Wednesday, 21 August 2013

How to Learn Math

I have been raving enthusiastically about this course currently being offered by stanfordonline and it was suggested today that I blog my experiences.  Duh!  It has been so long since I blogged that I had forgotten all about it!

So here goes.  Today I used for the first time some of the raft of techniques and ideas I have learned on the course.  I'm going to try and explain what we did, why and link to some of the key videos used in the course in case you are interested in learning more.  I don't know how long the videos will stay up for so apologies in advance if the links die at some future time.

The situation came up in conversation with the children that LB2 needed some new jogging pants.  I said we could go and look today but that I would wait until Sainsburys had a 25% off weekend before I would buy them (as it happened their discount is on now - yeh - but I digress).  LB2 asked why?  Why will you wait?

So having just watched this video last night I thought I would jump on his why and use it as a learning opportunity.  Not that I don't normally jump on whys - but today it was particularly in my mind:



So I started by asking what they thought 25% was .  LB2 was surprisingly intuitive.  He said he didn't know but he did know that 50% was the same as a half and that 25% was half of 50.  That led LB1 to chip in that he thought 25% must be the same as a quarter.  So, I asked, if those jogging pants cost £10, how much would they cost if they had 25% off.

Now I did think that was probably quite a hard question but I didn't want to start off by assuming they didn't know - I wanted to hear what they had to say.  And as I learned on this course there are no right or wrong answers.  Mistakes have been found to strengthen knowledge and they are all opportunities to learn.  And without embracing mistakes students will be too afraid to take risks with maths, to explore it and have fun with it.  This is what mathematicians do!  But for lots of reasons children often come away with the idea that you have to get things "right" and that in maths there is a right way and a wrong way with no room for manoeuvre.  Watch how the teacher in this video deals with mistakes.  I found this very inspiring!



Kids aren't daft - they know if you are leading them to a "right" answer and they can see through fakey "Nice try" comments.  I've struggled with that in the past - but this gave me a way of dealing with it.

So the boys came up with some answers.  LB1 first thought the pants would be £5.50 and then changed his mind to £6.50 but when asked he couldn't explain how he came up with that answer and got a bit cross with me.  So I explained to him that he was no longer a "Kid who Does Sums" but a "Mathematician"... and mathematicians can't just pluck numbers out of the air - they have to explain how they got them.  Being right isn't enough - you have to be able to explain it.  He still wasn't thrilled but he accepted this reasoning and said he didn't know.  LB2 was convinced the pants would now be free and he explained that was because 10 was less than 25.  This showed his thought process and as I know he has never studied percentages or fractions I thought it was a good try.

I passed no comments at all about their answers but simply wrote them on the board and said "Well I think we are going to have to work this out because I don't think all of the answers could be right."

Now one of the sessions I watched last night was about the "process" of maths.  Basically children need to be taught that maths is a process and that you are not supposed to be able to just look at a problem and get the answer right.  My analogy to the boys today was that you wouldn't expect to lift one tin of beans and then become a weightlifter - you need to work hard and build your muscles.  And it's the same with your brain - it needs to work and struggle with the hard stuff to grow big and strong.  (There was lots of amazing stuff in the course about research that has been done into brain development but I haven't got space to include the entire course!)  So this is the video where Professor Boaler describes her maths process which will give you an idea of what I'm on about:



So if you can't just look at a problem and get the answer right then there is a process you have to go through to really understand the problem.  So I began by trying to model my process for the boys and I started by writing down what I knew:


If you can't see that well I wrote down that we were starting off with £10 and the fact that LB1 had said he thought 25% was the same as a quarter.  And then I wrote down their guesses (and by this point LB2 had changed his mind and decided shops don't give stuff away so guessed it might be £3).

They were a bit stumped as to how to proceed so I suggested we draw a picture.  So I drew 10 pound coins.  I then asked how we could work out what a quarter of £10 was and this seemed to stump them a little bit too so I asked them if they knew how to find a half instead:


I had to mock up this diagram as we had to reuse the space later in the problem.  We took turns showing how to split the ten coins into two halves.  This was another point of the course - inviting ideas from everyone and looking for alternative ways of doing the same thing - really playing with maths.  So suddenly it becomes a creative thing with no right or wrong method.  LB2 went first and split it into two groups of five.  Then LB1 put a line down the middle and counted his half coins to make sure he had the same answer, and then the third example was mine.

Then I asked them how we could then work out a quarter and LB2 did a bit of a complicated bit of crossing out on the first model and concluded a quarter of 10 was 8.  So I then realised he didn't really understand what a quarter was.

Now maybe I would have realised that anyway - but I'm not sure.  Normally I would have written a bunch of stuff on the board (you know - the way I was taught) and drawn a few pictures and they would have nodded...  So I didn't say "That's wrong" (see learning here), I said - can you show me what a quarter looks like.  And then he carefully drew the circle you can see above.

And again I didn't say "That's wrong".  I said "How many pieces does that circle have?" and he replied "Six" and I clarified and "What does the quarter look like?"  and he pointed at two next to each other.  So again I learned something here.  I would have assumed from the picture that he thought there were six quarters in a whole but actually he thought there were three (maybe he didn't know how to draw three).

So I established that he thought there were three quarters in a whole.  And at this point I had to put him right because that was a key piece of knowledge that he didn't know.  At the same time LB1 had been drawing a very large picture of a circle with four quarters:


And so when I explained that there was something he didn't know and that was that there are four quarters in a whole LB2 got a bit upset (partly because LB1 was saying things along the lines of "Duh everyone knows that." - helpful!)

And so I used another technique!  No not everyone knows that.  People who have had the opportunity to learn that before - they know that.  But if you have never had that opportunity you would not know that.  Would you say someone was stupid if they did not know how to play Lego Batman on the Wii?  LB2 seemed really buoyed by this - I think because it seemed very fair to him.  "No" he said "They might never have seen Lego Batman before but that doesn't mean they are stupid."  Exactly so instead of getting upset that he didn't know something he felt pleased that he was learning.  Yeh it's working!

So then I asked him to draw another circle and show me his four quarters which he did as you can see above - albeit a bit wobbly.  So to make sure he had got the idea I said "I am going to show you a great trick that works with lots of things if you want to make quarters" and I showed him how to half and then half again.  I think he did know this but again he didn't take a huff at being shown something new - and that is a big thing for us because both my boys get really, really angry with themselves if they get something "wrong".  It's been one of my biggest worries since we started home education.

Phew - sorry this is turning into a bit of a marathon - in real life it didn't take as long as it sounds - I'm just trying to explain my thinking.

So next step, now we all knew what a quarter was, is to find a quarter of £10.  Another picture:


They told me where to draw the lines (but couldn't reach themselves!) and then we counted each quarter to make sure they were the same - because that's the important thing about quarters.  And then...


we worked out that two and a half pounds was £2.50 and that as we had said earlier a quarter was the same as 25%, then 25% of £10 was also £2.50.  All of this was coming from the boys and I was pushing them to justify each step to me so when I asked LB1 how he knew 25% of £10 was £2.50 I got him to explain quite clearly that we had discussed earlier that one quarter was the same as twenty five percent and we had written it down so that we wouldn't forget.

Pushing students to communicate like this is also discussed in the talk and I could see the benefit of it.  In a similar way that we model writing stories for children by reading to them we should be modelling and encouraging them to be "talking maths".  How can we expect them to later write a coherent proof if they are not experienced in justifying their ideas?  In the video above the children taking part are pushing themselves by not only coming up with right answers but also having to explain them clearly.

At this point LB1 made a jump and said the answer must be £7.50 but still I held back!  This was an opportunity for LB2 to learn from him so I asked him how he had come up with that answer and he showed us this:


That is £7.50 + £2.00 + £0.50  = £10.00

LB2 looked at this idea and worked it out for ourselves and he agreed that LB1 had come up with the right answer.


Well I hope you are still awake!  As I said that is just a part of the course but the key ideas are

  • encouraging mistakes and persistence
  • encouraging the students to come forward with ideas and justify them (or not)
  • explaining that maths knowledge comes from having had an opportunity to learn in the past (rather than an innate skill in maths we are born with)
  • pushing students to do the work themselves and refusing to walk them through the answer
The course also discusses (which I haven't here really) teaching and supporting a growth mindset which is really supporting the belief that you can grow and learn how to do something - as opposed to believing that if you fail it's because you never will be able to do it.  Also moving away from maths being a series of procedures that we learn (like column addition) and towards it being something that has lots of methods and that you can be creative with.

For me it was all common sense and really resonated so it was something that I did already know but just hadn't been able to articulate.  I also didn't really know what to do about it but the course has given me lots of things to try based on actual research and measurements of results.  Other key ideas are that tests are BAD!  Bad, bad, bad and no use.  We must get away from the idea that maths has a time limit and that fast = good.  Great mathematicians are not fast at all!  Also grades are BAD!  Feedback is not only more useful but encourages the recipient to improve.  And finally collaboration is GOOD!  No working on your own in silence.  Ideas, ideas and more ideas - from everyone - will improve everyone's knowledge and skill.

I hope I've given you a flavour of the course and if you're interested sign up and just watch the videos if you don't have time for the reflective practice. Perhaps this was stuff you already knew.  I'm not saying my teaching completely the opposite before this but I do think it has given me the confidence to know what to say as opposed to feeling I was fumbling around a bit - albeit with good intentions!

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