Archive

Archive for June, 2019

Interesting radius problem

June 29, 2019 1 comment

Last week a friend of mine sent me this interesting puzzle from Ed Southall (@solvemymaths). I was out and about when I received it so was unable to make a start on it right away but did have some initial thoughts.

Initially I assumed that the chord on the vertical side of the square was half the side and this led me to assume that it would be a problem I could solve using either similar triangles or chord bisectors. However, when I later sat down to have a go at it I realised that this assumption was incorrect, there was nothing to indicate that was what was happening on that side of the square and if this info were needed to answer the question Ed would have included it.

My next thought was that I could place the construction onto a set of axis and use a bot of coordinate geometry to solve it. At this point I decided this was impossible as there would be too many variables (3) to solve with 2 points. *

Today, still having no luck, I had a bit of time so I thought I would try to construct it and see what happened. First I constructed a square and marked the mid point and found a point that was equidistant to draw the circle.

Then it hit me – any point on the bisector of the line would give me a circle through these 2 points. How could I construct it? So I looked back at the initial puzzle and the answer was there, shouting at me. I had missed earlier (*) that there is another point on the edge of the circle. This meant we were dealing with a circle through three given points, not only would I be able to contstruct it now, I wouldn’t even need to as I could go back to the coordinate geometry approach I had chalked off before.

First I set the bottom left corner of the square as the origin, this gave me the three points on the circumference as (0,0) , (0.5,0) and (1,1).

Start with the equation of a circle:

(x-a)^2 +(y-b)^2 = r^2

at (0,0)

(-a)^2 + (-b)^2 = r^2

a^2 + b^2 = r^2 (1)

at (0.5,0)

(0.5-a)^2 + (-b^2) = r^2

0.25 – a + a^2 + b^2 = r^2

Sub in (1)

So 0.25 = a (2)

at (1,1)

(1-a)^2 + (1-b)^2 = r^2

a^2 – 2a + 1 + b^2 – 2b + 1 = r^2

Sub in (1)

2 – 2a – 2b = 0

1 – a = b

Sub in (2)

b = 0.75

At this point I realised fractions made more sense, so using (1):

(1/4)^2 + (3/4)^2 = r^2

r = rt10 / 4

There you have it. A nice little puzzle that took me far longer than it should have. Often when I solve these puzzles others then pop up with nicer more concise solutions. If you did it a different was I would love to know.

After I had finished I thought I would try doing the construction anyway, I’m not amazing at construction and I’ve been working to improve that so I had a crack. Pretty close correct to 2dp which is as close as I could measure:

Advertisements
Categories: Maths

Modelling in class

June 28, 2019 Leave a comment

Recently the idea of modelling is one that appears to be following me around everywhere. I mean that in the sense of modelling as a teaching strategy, not that Calvin Klein is stalking me and urging me to take to a cat walk for him.

The repeat appearance of discussion around modelling has got me thinking about it a lot. At the recent #mathconf19 many of the sessions discussed modelling. Ed Southall (@solvemymaths) did some great modelling on constructions and suggested many ways use it to improve outcomes, Kate Milnes (@katban70) talked through modelling a mathematical thought process and using it to help students achieve their own and Pete Mattock (@MrMattock) looked at visually modelling abstract ideas to make sense of it.

A recent CPD session I attended split the group into two and a different teacher taught each group how to construct an origami crane. One taught using modelling and instructions while the other went out of her way not to and then the different outcomes were discussed.

The trust I work in sees modelling in the classroom as best practice and it is encouraged in all lessons. This is similar to stories I hear from friends in other schools and trusts in the local area.

Then I read this piece by (@mrgmpls) which spoke about the “norm” in lessons being that students are given problems and expected to struggle their way through with minimal input because “without struggle there is no learning”. The blog post was arguing that this is not the best way to teach and pointed to many examples of recent posts about “desirable difficulties” and the such as evidence that this anti-modelling feeling was very prevalent in education today.

This got me thinking on a few levels, firstly it made me think about struggle in the classroom. I’m a firm believer in the idea of modelling. I think that modelling how to do sometime a good solid worked examples should be a staple of any teaching. But I also see the need to struggle in the classroom. If we model processes and have students then complete basically the same question following the model and never get them thinking about it again then we open them up to the possibility of becoming very unstuck in an exam if a topic is examined in a different way.

For me, this means that students need to learn the processes and the conceptual understanding of the topics together. I would also argue that completing exercises of similar questions to embed these is a very good idea. However, there needs to be some point when students need to learn to apply their processes and knowledge outside of their comfort zone. For instance, when teaching trigonometry I would teach non-right triangles and right triangles separately. I would teach sine and cone rules separately, but I would always incorporate some lesson time at the end to a series of problems where they have to deduce which process, or processes, they need to use to be able to answer the question. I might even model my thought process for them, but they will then need to think about why they are doing this and apply it.

I thought that this would be a common theme in all classrooms, so the second question I had from the article was “is there really a feeling of anti modelling at play?”. Having discussed it a bit with the author I discovered that he is based in the US, and I got to thinking that maybe it might be a US vs UK idea, or that perhaps it was even just an idea limited to the state he teaches in, so I tweeted out asking if any on edutwitter were in schools where modelling is discouraged.

I was surprised to discover that actually there are some UK based teachers who are discouraged from modelling in the classroom. This makes me wonder how widespread this is, and what the rationale is for discouraging modelling. If you are in a setting that discourages modelling, or are against modelling, I’d love to hear about it and the reasons behind it. Please get in touch via the comments or on social media.

A perfect circle

June 27, 2019 Leave a comment

Well maths edutwitter has taken up a new collective hobby over the last few days, and it’s drawing freehand circles.

It has come about since Rob Smith (@RJS2212) tweeted this link to a webpage on which you draw a freehand circle, click analyse and get rated.

I’m not very good:

My first attempt scored 156, after many practices I’m scoring around 150 000 but I’m seeing others hot well into the millions. It’s a fun app that’s a tad addictive and I’d urge you all to have a go.

Having played a bit I was in some discussions on Twitter and one thing that I found interesting was that right handed folks appear to instinctively draw circles anticlockwise where left handed folks instinctively go clockwise. If you tried it I’d love to hear if you followed this apparent pattern. When I noticed this I tried drawing it clockwise and it turns out I score better going that direction.

I’m intrigued by the pattern though. I wonder if it could be due to a preference of the dominant side of the brain to a specific direction, or if it’s due to other factors? Maybe the way we write etc. Do left handed people write the letter o in a different direction? My o is always written anto clockwise- if your left handed I’d love to know how you write o. I’d also love to hear any science that can explain this phenomenon or even any theories you may have.

Categories: Maths

#mathsconf19

June 22, 2019 1 comment

Today I went to #mathsconf19 in the lovely surroundings of Penistone Grammar School. It’s been a while since I last went to a #mathsconf (Jo Morgan, @mathsjem, reliably informs me it has been 5 years) and I was glad to get the opportunity to go. I saw some great sessions, caught up with some people I hadn’t seen in a long time and met some people for the first time. I thought I’d jot some notes here about my initial thoughts after the conference.

Keynote

The day started with a welcome from Mark McCourt (@EmathsUK), and a short talk from Andrew Taylor (@AQAMaths). It was a nice start to the day, Mark shared some good ideas around dynamic questions and other things and Andrew showed some questions from maths papers for 16 year olds from the last 80 years which was a very interesting thing to look at.

Making Mathematicians

The first session I attended was about helping students to think like mathematicians. It was run by Kate Milnes (@katban70).

She took us through what she thinks it means to be a mathematician and how she thinks this can be achieved. She shared a couple of tasks she uses and stories about her classes.

This is a topic I’ve been thinking about recently and I feel it helped frame my thinking. In order to gelp student become mathematicians we need to get them engaging with tasks in a way that engages their brain.

Students, when presented with a question need to be able to process it. Ask themselves questions, spot patterns, conjecture, test and make mistakes. A simple task with many extentions that Kate went through was based on the 1089 problem. This itself is a rich task that i have thought a lot about recently as I read an article about it in a recent MT journal. I’ve thought of mamy extentions but Kate looked at different bases which was a direction I had not considered before.

The session left me wondering how much I do to help my students become mathematicians. I think i do a fair bit with some classes, but certainly could do more – especially with younger students and lower prior attainment students. I need to think more on how I can achieve this, how I can start earlier and how I can support colleagues to do this better in their lessons.

What does it mean to be abstract?

This was led by Pete Mattock (@MrMattock). I chose this session for a couple of reasons. Firstly, I enjoy abstract thinking. Secondly I know Pete enjoys visible and manipulative maths teaching and this is an area I feel I could improve at. Thirdly the promise of developing multiple interpretations of a concept sounded good.

The session explored addition and subtraction looking at different interpretations and different visualisations. It made me think deeply about these topics and allowed me to explore some links between primary maths and what comes muxh much later.

I left the session with a much better idea of how I can use physical and visual cues better in my teaching, which I hope to start to incorporate more.

The main thing that has stuck in my head is the idea of using 1 dimensional vectors and numberlines for addition, subtraction and multiplication etc. It’s a simple representation that I feel has so many uses and could be extremely powerful as a teaching tool, especially when dealing with directed numbers. As Pete said afterwards “the term is directed numbers and the vectors are literally directed”.

The evolution of vocabulary in maths education

This session was run by Jo Morgan (@mathsjem) and was one I chose mainly out of a desire to expand my knowledge. I didn’t think there would be much that would affect my day to day practice, but hoped there would be things I’d find fascinating. And there was.

Jo has been researching old maths textbooks and has amassed quite the collection if them. I too have an interest in this, and thought my 50 spanning 127 years was quite inpressive collection. But it pales by comparison! Jo has also spent much longer than I have investigating them.

I was fascinated to learn “new” terms that have fallen out of use and how they fit. I was particularly interested in the terminology of quadrilaterals.

Desmos and large data sets

This session was led by Tom Bennison (@drbennison) and Ed Hall (@edhall125). I chose this one as I feel I didn’t spend much time looking at large data sets with my A level classes this year, and I have little knowledge of Desmos and have never used it in lessons. I was hoping to gain some knowledge that could help me improve both.

On the first point, I feel Tom summed.up.my attitude with a comment early on. He said “If you are only looking at large data sets because they are om the exam then whats the point? It’s only 2 marks.” This was certainly my opinion before the session. He then went on to outline his reasoning for using them. That they give a real life context that can be important going forward. He spoke about using large data sets other than the exam boards and how it can enrich the stats teaching. This made me think differently about them and I need to ponder more.

On the second point, I feel I have a better basic grasp on Desmos and what it can do. I have learned some things I can immediately apply in lessons and I hope to get time to play with it more and find more out.

Yes, but constructions….

This session was led by Ed Southall (@solvemymaths) and I chose it because I have never really enjoyed teaching constructions and hoped to pick up some tips to improve my practice.

Ed gave some advice on being prepared, which I feel is massively important, and showed some great tasks and activities that I feel will improve my teaching of the topic. I am teaching it to year 10 in a week or two and I intend to try some of Ed’s tasks out.

In summary

I had a thoroughly enjoyable day, learned some new things and left with a number of questions about my own practice I hope to explore, which is certainly the point of a day like this. I hope to write further on some of thw topics mentioned here as I work through those questions.

If you were there I’d love to hear your views on the sessions you attended. If you weren’t I’d love to hear your thoughts in the topics mentioned. Do get in touch in the comments or on social media.

Categories: Maths

Social Mobility or Social Justice

June 11, 2019 Leave a comment

Last week I was doing some research and I happened across and interesting report from the education select committee reviewing the work and the future of the social mobility commission, following the resignation of all the commissioners. The report itself had some damning things to say about the government’s treatment of the commission and the distinct failure of the government to work to achieve a higher standard of social mobility, despite the prime minister stating that social mobility would be a priority of her government.

The thing that interested me the most was the discussion about social justice vs social mobility. The education select committee expressed a feeling that social mobility seems to focus on raising people up the ladder of opportunity, and can sometimes leave people struggling to get onto that ladder. They discussed that the current focus seems to be on picking a few out of poverty and giving them an opportunity to attend a good university. Their recommendation was that the name of the commission be changed from social mobility to social justice and that their focus be to look at all policy changes from a social justice viewpoint to ensure that it was working for all. These recommendations appear to have been rejected by the government.

Roll forward a few days and I read an article about the opposition policy announcement that they would alter the name and focus of the commission from social mobility to social justice and switch its focus from picking a few to lift out to a radical new way of thinking which aims to help everyone. When I read the article I could see that the opposition had clearly read the education select committee’s report. That they too feel that after decades of failure by consecutive governments from both sides of the house to achieve a more equal society a radical overhaul in the approach was required.

To me this seems a sensible policy. Tweaking has failed, we’ve rehashed the same policy ideas over and over and all we have seen is a greater inequality than we had before. Surely it’s time to rethink? But then I read the backlash. The education secretary spoke out against the idea saying it was “downgrading the importance of social mobility”. Let that sink in, the current government have downgraded the importance of social mobility so much that the entire commission resigned due to government actions and their education secretary is accusing this policy of downgrading the importance. The hypocrisy is ridiculous and there is also a condescending overtone to those who do not want to move towards a graduate career. To write these people off as being “without ambition” is wholly wrong. A university education is not the only measure of success.

Then there is the idea that getting students from disadvantaged backgrounds into university is even a good indication of social mobility and reducing inequality. In a world where unpaid internships and old boys networks are the biggest steppingstones to the top jobs getting to university is only half the battle. A shift of focus from social mobility to tackling the inequality in society at all levels is, for me,  a welcome one.

Further Reading:

Education Select Committee report mentioned above, The future of the Social Mobility Commission: https://publications.parliament.uk/pa/cm201719/cmselect/cmeduc/866/86602.htm

TES report on Labour policy and Hinds’ response: https://www.tes.com/news/labour-swap-social-mobility-social-justice

Letter from Prof Reay (Cambridge University) on social mobility: https://www.theguardian.com/society/2017/jun/28/social-mobility-is-the-wrong-goal-what-we-need-is-more-equality

 

When will I use this?

June 4, 2019 Leave a comment

Recently I read a rather interesting article from Daniel Willingham about whether there were people who just cant do maths. It was a very good read and I hope to write my thoughts on it later, when I’ve had more time to digest the material and form some coherent thoughts, but there was one part that set me off on a train of thought that I want to write about here.

The part in question was discussing physical manipulatives and real life examples. Willingham said that there is some use in them but that research suggests this can sometimes be overstated as many abstract concepts have no real life examples. He then spoke about analogies and how they can be very effective in maths of used well.

This got me thinking, earlier on the day a year 12 student had asked me “when am I ever going to use proof in real life?”. This type of question is one I get a lot about various maths topics, and my stock answer tends to be “that depends what career you end up in”. Many students, when asking this, seem to think real life doesn’t mean work. A short discussion about the various roles that would use it and that its possible they never will if they chose different roles but that the reasoning skills it builds are useful is usually enough and certainly was in this case.

It does beg the question though “why do they only ask maths teachers”? Last week when a y10 student asked about “real life” use of algebraic fractions I asked him if he asked his English teachers when he’d need to know hiw to analyse an unseen poem in real life. He said no. I asked if he thought he would. Again no.

So why ask in maths?

The Willingham article got me thinking about this. There has been, throughout my career, a strong steer towards contextualising every maths topics. Observers and trainers pushing “make it relate to them” at every turn. But some topics have no every day relatable context.Circle theorems, for instance, are something that are not going to be encountered outside of school by pretty much any of them. So maybe thats the issue. Maybe we are drilling them with real life contexts too much in earlier years, and this means when they encounter algebraic fractions, circle theorems or proof and don’t have a relatable context the question arises not from somewhere that is naturally in them, but from somewhere that has been built into them through the mathematics education we give them.

Maybe we should spend more time on abstract concepts, ratger than forcing real life contexts. Especially when some of those contexts are ridiculous – who looks at a garden and thinks “that side is x + 4, that side is x – 2, I wonder what the area is?” (See more pseudocontext here and here).

What do you think? Do you think we should be spending more time lower down om the abstract contexts? Please let me know in comments or via social media.

%d bloggers like this: