Are your children reasoning in the lesson? Not sure? Or maybe you want to prove that they are?
Here is a simple activity that is fun for the children and shows you – or anyone else – how they are reasoning. Before I go on to explain it, it may be useful to give a definition of reasoning that we can use here – just so we know what we are talking about.
Reasoning is described this way by ACARA, the Australian curriculum authority:
‘Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.’
A simpler, neater, definition is that when we reason, we use information that we already have to prove information that we don’t have. So, for example, we use DNA evidence to reason that someone must be guilty. We use the lengths of a polygon’s sides to calculate its area.
It also helps to define something negatively – in this case, to say what is NOT reasoning. And here are some examples:
Not that those things are bad in themselves. Sometimes we have to guess to get started. Sometime the best thing to do in a given situation is just copy someone who seems to know what to do. And sometimes it is great to realise that you know the answer because you remember it from before. That’s all fine. It is just that where we rely on these strategies we are not, at that point, reasoning.
Here is the activity…
The simplest way to explain is to say that it’s 20 Questions, but with numbers. So 20 Questions goes like this:
• One individual thinks of a type of object at random – such as ‘chair’.
• The rest of the group can ask him/her questions. The individual will only answer Yes or No.
• If the group can guess the word in 20 questions or less they defeat the individual.
For maths, the individual thinks of a number instead of an object. Usually I say it has to be between 0 and 100. With Year 2 or lower, you might want to set it at 1-20. Also, instead of a limit to the number of questions, our goal is to get the answer in the lowest number of questions, improving our score with practice.
If you play this game you will be able to see the reasoning of your class and the people in it. At primary level, the person who finally guesses the answer often wants to claim maximum credit from the others even though he or she may have contributed very little to the hunt. That is a sign that reasoning – using reasons to move from one question to the next – is not taking place, at least with that person.
Between each game, I ask the children to say which questions were good and why (so it’s a good idea to write the questions, or short forms of them, on the board as you go). At the beginning, some may say that ‘Is it 17?’ was the best question because the answer was 17. But gradually the children will start to spot that the final shot was the easy one as all the other numbers had been eliminated. And if not enough numbers have been eliminated then glory-seeking stabs in the dark like ‘Is it 17?’ are a waste of a question (unless the questioner happens to get lucky). The class often takes a while to articulate the idea that a good question ‘narrows it down’. In other words, it reduces the possibilities to a narrower range.
Children also fail to realise, to begin with, that a ‘No’ answer is not worse than a ‘Yes’ answer, necessarily. If you ask ‘Is it an odd number?’, then either a Yes or No answer gives you exactly the same amount of help. And there are some Yes answers that tell you very little. For example, if you ask ‘Is it a two-digit number?’ the answer will probably be ‘Yes’ but it doesn’t get you far because you still have so many possibilities left. Now, you may still feel that ‘Is it a two-digit number?’ is a pleasing answer for you the teacher, because it shows the children recruiting prior knowledge to apply to the task (i.e. they’ve been learning about one/two/three-digit numbers, have remembered it, and are trying to apply that knowledge here). I couldn’t agree more, so you may want to praise some questions while preparing to nudge the questioner towards a more effective strategy.
After a few runs through, I ask the children if they can come up with a strategy that will always get them the answer in a set number of questions – so, can they guarantee to get the answer in 10 questions, or 5? Children then explain their strategies and we try them out. This is crucial because they are now thinking about their reasoning.
Most classes latch onto 0-50 as the first question before long. Either that or ‘Is it even/odd?’. However, it can go in two ways from there. Usually, you have two different strategies being used within the class. For example, after ‘Is it between 1 and 50?’ with the answer No, we might get:
Is it odd? Yes
Is it between 50 and 75? Yes
Is it in the 3-times table? No
Now it is quite hard from here to work out which numbers that leaves. Try it yourself! It’s better to stick to one strategy. So:
‘Is it between 0-50?’ halves the possibilities should be followed by a question that halves what’s left in a predictable, memorable, way. So if the answer was No, then the next question should be either 50-75 or 75-100, and so on, halving each time. For example:
The only remaining number is 59.
Using the ‘halving’ method outlined above, the class should be able to guarantee to find any number within 7 or 8 questions. You may able to refine this further. I’ve only had a couple of classes who got that far (I generally teach primary).
When I’m playing the game, though, I’m content for the children to circle gradually closer to a strategy like this, and don’t worry if they never quite nail it. What I’m interested in is them looking at a task and saying to themselves: ‘How do I make this simple? How do I work steadily towards the answer?’.
One side issue that comes up here is a problem with the edge of the range. For example: is 50 itself between 0 and 50? You might be sure that it is. But imagine if you have a group of children sitting in a row in this order:
Floriana, Luke, Chester, Adibola, Polly.
We wouldn’t say that Floriana is sitting ‘between’ Floriana and Polly (we’d say that Luke, Chester and Steven were). So the word ‘between’ can be applied differently. You can introduce the word ‘inclusive’ here to help the children, (‘Is it between 0 and 50 inclusive?’) and they will have learned a valuable lesson about the definition of a range. In a similar way, by the way, the question ‘Is it below 50?’ doesn’t make it clear (for some children) whether 50 itself should be judged above or below! You could argue that ‘below’ is not ambiguous at all – 49 is below 50; 50 isn’t – and that’s exactly the kind of precision in the use of terms that we want the children to learn.
If, instead of the ‘halving’ method, children try to extend the ‘Is it odd?’ question into a strategy, another set of problems is thrown up. Because to extend that strategy means using times tables:
• Is it odd?
• Is it in the 3xTable?
• Is it in the 4xTable?
This is much harder to operate, and soon challenges the reasoning of the child. For example, if the answer to ‘Is it odd?’ was Yes, then there is no point in then asking if the number is in the 4xTable (or the 6, 8, or 10xTable, come to that) as odd numbers won’t feature in any of those times tables.
Another issue is that it is very hard to know what numbers are left if you eliminate them through timestables. You could do it with a number line or square, crossing out the eliminated numbers, but few of us could do it in our heads.
And finally, and most fascinating, is that if the person choosing has chosen a prime number, then it is not in any times tables – except of course its own. So you’d have to wait until you got to ‘Is it in the 97xTable?’ to eliminate 97.
All of these knotty problems are rich pickings for reasoning. Lead the children again and again back to a discussion of their strategies. Do that by asking ‘What did the answer to that question tell you?’ to develop the children’s logical thinking. Try not to jump in and tell them – stick to questions. If you feel they’re falling short of what you’d hope, just keep encouraging them – that’s more valuable than getting them to the best strategy fast. After all, it’s only a game – not a SAT.
Trust me, you can spend a whole hour on trying to crack this. Alternatively, you can use it to warm up/down at the beginnings or endings of lessons or weeks. Just remember that its main value as a teaching tool – rather than a mere time-filler – is in developing awareness of reasons, and how a chain of reasoning can solve a problem.
Once they have exhausted the possibilities of this game, you could try some of the games on the NRICH website. I particularly like Strike It Out
…and Got It – where the whole class can play against the computer
I am indebted to Peter Worley at The Philosophy Foundation for showing me this game, and Andy West (also TPF) who reminded me of its value.
For more on how to introduce Enquiry into your set of teaching skills, try my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.
You can also buy it here: http://www.philosophy-foundation.org/resources/philosophy-foundation-publications/the-numberverse