Reflections on maths, learning and maths learning support, by David K Butler
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”.
The titles of the five blog posts are:
In maths, or at least university maths, there are a lot of statements that go like this: “If …., then …” or “Every …, has ….” or “Every …, is …”. For example, “Every rectangle has opposite sides parallel”, “If two numbers are even, then their sum is even”, “Every subspace contains the zero vector”, “If a matrix has all distinct eigenvalues, then it is diagonalisable”. Many students when faced with statements like these automatically and unconsciously assume that it works both ways, especially when the subject matter is new to them. This post is about a way of helping students see the problem.
I had a meeting with an international student in the MLC on Friday who has having a whole lot of language issues in her maths class.
She was from the USA.
Yes, the USA. Her problem wasn’t the everyday English; it was with the different terminologies for mathematical things here compared to her experience where she comes from. Her only experience with vectors was in physics where a vector is a quantity with magnitude and direction, whereas in maths class here, a vector is usually just a list of coordinates. She knew how to find a derivative, but had never heard the word “differentiate” used for that action. She had only ever used the word “anti-derivative” for what we call an “integral” most of the time.
I was talking through the transpose of a matrix (something else new to her) and how it interacts with other operations on matrices, and how since it’s normally written as a power it takes precedence over nearby multiplications like powers do. She asked me if, here in Australia, we still use the same order for what is supposed to be done before other things. I said, yes, we do, and told her that most local high-school students use the acronym BEDMAS to describe that order.
She wrote underneath the acronym familiar to her: PEMDAS. First she focussed on the fact that one had DM and the other MD, but reconciled that quickly saying, “Well I suppose they go in the order they come and so it doesn’t matter which way around they are.” But she had no idea what to do with the B.
I told her the B stood for brackets, and I drew what brackets look like: ( ), [ ], { }. And then she freaked out. To her, those things are called parentheses ( ), brackets [ ] and braces { }. I said, yes, those are the official titles of those things, but here in South Australia they’re thought of as different kinds of brackets. If we want to distinguish between them we’ll call them round brackets ( ), square brackets [ ] and curly brackets { }.
And suddenly a light came on for her and a whole lot of stuff people had said this week made sense. She also suddenly understood the very odd look her class tutor gave her when she mentioned the word parentheses. “Yes,” I responded. “Most maths students and tutors here would never have heard the word parentheses.”
And what happened next? Well we’ll have to wait and see. I think we made some excellent progress, and we agreed to keep meeting across the semester to help deal with anything more that might come up.
For me, I’m so glad I knew a little about the differences between Australian and American mathematical-English. (Thanks MTBoS!) And perhaps if anyone is reading this, then you will know too.
| Australia | USA | |
|---|---|---|
| ( ) | round brackets | parentheses |
| [ ] | square brackets | brackets |
| { } | curly brackets | braces |
PS: I find it interesting how the Australian acronym BEDMAS references a general term “brackets” which covers all shapes of bracket, whereas the American acronym PEMDAS references a specific term “parentheses” which only covers one shape of bracket.
These comments were left on the original blog post:
Claire 31 July 2016
This is very interesting. I teach high school in Southern California. When I teach the Order of Operations, my students have seen it before from middle school. They were either taught PEMDAS or Please Excuse My Dear Aunt Sally to learn the order. I always use these acronyms as a way to highlight the limitations of some of the “tricks” they use to memorize math concepts.
It’s interesting that the parenthesis, brackets, and braces are all called “brackets” where you are. Having 1 word for that is super helpful. I tend to say that they are all “grouping symbols” and if it were up to me, the acronym would be GEMDAS. Because then, for example, absolute value symbols are grouping symbols.
We also discuss why MD doesn’t matter the order… That they are inverse operations and part of a mathematical family. Same with AS.
This family idea is helpful to explain where roots and logs might go in the order of operations.
This post gave me a lot to think about.
Geoff Coates 1 August 2016
I helped an African student once who learned his high school maths in French. They used commas where we use decimal points and full stops where we use commas in large numbers. He was very confused for a while …
David Butler 2 August 2016
This came up in the Chemistry labs last semester, when all the pipettes had their volume listed as “0,25mL”.
A few months ago, I learned a new word: “mansplaining”. You may have heard it before, but I never had until this year.
The general idea is that very often, a man will explain something to a woman in a way that seems to be based on the assumption that the woman is incapable of understanding the concept themselves and requires the man to rescue them from their misunderstanding. Often it is very explicitly patronising or condescending. This is a mansplanation.
In recent weeks, I have seen people I greatly respect being treated this way in the online space, and they have called out the man in question by telling him that he was mansplaining. Quite often, he has responded with quite a bit of vitriol, claiming that the word “mansplaining” is in itself sexist and they were just “trying to help”. This very vitriol is of course really not supporting the man’s case, and tends to show that his assumptions actually are that the woman did need to be rescued from her ignorant state. You can see some classic examples of this sort of assumption in Fawn Nguyen’s excellent blog post “Baklava and Euler “.
I had formed the idea that mansplaining was really just assholesmanplaining, and it didn’t have all that much to do with your general everyday respectful man.
But then something happened that hit me in the guts. Megan Schmidt started a conversation on Twitter about notation, and it had a flurry of responses, all from men, one of whom was me. She tweeted separately that “the mansplaining game is strong right now”. I was not consciously responding from an assumption that Megan needed to be rescued from confusion, and yet the conversation was called mansplaining. Clearly Megan’s use of the word didn’t fit with my understanding that only assholes mansplain.
It was time to get to the bottom of this, so I asked Megan to help me understand what she meant and how she felt about it. I have to thank her a hundred times for the thoughtful and gracious responses that she gave. I hope I will do justice to what you taught me, Megan!
I learned that there are times when offering an explanation at all is actually mansplaining. Not because the explainer is an asshole, or because they meant to be condescending or sexist, but because the explainer is unwittingly playing to a wider cultural assumption that the woman needs an explanation at all.
When a woman expresses frustration or anger or worry at something, a man’s common response is to offer an explanation to clear up confusion. Do you see the disconnect there? The man is rescuing the woman from confusion, but the woman wasn’t expressing confusion. She didn’t need an explanation – she didn’t need to be “rescued”. It’s most likely that she actually does understand the nuances of the concepts involved. Indeed, she would usually have to understand in order to have the emotional response she is having.
An unfortunate part of it is that the majority of men in this situation, especially in a professional setting, actually do realise that the woman does have the same or greater experience and training. It’s just that they are culturally conditioned to offer explanations in response to frustration. Indeed, it seems to be that men in professional settings are expected to engage in more “academic” conversations than “emotional” ones. Yet by doing so, we are still mansplaining.
The problem is that it opens the door for assholemansplanations, which are sure to follow. Even worse, it is adding to the hundreds of tiny sexist events that occur for a woman every day. And it reinforces the very cultural norm that produces those daily tiny sexist events. It’s important to give the experience a loaded name like mansplaining to make sure that those of us who do care have our attention drawn to these problems.
But how, as a man, can I fight back? Well, I can certainly call out others when they are mansplaining. Assholemen need to hear it from other men to have a chance of hearing the message – they’ll never listen to a woman. Ordinary men need to know about the damage they do unintentionally.
And what about my own daily actions? All I can think of is to be more aware. I can listen to the actual words people are saying and notice the emotional part of what they say. I can choose to respond by asking for more information first, rather than launching into an unwanted and unnecessary explanation. It takes a lot of energy to watch your own words and actions, and sometimes I will slip (sorry in advance) but with practice I’ll get better at it. And then one day maybe I’ll find I never offer a mansplanation again.
When doing algebra and solving equations, there is this move we often make which is usually called “doing the same thing to both sides”. Quite recently this phrase of “both sides” has begun to bother me.
There are two terminologies in probability which many students are confused about: “independent” and “disjoint”. The other day I was working with a student listening to their thinking on this and I suddenly realised why.
You can read the rest of this blog post in PDF form here.
The above post has a special diagram to explain independence, which you can download in various formats:
The Complex Numbers are unfortunately named. Most people take the word complex to mean “difficult to understand”, so the very name we give this family of numbers sets students up to think it’s going to be a lot of hard work to understand them. This is sad, because they really are very very cool and not quite as difficult as people make them out to be.
It turns out, though, that the word complex has only recently attained the connotations of confusion. The word complex according to my dictionary means simply “composed of multiple parts”, which is plainly true of complex numbers: they have precisely two parts – a real part and an imaginary part.
My dictionary has another meaning for the word complex. The meaning above is the one used when we’re using it as an adjective, but you can also use it as a noun. In that case a complex is an object which is composted of multiple parts. For example, a cinema complex is a building composed of multiple cinemas. In this sense, the phrase “complex number” is much more akin to “house boat” – a number which is a complex, like a boat which is a house. (I recently read this idea in “The Joy of x” by Steven Strogatz.)
I quite like both of these ideas. When people think of them as complicated numbers, it feels like they are making a value judgement, but with this older meaning of complex, it is value-free. It’s a simple statement of fact about the structure of the numbers themselves. Sometimes you need to reclaim language to have a better perspective on the mathematical meaning.
We had students the other day from Maths for Information Technology and their task was to form the contrapositive of a several statements. Given a particular statement of the form “If A, then B”, the contrapositive is “If not B, then not A”, so mathematically the problem is not actually very difficult. However grammatically the problem is much harder than it looks.
Consider this statement: “If it is raining, then there are clouds.” If we compare this to my generic example above, we see that A is “it is raining” and B is “there are clouds”, so by my own rule, the contrapositive ought to be this: “If not there are clouds, then not it is raining.” This is obviously not a grammatical English sentence! A correct version is, “If there are not clouds, then it is not raining.” By giving people a generic rule, we are getting them into trouble with their grammar. This may seem like a small thing, but there are plenty of students for whom English is not their first language, and even students whose first language is English often don’t have a very good command of the rules of grammar!
There’s no easy way around this in the way we present these generic rules, except to make them aware that they need to think about the grammar of the sentence that they write when they do this, in particular, where the word “not” has to go.
But it gets worse! Consider the statement “If f: R → R is differentiable, then f is continuous”. According to my above rule, A is “f:R → R is differentiable” and B is “f is continuous”, so taking into account the grammatical placement of the word not, we get the contrapositive is “If f is not continuous, then f:R → R is not differentiable.” The students we worked with did this exact thing, and they could tell there was something odd about it, but they couldn’t quite figure out what it was.
The problem is that the part saying “:R → R” is not technically part of the if-then construction. It could have been stated in a completely different sentence like this: “Let f: R → R. If f is differentiable, then f is continous.” Then that lead-in sentence isn’t an if-then construction, so it isn’t part of the contrapositive.
And here’s where the grammar gets particularly tricky. The fact that this little bit of the sentence can be pulled out into a sentence of its own means that grammatically it is called a “relative clause”. A relative clause gives more information about a noun in a sentence, without interfering with the verb. You see it in sentences like “My brother, who is in Canada at the moment, says hi.” I could have said: “My brother is in Canada at the moment. He says hi.” Of course it wouldn’t have had quite the same impact as the first sentence, which is why we say it the way we do. Another example is “If Catherine, your wife, is a kindy teacher, then she is clearly awesome.” The relative clause here is “your wife”, which is telling more information about who Catherine is before you go on to say stuff about her. This sentence is closer to the maths sentence above, but it has one very important difference. In the maths sentence I mentioned f twice; in this English sentence I didn’t mention Catherine twice. Instead, I used the pronoun “she”. We could have done the same in the maths sentence too: “If f:R → R is differentiable, then it is continuous” would become “If it is not continuous, then f:R → R is not differentiable.” It would have been much more obvious what was wrong with this sentence – we haven’t told the reader what f is, or indeed even mentioned f at all until the end. This makes it obvious that we ought to move the relative clause to the first part of the sentence when we form the contrapositive.
Other than the strange tendency of mathematicians to not use pronouns, there is something else that prevents us from seeing “:R → R” as a relative clause: the flexibility of the notation itself. Maths notations that include a verb in their meaning can be read aloud in multiple ways depending on context. Many students do not actually realise this, mainly because they never read their maths aloud. For example, consider this bit of maths: “Let x = 12. Then x = 4×3 = 2×2×3.” This is read aloud as “Let x be equal to 12. Then x is equal to 4 times 3, which is equal to 2 times 2 times 3”. That “=” was read loud as “be equal to”, “is equal to” and “which is equal to”. In our contrapositive example, consider these three sentences: “Let f: R → R.”, “Suppose f: R → R.” and “If f:R → R is differentiable…”. The first is read aloud as “Let f be a function sending R to R”, the second is read aloud as “Suppose f is a function sending R to R”, and the third is read aloud as “If f, which is a function sending R to R, is differentiable…”. But they all look the same!
The flexibility of our maths notation makes for easy writing, but sometimes it makes for difficult grammar, especially when it masks those pesky relative clauses!
This comment was left on the original blog post:
John Baez 8 September 2014:
Very interesting analysis. A digression: you write “Other than the strange tendency of mathematicians to not use pronouns….” This reminds me of something the computer scientist Tom Payne told me: “mathematicians are people with an extraordinary ability to keep track of many pronouns”. His point was that variables in mathematics serve as pronouns. Instead of saying “he”, “she”, and “it”, which breaks down when you have more than one he, she, or it, we introduce new pronouns (variables) as needed. Ordinary people lose track of all these pronouns.
There is a phrase people use when talking about statistics that really bugs me. It’s “non-parametric data”. I see it all the time in statistical teaching materials and I hate it because I know what they mean, but what they’ve said is simply wrong. Whoever writes this phrase has a tenuous grasp of what the word non-parametric means. If they really understood what it meant, they would realise that the word non-parametric can only be used to apply to a statistical procedure, not to the data itself; the words “non-parametric” and “data” just can’t be put together like that.
Ok, so now I’ve had my rant, let me tell you about the word collocation. I was looking at what was on the other side of our scrap paper recently (this is always a good way to procrastinate), and I found myself reading drafts of the PhD thesis of Julia – one of the Writing Centre team. The bit I was reading concerned whether people understand certain idioms and how you might include information on what these idioms mean in a dictionary. As is appropriate for a scholarly work, Julia spent quite a bit of time discussing what is meant by an idiom, and here is where collocation comes in.
One of the features of an idiom is the fact that it contains certain words which need to go together in a certain order. For example, the phrase “a piece of cake” has to contain both the word “piece” and “cake” or it just doesn’t mean the same thing. This phenomenon is called a collocation – some words just go with other words, and other combinations either just don’t happen or have a different meaning.
Julia pointed out that part of learning a new language is learning which words go with which words in order to make a collocation. For example, you need to know in English that “piece” collocates with “cake” in this way, but that you can’t say “a slice of cake” and mean the same thing. As another example, you need to know that the word “hand” can be used in collocations such as “hand in”, “hand up”, “hand over” and “hand out”, but not “hand under”. And as a final example, you need to know that the word “fro” can’t stand on its own as a word but must be used in the specific collocation “to and fro”. These are difficult things to learn because most native speakers don’t even realise they are doing it; it’s just natural to them. And the native speaker can often tell there’s something wrong with what a non-native speaker said, but sometimes can’t quite figure out why it sounds wrong. So the teachers of language have to point these things out explicitly as new words are learned.
And it occurred to me while reading this that collocations are really important to know about when learning maths too. In geometry we say that a point is “on a line”, but we say a line is “in a plane”. In set theory we say that an element is “in a set”, but a smaller set is “contained in a set”. (We even use different symbols for element-in-set, and set-contained-in-set.) In calculus we say that this is the Taylor series “for this function”, not “of this function.” And in statistics the word “non-parametric” collocates with the words “test” and “statistics”, but not the word “data”.
Yet somehow we expect our students to pick this up all on their own. I think we need to learn something from our esteemed colleagues teaching language…