This is an account of the third session with L on proving the statement 'All primes >3 are of the form 6n+-1'. Since our last session he had created the following table:

He had highlighted the numbers 119 and 121 as examples that proved the statement to be false. After a while, it became clear that he thought this because there was no prime one less or one more than 120.

This is logically very tricky, I think. To disprove the statement we would need to find a prime that is not of the form 6n+-1, not a non-prime of the form 6n+-1. This would mean extending the search beyond the numbers shown.

We talked about this for a while, sometimes using the statement 'All cows have 4 legs', which was sometimes useful and sometimes not. At some point it became clear that what we had only checked 4-legged animals, and would need to check other-legged animals to disprove/prove the statement.

As we talked about the numbers we had checked, L said: "The numbers are in a funny order." He moved the 6n-1 column to the left of the 6n column, and then spontaneously starting filling in the other numbers (8,9,10,14,15...), as shown below:
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This was an important moment. L looked at the new numbers and said, "These are not primes!" I replied: "How do you know?" He started checking some of the new numbers for primeness: 

L checked five numbers (8, 34, 22, 27, 39) for primeness. This checking is an important part of getting-a-sense-of, but I felt an intervention was required in order to guide him towards the idea of looking at the whole, rather than continuing to check individual examples. He noted that the columns contained evens and odds, and it was at this point that he entered the column headings shown above. 

​As he was about to check 45, I asked: "Can you make a conjecture about whether this number is prime?" He said "It's not prime, because of 27 and 39." We talked about how each number in the 'odds' column was divisible by 3, so there were no primes in any of these new numbers. 

At this point, L asked: "So, where are we at?" This is what I call 'returning to the problem'. I invited him to look at the statement, and look at what he had done. He felt it would simplify everything to delete all references to cows, during which he spontaneously started relabelling the even/odds columns with algebraic expressions 6n+2, 6n+3, ... as shown below:
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This felt significant. After a valuable discussion about 6n+5 being equivalent to 6n-1 (with different values for n), he arrived at this:
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At this stage, he asked a 'what-if' question: "What if the statement we were trying to prove was ... of the form 6n+-2?". He subsequently showed that this statement would be false. This is in some sense a red herring, but perhaps not; what-if questions are certainly something I would encourage a mathematician to create.

He then returned his attention to the example of 119 and 121 that we started with. At this point, I felt some guidance was required, in order to avoid going in a circle. I asked him to look in the 'primes' column (on the left) and invited him to "Find a prime that is not of the form 6n+-1". He could see that all of the primes 5 to 139 were in the two columns 6n +- 1, but what about some other prime? He tested the (known) prime 8293 and and found it was one more than 6 x 1382. He then tested 10169 and found it was one less than 6 x 1695.

Then something suddenly clicked. He exclaimed: "119 and 121 didn't matter!" He could see that they did not constitute a counterexample to the statement. I asked him if he thought the statement was true. He said that he was convinced it was, mostly because of the two "random examples" that he had just tested. Although these two examples do not make a proof, they were valuable in that they convinced him of the truth of the statement, and allowed him to move on from his fixation with 119 and 121 as counterexamples.
   
At this point, L pondered whether it would be fruitful to test the prime 56,597, but I think he also realised that this would not be of any use as a proof. I asked him what column he thought 56,597 would be in. He thought it would be in one of the two columns 6n+-1. I left him with the question: "How could you know that 56,597 - and indeed any prime - must be in one of these two columns?"

I am working with a group of teachers on noticing. Recognising the difficulty of recording accounts, we set ourselves to notice something very simple that we do not normally notice, and make accounts of it. I set myself to not make lists, and to recognise when I felt the desire to make a list. 
 
As soon as I set myself to do this, I went to write 'NO LISTS!' on my hand, and then stopped. I started making a list of the things that the other teachers in the group had set themselves to do, and then crossed it out. In the days that followed, I went to make a list a number of times, either on my hand or on paper, but stopped. 

I realised that I became very anxious that I would not remember something. I found myself frantically going over in my head what was coming up over the next day or two. After a couple of days this went away. I started to question why I had this tendency to go over what I needed to do so frantically. As the worrying subsided, I felt that I became more present in the moment, rather than worrying about the future.

I have become more relaxed. I have found myself spending more time with my partner in the evenings, instead of doing things on my long-term list. A few jobs that I was planning to do have slipped off the bottom of my non-list. I have found that things that matter nearly always come to my mind when needed. I had to phone my partner at 4pm, I remembered to phone her. I had to take a pen-drive to a colleague, I remembered to take the pen-drive.  

I found myself doing things as they came to mind, rather than writing them down and doing them later. I had to take some books back to the library, so rather than write it down, I put the books in the bag that I was going to take. There is something paradoxical about listing: I make a list in the perception that it will relieve anxiety, and yet it allows me to procrastinate and nags away at me, increasing my anxiety.      

I have broken my vow not to make lists three times. Once, to write something on my hand that I had forgotten that sprang to mind; I didn't want to forget it again, as there was a deadline coming up. Upon writing it on my hand, I ignored it a number of times before actually doing it. The second time I made a list was a list of books I was interested in reading. I buy a lot of books, more than I can read, and wonder if I would be better off without lists of books!

The third time was a Christmas shopping list, which I think was useful. There are different types of list, from short-term lists (such as shopping lists), to long-term lists with things that sit there nagging away at me to do. It is these types of lists that I will not be making any more.   

There are possibly other types of list. I get ideas about planning lessons that I used to note down, but now they either stay in my mind or they don't, until I sit down to plan. A lesson plan is a list. I have been more disciplined in making accounts (as part of noticing) since not making lists. Accounts are in some sense a list of past events, to revisit some time in the future.  

Doing something, such as making a list, even though I have set myself not to, or the opposite - not doing something that I have set myself to do - is a phenomenon that I am very interested in. New Years' Resolutions are a great example of these. 

I have forgotten things, but not important things, and it turns out that forgetting things is often not as problematic as I feared. I had to make a presentation last week, and had to make it up as I went along. I think it went OK. 

People have reacted strongly when I say that I am not making lists. Some say things like "Oh. yes, but I need them for my work!' People talk about making lists of lists, while others laugh and say, "I never make lists!"

This is a post describing some ideas for introducing 2D and 3D vectors that I used with my students this week. Each of the tasks contain an element of physical construction. It is my conjecture that the time constraints at A-level and Higher lead to teachers feeling as though they do not have enough time to spend working with vectors in a physical way. In my experience, this leads to learners having difficulties visualising vectors problems, particularly those in three dimensions. 

The students then worked on a variant of this task, but with the two vectors i and j, to gain familiarity with this notation  whilst exploring collinearity. Here is a recording of the screen as they worked on this task. 

We worked on a number of other interactive tasks in 2D before extending to 3D. ​I wanted the students to be able to see and feel what was happening in three dimensions, so I (and they) posed a number of problems about a 2 x 2 x 2 cube, as can be seen in this video. Here is a screenshot:

There are some lovely moments in this video, particularly when the students start posing their own problems, and explaining what they are doing and why.  

After the video, the students were working on some problems (see below), for which the cube proved useful. During this, I decided to ask one of the students whether he could find two points on the cube with a length sqrt(7). He replied: "For that you would need 1 ,2 and 4 [as the squared components of the vector], but that's not possible because of the 2."  

The students worked on a set of exam questions. For each of the 3D problems, I asked them to construct the image in geogebra before solving. One of the problems was as follows:
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Here is a video of the students constructing the image. This felt like a very useful exercise, as it provided a visualisation, and provided a means of testing conjectures about the position of the coordinates. 


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​Yesterday, I introduced J and K to recurrence relations. I set a few problems for homework. This post contains descriptions of one student's (J) response to three of these problems. Here is the first one:

J (unsuccessfully) tried to solve part (a) using trial and error. This is a common approach when unsure what to do. There is some value in trial and error approaches for getting a sense of a problem, but often a shift towards an algebraic approach is required. How might J learn how and when to make this shift?

It is something that many learners find difficult to do. I think there are a number of reasons for this. To solve this problem you can substitute the information about the terms into the recurrence relation to create two simultaneous equations, that can then be solved. How might a learner realise this? Recognising the 'type' of question (due to familiarity) might be useful, but this will not always be enough: it is necessary to prepare for variations in what is possible (see below). 

Something like a leap of faith is required. It might be necessary to form an equation without knowing if or how it is going to be useful. Often, it will not be clear how the equations may be helpful (or whether they will be solvable!) until they are formed. Then, solving the equation that we have formed requires working on a level of abstraction at least once removed from the original problem.

Here's a second problem we worked on:
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Again, part (ii) requires the formation and solution of an equation in order to solve the problem. However, this time the circumstances are different, not least because it is a quadratic that is formed rather than a pair of simultaneous (linear) equations.

Once J had formed the quadratic equation, it was very easy for him to know what to do - solve it! I think our work on the previous problem had prepared him well. Flexibility is required, something along the lines of: it might be useful to use the information to form some kind of equation, and then solve whatever is formed, whilst bearing in mind its relevance to the original problem. Had he memorised the previous question as a type - perhaps 'form a pair of simultaneous equations and solve them' - he may have been puzzled. 

Here's a third - rather challenging - problem we worked on:

It was not clear to J what to do here, not least because the coefficients in the recurrence relation are themselves variable! It takes a quite large degree of faith to substitute the expression 1/2.sin(x) into the recurrence relation in order to form a trigonometric equation in x that can then be solved to find the variable x! J was able to solve this problem, but only after all of this work on forming equations, and after a discussion about how we 'normally' form an equation when trying to find the limit.

Here are J's responses to the three questions; the equations formed are boxed (click to enlarge):

​Today, J and K used this spreadsheet alongside these tasks to explore recurrence relations. The first two tasks were relatively straightforward, designed to help them gain familiarity with recurrence relations for a variety of linear and non-linear sequences. ​The third task was an exploration of recurrence relations of the form:

Here are the questions that I invited them to explore:
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Below is their record of the first few values of u(1), k and d that they tried:
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The example in the red box was the first sequence that tended to a limit. The examples in the green box are special cases. The breakthrough came with the first example in the blue box. Here is a screenshot of the spreadsheet for this example, to give you an idea of what they saw):

It was clear to them that the sequence was tending to some limit. At this point, K said: "k makes the limit." (see writing in blue box). They tested this by changing k to 9, as can be seen, presumably to test some conjecture not yet made explicit, before changing it back to 0.3 and then changing d instead (see also blue box). This confirmed to them that the limit was a function of k and d, but that it was the value of k itself that determined whether there was a limit or not.    

​K then made the conjecture that k must be <1, and J mentioned that it must also be greater than -1. This can be seen in the red box in the next sheet they worked on:
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The numbers in the red box (-100, -2, -1.3) are some of the values of k they used to test the -1 conjecture. K was excited about the boundary example in the green box, which alternates between 0 and 1.  

Then they only altered k for values between 0 and 1 (keeping the first term and d constant), and looked at the changing values of the limit L. They could see there was some connection between the value of k and L, but could not derive it, at least in the time available. I think it would be very difficult and perhaps not that productive to discover how to work out L,  I described an algebraic approach for finding L, using some of their examples to illustrate the method.

It felt as though this approach, of using a spreadsheet along some explorative tasks, was a nice way of introducing recurrence relations and the idea that some of them tend to a limit. We will look at graphical representations tomorrow.  

This post is about a particular aspect of a session with two Higher maths students (J and K). 

​We were working on analysing some exam questions that the students had tried for homework, but found difficult. This is the first time we have tackled some of the difficult questions at the end of Higher papers (English A-level teachers might find these papers a source of interesting / challenging problems).

I decided to break the problem down into a set of questions and prompts for the students to work on, rather than through. A loose structure for these prompts started to emerge that might be interesting to develop:
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• Some reflection on a first attempt at the problem. What did you try? Why? Why was it difficult?
• Some 'breaking down' of the problem with isolated / simpler / numerical examples, designed to give some insight into structure.
• A return to the original problem in light of this breaking down.
• Then, an invitation to try a similar (exam) problem, and then create a problem 'like this'.
• Reflection on what was learned about the content, the 'type' of problem, and/or solving problems more generally.  

​Here are some of the responses (click to enlarge): 

There were further reflections on this problem that were very interesting - more on this below.

I had also created a similar set of questions related to this problem, along the lines of the structure described above. Here are some responses (click to enlarge): 
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Reflection:
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At this stage, I asked J and K what they had learned from working on these problems. Their responses were generally about the content, logs. Then J had a real insight. ​He said: "We can change something we don't know into something we know about by re-naming it, and then replacing it later." He gave the example of using A and B in place of log(3,x) and log(3,y) in problem 2 (see axes, and equation):
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Re-naming log(3,x) and log(3,y) as A and B allows us to solve the problem as if it was about straight line graphs, before replacing A and B with what they represent - log(3,x) and log(3,y) - in order to find the relationship between x and y. 

We then realised that something similar had happened in problem 1! Renaming log(3,x) with P [and log(9,x) with 1/2.P] allowed us to solve the equation in terms of P, before replacing P with log(3,x) to solve the original equation (see circle):

This is of course not new for mathematicians. It is a form of substitution, but is not only substitution. It is rather a type of working as-if, a suspension/simplification of the original problem by re-naming something, thus making it easier to work with, and then re-placing that which was re-named in order to solve the original problem.

This is an exciting realisation / strategy, and now that it has come into our conscious awareness and we have named it, will be something we can come back to. 
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I am teaching maths once a week to a class of 8-11 year olds. This is the ninth (and last) in a series of lessons on arithmetic in different bases using Cuisenaire rods. 

Today started with children giving feedback to each other about the clarity of the explanations in these videos they made last lesson, which they did with sensitivity and insight. I considered spending this lesson making second versions of the videos, but decided not to, mainly because I have been asked to teach the children about 'time' next week, and wanted to spend the last lesson on this sequence exploring orange rods (base ten). The feedback will be useful for the next time we make videos. 

After we discussed the videos, the children worked on the task below in pairs, comparing subtraction calculations in blue (base nine) and in orange (base ten):
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A key difference in today's lesson was that I did not give the children rods, to see if they could perform the calculations mentally. Here are some examples of the children's work: 

There are some lovely calculations here, and a beautiful explanation at the bottom of one of them: "If you don't exchange the blues or the oranges for whites you get the same answer." 

I had planned to give the children around 40 minutes to do this task (and a similar one involving multiplication), but I sensed a loss of energy in the room after around 25 minutes. This has not always been the case. It was not clear to me whether it was the task, a shared mood among the children (it was the 1 December?), or whether it was over-familiarity with a particular way of working

I decided to bring the class together, to guide exploration on the subtraction task, using the board as a focal point, in order to alter the structure and pace of the lesson. I decided in those few moments to invite each pair to offer an example of a calculation that was different in blues and oranges, which I wrote on the board. I considered also inviting them to offer a calculation that was the same, but felt at the time that this might add to complexity and loss of energy.

Here are the examples the children offered:  

I asked if anyone had anything to say about what had been written. One of the children then made the conjecture "Oranges are always one more than the blues". There were a few qualifications to this conjecture: (1) "If these [calculations] are all correct", and (2) "It's not always true, if you don't exchange," as can be seen at the bottom of the board in the following image:
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There followed much uncertainty about the calculations for 1 0 0 - 8 8, so I gave each pair a few minutes to work on it. I wrote all of the answers to their calculations on the board.

The centre of gravity for the lesson was unusually at the board, in order to generate a shared focus/energy. It was as though I was a minute-taker for the children's ideas, writing down conjectures without evaluation. The board was the medium through which ideas were reflected back to the children:    

Around half of the pairs had both calculations correct. I invited someone to explain how to do it using the rods, and was pleased when one of the less confident children offered what turned out to be a lovely clear explanation. The rest of the class gathered round, and affirmed what she had done. 

All of this work brought the following amendments to the conjecture: "At least one of the numbers in the orange calculations is one more than the blues unless you don't exchange!" as can be seen below: 
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The (green) writing on the board is a reflection of the evolution of the children's ideas.

It was noticed at this stage that the calculations for 1 1 3 4 8 - 8 2 4 were not 1 apart, but rather 1 0 0 apart, another counter-example to the earlier conjecture. There were a few minutes left, and my decision to centre the lesson on the board had resulted in mounting energy; it seemed to me as though all of the children were now 'active'.

This building of excitement led me to propose a final challenge: Create a calculation in which all of the digits of the answer change when calculated in blues and in oranges. There was a real buzz as all the pairs worked to find an example. Here is what they came up with:
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Some lovely examples here, and for me some evidence that they understood what we had been doing with the rods.

​We rounded off this sequence of lessons with a discussion about calculations using orange rods. A number of children had noticed that calculations in oranges were just like 'normal' calculations, and some of them realised they could use their calculators again to check their answers! 

Next week, time.