Here is a diary of a Higher (Scottish ASlevel) maths lesson in which I highlight five things students did when presented with a set of unfamiliar problems.
Here is a diary of a Higher (Scottish ASlevel) maths lesson in which I highlight five things students did when presented with a set of unfamiliar problems.
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This is a description of my third maths lesson with 'upper primary' children (ages 811) at my local school in Westray. It lasted just over an hour. I started by reminding the children what we did last week by showing them this, and praised the quality of their descriptions. Then we started work on this task, designed to help the children gain familiarity in converting between base four  part of our ongoing exploration of different mathematical tasks in different bases  and base ten. Here are some responses to the (reasonably straightforward) questions on the first page. Notice the different representations/language used: Here are some responses to the questions at the top of page 2: Again, there were a range of names used, particularly for the cube (64 whites), such as "ultimate rubix cube", and "4 squares". The response "1 0 0 0" shown above came after a lovely moment: the children in question noticed that 63 whites was "3 3 3", but were not sure how to express 64 whites, suddenly realising that an extra column was required. We had a class discussion about the 'objects' represented by this extra column: I felt that these questions were helping students gain a real familiarity with working in base 4. However, this lesson really 'caught fire' when students started creating their own questions: After a few minutes of making up questions (which they had to work out the answers for), it dawned on me that it would be fun to invite them to write their 'best question' on the whiteboard: This is what they wrote: There is a lovely range of questions, and points for debate, including:
The children then spent the last 20 minutes of the lesson having a go at each other's questions; there was a great deal of energy, excitement, and mathematical thinking : There is so much here that we can discuss next week, before doing this. I think the way the children respond to the 'patterns' part of this task will be really interesting. Here is a question from a Scottish Higher (equivalent to AS) exam paper: Here is one student's response: How would you help the student take a step beyond, in this case? What is required (in this case, and more generally) for students to become able to take such steps beyond for themselves? This is an account of a lesson I taught today, with 19 children aged 8 to 11. The aims of the lesson were: (1) Looking at, and describing. (2) Working with other people. This is part of a unit of work aimed at developing childrens' understanding of place value. This is one of a series of lessons in which we will explore various bases using Cuisenaire rods. The children worked in pairs of their choosing, and one three. I started by presenting this video. We watched it twice, and then I asked them to describe what they had seen, continue the pattern (using rods of their own), and describe anything they noticed. Here are their responses: There were some lovely descriptions, including:
One child who was not particularly engaged last week seemed to enjoy using the rods to mimic my actions on the video almost exactly, and became increasingly engaged in this lesson. I invited children to share any descriptions with the whole class, and a number were willing to do so, and gave extraordinarily clear descriptions. It was also interesting that all of the other children seemed to be listening. We then watched a second video, and again, some lovely descriptions: There were some very detailed descriptions of the video. Some highlights include:
A third video, this time counting up with browns (and reds) instead of greens, and more descriptions: Highlights includes:
The next video introduced the idea of adding columns, this time using green and white rods. I changed the questions slightly, as can be seen in these examples of children's responses: I love the sensemaking in these responses. There was great excitement when the children discovered that we needed extra columns, with a number of children spontaneously making 'cubes' by putting 3 squares on top of each other. It was also exciting to realise we only needed three numbers (0, 1 and 2). Finally, I asked the children what they thought would happen if we had pinks and whites (4 whites in a pink), not greens and whites. They did not see a video or have cubes, instead having to rely on their imagination. Here is what one pair did: This quality of response was not unique, but some students also began to become confused, which makes a lovely starting point for next lesson. This took us up to an hour and twenty minutes, and time for break. Next lesson we will review this, perhaps with a few questions that require conversion between numbers of whites and the number in base 3 and 4. Perhaps I might ask students if they would like to make videos of their own, with rods of their own choosing. Perhaps we will follow this with some addition and subtraction in various bases (perhaps moving into binary, and/or extending to base 5), and then some times tables (perhaps in base 6), depending on how the children get on with it. Finally we might do some work in base 9, 10 and 11. At some point I think it would be fun to do some work on logic gates, and maybe some programming. Imagination takes different forms for different people. In our minds, we can form and manipulate images of past, future or purely abstract objects and occurrences. In The Mind Teaches the Brain, Gattegno describes the importance of imagination: The imagination says that we can make the potential actual… Work with the imagination may be one of the most important aspects of mathematics education. But how often are students allowed time to imagine? If a student is sitting looking into space, it is often assumed they are doing nothing. I have often heard teachers advise seemingly 'inactive' or 'stuck' students to 'do something', meaning to write something down. But this might be counterproductive when solving problems. In his book Creative Thinking, JG Bennett suggests putting pen to paper may inhibit creativity, the ability to think of something new: Sometimes when people set themselves to understand something… they proceed to get all the relevant information down in front of them. They write it down on a sheet of paper… But this is not at all sufficient. It is necessary that one should do that work of selection, elimination and assembly (as far as possible) inwardly… Bearing this in mind (!), try to solve the problem below without putting ever pen to paper, even if you are very tempted! [If possible, you may also find it interesting to notice how you form and manipulate images in your mind] Three fifths or four sevenths? Continue the pattern... could this help you decide? Is this approach always possible? In Being Peace, Thich Nhat Hanh asks: Can we be a river and experience the hopes and fears of a river? Before I arrived, the river was undifferentiated energy, like myself. I come near to the river and dwell. In Conversation on a Country Path, Heidegger describes thinking as a ‘lettingoneselfintonearness’. I watch the water flow along its path, I listen to the gentle bubbling sound. In this act of attention, there is differentiation: between the river and its surroundings, and between the river and me. But there is also an attraction, a sense of wholeness. In The Principles of Psychology, William James states: …each of us literally chooses, by his ways of attending to things, what sort of a universe he shall appear to himself to inhabit. A river is never the same, but we consider it the same if it follows a recognisable path. With recognition comes identity. We may gain our identities in a similar way, as described by James: A man has as many social selves as there are individuals who recognise him. Can we be a river and experience the hopes and fears of a river? Does the river have hopes and fears, a sense of self? From where do we derive our sense of self? William James says ‘the stream of our thought is like a river’, and describes consciousness as: … a succession of perishing thoughts, endowed with the functions of appropriation and rejection … Thoughts flow past, but sometimes we bring our attention to the flow itself. This may be regarded as stepping out of the flow of time and becoming aware of time itself. We cannot directly experience the past or future, only the present. But even the present is elusive, as James describes: Let any one try, I will not say to arrest, but to notice or attend to, the present moment of time. One of the most baffling experiences occurs. Where is it, this present? It has melted in our grasp, fled ere we could touch it, gone in the instant of becoming. He suggests that the present moment is an abstraction, and that we experience the present not as a moment, but as an extended period of time which he calls the ‘specious present’: These lingerings of old objects, these incomings of new, are the germs of memory and expectation, the retrospective and the prospective sense of time. The brainprocesses that create this feeling of succession are those that give our sense of time, and ultimately our sense of a continuing self: Resemblance among like parts of a continuum of feelings (especially bodily feelings)… constitutes the real and verifiable personal identity which we feel… 
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