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I have had many people say to me over the years, “But algebra is easy: just tell them to do the same thing to both sides!” This is wrong in several ways, not least of which is the word “easy”. The particular way it’s wrong that I want to talk about today is the idea that doing the same thing to both sides is somehow the only move in algebra, because it’s not even the most important or the most common move.

I think the most important and most common move in algebra is this:

Replace something with something else you know it’s equal to.

This rule isn’t even an algebra rule, it’s a rule you’ve used in plain arithmetic. Look at this working:

(3+4)×5-6
= 7×5-6
= 35-6
= 29

All of those steps were replacing something with something else I know it’s equal to. First, I replaced the (4+3) with 7, and then I replaced 7×5 with 35. Finally I replaced 35-6 with 29. The reason I was able to write the “=” signs there was because I knew each expression was produced by replacing something with another thing it’s equal to. Of course they are the same. If you have ever written your working in the way I did there, then you were using the replacement move.

Indeed, this move is the heart of how mathematicians write calculations. We always move from one step to another by replacing something with something it’s equal to. Knowing that this is what we do, we read other people’s working by comparing each expression in a chain to the next to see what it is that has been replaced.

It’s interesting that nobody has ever told me explicitly that this is how to read maths working that’s written with a chain of equal signs. I just somehow figured it out. I am sure quite a few of my students don’t actually know this strategy.

This idea that anything can be replaced with something it’s equal to is for me the major thrust of all the algebra laws and identities.

For example, the distributive law―a(b+c) = ab+ac―doesn’t tell me how to “expand brackets”. Instead it tells me any time I see a(b+c), I’m allowed to replace it with ab+ac, and every time I see ab+ac, I’m allowed to replace it with a(b+c). The same goes for all the algebra laws, even though some of them seem very complicated.

For me, completing the square illustrates this very well. For example:

3x² + 24x + 7
= 3(x² + 8x) + 7
= 3(x² + 8x + 16 – 16) + 7
= 3((x+4)² -16) + 7
= 3(x+4)²- 48 + 7
= 3(x+4)² – 41

The first move was to replace 3x² + 24x with 3(x² + 8x) by the distributive law.
Then I replaced 0 with +16-16.
Then I replaced x² + 8x + 16 with (x+4)².
Then I replaced 3((x+4)² -16) with 3(x+4)² – 48.
Then I replaced -48 + 7 with -41.

The second pair of moves are usually extremely surprising to students, and even I have to stop and look closely when I read already-written-down completing the square working. But they make sense when I compare the line of working to the one above and see which parts are the same, so that I can deduce that the parts that change must have been replaced because they are equal.

Incidentally, this is why I am such a stickler for keeping things lined up when you do maths working if you can. Changing the order makes it hard to deduce what’s been replaced from one line to the next. Of course, you can change orders, but I prefer to do that as its own move so you can see when it happened.

Finally, the replacing move is actually used when solving equations too, even though it’s usually hidden. Look at this working that I did in a chemistry lab with a student once.

ρ = m/V
ρ × V = m/V × V
ρ × V = m

The move from line 1 to line 2 was multiplying both parts of the equation by V, but the move from line 2 to line 3 was replacing m/V × V with m because you know they are equal.

The student who I did this with was ok with the multiplying both parts by the same thing, but they were not ok with the replacing. They complained that I hadn’t done the same thing to both sides. And this was when I realised that there was another move in algebra and it’s much more fundamental than doing the same thing to both parts of an equation. And we need to tell students explicitly about the existence of this move.

So there you have it. Algebra is not all about doing the same thing to both sides, it’s very very often about replacing something with something else it’s equal to. Keep an eye out for it next time you read or do any maths working, and maybe explicitly remind your students every so often when it happens.

Actually just a bit of an epilogue: the replacing move is really rather fun to do “in reverse” as it were. Usually we do it in arithmetic by replacing an expression with a single number, but there’s nothing stopping you replacing a single number with an expression and so finding some rather complicated expressions for familiar numbers. This first one is based on the fact that if 1 = 2-1, then any 1 can be replaced with 2-1 at any time. (If you click on the tweet you can see me replacing 1 to 9.)

May you enjoy your replacing too.

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I think asking students questions is an important part of my job of helping students succeed. Good questions can help me see where they are in their journey so I can choose how to guide them to the next step, or can help to make clear the skills they already have that will help them figure things out for themselves. But there is a class of questions that shuts all of this down immediately. Here are some examples:

  • “Did you go to the lecture?”
  • “Have you started yet?”
  • “How many of the exercises have you done?”

These questions all have answers that are morally Right or Wrong. The answers a student gives make the student out to be a Good Student or a Bad Student. And if a student has the Wrong Answer, they will feel ashamed.

I know many people who believe it is very important to send students the message that they should go to lectures, start assignments straight away, and do all the exercises. While these are all things students could do to help themselves, they’re not the most important thing to focus on when they are here seeking support from me. They can’t change any of those things right now, so all a question like those does is make them feel ashamed. And, as Turnaround for Children CEO Pam Canto says in this blog post, “shame is toxic to positive outcomes”.

Shame is the feeling that you are a bad person, that there is something wrong with you. Guilt is a bad feeling about your actions, which is unpleasant, but may make you want to change those actions in the future. Shame is the next level, where you feel you have been exposed as the horrible person you really are. A person who feels shame won’t try to change their actions, they’ll just try to avoid situations that expose them, which will just make the problem worse. I don’t want this to happen to my students, and I certainly don’t want them to think that seeking support from me will expose them to shame, or they will decide not to seek help.

Once upon a time, I realised that I was causing a student shame, and I decided that I would give myself a new principle.

Never ask a question that has a morally wrong answer.

This is one of the rules I use to evaluate if my question is useful and choose a better alternative.

For example, I could ask “Did you go to the lecture?”, but there is definitely an answer to this question that is morally wrong and having to give that answer will cause shame. Do I really want to know if they went to the lecture? How will that help? Maybe what I really want to know is what the lecturer has to say about the topic, since that might be useful. In that case, I could ask “What did the lecturer have to say about this?” The student doesn’t have to reveal their attendance status to answer this question, thus avoiding the shame. Even better would be to avoid the awkward moment where they have to reveal they don’t know, and say, “It would be useful to know what the lecturer says about this. Can you tell me what they said, or tell me where we might go looking for that?”

For my second shame-inducing question of “Have you started yet?”, the first simple fix is to remove the “yet”. That implies they should have started already. The second fix is to think about why I want to know this? Maybe I want to know what they’ve done already so we can build on it. In that case I could just ask “What have you done so far?”, since that’s directly asking for the information I want. But there is still an implication that they should have done something, so causing shame if they have to reveal they’ve done nothing. So instead I could ask “What are you thinking about this problem?” or maybe “How do you feel about this problem?”. These let me get into their head and heart and I can help them move on from there. I might be able to ask them about what they’ve done so far later, or it might not even be important because they’ll tell me what they need to help themselves.

This second example highlights another principle, which is to ask open ended questions, preferably about student thoughts and feelings. This makes it much easier to ask questions without morally wrong answers, because there are no specific predetermined answers in particular! (Asking open-ended questions is actually one of the factors in SQWIGLES, the guide for action I give to myself and my staff at the MLC.)

So, I urge you, think about whether the questions you ask have a morally wrong answer, and if so, try a more open-ended question that is less likely to cause the shame that is so toxic to success.

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I conscripted the game Numbers and Letters seven years ago to help promote the Maths Learning Centre and the Writing Centre at university events like O’Week and Open Day. Ever since then, it has always bothered me how free and easy participation in the Letters game is, while the Numbers game is much less so. People seem determined to only put answers up on the board if they are right, and even rub off answers that don’t produce the target after all. And it’s a real hard sell to get people to try for an alternative solution too. Once a solution is found, people stop. I have always been worried that the game sends the wrong message entirely.

This Open Day I had a remarkable idea: instead of stating in the rules that the goal is to achieve the target, and trying to encourage people to take a different approach, what if I just changed the stated goal! I don’t know why I didn’t think of it before, to be honest!

So this is my new version of the Numbers game instructions:

NUMBERS

Help to make calculations that produce as many numbers near the target as possible, each calculation using some or all of these two big and four small numbers, and any combination of +, -, ×, ÷, and brackets. (Numbers can only be used in each calculation as many times as they are in the list.)

For posterity, I choose the numbers randomly from pop-sticks that I painted:

  • the small numbers are chosen from a set with two each of of the whole numbers 1 to 10
  • the large numbers are chosen from a set with one each of 25, 40, 50, 60, 75, 100, 120, 125.
  • the target is chosen from a set with two 0’s and three each of the digits 1 to 9.

Since coming up with this on Open Day, I’ve put a daily Numbers game on the board in the MLC Drop-In Centre, and it’s been a delight to have students and staff join in and add their solutions.

After explaining to students that the goal is now to get as many numbers near the target as we can get, all of a sudden they start just saying things to write on the board. Mostly we don’t even get the target until quite late in the piece, because people are excited that they can modify what’s there to get other numbers. Even if the target is produced early, there is still a desire to fill in the numbers on either side, and then the whole set of 10 that contains the target.

That modification of existing answers is my favourite part. I used to work it in before when I would put a wrong answer up and fiddle with it to get closer to the target. But now it just happens naturally because the goal is just to make as many different answers as possible.

Here are four games from the last couple of weeks:

As a testament to how absorbing this new version is, I came back to work on Monday last week, after having worked from home on Friday and a student arrived early to see if there was a game yet, as well as to show me all the ones they came up with when I wasn’t there on Friday.

I am so very pleased that this new version of the game works so well, and a little ashamed that I didn’t think of it earlier. As much as my catch cry is “the goal is not the goal, the end is not the end”, really sometimes you have to explicitly change the goal.

PS: Every day, people have come to look at the board when I am starting the game, and they say “Ohh, this is like Countdown!” My response is always that yes it is like Countdown, but it’s BETTER because of three major differences:

  1. There is no clock, so it’s not stressful.
  2. We are working together, not competing, so it’s friendlier.
  3. The goal is to get all different answers instead of just one, so we get to keep playing.

 

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This blog post is about a game I invented in February 2020, the third in a suite of Battleships-style games. (The previous two are Which Number Where and Digit Disguises.)
NUMBER NEIGHBOURHOODS: A game of analytic deduction
Players:

This game is for two players, or two teams.

Setting up:

Each player/team choose six different numbers between 0 and 10 (not including 0 […]

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Introduction
In 2016 I created the iplane idea, which allows you to locate the complex points on a real graph.
In case you haven’t heard of it or you need reminding. The idea is that at every real point (p,q) of the real plane, there is a planes-worth of complex points attached, all of which have coordinates […]

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Once upon a time, I met a His Royal Highness the Duke of Kent.
The story of how that happened was pretty cool from my perspective, but every so often I wonder about it from his perspective. The Duke is the patron of the Royal Institution of Australia, and was in Australia just as they were […]

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One of my favourite puzzles is the Twelve Matchsticks puzzle. It goes like this:
Twelve matchsticks can be laid on the table to produce a variety of shapes. If the length of a matchstick is 1 unit, then the area of each shape can be found in square units. For example, these shapes have areas 6, […]

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Quarter the Cross is one of my favourite activities of all time, whether in maths or just life. I learned about it way back in 2015 and have been mildly or very obsessed with it ever since. You can read about my obsession in my first Quarter the Cross blog post, and you can read […]

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I’ve just started teaching an online course, and one module is a very very introductory statistics module. There are a couple of moments when we ask the students to describe how they interpret some hypothesis tests and p-values, and a couple of the students have written very lengthy responses describing all the factors that weren’t […]

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I have learned a lot from Twitter about how to treat my students, and most of it has been through being treated in ways I do not like. Recently I have been searching my own tweets to find things I’ve said before, and as I’ve dipped into old conversations, several unpleasant feelings have resurfaced when […]

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