There is a procedure that people use and teach students to use for finding the inverse of a function. My problem with it is that it doesn’t make any sense, in two ways.
Category: Reflections
Reflections on learning and teaching and research and life.
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How I choose which trig substitution to do
Trig substitution is a fancy kind of substitution used to help find the integral of a particular family of fancy functions. These fancy functions involve things like a2 + x2 or a2 – x2 or x2 – a2 , usually under root signs or inside half-powers, and the purpose of trig substitution is to use the magic of trig identities to make the roots and half-powers go away, thus making the integral easier. One particular thing the students struggle with is choosing which trig substitution to do.
You can read the rest of this blog post in PDF form here.
The blog post references a YouTube video with worked examples, that you can watch here:
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Holding the other parts constant
It seems like ages ago – but it was only yesterday – that I wrote about differentiating functions with the variable in both the base and the power. Back there, I had learned that the derivative of a function like f(x)g(x) is the sum of the derivative when you pretend f(x) is constant and the derivative when you pretend g(x) is constant.
Since then I have realised that this idea actually dictates ALL of the differentiation rules where two functions are combined through an arithmetic operation! It’s everywhere!
You can read the two blog posts in this series in PDF form here.
The titles of the two blog posts in the series are:
- Differentiating exponentials: two wrongs make a right
- Holding the other parts constant: it’s everywhere!
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The Zumbo (hypothesis) Test
Here in Australia, we are at the tail end of a reality cooking competition called “Zumbo’s Just Desserts “. In the show, a group of hopefuls compete in challenges where they produce desserts, hosted by patissier Adriano Zumbo. There are two types of challenges. In the “Sweet Sensations” challenge, they have to create a dessert from scratch that matches a criterion such as “gravity-defying”, “showcasing one colour” or “based on an Arnott’s biscuit”. The two lowest-scoring desserts from the Sweet Sensations challenge have to complete the second challenge, called the “Zumbo Test”. In this test, Zumbo reveals a dessert he has designed and the two contestants try to recreate it. Whoever does the worst job is eliminated.
I find it very interesting that the Zumbo test is the harder of the two tests. In the Sweet Sensations challenge, the contestants can choose to use whatever skills they are already good at, and design their dessert in a way that they can personally achieve. In the Zumbo Test, the contestants have no control over the techniques that are required, and must try to do things they are not familiar with in ways they may not have seen before.
And why am I talking about this? Because my medical students find themselves in similar situations. Our medical students have two projects to do as part of their research curriculum during their third year. One project is a research proposal: they work in a group with a supervisor to plan a hypothetical research project, including ethics, literature review and (this is where I come in) statistics. The other project is a critical appraisal: they work in pairs to analyse a published article, including where it fits in the research, the writing, the importance and (again where I come in) whether the statistics is appropriate.
I have noticed over the years that in terms of statistics, the critical appraisal is harder than the research proposal. A meeting with students about the critical appraisal usually takes twice as long as one for the research proposal, and twice as much preparation for me. Many more students come to me to talk about the critical appraisal, and the ones who do come are more worried about the statistics they find in the critical appraisal than the statistics they need in the research proposal. Why is this?
When watching Zumbo’s Just Desserts, it occurred to me that the reason why is the same as the reason the Zumbo Test is harder than the Sweet Sensations challenge.
When doing your own research you can choose to only investigate questions in such a way to use the statistical methods that you understand. Even if you need a new statistical method, you just need to learn that one. Either way, you have complete control over your own decisions and know the things you are measuring and what they mean. It’s just like in the Sweet Sensations challenge the contestants get to make all the choices and use methods they are familiar with.
On the other hand, when reading someone else’s research, you have no control over the wacky statistical methods they choose to use. Even if they are the appropriate ones (they often are in medicine, actually), the paper almost never describes how the researchers decided to use them – it just says what they used. And they often measure new things in new ways that you don’t deeply understand. It’s just like in the Zumbo Test the contestants have to do things that are new to them in ways that are new to them.
It’s much much harder to understand the statistics in someone else’s research than it is to make your own.
Let’s just hope we don’t eliminate all the students by asking them to do it with less support.
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Problem strings and using the chain rule with functions defined as integrals
In Maths 1A here at the University of Adelaide, they learn that says that, given a function of x defined as the integral of an original function from a constant to x, when you differentiate it you get the original function back again. In short, differentiation undoes integration. And then they get questions on their assignments and they don’t know what to do. They always say something like “I would know what to do if that was an x, but it’s not just an x, so I don’t know what to do”.
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(Holding it together)
Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff. One of the more important problems was about how the calculator interprets what they type into it, which is really in essence about the order of operations.
The titles of the five blog posts are:
- The reorder of operations
- (Holding it together)
- The Operation Tower
- Replacing
- Sticky operations
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One reason I’ll still use pi
Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, I think.
But for a while I have wanted to add just this one thought to the conversation and now is as good a time as any.
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All dogs have tails
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.
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Brackets
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”.
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Mansplaining
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.