## A new term, and a new school

I’ve started a new job this year at a new school. This is the second time I’ve moved schools and I have to say it has been a much smoother transition than it was the last time.

This school is very close geographically to my last school and as such has a similarge make up of students.

I’ve now got to the point where I can remember most of the names of my students and we are working hard to put some real progress in the classroom.

It’s been a time of change all round really, my daughter started school this year too, and as such I have now become the parent of a school kid. That’s been weird all round but she’s enjoying it and I think we picked a really good school. My wife and I have been invited to attend a meeting there next week when the will tell us how they teach English and maths in reception. I’m interested to see what they say about it, particularly in mathematics!

Also this summer I finished my MA and I’m awaiting results for the dissertation. I think it went OK, but I won’t know until the brown envelope arrives with my feedback and grade. The dissertation was entitled “Investigating problem solving as a means to improving understanding in A level mathematics” – catchy I know. I enjoyed writing it and I may share a summary on here at a later date.

All in all its currently a time of change and that brings with it excitement and challenges.

*How has your start to the new year been? How are your new classes? Have you started a new job? I’d love to hear about it in the comments or via social media.*

## Circles and Triangles

Regular readers will know that I love a good puzzle. I love all maths problems, but ones which make me think and get me stuck a bit are by far my favourite. The other day Ed Southall (@solvemymaths) shared this little beauty that did just that:

I thought “Circles and a 3 4 5 triangle – what an awesome puzzle”, I reached for a pen an paper and drew out the puzzle.

I was at a bit of a loss to start with. I did some pythag to work some things out:

Eliminated y and did some algebra:

Wrote out what I knew:

And drew a diagram that didn’t help much:

I then added some additional lines to my original diagram:

Which made me see what I needed to do!

I redrew the important bits (using the knowledge that radii meet tangents at 90 degrees and that the line was 3.2 away from c but the center of the large circle was 2.5 away):

Then considered the left bit first:

Used Pythagoras’s theorem:

Then solved for x:

Then briefly git annoyed at myself because I’d already used x for something else.

I did the same with the other side to find the final radius.

A lovely puzzle using mainly Pythagoras’s theorem, circle theorems and algebra so one that is, in theory at least, accessible to GCSE students.

I hope you enjoyed this one as much as I did!

## End of term emotions

What an emotional few weeks. This time of year is always emotional, but this year that has been ramped up to a whole new level. There is all the usual emotion of Y11 and Y13 classes finishing the year, and this year that has been compounded by the fact that I am leaving my current school at the end of term.

I’m sad that I won’t work. With some of my colleagues anymore and I’m sad that I won’t get to teach some of my classes next year. On the flip side, I’m excited by the challenge that lays ahead and I’m excited by the fact I’m going to be working with some former colleagues and friends again.

Then I’m devastated by the referendum result. I thinks it’s a disaster for the country for so many reasons. The economy will suffer, the rich diverse culture that we have in Britain will suffer, it will affect touring musicians which may mean many UK based ones will give it up and less overseas stars grace our shores.

Then there’s the rise in hate crime. In the first week after the referendum there were 300 reported hate crimes against non brits. Up from 60 on a normal week. I find both those figures abhorrent, but the larger one particularly so. To me it shows that the racist and xenophobic underbelly of our society now feel they have been legitimised. It was always going to happen they way Nigel Farage and his cronies have spent the last two decades selling the EU debate as “we want our country back”.

## Another Multiplication Technique

I’ve written a few posts over the years on different multiplication techniques (see this and this), there are many and each has its own appeals and pitfalls.

Today I discovered another. I was looking over Q D1 exam paper and came across this flowchart:

The questions were all fairly reasonable and one of my students was completing the question to see if he had for it right. Afterwards I asked if he knew. What the algorithm was doing, he wasn’t sure at first but when another student explained it was finding the product of x and y he realised.

Then he asked, “*but why does it work?*”

I looked at the algorithm and initially it didn’t jump out at me. I tried the algorithm with 64 and 8.

I could see it worked through mocking factors of 2 from the left to the right but this time there was no odd numbers, so I picked some other numbers:

And that’s when it all made sense. Basically, what’s happening is you are moving factors of 2 from x to y thus keeping the product equal. When x is odd, you remove “one” of y from your multiplication and put it in column t. Your product is actually xy + t all the way down, it’s just that until you take any out your value for t is 0. T is a running total of all you have taken from your product.

The above becomes:

*40 × 20 *

*= 20 × 40 *

*= 10 × 80*

*= 5 × 160*

*= 4 × 160 + 160*

*= 2 × 320 + 160*

*= 1 × 640 + 160*

*= 0 × 640 + 160 + 640*

*= 0 × 640 + 800*

*= 800*

I tried it out again to be sure:

This is an nice little multiplication method that works, I’m not sure it’s very practical, buy interesting nontheless.

*Have you met this method before? Have you encountered any other strange multiplication techniques?*

## Understanding students’ ideas

I read a really interesting article today entitled “Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solving” by L.B. Warner and R.Y. Schorr. It is a great report that looks at three teacher’s responses to their students’ solutions to a problem, and it discusses in detail how the teachers reflected on them together. It is well worth a read for all maths teachers.

The teachers were middle school maths teacher and they were presented with a problem to solve by the researchers they then presented their classes with the problem and debriefed afterwards. It was clear that the teachers didn’t have the thorough subject knowledge of a high school maths specialist and this lead to them failing to pick up some misconceptions and not allowing students to explore their own methods if they didn’t understand it, rather moving them on to a method that was more familiar to the teacher. The reflections of the teachers are interesting, they all appear to become frustrated with themselves when analysing their responses and are able to reflect on this by offering alternatives. It does show that deeper subject knowledge is important to allow that exploration to take place. The study showed that in this context when the teachers just told students how to fix their mistakes, rather than question students as to why they had made them, this led to student confusion. This suggests that we should be striving to understand our students thinking whenever possible and using that to combat their misconceptions so they don’t fall into similar traps again. This will also allow students to see why they are coming up with these misconceptions.

There are many teachers who, at times, fail to understand the lines of mathematical thinking taken by their students when solving problems. This can lead to not giving the proper amount of credit to valid ideas and it can lead to teachers failing to spot misconceptions. Some students may have a perfectly valid method but as the teacher may not see where they are going they can sometimes block this route off. This has deep links to “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov , which I read recently (you can read my reflections here). I feel that it shows that deep subject knowledge is important, as is allowing students time and space to work through the problem on their own. Rather than saying, “No, do it this way” we should, be encouraging students to follow their nose, as it were, and see if they can get anywhere with it. It is always possible to show the students the more concise method when they have arrived at the answer to bui8ld their skill set.

Warner and Schorr believe that subject content, as well as pedagogical content is vitally important to teachers to enable than to know how to proceed when a student is attempting a problem. They look at relevant literature on this and quote Jacobs, Philip and Lamb (2010) who suggest that this is something that can be achieved over time and Schoenfield (2011) who says that teachers tend to be more focussed on students being engaged in mathematics and replicating the solutions of the teacher rather than allowing students to meander their own way through so the teacher scan identify their understanding and misconceptions. The latter would, in my opinion, be a much better way of developing, and I agree with JPL that this is a skill one can develop over time.

**References**

Jacobs, V. R., Lamb, L. L. C., and Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. *Journal for Research in Mathematics Education*, **41**, pp 169–202.

Schoenfeld, A. H. (2011). Toward Professional Development for Teachers Grounded in a Theory of Teachers’ Decision Making. *ZDM, The International Journal of Mathematics **Education*, **43 **pp 457–469.

Von Duyke, K. and Matasov, E. (2015). Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency.* Pedagogies: An International Journal.* **11**:1. pp 1-21

Warner, L.B. and Schorr, R.Y. (2014). Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solvin. *Proceedings of the 7th International **Conference of Education, Research and Innovation, *Seville, Spain, pp 669-677.

## Impossible scorelines

Yesterday I was watching Exeter Chiefs vs Saracens in the premiership final. During the first half I was thinking about scorelines and how they are constructed and I thought that you could come up with some interesting activities around scorelines.

My first thought was “what scorelines are impossible?” – in Rugby Union there are a variety of ways to score, you can kick a penalty goal or a drop goal for 3 points each, you can score a try for 5 points and if you score a try you get a chance at kicking a conversion for an extra 2 points. From this we can see obviously that 1, 2 and 4 are impossible but I wondered briefly if any others were. I don’t think there are as you can make a difference of 1 between an unconverted try and 2 penalties, however that’s not really a strong proof. I may think about how to prove, or disprove, it later.

I then thought about the 4 4s challenge, and the variety of related challenges based around the year etc. I thought this might be interesting to attempt with rugby scores. It would be nice to investigate how many ways there are to make each score too, and to see if there were any patterns to it.

My thoughts turned to rugby league, the scores in that are 1 for a drop goal, 2 for a kicked goal and 4 for a try, thus all scores are possible, but it still might work for a 4 4s type challenge or an investigation into how many ways each score can be made.

I considered other sports too, football would of course be pointless, basketball would provide a simpler version which could be good for embedding the 2 and 3 times tables and that was as far as I managed.

*Have you considered any of these activities or similar? Do you know of any other sports with interesting scoring systems that could be investigated? I’d love to hear in the comments or on social media if you have.*

## Infuriatingly impossible exam questions

Today I was working on some Vectors exam questions with my Y13 mechanics class and I came across this question:

A student had answered it and had gotten part d wrong. What he had done was this:

I have recreated is incorrect working.Obviously he had found out when the ship was at the lighthouse, instead of 10km away. I explained this to him and started to explain how he should have tackled this when a sudden realisation angered me.

Now for those if you that didn’t work through the question, here is the actual answer:

Can you see what had me infuriated?

This is an impossible answer! If the lighthouse is on the trajectory of the ship and it will hit said lighthouse at t=3 then that would stop the ship! At the very least it would slow it down!!!! In reality it would have to avoid the lighthouse and change trajectory. Meaning the second answer, T=5, would not happen under any circumstances!

My initial thought was:

“are they expecting students to spot this and discount the second answer? That’s a bit harsh.”So I checked the markscheme:

Nope, they are looking for both answers. Argh! I can understand using a real life context in mechanics, I really can. But why not check for this sort of thing!

What do you guys think? Is this infuriating or am I just getting get up over nothing? I’d love to hear your views in the comments or via social media.## Share this via:

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