American approach is to introduce different concepts individually, whereas the Chinese approach is to develop multiple approaches to the same problem at the same time – which seems confusing at first, but that’s only because we are so used to working in that individualised way. Actually, the more I see of the multiple method approaches, the more I like it – I’m just not sure how to introduce it to my children who have been conditioned to learning in a different way. Sun believes that the one-thing-at-a-time approach limits children’s chances of making connections between concepts and lead to gaps in learning.

The two articles below explore strategies to develop fluency and flexibility in the knowledge and application of number facts.

  • Kling, G., & Bay-Williams, J. M. (2014). Assessing basic fact fluency. Teaching Children Mathematics, 20(8), 488-497
  • Baroody, A. J. (2006). Mastering the Basic Number Combinations. Teaching Children Mathematics, 23.

Kling and Bay-Williams discuss the negative impact of timed testing on children – noting that anxieties around these aren’t generally related to the time aspect of the test. They assert that fluency is often misunderstood to be based on speed recall of facts, that timed tests are used to prepare children for tests yet to come (the ghosts of SATs future!) and that they are used because there are seemingly few other alternatives out there. Instead, they advocate the use of interviews (questioning techniques within lessons to assess fluency, flexibility, strategy selection and use of appropriate strategies), observations of pupils during maths activities, journaling (making use of written explanations of their reasoning in books) and quizzes to assess understanding of ‘foundational facts’ – one- and two-more-than, bond to 10, doubles etc. – which are short, untimed and strategy focussed. Regularly weaving these into lessons would appear to be beneficial and may replace the need for a timed test, or at least, provide more meaningful assessments.

Baroody’s article discusses some different approaches to aiding children’s basic fact recall. This made me think about the way games are used in the classroom. I used to use games to ‘motivate’ learning of facts but have shied away from it in recent years. I began to notice that it was always the same children who did well at the games and, similarly, there were groups of children who did less well. I found it didn’t really have the desired effect, so I just stopped using them. I wanted to embed regular practice and reinforce the fundamental/foundation facts by using these games but as I learnt more about mathematical learning, more about the concrete, pictorial, abstract approach, the more I realised it didn’t develop conceptual understanding – the games only benefitted those who already had that underlying understanding and made those who didn’t have it feel not too great about their mathematical abilities. The article reaffirms my personal findings and discusses how the brain works to memories such facts. Isolated facts are more difficult to remember than interrelated ones – leading to forgetfulness as they are only shallowly learnt in the first place. The article also discusses how inflexibility in teaching – when there is little encouragement to reason or construct concepts independently – can inhibit children’s ability to invent their own reasoning strategies and are less likely to spot general patterns that could be applied in wider situations. Encouraging a child’s own number sense develops this skill; inviting children to discuss their thinking regularly in order to make connections and therefore refine strategies. Providing small step opportunities to learn helps as does embedding practice through drill – but ensuring at the same time that drill is used wisely.

  • Skemp, R. R. (1978). Faux Amis. The Arithmetic Teacher, 26(3), 9-15.

This article discusses the differences between relational and instrumental understanding. Relational mathematics involves making links between objects – the concrete and the abstract. Skemp argues that most teachers teach instrumentally without the underpinning security provided by relational mathematics. In essence, instrumental mathematics teaching depends on teaching children ‘tricks’ – “to divide by a fraction you turn it upside down and multiply” being a key example – but not the underpinning reasons why the tricks work. These can have some benefits, in that successful lessons look like pages and pages of correct answers, boosting confidence. That learning though is quite shallow, and for some hard to transfer that new skill to other, related, elements of maths. Relational mathematics is generally easier to remember – the rules are easy to recall, but often harder to learn. There is more to learn, but the result is longer lasting. They are harder to teach, as they require lots of elements to come together to, in essence, discover the rules that would have normally been shared instrumentally. Skemp also argues that relational understanding allows children to get greater satisfaction from their learning, and thus enjoy maths more.

As well as the readings, I was asked to watch the video below, observing the small steps throughout the lesson.

The lesson starts with the teacher discussing where the children have seen circles before. The first activity is then to make a circle by cutting it out of a piece of paper. The teacher discusses the methods of cutting used and the pitfalls of each method with the children – some cut freehand, some folded before cutting a semi-circle. The teacher then discusses how this method could be improved by folding into quarters and eighths. He shows two children who have folded paper into eighths – one cuts a curved edge, the other a straight edge. He elicits from children that the side of a circle should be curved and not straight. All the children believe that the curved cut should produce a circle. They are given time to cut and explore – the teacher circulates. He describes their work as making “a very important finding” – a phrase I love – as they discover the curved cut produces a flower-like shape and the straight cut produces a circular shape. He asks the children to discuss their findings and they feedback their thoughts. One child uses symmetric to describe the shape, saying the circular one is but the flower isn’t (it is), and another points out the difference in length of the edges from the centre of the folded segment, noting that the distance is longer on the flower-like shape on one side, but the same on the straight cut segment. He then details this and asks the children to prove they are different or the same using a ruler. He goes on to point out that the creased edge becomes the radius of a circle once unfolded – a word some of the children already knew. He refers to this sequence as observation, analysis and exploration. The next step is to offer advice to the curved cutter to improve his circle, and also to look closer at the almost-circular shape made in order to make it closer to a circle, building up to state that he needs infinite points from the centre in order to make a perfect circle – but also that they need to be the same distance from the centre. He then discusses how they children focussed more on the curved edge – an extrinsic characteristic – rather than how the circle is made: a series of fixed points a fixed distanced from a fixed centre – the intrinsic characteristics. He goes on to look at how a compass is used to draw a circle using a visualiser to show this. He asks the children to explain how the compass creates a circular shape – some time is taken here as he looks carefully at each aspect of the intrinsic characteristics discussed earlier. He shows an ancient Chinese sentence describing a circle and compares it to the learning they have done today. He continues by sharing images of people drawing circles – including figure skaters, and what looks like football pitch markers. They look for the fixed points and fixed lengths in each situation. He discusses wheels on cars and why square wheels are a bad idea and why circular wheels provide a smoother experience – the circle is always the same distance from the ground; the square has varied distances from the centre.

I enjoyed the lesson – did all the children learn? It’s hard to tell. I think so. Did they all get something out of it? Possibly.  Would this work here? I don’t know. The repetition helps. I haven’t seen many lessons where there is this much repetition of facts. It probably links best to Skemp’s view of relational mathematics. There is no doubt the lesson above makes great use of fundamental knowledge and the very small steps allow everyone to  both keep up and push their understanding too.

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