Archive for the ‘Recommended Reading’ Category

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.

This reading discusses how an average student can benefit from a structured and ordered approach to the teaching of fractions. It could probably be applied to any mathematics topic.

The approach taken was one of understanding the needs of the child first and having an in depth idea of where her weaknesses were. There was a strong focus on the language of fractions – this relates to some of the findings I wrote about yesterday. Fractions are a relative part of maths and their outcome depends entirely on whatever the whole is.

However, we initially observed Audrey having problems in comparing the equivalent fractions 1/3 and 2/6. We therefore used the fraction-lift to clarify equivalent fractions, by making them fractions living at the same floor of the fraction-building and by introducing the metaphor of “roommates” for fractions at the same position on the number line. We now observed how Audrey used the strategy of doubling both the numerator and the denominator to generate equivalent fractions, for example by replacing 2/3 by 4/6 to compare the latter fraction with 5/6.

This gives Audrey a hook to hang her ideas on, something concrete that she can build her learning on. I’m convinced that children need to have the basics of any topic before moving on. I also think that the current approach we have to teaching in this country, based on the current Mathematics Framework, is far too fleeting and jumpy. As practitioners, I feel we need to consider the needs of our children – group parts of topics together so that they have more time to practise and consolidate their learning.

Link (the full article): Ronald Keijzer and Jan Terwel: Audrey’s Acquisition Of Fractions: A Case Study Into The Learning Of Formal Mathematics.

Angle with highlighted vertex
Image via Wikipedia

The main reading for this study block (linked below) is a tricky and detailed account of current teaching relating to angles. It’s main findings are that not enough is done to develop the concept of ‘turn’ with children – that an angle can be defined at the amount of turn from one position to another, and that, if it is taught, the main focus is on right angled turns.

Beebots and roamers could be used in school to investigate the idea of turn, but why not use the school grounds? Create obstacle courses in the playground for children to be directed around – making sure that the turns aren’t always at right angles.

Mitchelmore and White state that children need to experience a wider range of angle concepts. They believe that teacher move too quickly on to the abstract idea of an angle – as shown here.

Their research looked at whether children could represent the movement of objects such as a door, or wheel in terms of diagrams and still understand what was happening – whether they could move from the physical to the abstract in one move.

For instance, the angle shown here could represent the movement of the blades of a pair of scissors, or the opening of a handheld fan, things that children could see happening, and represent in a diagram like this. However, children referred to these movements as ‘opening’, not ‘turning’.

Children were asked to represent the angles using bendy straws to demonstrate the movement and the associated angles. If children could see the angle of movement, and explain what was happening, the researchers were happy.

The researchers also looked at children’s ideas of slope, with the idea that this is an area that is overlooked in schools. I would consider this to be the case simply because of the difficulty of representing it as well as not always being able to see the angle that a hill slopes at, for instance.

…most students had some global concept of slope but that many did not quantify it by relating the sloping line to a fixed reference line. Unlike the wheel and door, where the second line may be suggested by the initial position and there is a global movement which can be copied, there is in fact very little to help a naïve student interpret a slope in terms of a standard angle.

We conjecture that many students have a global conception of slope as a single line and do not conceive it in terms of angles. Had the physical model of the hill consisted simply of a sloping plane without any supporting edges, it is likely that far fewer students would have indicated a standard angle interpretation.

Mitchelmore and White discuss how children find it easier to see the turns, angles and slopes when both elements are easily visible (the scissors, fan, etc.) and this is likely to be because it fits more readily to the idea of an angle as drawn above – something they are likely to encounter in class. They go on to say that, “the fact that the standard angle was used more frequently for the door and hill (where one arm must be constructed) than the wheel (where both arms must be constructed) supports the view that the crucial factor accounting for the rate of use of standard angles is the physical presence of the angle arms… 88% of the students used standard angle modelling when both lines were visible, 55% when only one was visible, and 36% when no line was visible.”

There are three main findings to this piece of research.

  1. That angle work can be related to the everyday concepts of corner, slope and turn.
  2. The fewer arms that are present in a particular angle context, the more that has to be constructed to bring it into relation to other angle contexts and, therefore, the more difficult it is to recognise the standard angle. It is only in exceptional cases that the relevant line has to be discovered. In most cases, it has to be invented through conscious mental activity.
  3. That many children form a standard angle concept early, but that this concept is likely to be limited to situations where both arms of the angle are visible. If the concept is to develop into a general abstract angle concept, children will need more help than is presently given to identify angles in slope, turn and other contexts where one or both arms of the angle are not visible. The slope and turn domains are particularly important for the secondary mathematics curriculum, the former because of the frequency of angles of inclination in trigonometrical applications and the latter because it provides a valuable aid in teaching angle measurement.

Again, the more hands on practice children have at experiencing  the different elements of angle, the stronger their knowledge is likely to be.

After Study Block 1, taught session 2, we were given a reading to look through and digest.

We were given Chapter 5 (pp. 79-104) from “Children’s mathematics 4-15: learning from errors and misconceptions” by Julie Ryan, Julian Williams. (McGraw-Hill International, 2007)

The mistakes children make in mathematics are usually not just ‘mistakes’ – they are often intelligent generalizations from previous learning. Following several decades of academic study of such mistakes, the phrase ‘errors and misconceptions’ has recently entered the vocabulary of mathematics teacher education and has become prominent in the curriculum for initial teacher education.

The popular view of children’s errors and misconceptions is that they should be corrected as soon as possible. The authors contest this, perceiving them as potential windows into children’s mathematics. Errors may diagnose significant ways of thinking and stages in learning that highlight important opportunities for new learning.

This book uses extensive, original data from the authors’ own research on children’s performance, errors and misconceptions across the mathematics curriculum. It progressively develops concepts for teachers to use in organizing their understanding and knowledge of children’s mathematics, offers practical guidance for classroom teaching and concludes with theoretical accounts of learning and teaching.

Children’s Mathematics 4-15 is a groundbreaking book, which transforms research on diagnostic errors into knowledge for teaching, teacher education and research on teaching. It is essential reading for teachers, students on undergraduate teacher training courses and graduate and PGCE mathematics teacher trainees, as well as teacher educators and researchers.

Anyway, I found an online version of the text via Google Books, which is embedded below. However, pages 82, 83, 87, 88, 94, 95, 101 and 102 are not included in the preview due to copyright reasons. Although the content that is there gives a reasonable account of the subject and includes many examples of good practice. Clearly, the lack of the full chapter really doesn’t help!

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