BLOGS WEBSITE

I don’t like BODMAS/BEDMAS/PEMDAS/GEMS/GEMA and all of the variations on this theme. I much prefer to use something else, which I have this week decided to call “The Operation Tower”.

In case you haven’t heard of BODMAS/BEDMAS/PEMDAS/GEMS/GEMA, then you should know they are various acronyms designed to help students remember the order of operations that mathematics users around the world agree upon so that we can write our mathematical expressions simply and unambiguously. For example, the most common here in Australia is BEDMAS, which stands for “Brackets, Exponents, Division, Multiplication, Addition, Subtraction”. The idea being that things in brackets are calculated first, followed by the rest. There are some technical details there, where Division and Multiplication are not strictly in that order, and instead ought to be done in whatever order they come, with a similar rule for Addition and Subtraction. This is one of the reasons I don’t like these acronyms, because if people only remember the list, then they tend to think that addition is always before subtraction. Indeed, I observed someone doing this within the last month.

The acronym GEMA is supposed to get around this by not explicitly mentioning the Division or Subtraction, and by using Grouping symbols to cover all grouping-type things. (If any Australians are wondering what the P in PEMDAS stands for, it’s “parenthesis”. People in the USA actually do use this fancy-sounding word and in their language they think that “brackets” means only the square ones.)

But this still makes me uncomfortable. The first way it makes me uncomfortable is that all of these acronyms seem to imply that brackets (or parentheses) are operations. But they aren’t. I mean, some types of brackets are operations, sure, but at the life stage when you usually learn this order, they’re not. Even if you do know about something like the absolute value, when you do calculate |-4+8| you don’t do the absolute value first do you? You do the thing inside the absolute value brackets first and it would be just wrong to do the absolute value to each piece first.

For your ordinary brackets this idea of focusing your attention on the thing inside is their whole purpose. In general brackets aren’t operations but just a way of holding things together so you do the actual operations at the proper time. It really bothers me to have them in a list of operations at all. I do understand they need to be mentioned somewhere, but still it bothers me.

The second thing that makes me uncomfortable is naming the concept by the acronym you use to remember it. I can’t count the number of people who I have heard say “should I use BEDMAS?” when they ask how to evaluate or read an expression. The concept is called “the Order of Operations” or “Operation Precedence” people! Get it right!

Anyway, this makes me less uncomfortable than it used to. It feels more like a mild transient little itch than a raging angry rash. I mostly made my peace with it by imagining they are saying “should I use BEDMAS to remember the right order?” which is perfectly fine. While there is reasoning behind why we do it in the order we do, it is more-or-less an aesthetic choice that could have been made a different way in an alternate reality, and aesthetic or arbitrary choices often need supports to help remember them. Also I realised that people need names for things, including mnemonic tools, and one of the nice things about acronyms is that they are already a name. It’s natural for the acronym, which has a ready-made name, to end up being the name for the thing itself. It doesn’t mean the itch isn’t there though!

So how do I reconcile all of this when I teach people about the order of operations? I use the Operation Tower.

It’s a visual representation of the order of operations, that keeps the same-level things at the same level, and carefully separates the brackets and other grouping symbols out from the operations themselves to make the point that they are different things. The innovation in the last couple of days is giving this thing a name, so that students can talk about it both to themselves and to others (thank you to a friend online for pushing me to do so!)

Here is how I usually introduce it: I start at the bottom, drawing the + and – first, saying that they are the most basic operations. Then I draw the × and ÷ above that, saying that they have to be done before + and – nearby. Then I draw the ^ and √ above that, saying they have to be done before any of the lower ones nearby. Finally I draw the box on the side with the (), [], ___, saying they are designed to hold things together in order to override the usual hierarchy. I also point out that the horizontal bar is usually seen as part of the root symbol, which is why it holds things in.

(Note I have been careful to say “nearby” in that previous paragraph. It’s not actually true that all the multiplications have to be done before all of the additions. Only the ones that are near to additions. Indeed, the whole point of the brackets is to put new boundaries on what “nearby” means!)

I also use the operation tower to help students remember some other rather nice properties of operations. Notice how the operations in each box distribute across the operations in the box below. For example √(4*9) = √4*√9 and (2+10)*4 = 2*4 + 10*4 (though you have to be a bit careful with division). Also notice how the higher things inside a log turn into the lower things outside the log. That makes the Operation Tower a reusable tool for multiple different things, which I rather like.

But the thing I like the most is how students respond to it. It really seems to make sense to them to organise the operations spatially as they learn for the first time that there is an order mathematicians prefer to work in, and they seem to understand that the brackets are a different sort of thing and appreciate having them listed separately. For those who are like me and have serious trouble remembering the correct order of letters, it’s much easier to process than an acronym, too.

I hope you like using the Operation Tower yourself.

PS: I have actually written about this before, and you can track the change in how I have used it over time if you read the previous blog posts about it:  The Reorder of Operations (2015) and Holding it Together (2016),

Posted in How people learn (or don't), Thoughts about maths thinking | Tagged ,
Leave a comment

This blog post is about a game I invented this week, and the game is AWESOME, if I do say myself.

DIGIT DISGUISES: A game of algebraic deduction

Players:

  • This game is designed for two players, or two teams.

Setting up:

  • Each player/team has two grids with the letters A to J, one labelled MINE and one labelled THEIRS, like the picture below.
  • Each player/team writes the ten digits 0 to 9, each in a separate box in the MINE grid, keeping the grid where the other player/team can’t see it. (I actually have a printable version with the rules on them that can be turned into battleships-style game stands here.)

The goal:

  • Each number has been disguised as a letter. You need to find out which number each of the other player/team’s letters is, by finding out what calculations with the letters produce.

Your turn:

  • On your turn, ask for the result of a calculation involving exactly two different letters and one of the operations +, -, ×, ÷. Some examples are A – B or C÷J or H+D or E×G.
  • The other player/team answers your question truthfully, either telling you which of their letters is the result, or telling you the result is not a letter. To be clear: players do not ever say a number in response to a question. They only ever say a letter or “not a letter”.
  • You can write notes to help you figure out what you know from the information you have so far.
  • Now it is the other player/team’s turn.

Ending the game:

  • Once during the game, instead of asking a calculation, you can say you are ready to guess. Then you say what number you think goes with every letter. The other player/team tells you if you are right or wrong.
  • If you are right for all letters, you win! If you are wrong for any letters, you lose and the other player/team wins! Either way the game is over.

 

Thoughts from Digit Disguises games I have played

This game was inspired by a puzzle I wrote called “The Number Dress-Up Party“. In that puzzle, all of the numbers are at a dress-up party and you have to find the identity of just a few of them by asking them to perform operations. I was reminded somehow of the puzzle and I was wondering about modifications of it. One thing I was wondering was if I had only a few numbers at the party, how long it would take to find out what all of them were, and so this game was born. Later that same day I played it across Twitter with a friend, on either side of a whiteboard at One Hundred Factorial, and with my daughter at home. I was then completely obsessed. 

I learned a lot from these three early games.

In the first game with Benjamin, I was struck by how quickly the logic got complicated, and then how quickly it all cascaded into finding everything when I finally got a few different numbers. It was interesting thinking about what it meant to get the response “not a letter”. I loved finding ways to keep track of the information I knew so far, and making sure I was using all the information I had so far. The presence of the 0 really made for an interesting ride on Benjamin’s side.

When I played at One Hundred Factorial, we played in teams on either side of a whiteboard and it was way more awesome! Firstly, it was heaps more fun to play in teams — talking through the logic so far and finding ways to represent it so the rest of the team understood was a really pleasurable experience. I loved hearing other people’s thoughts about what we knew so far and what we should do next. Not to mention having people notice when we had made a mistake in our logic. Secondly, having a huge space to write all of our reasoning was really nice. It was fascinating to see the other team’s approach, which was to have a big grid of which letter could be which number and slowly cross off the possibilities. I had never even considered doing something like that!

When I played with my daughter C (she’s just turned 11 years old), it was a whole different experience. C had no trouble answering my questions using the letter/number correspondence. In fact, she really enjoyed that part. (Indeed, when I offered to play with C the next day, she was happy just to be the keeper of which letter was what number and let me play Mastermind style.)

I found it rather fascinating that even though they haven’t really done algebra at school, it was perfectly natural for her to refer to these numbers by their letter disguises and to write stuff down on the page in terms of the letters. That is, she didn’t need to be taught about using letters for numbers to really get into the idea of the game. I think that is pretty awesome, actually!

What C was having trouble with was making sense of the information she was getting. She immediately realised that getting something like A-B = “not a letter” meant that A was smaller than B, but she didn’t really know what to do with the information that B-A=C. A little discussion helped her realise that this meant B had to be A+C and both A and C had to be smaller than B. But then it seemed like a huge task to find any letters!

It took a few tries to come up with a representation that was helpful, and a bar model really worked to make sense of it, and even allowed us to pull out information such as I=2F. It was still a little difficult for her to understand how knowing I=2F and I/F=C meant she could know that C=2.

So I reckon that for kids her age, definitely playing in teams is a good idea, or having the whole class be a team against a mastermind, or even the very first time with the teacher against the class as the mastermind with the teacher describing their thought process, to get an idea of the strategies involved. I’m also imagining a version much more like the original dress-up party puzzle where a collection of students are the numbers and they know which student is which number and the rest of the class have to figure out which student is which number by asking them to combine with operations.

On the topic of “mastermind”, my friend Alex has created a little python script that will allow you to play the game against it, mastermind-style. If you prefer to play in teams but you only have a few players, then you can play with all of you on one team against the computer!

Mathematical wonderings

The other thing that is so very awesome about Digit Disguises is how many mathematical wonderings flow out of it so very easily. On top of the usual things that come up during the game, such as what getting an answer of “not a letter” tells you about the numbers involved, you can go a long way wondering about the game as a whole and what happens if you change it.

I have wondered all of these things, but only investigated some of them. I won’t ruin the answers for you for the ones I have thought about already.

  • How many questions do you need to finish the game? Is there an algorithm that will finish in the smallest number of questions possible?
  • What if the goal was to correctly identify just one number? How quickly can you find the first number?
  • Which number is the easiest to find? Which number is the hardest?
  • Can you identify all the numbers using only one operation, such as only using subtraction? What is the smallest number of questions to finish the game if you’re only using each operation?
  • What if the operations were mod 10? You wouldn’t be able to use subtraction to tell you greater or less than in this case, but would it still be possible to find everything?
  • What if you had 0 to some other number, like 0 to 25 or 0 to 5. How long would the game be then? Would your strategy be any different?
  • What if you didn’t have 0 or 1? What if you had some negative numbers? What if you had some completely other collection of numbers?
  • Is it even possible to find all the numbers if you have a different collection? Are there collections of numbers any size you like where you can’t find any of them? Are there collections of numbers where you can find some but not others?
  • For the previous question, if there are only a small number of numbers (like two or three), what are all the sets of numbers that can all be identified?

Have fun!

I hope you like my game of Digit Disguises. I think it’s AWESOME! If you do play it yourself or with your students in a classroom, or you have thoughts about the answers to my wondering questions, or have anything you are wondering about yourself, please do let me know.

 

Posted in Isn't maths cool?, One Hundred Factorial, Thoughts about maths thinking
4 Comments

Context fatigue is a particular kind of mental exhaustion that happens after having to make sense of multiple different contexts that maths/statistics is embedded in. I feel it regularly, but I feel it most strongly when I have spent a day helping medical students critically analyse the statistics presented in published journal articles.

The problem with maths in context is that the contexts themselves require understanding of their own in order for the maths to make sense. This is nowhere more true than in statistics, where you have to use your understanding of whether you expect the relationship to exist, what direction you expect it to be, and whether you think this is a good or bad thing. The classic one in my head is an old first year statistics assignment where they used linear regression to investigate the relationship between manatee deaths and powerboat registrations in each month in some southern coastal American city. You have to know what a manatee is, what it means to register a powerboat, and why those things might possibly be connected in order for the statistical analysis you’re asked to do to make sense, not least because at least one part of the question will ask you to interpret what it means. When helping students read published articles recently, I’ve had to find out what’s been done to the participants in the study, how things have been measures, what kind of measurements those are, why they’ve been measured that way, and all sorts of little details to decide how to interpret the numbers and graphs that are presented.

Even ordinary everyday word problems are a minefield. Across two recent assignments, some financial maths students had to cope with album sales for AC/DC, flooding of the land a factory is built on including insurance, bull and bear markets, machines in a mining operation, committees with various named positions, road testing electric cars, contraband being smuggled in shipping containers. This is a lot of context that has to be made sense of before you can get a handle on the maths, and there is nothing in the question itself to tell you what any of this context means if you don’t already know. Even if you are already familiar with the context, you actually have to suspend some of your understanding in order to do the maths problem, because it’s much simpler than the actual situation any of the questions are talking about.

All of this interpreting is exhausting stuff! It just tires you out if you have to even a moderate amount all at once. You just feel like you don’t have any more energy to deal with any more today. That feeling there is context fatigue. Yesterday the first year maths students were doing related rates and every question was a new context with little nuances created by the context that had to be dealt with. Those poor students were exhausted after just one problem, letalone three or four.

As teachers, we need to realise that as the people writing the assignment questions, or at least people who have dealt with them before, we are much more aware of the details and nuances of the context than the students are, so we don’t have to work so hard to make sense of them. Not only that but we’re usually simply more experienced in both life and language than most of our students so it’s easier for us. Imagine the context fatigue you would get reading ten research papers in an unfamiliar area in one day (I feel this in real life regularly). That’s the sort of context fatigue your students have just from your assignment questions. Cut them a little slack, and make sure there is adequate time to process the context with appropriate rest time between context-interpretation. Also it wouldn’t be the worst thing to explicitly teach them strategies for making sense of context, such as ignoring the goal, and finding out about what some of the words mean. Strategies can make the work less intimidating, especially in the face of knowing how tiring it is already!

PS: If you’re in charge of tutors in a drop-in support centre, especially one that deals with statistics, please be kind. Context fatigue is real and tends to wear us down some days!

Posted in Being a good teacher, Thoughts about maths thinking | Tagged ,
Leave a comment

I am often saying how important it is to listen to students, and that I am fascinated by student thoughts and feelings. When students say I am a good teacher my usual response is to say it’s because I have spent the last eleven years in a situation where I get to listen to lots […]

Posted in Being a good teacher | Tagged ,
Leave a comment

Three years ago, my university’s Student Engagement Community of Practice collectively wrote a series of blog posts about various aspects of student engagement. I thought I would reproduce my blog post here, since it is still as relevant today as then.
There is a lot that staff can do to engage students in the university […]

Posted in Being a good teacher | Tagged
Leave a comment

I firmly believe that all students deserve to play with mathematical ideas, and that extension is not just for the fast or “gifted” students. I also believe that you don’t necessarily need specially designed extension activities to do exploration — a simple “what if” question can easily launch a standard textbook exercise into an exploration.
This […]

Posted in Being a good teacher, Thoughts about maths thinking | Tagged ,
Leave a comment

Once upon a time, I decided I would be vulnerable on Twitter. As part of that, when someone posted a puzzle that I was interested in, I decided that I would not wait until I had a complete answer to a problem before I responded, but instead I would tweet out my partial thinking. If […]

Posted in Thoughts about maths thinking | Tagged ,
Leave a comment

The Story
Sometime in the past, I was approached by academics in the Faculty of Arts to discuss the numeracy skills of the students in their faculty. They wanted to discuss how they might include numeracy skills in some of their courses across all the degrees they teach. It was a lot bigger than the MLC […]

Posted in Other MLC stuff, Thoughts about maths thinking
2 Comments

While I am thinking about SET, it is high time I wrote about a version of the game SET that was invented at One Hundred Factorial back in 2017, but has never been recorded anywhere for posterity. It is prosaically named Team SET.
In case you don’t yet know, the game SET is a game of visual […]

Posted in One Hundred Factorial | Tagged
Leave a comment

Amie Albrecht recently posted a most wonderful blog post about SET, and it reminded me there were some SET-related things I should post too.
The first is this little reflection on how I go about teaching people to play SET. Amie talks here about a very excellent way to do this, which is to get people […]

Posted in Being a good teacher, How people learn (or don't), One Hundred Factorial | Tagged
Leave a comment