Are Our Students Thinking Too Much?

"It is a profoundly erroneous truism that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of operations we can perform without thinking about them." 

-Alfred North Whitehead, An Introduction to Mathematics

How many times in your life as a maths teacher have you said to your students, "think about what you're doing!"?

And how many times in your life as a maths teacher have you thought to yourself, "how do you still need to use your fingers to know 4 times 5 is 20?"

I know that I have these two conflicting belief systems. I believe that knowing times table facts, number bonds, the values of sine, cosine and tangent for common angles, the quadratic formula, and probably loads more things are very very useful. I would go so far as to say important. Maybe even so far as to say, your exam success depends on it.

But at the same time, I believe that rather than relying on memory, we should have a solid understanding of maths and think about what we are doing and why at each stage of a process. This is why I am conflicted.

A good example is long division. Students in UK Primary schools are generally taught chunking as their primary method of long division. On the one hand, I'm in favour of this. Chunking is my preferred mental method, and I have even started doing it as my main written method. However, students (especially those who do not know their times tables very well) are slow with chunking. It also takes up a lot of space. In secondary schools, I think most teachers teach the traditional bus stop method of long division. Students find this difficult at first but after much practice they can use it efficiently and accurately. But when they are used to this method, they do it without thinking about it. Is this a good thing or a bad thing?

Does it make sense to "save your brain" for the more challenging bits and rely on your subconscious to do the easy bits? Or should we be mindful at all times?

Should we memorise the quadratic formula or derive it by completing the square every time we want to use it? Should we memorise that cos(30) is root three over two, or draw a 30/60/90 triangle every time? Does it free up space in our memory if we don't memorise these? Do we free up space in our working memory or unburdon our central executive if we do memorise them? Which is more beneficial?

I think that in the UK, we place much much less emphasis on arithmetic fluency compared to other countries. I'm frequently told by friends and colleagues who went to school in Asia or Europe that you simply did not move on to the next grade or year group until you had mastered addition, subtraction, multiplication and division. And mastered means you can do these things efficiently and accurately. Over here, we move on to algebra before students know their number bonds. We teach fractions and percentages before they know their times tables. We say, it's OK if you're no good at arithmetic, you might be really good at shape and space. Is it the lack of arithmetic fluency that is holding UK students back? When it comes to maths, are our students thinking too much?

[Is it just me, or is that last sentence totally Carrie Bradshaw-esque?]

The GCSE for maths now has a much larger emphasis on functional problem solving. Teachers say this will be a challenge because our students are bad at problem solving. I think they're wrong. I think our students are brilliant at problem solving. But they do badly on these questions because the arithmetic involved in these problems poses a much bigger problem to our students than, say, Korean students. If you know how to do arithmetic automatically, without thinking, you only have the problem to think about. If you also have to think long and hard about the arithmetic, it interrupts your thinking process and tackling the problem becomes much harder. It also means you won't as easily spot neat tricks or patterns that simplify the problem.

For example, there is a Foundation GCSE question with a rectangular patio 3m by 3.6m. Henry has 32 square paving slabs with side length 60cm. Does he have enough slabs to cover his patio? If you are good at arithmetic you may spot that 60cm goes into 3m and 3.6m nicely. If you don't know your six times table, you won't notice this. That means you may choose a longer, more complicated method, like finding the area of the patio and the area of the slab and doing a division. This was a non-calculator paper so now students will be having to calculate 3 x 3.6 and 0.6 x 0.6, then use long division. It is much, much simpler to see that you need 5 rows of 6 slabs, which makes 30.

So although I'm a big fan of exploring, discovering, reasoning, and understanding, I'm starting to think that a bit of rote learning of efficient, accurate methods, and drilling of times tables (and their related quotients) would actually improve students' ability to solve problems a lot more than simply getting students to practise problem solving.

What do you think?

Emma x x x

Show Me Your Knowledge - An Alternative to Tests

I had become fed up of marking my year 10s' fortnightly assessments. They are aiming for a grade C, and following the Higher syllabus, which means the assessments are quite challenging for them. Marking takes a lot longer when the student has got almost everything wrong.

One day I decided to do something different. I offered my class the choice. They could either do the assessment as normal, or they could Show Me Their Knowledge (Patent Pending). I explained what I meant by this. I wanted them to pour out the contents of their brains on that particular topic (it was averages) onto an A3 sheet of paper. I told them they needed to make up their own examples to demonstrate all of the skills. I gave them a list of skills so they wouldn't forget anything: mean, mode and median from a list, mmm from a frequency table, mmm from a grouped frequency table.

What was quite interesting was all of the boys chose to take the normal test, whereas all of the girls chose to Show Me Their Knowledge. And the girls loved it, they were begging me to do this every assessment. This gender difference is worth exploring in more detail, so I will save that for a separate post.

The pages that were created were impressive. Most girls did an example for each of the three sections, and heavily annotated the solution with explanations. For example, "You find the midpoint here because you don't know the actual number" then an arrow from this led to, "To get the midpoint add them together and half it or just work it out by looking at it".

Some girls put up their hands at various points and said they didn't know how to explain a certain bit. I told them that means they maybe didn't understand that bit. I told them to leave a bubble or box where they wanted to write the explanation, so that they could see there was a gap in their understanding. I told them I would help them fill that gap in the follow-up lesson.

When it came to marking, I read through the explanations and tweaked some of the wording if it wasn't quite right. I did this very neatly because girls don't like you defacing their work (male teachers please take note of this). If I thought some bits needed further explanations, I drew bubbles or boxes (copying their style) but left them blank, or started the sentence for them. For example, I might have written, "If there are two numbers in the middle..." or "The question will usually tell you to r____ your answer...".

When I gave these back, they filled in the bits they could by themselves, then I talked to them on each table to explain any bits they didn't understand. They also talked to each other to fill in the gaps. By the end, they all had a poster explaining the whole chapter. That is now a very useful resource for their revision. The best looking one I could have copied and had laminated for future classes to use.

I will continue to use this method of assessment, although I will still do some normal tests too, as at the end of the day, they will have to sit a traditional exam, and they need to be prepared for this. I can't help but think the fact that girls prefer this method suggests that maybe they are disadvantaged by our exam system and that girls could do much better if we changed this. But again, that's a post for another day.

Emma x x x

Run a Reverse Pilot

It's the start of a brand-new term, and lots of us teachers will be returning to work with more energy and a renewed sense of purpose. It is when we are in this frame of mind that we are eager to read about new teaching techniques and "tweaks" and we are thinking about adding a few new things to our lessons.

Well, here's a new idea for you. Instead of starting to do something new this half term, do the opposite. Stop doing something old.

You will have all heard of the idea of a "pilot", where you trial something on a small scale to see if it will add value and if it does, you run with it.

Daniel Shapero, a director at LinkedIn, coined the term "reverse pilot" to refer to the exact opposite: removing an activity or initiative to see whether there will be any negative impact.

Is there anything you are currently doing that is taking up a lot of your time but you suspect is not actually adding any value? For example, maybe you have developed an elaborate points system with prizes and a "star of the week" award. This might have been effective at the start of the year, but maybe it is no longer doing anything. Try getting rid of it for a trial period, and see if it has a negative impact. If it doesn't, you have just saved yourself the time it used to take you.

Does your faculty have a weekly lunchtime meeting (sorry, not "meeting", calling it that would make the unions unhappy, so let's call it a "gathering") that takes up 15 people-hours and accomplishes very little? Try scrapping it, and see if anything bad happens.

Do you make sure to write "next steps" or similar after marking your students' books? Stop writing these, save yourself 3 minutes per book (an hour and a half per class per fortnight) and see if your students' progress starts to decelerate.

Are you currently running a weekly after-school revision class for your exam class? Have you done this every year since you can remember? Don't do it this year. See if the results are worse than normal. If they're not, then you've just saved yourself an hour a week for the rest of your life. You are welcome.

Of course, it could be that everything you do at the moment does make a positive impact on learning. But we shouldn't just assume that they do. To do so would be wasting precious resources. I flippantly said in the above paragraph that you could stop doing that revision class to save yourself time. Well think about redirecting that time towards something else - better quality feedback, or one-to-one tuition. These might have a bigger overall impact. That lunchtime meeting (sorry, gathering) could be spent producing the world's best corridor display. Or it could be spent, I don't know, eating?!

There are several cognitive explanations for why we are probably wasting a lot of time on activities that might be adding little or no value. These are loss-aversion, the sunk-cost fallacy, and the status-quo bias.

It has been shown that human beings are naturally loss-averse. We think that losing something will have a bigger negative impact on us than gaining something of the same value will have a positive impact. So it seems to you that stopping sending out your weekly teaching and learning newsletter (which takes three hours to write) would have a big negative effect, but gaining three hours of extra time per week would have a smaller positive effect, not enough to balance it out. When you write it down like this, it might seem that the choice is obvious, but most of us don't write it out like this, and hence we don't question whether carrying on doing the things we're doing is the best use of resources.

You spent hours setting up your praise and reward spreadsheet, with built-in mail merge facility and automatic colour coding. So spending an hour a week filling it in for all of your classes is definitely worth it, right? After all, you spent so long on building it, it would be a waste not to use it, right? This is the sunk-cost fallacy in action. The time you spent making the spreadsheet is just that, spent. You cannot unspend it. Therefore whether you use it or not makes absolutely no difference. Do you still have a dress in your wardrobe you've never worn, but you can't get rid of because it cost a bomb? Whether you keep it or not, you have already wasted the money. Let it go. Spent forty minutes waiting for a bus so there's no way you're paying for a taxi now? That forty minutes has been lost either way, do what you want. The only time the sunk-cost fallacy is your friend is with your gym membership. "I've paid all that money, I really have to use it!" this is a fallacy, but a healthy one. Don't fight it.

The "status quo bias" is where we have a tendency to keep doing something simply because we have always done it. Why do we write the learning objective on the board at the start of the lesson? Because we've always done it (and in my case, because my own teachers did it). Why do we begin each lesson with a starter activity? You can probably think of lots of examples.

I challenge you this half term to stop doing one thing. I suggest at first you don't replace it with anything else, just stop doing it. Feel the relief of having one fewer thing to do. If you observe no impact, then keep not doing it. If you observe a negative impact, instead of going straight back to doing the same as before, tackle the problem from scratch. You might find there is a better solution.

To read more about the cognitive biases I have mentioned in this post, I recommend The Art of Thinking Clearly by Rolf Dobelli. 

To read more about the idea of doing less to achieve better results, I recommend Essentialism: the Disciplined Pursuit of Less by Greg Mckeown. 


Emma

x x x

And by the way the weekly lunchtime meeting referred to in this post is a work of fiction and bears no resemblance to any real weekly lunchtime meeting, living or dead. Any resemblance to the weekly lunchtime meeting my own faculty has on a Thursday is purely coincidental.

The Chaos Game

Draw an equilateral triangle on a bit of paper. Draw it nice and big. Now pick a corner to start at. Next, get a die (a D-6) and assign two numbers to each corner. For example, the top corner can be 1 and 2, the left corner can be 3 and 4, and the right corner can be 5 and 6. Write these numbers near the corners on the outside of your triangle so you remember them.

Roll your die. The number you get tells you which corner you are heading towards. Find the midpoint of your current location and the corner you're heading towards. Use a ruler for this and try to be as accurate as you can. Mark this new point with a good-sized dot, that is your new location.

Repeat.

After about fifteen minutes your triangle should have lots of lovely dots, and at this stage you might even see a pattern emerging.

A pattern! (I hear you cry) How can there be a pattern, when I am moving randomly! Surely the dots will be scattered in a random manner, looking like the freckles on a pasty Irish face. But there is indeed a pattern. A very nice one in fact. A very familiar one, actually.

SPOILER ALERT

Please actually carry out this experiment before looking ahead.

I will now insert some line breaks to stop you seeing the pictures below. Don't scroll until you're ready.

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I didn't rub out my construction lines because I've been taught well :-)

Can you tell what it is yet?

Holy fractal, Batman! That there is Sierpinski's triangle! Or as my students would say: Illuminati confirmed.

OK, it doesn't look exactly like Sierpinski but it's definitely getting there. I need to do another thousand or so iterations.

I am a mathematician, which basically means I have turned being lazy into a career. So at this point I started thinking, why am I drawing this s*** when I could be running a simulation instead?

Here is the spreadsheet I used to simulate the Chaos game. It was kind of fun to set up, so I suggest you try it yourself before reading my formulae. You can probably find a much more elegant way to do it, but I'm a mathematician, dammit, not a programmer!

Teachers: this is a cool way to kill an hour with a group of students who know how to use a ruler and divide stuff by two. You could do it as I have suggested, with a triangle drawn on blank paper, but you could instead do it with a triangle drawn on a coordinate grid, with the corners having the coordinates I used in my spreadsheet. This way, students can practise finding the mid-points of two points (a skill that is needed for Higher tier GCSE and for AS Level). However, the numbers get nasty pretty quickly (as there is a root 3 involved). Even if you just use this as an exercise in measuring with a ruler, I think students can get a lot out of it. The pattern is so cool and unexpected, your students may even have that "wow" awe and wonder moment.

Have fun!

Emma x x x

Cheryl's Birthday

Photo credit: Kenneth Kong/Facebook. 

Imagine my delight when I found out last night that loads of people on the internet were talking about a maths puzzle! Some people are even saying #Cheryl'sBirthday was the new #TheDress. Not being a Twitter user (is it OK that I just don't get it?) I arrived a bit late to this party but I hope you don't mind.

So, this puzzle comes from a Singapore Maths competition which I think is probably comparable to the UKMT Intermediate Maths Challenge follow-up round (the Pink Kangaroo) which means it is supposed to be challenging. Can I just take this opportunity to brag and say I got the answer right in about three minutes? Please, if you haven't already, pause and work out the answer for yourself.

There are lots of explanations of the solution out there on the internet, but I'm a maths teacher and I can't help myself, I love explaining stuff!

Statement one:

Albert knows the month. This means he can't know when the birthday is, as there are no months with just one possible date.

However, Albert has deduced that Bernard cannot know either. The only way Bernard would be able to know is if the number he was told was 19th or 18th because they are the only dates with one possible month. So for Albert to know that Bernard does not know, these two options must not be possible. Albert only knows the month, so for him to know these are not possible, the months of these must not be correct. Therefore it is not May or June.

Statement two:

Now that Albert has said that, Bernard has deduced that it is not May or June, just like we have. This information is enough for him to know the correct date. This means it can't be the 14th because there are two months with the 14th. So it must be 16th July, 15th August, or 17th August. .Bernard knows which one of these it is because he knows the number. We do not know.

Statement 3:

Albert has deduced the same as us, and narrowed it down to those three dates. But Albert knows the month, and by knowing this, he knows the answer. So it must be July, as if it was August he still wouldn't know.

Therefore the answer is 16th July.

What a wonderful question!

This question is similar to those questions where nobody knows anything but by saying "I don't know" enough times everyone suddenly knows everything. Do you know the kind of question I'm talking about? My favourite is probably the one with the island full of brown eyed and blue eyed people: The Blue Eyes Logic Puzzle. This is so difficult to wrap your head round it but once you do, you feel like you have understood the secrets of the universe and your brain suddenly enters this state of ultimate clarity. Unfortunately this only lasts for a few minutes and then you stop understanding it again. I have read about this puzzle so many times now I can hold onto this state for a whole evening. I always wake up ignorant again.

Maybe I will do a full post on the Island puzzle one day when I'm feeling brave and I have a large supply of stimulants to hand.

Let me leave you with a link to my favourite place to find logic puzzles. These are great for teaching the Logic chapter in D2. They are also nice and short so you can do one at the beginning of every maths faculty meeting just to get everyone's brains warmed up.

And also, a maths joke: Three logicians walk into a bar. The barman asks “does everyone want a drink?” The first logician says, “I don’t know”. The second logician says, “I don’t know”. The third logician says, “Yes”.

Ha ha ha ha ha ha!

Emma x x x