This is the third instalment of a record of some of my thoughts and actions in response to the #beingmathematical event on November 22 2018, part of series of fortnightly tasks and discussions for maths educators hosted on Twitter by ATM. I will continue to explore the relationships between how both novice and experienced learners might approach mathematical thinking and performance.
Here is another thread that caught my attention.
I offer below an explanation of how the the doubling sequence comes about. While trying to address Mary’s question about how very young learners might express their understanding of this phenomenon I will, like Ashley, inevitably bring my own ways of thinking and seeing to bear. Picking up on Helen’s point, I’m also interested in what other areas of maths open up when we start to think about repeated doubling.
Recall that the sequences of triangles shown above are assembled from unit isosceles right-angled triangles (IRATs). Here’s a reminder of how to make IRATs using paper or card.
While Mary’s triangle images – themselves scaled-up IRATs – have been created electronically, it is informative to use card versions to consider how different ways of combining unit IRATs lead to her doubling and squaring sequences. Labelling the sides a and b (see Part 1 of this blog series), helps illustrate this distinction. I have also found it helpful to start each sequence at 1 (i.e. with a single unit IRAT)
Separating individual and composite IRATs may help highlight the doubling process.
Here, for comparison, are the first three (1, 4 & 9) of the sequence of square numbers that was the focus of Part 2 of this blog series and which are constructed in a different manner to the “doubles”.
Here’s a different, inverted, perspective on 9.
Some brief comments on orientation. The “upright” form of the right-angled triangle – with the right angle at the bottom left or right – that results from both sequences seems to me to be the norm, for example in textbooks. However, it has been suggested that the “inverted” form – with the right-angle uppermost – may have been the standard representation in ancient Greece. This would explain the word hypotenuse, which can be read literally as stretching under (the right-angle) – this is also the meaning of the word subtending.
Returning to the doubling sequence, in my explanation below I have used drawings (with instructions) rather than physical triangles. I have also changed the colours of individual unit IRATs in progressing from smaller to larger triangles to emphasise how the doubling arises. Here’s a grid I produced for the original #beingmathematical event.
And here’s the way I represent the doubling sequence.
Note that we could equally have started with the first (unit) IRAT in the “upright”, rather than the “Greek” position.
Let’s imagine a young learner has replicated Mary’s discovery of the doubling sequence. Through previous experience of this task, or through careful planning, suppose the teacher has anticipated this and has a version of my instructions at the ready. Perhaps calling all the learners together, (s)he has them count the units shaded orange and green separately and then either count or calculate (depending on age and stage) the total number that make up each member of the sequence. This seems to me a useful way to bring attention to the concept of doubling and to discuss the equivalence of adding a number to itself and multiplying by two (again taking account of prior learning).
Quite often the first task I give on meeting a new class in S1 (or, perhaps, Primary 7 pupils on a transition visit) is to simply write 1, 2 on the board and ask for the next number: “3”, of course, is the first answer I receive. Asking if it could be anything else will normally elicit “4” from someone. I write up 4 and ask for the next term. It’s not uncommon for 6 to be suggested – perhaps the distinction between adding 2 and multiplying by 2 is not always clear, even in the minds of these older learners than the ones Mary was wondering about. But I’m confident of getting “8” soon, along with a justification and a refutation of 6. Once the rules of engagement are established, some pupils will happily continue the sequence indefinitely in writing, while others gradually give up either through boredom or – more likely – becoming overwhelmed as the level of challenge quickly ramps up: for a seemingly straightforward task, mistakes are common. I usually end up getting everyone to stand up before taking answers round the class, instructing anyone who has either made a mistake or reached the end of their list to sit down. Once there are only a few left standing I bring them to the front of the room for a “knockout” round from which a doubling champion will eventually emerge. Contenders generally find it easier to write their answers on the classroom whiteboard, rather than attempting to continue verbally.
With pupils of this age, there are a number of ways to take this learning forward. One (taking up Mary’s question again) is to invoke the concept of powers and power notation. It is, I believe, clear to learners that the number 2 is fundamental to the doubling process involved in the activities discussed so far. Here is a written sequence that capitalises on this observation.
I would conjecture that learners younger than P7 age (11-12) could make sense of this exposition and rise to the challenge of continuing the sequence. How much younger? It would be interesting to investigate.
Another activity I have used with S1 classes is to have them continue the sequence 16, 8, 4, … Many learners choose to express their answers beyond 1 as decimal fractions, readily obtaining 2, 1, 0.5 and 0.25. However, it becomes increasingly difficult to continue the sequence in decimal form – try it yourself! Picking up on Helen’s point about finding out about what they are attending to, it is not difficult to elicit the word “halving” (perhaps, more likely, “halfing” – the linguistic demands can be just as high as the mathematical ones) which, in turn, can help direct them to representing their answers as proper fractions. The further they get with the sequence 1/2, 1/4, 1/8, …. the more likely they are to realise – with some gentle prompting if needed – they can use the terms in their doubling sequence to continue the halving one. This can also – perhaps at a later stage – provide a lead into dividing any fraction by any whole number.
My current S4 class have recently been learning about indices. As it happens, I used the tasks described above with them when they arrived in S1. I reminded them of this and the halving sequence proved the perfect context for introducing non-positive integer exponents.
Could they have done this in S1? I wouldn’t rule it out, although this might have told me more about performance than learning – some of them have struggled conceptually with aspects of the indices topic this session, particularly in relation to maintaining performance levels over time.
In June of last year, with my visiting P7 class, I repeated the doubling task, but provided a context in the form of the Rice (or Wheat) and Chessboard Legend. This is a story about how the game of chess had impressed the emperor so much that he invited its inventor to choose any reward he wished. Being such a clever person (and good at maths, of course!), the inventor asked for one grain of rice to be placed on the first square of the chessboard, two on the second, four on the third, and so on. Pupils were then challenged to continue the doubling sequence for all 64 squares. After the class had spent an appropriate amount of time on the task (including the contest to find the class champion) I showed them this You Tube clip. (There are a number of versions of this story on You Tube you could choose to show instead.)
Not only does this animation give a sense of the speed and extent of the doubling process, but it introduces another question or, as Helen might say, something we can “hook some more maths onto”: what is the total number of grains – for all 64 squares of the board – required to fulfil the reward?
This is the kind of question I might find myself teaching at the highest level in secondary school: the sum of a finite geometric series. However, much younger learners could solve this particular question easily (assuming, of course, they can successfully complete the doubling task). Once again, a table can help us spot the pattern that will provide the answer.
square number 1 2 3 4 5 6 7 8
grains on square 1 2 4 8 16 32 64 128
total so far 1 3 7 15 31 63 127 255
Can you see how we can obtain the running totals without having to actually add the grains on each individual square?
Here are some pictures of Cuisenaire rods that can help illustrate what’s going on here in a more concrete way. The first two show the doubling sequence as a “stack”, with confirmation that each level is double the level above.
In the next picture we see the series growing: 1, then 1 + 2, then 1 + 2 + 4, followed by 1 + 2 + 4 + 8.
Next, we see each stack re-configured as a line.
In the final image, the rod at the bottom of each stack has been doubled and placed below each line for comparison.
Given that the number of grains required for the 64th and final square is
9 223 372 036 854 775 808
what, then, is the sum of the number of grains required for each square?
In this series of posts, I have been exploring how thinking about approaches that young, inexperienced learners might take in response to mathematical tasks has influenced my own thinking on learning, teaching and doing maths. This particular post has motivated me to seek ways to share these approaches with some of my older, high-attaining learners. These young adults, some of whom will be studying maths at University a year from now, have experienced regular success working with abstract, symbolic mathematics. I wonder if their understanding of some of these concepts will be deepened – as mine has – by taking a simpler, more concrete perspective.
Thank you, as ever, for spending some time with me and my thoughts; I hope you’ve found it worthwhile.