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:

**varying a problem**: extending the original problem by varying the conditions, changing the results and generalisation – this is the fair test method mentioned above.**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**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.

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.

]]>It is with great pleasure that I can resurrect this site with news that I have recently accepted the opportunity to work with the White Rose Maths Hub and train as a Primary Mastery Specialist.

This site is, and always has been, a little diary of my work as I went through the Primary Maths Specialist Teacher (MaST) course a few years ago and I intend for the same to happen here, although there is a long way to go before I can fully update things. I have no idea where or when training will be happening yet, as it is very early days.

Exciting times ahead!

]]>I am now a qualified MaST Specialist and use my skills with colleagues on a regular basis. I guide my work mates through many aspects of maths.

At some point, I will upload some highlights of my more recent maths work. With the changes in curriculum coming up, I have a lot to organise in my current school.

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The hardest to get my head around was Babylonian which uses a base 60 system, which we still find in our time units today. Once I began thinking like that, I began to be able to see the beauty of the system and its inherent usefulness, however clumsy it is to work out at first!

The main thing I took from this was the urge to share some of this maths from other cultures with my class – we are currently studying Ancient Greece as our main topic, and were about to start a Literacy unit called stories from other cultures in which I focus on tales from the middle East (Mesopotamia, now Iran and Iraq – precisely where the Babylonian system originated). This seems like an opportunity not to be missed to create a link between Literacy and Numeracy as well as discover a little about the history of the numbers we use today.

I planned two lessons based on the work we covered – one looking at Ancient Egyptian numbers and the other based on the Babylonian system.

My class were able to understand and use the Egyptian system well – by the end of the hour they could add, subtract and complete multiplication sums using their symbols. I deliberately didn’t teach them the Egyptian way of multiplying as I was unsure whether their knowledge of the systems we use today was strong enough and wanted to avoid confusing them! The children were able to see links between the Egyptian system and ours easily, with them both using base 10, and could therefore think in English numbers, as it were, and convert into Egpytian…

The Babylonian lesson went less well. The process of converting from our numerals into Babylonian ones a simple process up to 59. After that, it becomes a little tricky as my class made clear with exasperated cries of frustration!

I still think there is merit in discussing these systems with the children – if only to look at the practicality of different ways through history!

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Other tasks we have been asked to complete include looking at how numerate my school is and to consider what children know about numbers by asking them to consider what the biggest number they know is.

**Numerate school**

It turns out that the classrooms at my work place are generally well set up children to encounter numbers. There were many examples of number lines, multiplication squares, and so on. It was also clear that these opportunities occur far more often the lower down school children are – with the upper Key Stage 2 classes having far fewer examples of numbers around the rooms. The other areas particularly lacking in number were the additional parts of the building – things such as corridors, the hall, the reception area and the whole outdoor area. Plans are now in place to make these areas a little more interactive number-wise. There is now a dedicated numeracy display board in the hall, there will be a series of number challenges around the corridors and there will be an outdoor number trail created around school when the weather is a little nicer.

**“What’s the biggest number you can think of?”**

I asked this question to children in all classes from Reception upwards with the results showing a wide range of ideas. The concept of billions appeared as early as Year 1, but even in Year 2 there was a little confusion over the order of numbers (2000 and a billion being one answer there). In Year 3, they struggled to think of anything larger than one million. Infinity made its first appearance in Year 4, although I also got the answer “20 thousand, 5 million and 2 million” from a child in this class. Year 5 produced some spectacular answers, each trying to outdo the previous child. After “a trillion 9s”, we had “infinity times a trillion”. I couldn’t really argue with that…

This exercise shows me that there are some issues with place value in school – many children gave garbled answers which I had to reorder to make into a number – but also that children struggle to read larger numbers when written down. This was the extension task I gave them and the children were OK when reading four digit numbers, but problems crept in with the majority of children once more digits were used.

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The main reason is that I have just moved house, and other factors are an increased work load lately as well as a distinct lack of time to work with children in school on MaST type things.

So, things are just about settled down now, so I feel able to post here a little more. I also have things to write about based on my work in school and from the most recent study block.

But, thinking about pendulums – and chaos theory (I have been reading a lot lately about oscillations and magnitudes of vibrations creating wonderful changes as discussed by David Acheson) – here is a fabulous video.

Wonderful!

]]>A tricky one this.

Have barely written much in quite a while. In fact, since I completed my first assignment – something I will post here once the course is over.

We’re going back to geometry with study block 7. For me, shape and geometry is fascinating. I haven’t particularly studied this aspect of maths in a long while, but my degree’s dissertation was based around the Fibonacci sequence, the golden ratio and its appearance in art and nature.

Absolutely wonderful stuff, with some complex maths.

My recent fascination, a clear one given the theme of this site, are fractals.

Given 4 shots of a fractal, can you order them from

least zoomedinto most zoomedit? The point is that this ishardto do. Fractals are objects which areequally complex and look similar on all scales– therefore it is inherently difficult to tell how zoomed in you are. If you looked over someone’s shoulder, and saw them looking at a shot of the Mandelbrot set, it is entirely possible that at their zoom-level the entire set would span the size of the observable universe!

The above quote and image are from Matt Henderson’s maths and science blog.

Again, the maths is complex, literally, and not something I totally understand (that knowledge has somewhat left my mind). However, I feel that something like this would be a good thing to explore in the primary school setting. Not only are they incredibly beautiful, they can provide a stimulus for ordering exercises, they provide a new and exciting set of shapes to explore, they can make great displays.

Let’s expand:

- Sierpinski Triangles can be made using equilateral triangles – which in turn can be created through paper folding.
- Investigating the Sierpinski Triangles can lead to such questions as: what fraction of the triangle is left after one step, two steps…? Is there a pattern?
- Linking Sierpinski’s Triangle to Pascal’s Triangle, see below. This involves investigating the pattern in Pascal’s Triangle and shading all even numbers.

To me, this would make a good series of a couple of lessons at the upper end of Key Stage 2.

Also, part of the Eiffel Tower is similar to a fractal!

- Understanding Fractal and L-System (solid1610.wordpress.com)
- Breathtaking Mandelbrot Videos – Now in Brilliant 3D (techie-buzz.com)
- Fun with fractal squiggles (i-programmer.info)
- Fractal Menger sponge made from Post-Its (boingboing.net)

- This article is full of ideas to support my action research with children when the time comes. I intend to use some of the ideas within to help structure my input and form the base and review tasks.
- This article also grabs me as it contains citations of other authors I have read so far – not necessarily the same articles/sources as the Kaput writing I have read is a later date than this article, however the 2008 Kaput source seems to be a development of the 2001 piece used here. These connections confirm to me that my thinking and research is along the right lines and hasn’t been as unfocussed as I first feared.
- “Difficulties occur with adolescent students stem from a lack of early experiences in the elementary school” – relates to functional thinking and how students find it difficult to spot generalisations easily. They lack apporpriate language to describe what’s happening, generally focus on a single data set rather than comparing information and have “an inabilty to visualise spatially or complete patterns.” (Warren, 2000). The researchers found that children had limited experience with visual growth patterns and had rarely used arithmetic for anything other than finding answers.
- It continues to state that recording data in a table inhibited the children’s thinking, encouraging “single variational thinking, finding relationships along the sequence of numbers instead of find the relationship between the pairs.” … “The patterns chosen here were those where links between the pattern and its position were visually explicit…to focus in particular on the relationship between the position number and the pattern.” The article gives examples of the patterns used (shown below) and describes the aims of the questions in detail.

**Main Reference**:

- Warren, E., Cooper, T. (2007) ‘Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking’
*Educational Studies in Mathematics*, Vol. 67, No. 2, pp 171-185

**Citations**:

- Kaput, J., Blanton, M. (2001) ‘Algebrafying the elementary mathematics experience’ in Chick, H., Stacey, K., Vincent, J. and Vincent, J. (eds.)
*The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference*. Melbourne: ICMI, Vol. 1, pp. 57-94 - Warren, E. (2000) ‘Visualisation and the development of early understanding in algebra’ in Nakahara, T. Koyama, M. (eds.)
*Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education.*Hiroshima. Vol. 4, pp.273-280

**Session 1**: Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding. These may not be accurate at this stage – but that’s ok (and the whole point of the sessions…)

**Fir Tree investigation**: uses pattern block triangles to create growing fir trees. Children must extend the pattern, complete the table of values and describe the 10th tree.

**Session 2**: use ideas from algebraic infants. Modelling how a family of dogs can be constructed. Legs, shoulders, bum, body, head – with only legs and body growing, Children need to show each stage in a table for the dog. They then create their own animal, growing it in the same way. Multilink cubes needed.

**Session 3**: **Tables & Chairs investigation** challenges students to find a rule to describe the relationship between the number of small square restaurant tables placed together in a line and the number of diners that can be seated at the larger table if only one person sits on each side. Can model with shapes if needed (squares and circles). As before, children make a table to shapes used and differences to aid thinking.

**Session 4**: Part 1: **Ships Ahoy**. Children look at a simple pattern of horn blows and predict future patterns.

Part 2: **Rockets**. Children look at a pattern for building rockets. They are challenged to see how many parts will be needed for the 50^{th} stage. Whiteboard/paper drawings probably best – stickers might be helpful though?!

**Session 5**: **Task Cards**. Children choose a task card then construct it using squares or triangles. Encourage them to look at the patterns and decide how it is growing. Create and complete a table to show the growth pattern. Use the tiles to make the next two shapes in the pattern. Be prepared to explain how the pattern is growing. If there is time, choose another task card. Cards attached at the end.

Session 5 – Challenge Cards (PDF)

**Session 6**: Same as session 1. Can the children identify the growing pattern and express the rules algebraically?

Show a pattern made from multilink cubes. 1^{st}, 2^{nd} and 3^{rd} stage of the pattern. This pattern must be similar but not the same as session 1. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding.

**Hexagon dragon investigation**: dragons made from equal numbers of hexagons and triangles, each new term adding on more. Requires children to extend the pattern, create an input/output table to describe the growing pattern, then draw and/or describe the 10th dragon in words. An extra challenge asks students to generate a rule for this pattern so that Miguel can figure out how many blocks he will need to build a dragon of any size.

Each of these sessions will be completed during the children’s Numeracy time. I am likely to choose 4 or 5 children to work with from across ability groups and a mix of genders. I will take copies of their written work and record conversations to help me analyse their progress and thinking over the course of the sessions – I feel that this will give me the data I need to go further. Session 6 is a repeat of session 1 in order to set a baseline and see the progress, if any, the children have made in between.

Clearly, this is such a small scale research project that results can’t be read into too much. However, it is a beginning for my school to look at how we can use algebra in wider contexts. I have been careful to choose similar tasks throughout the sessions as I will only have at most half an hour to complete the tasks with them.

]]>- Algebra is a third of the Malaysian Secondary Mathematics curriculum.
- The article lists a large range of sources relating to misconceptions in the understanding of algebra. It states that, because algebra is an extension of previously acquired mathematical learning, it is linked to the use of symbolic representations and that “Warren (2003) felt that understandings of basic arithmetic operations could assist sucessful transition from arithmetic to algebra.”
- “Students use the equals sign in both contexts: arithmetic and algebra. The concept of the equals sign in the framework of arithmetic is that of a ‘do something’ signal (Bodin & Capponi, 1996). Many students tried to add “= 0″ to algebraic expressions when they were asked to simplify (Kieran, 1997).”
- The article cites Booth, 1981, as a refernce for children not understanding the idea of a letter as a variable, and they rather see each letter as representing a digit.
- The article looks in detail at a range of errors (12 types) made by students. The examples given are from the secondary curriculum and so may not apply to any errors made by children in a primary school – but a worthwhile source to help identify the sorts of errors that could be made.

**Main Reference**:

- Lim, K. (2009) ‘An Error Analysis of Form 2 (Grade 7) Students in Simplifying Algebraic Expressions: A Descriptive Study’
*Electronic Journal of Research in Education Psychology*, Vol. 8, No. 1, pp 139-162

- The article discusses how algebraic equations can be shown as visual sentences. The example of
*x*+*y*=4, where*x*>*y*is used. The Reception aged children are given two rules to colour them in: “they have to colour in four snails, and the number of brown-coloured snails must be more than the number of yellow-coloured ones.” - The author states how remarkable it is that the children of this age can complete this algebraic idea and that staff argue it should make it easier for the children to manipulate equations later in life.
- The approach to teaching also requires children to discuss their ideas in groups, challenging each other’s answers, explaining their reasoning and arguing with the teacher who deliberately makes mistakes to generate such discussion.
- It seems that both children and teachers are capable of exceeding perceived expectations through innovative thinking. Clearly, this is just one example and it’s hard to draw conclusions but it would be interesting to see where those children of 2006 are now in Year 5.

**Main Reference**:

- Mansell, W. (2006)
*Algebra at the age of four.*TES. [Online] [Accessed on 17th May 2011] http://www.tes.co.uk/article.aspx?storycode=2264060

**Original Article**:

- TES Online: Algebra at the age of four (Published in The TES on 21 July, 2006. Accessed on Tuesday, 17th May 2011)