It Stands to Reason
“We need to do much more work with the most basic material to ensure that pupils grasp the relevant concepts. The last thing our more able pupils need is to be accelerated … able pupils may need challenges that are surprisingly basic, before they are confronted with material that is rich and sophisticated.”
Teaching Mathematics at Secondary Level, page 20, Tony Gardiner, 2014
The Advisory Committee on Mathematics Education (ACME) offers similar advice; one of ACME’s principles for mathematics education is that
“[all young people] experience mathematics in a deep, rich and connected way rather than being accelerated through a fragmented, test-driven curriculum.”
Maths Snapshots Issue 1: A blueprint for mathematics education, ACME, June 2014
What kinds of task can we give pupils that will challenge even the highest achievers and the most “rapid graspers” (to use the Ofsted term), and that will develop their reasoning skills while involving only ‘age-appropriate’ (i.e. not accelerated or pulled forward) mathematical knowledge and concepts? In this article we look at a particular two-way-grid-of-cells that can be used to stimulate pupils’ deep thought about any pair of related quantifiable concepts which can be applied to the same mathematical object – and the objects and concepts need not stray outside the expected content for the given pupils’ age / year group.
We will look at a specific example shortly. The generic grid is as shown:
Idea from John Mason, More or Less Perimeter and Area, 2015
A task in which pupils work on an object represented in a less-same-more grid will provide excellent opportunities for them to reason about, and deepen their understanding of, mathematical concepts that are exemplified in the object – in particular, how they are related and how they interact. In so doing, pupils will usually “try to make sense of what has happened, what structure has been revealed, what inner aspects have been noticed while working on the task”. (Ref: Studies in Algebraic Thinking No 1, Inner and Outer Aspects of Tasks, John Mason, 2006). In the same paper the author advises “… there must be more to tasks than the overt or outer aspect: what learners are asked to do”. In our main example of a less-same-more task we try to give some indication of how it can facilitate insights beyond merely the completion of the task. You will have this pleasure yourself if you work through any of the six further examples, including John Mason’s original More or Less Perimeter and Area task which we have reproduced at the end of the article.
Task: Slam Dunk!
Pupils focus on the concepts of numerator, denominator and fraction-size. They first work with fractions of the form , where and are whole numbers, before shifting attention to the simpler fraction form, . The mathematical concept should be introduced as an exploration about a human situation, for example:
Pupils are likely to decide that, in order to get a sense of what happens, they need to look at one example – at least! For example, they might increase/decrease the values of and given in the central fraction by 3 … (click to enlarge)
On the basis of this one example a pupil might make conjectures such as …
- if the number of successes is the same (as in the central fraction), the ‘success-fraction’ is bigger when the number of failures is less, but smaller when the number of failures is greater. (Is this ‘common sense’?)
- if the number of failures is greater (than in the central fraction), the ‘success-fraction’ is smaller – whatever the number of successes is. (Is this ‘common sense?)
- if both the number of successes and the number of failures are greater, the ‘success-fraction’ will be smaller. (Again, does this seem correct – is it best for a player to choose to have only a very few attempts?)
In order to get a better idea about the validity of what has been conjectured after looking at only one example pupils should feel the need to examine more possibilities with as the central fraction, and then to look at examples with other values of and in the central cell. For example, if they substitute , , and increase or decrease both and by 1 from cell to adjacent cell, they will see that the fraction in each cell, compared with the central fraction is …
whereas if they substitute , , and again increase or decrease both and by 1 from cell to adjacent cell, they will see
As pupils try more and more of their own examples, probably at first chosen without much reasoning behind the choices, there will be plenty for them to notice. For example, if they increase or decrease both and by the same amount from cell to adjacent cell, the denominators of the fractions on the SE-NW diagonal
will be equal because . So, whatever the values of and are, these three fractions are always easy to compare: they should be able to reason that, since , the fraction in the top-right cell must be bigger than the central fraction; and that, since , the fraction in the bottom-left cell must be smaller. In the context, this means that if the number of times that a player gets the ball in the basket is greater and the number of failures is less, then the basketball ‘success-fraction’ is bigger – in line with common sense! But if the number of successes is less while the number of failures is greater, the ‘success-fraction’ is smaller – again common sense!]
When they look at the fractions on the NW-SE diagonal
pupils are likely to notice that both the top-left and bottom-right cells contain fractions that are bigger than the central fraction for some values of and , but that are smaller for other values. So the question that arises naturally from THEIR investigations is: “For what values (if any) of and will the fractions in the top-left and bottom-right cells both be equal to the central fraction?”. In the context of basketball ‘success-fractions’ this question could be posed as “Suppose that Steve gets more ‘hoops’ and misses more times than Robert does, and Robert gets more ‘hoops’ and misses more times than Charlie, but all three contestants have the same ‘success-fraction’, what can be deduced about Robert’s numbers of hoops and misses?”
Pupils need the practice of choosing examples intelligently: they need to learn how to pick-out ‘significant’ examples – a lesson learned, often, after experience of not doing so! In the paper cited earlier John Mason writes “Paulo Boero (2001) drew attention to the vital role of anticipation in mathematics … You do not embark on random calculations; rather you anticipate something and then check it out.” Since the question about equality of the fractions on the leading diagonal involves contemplating a symmetrical arrangement of ‘smaller, ‘bigger’ and ‘equal’
and since the central fraction itself, , is symmetrical when written in this form only if , a pupil might anticipate that the fractions on the leading diagonal will be equal when , and so choose to explore an example in which and both and are increased/decreased by the same constant number when moving from cell-to-adjacent-cell, such as
The conjecture that the top-left and bottom-right fractions are equal when, and only when, , is reached by looking at examples (such as the previous one). Some pupils will want to prove this by reasoning algebraically, and we have included both arguments (“if” and “only if”) in the Appendix to this article. In the context of the basketball situation this is saying that “If Steve gets more ‘hoops’ and misses more times than Robert, and Robert gets more ‘hoops’ and misses more times than Charlie, and all three contestants have the same ‘success-fraction’, then Robert and Steve and Charlie all get the ball into the basket on exactly half of whatever is their total number of goes!”
This might prompt pupils to wonder whether the smaller/bigger alternatives in the leading diagonal corners depend (crucially) on whether or . If they choose examples to check this out pupils will obtain support for the conjecture that:
Having begun to get some idea of what happens to the size of fractions of the form as and are increased or decreased in systematic ways (such as interpreting ‘increasing or decreasing’ strictly as ‘adding or subtracting a constant number’), pupils might feel that they would gain a better idea of how to proceed to a more general understanding if they first looked at what happens with the simplest fraction-form, .
When pupils look at a few examples, they will find, as before, that if and are increased or decreased by the same amount, the fractions on the leading diagonal are equal when, and only when, . The algebraic argument is in the Appendix.
So they may wonder what happens if the values of and increase/decrease by different amounts: will it still be true that the fractions on the leading diagonal are equal only when ? With just a little exploration of any numerical example
pupils should be able to explain why this is not the case.
Same-less-more tasks exemplify the principles quoted at the start of this article, and they can be constructed in many other mathematical contexts. Here are five examples that provide opportunities for pupils to develop their reasoning skills and deepen their understanding of various key mathematical concepts – without any need whatsoever to rush ahead in the textbook, scheme of work or key stage planner!
Our last example is John Mason’s More or Less Perimeter and Area task, our inspiration for this article (click to enlarge)
What other examples of less-same-more tasks occur to you? Have you tried something like this with your pupils? Let us know: email firstname.lastname@example.org, or tweet us @NCETM.
Appendix: algebraic arguments
Conjecture 1: the top-left and bottom-right fractions are equal if, and only if,
(click to enlarge)
Conjecture 2: if and are increased or decreased by the same amount, the fractions on the leading diagonal are equal if, and only if,
Page header by Brian Hillegas (adapted), some rights reserved