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! Look:

Multiplication…

Oh my goodness – it’s the product rule!

Division…

Oh my GOODNESS – it’s the quotient rule!

Addition…

OH MY GOODNESS – it’s the addition rule!

So it’s everywhere! This is totally awesome!!

In the comments in the last post, it was mentioned that the reason all this happens is because of the multivariable chain rule, which is all about holding all but one variable constant and adding together the different results. When I realised this I thought to myself “duh!” – I was using the same words to tell myself what to do in both situations.

The issue is that when students learn these rules, they don’t know multivariable calculus so this can’t be used as a way to explain what’s happening. But I have a good feeling that it might be able to help make sense of the multivariable chain rule when they do learn it because it will connect perfectly so something they already know. I’ll let you know if I ever try.

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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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In Maths 1A here at the University of Adelaide, they learn the following theorem (this is taken from the lecture notes written by the School of Maths here):

It 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. In many places in the world, including in the textbook for this course, this theorem is called “Part I of the Fundamental Theorem of Calculus”. (Part II says that integration can be done by doing differentiation backwards. Here at Uni of Adelaide, only Part II is called the Fundamental Theorem of Calculus. Go figure.)

And then they get questions something like this 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”.

I have a way of thinking about this that makes sense to me and it has to do with blocking what I see into blobs and a mantra I say to myself when I am faced with certain derivatives (I will probably write about that soon). However, I know that this doesn’t always make sense to students when I tell it to them for the first time. When I was with a student grappling with this last week, I suddenly realised that I had come to my own way of thinking because I had noticed a pattern in the way derivatives worked and noticed that I could apply my pattern to this situation. So I decided to take a different approach and help the student notice this pattern for themselves.

And what do you do to make a pattern more obvious? You line things up!

I made a grid and said to the student I would write problems in the grid for him to do, and to bear with me because I did have a point I promise. So for the next several minutes I wrote a new problem in the grid and passed it over for him to do it, asked him what his thinking was in that problem, and then wrote another one for him to do, and so on. I was very proud of myself for relinquishing the pencil and letting him do all the writing – this is the part of SQWIGLES I often find the hardest.

These were the problems we did, approximately (it was almost a whole week ago now). I was particularly proud I thought of f(x) = sec(x), because its derivative has two x’s in it:

At the end I asked him what he noticed about the problems and if anything he noticed would help him with the one at hand. And he figured it out! He noticed that all of them had an x in one version and an x2 in the second version, and he noticed that in every one, you did exactly the same move on the bottom row as the top, but multiplied by 2x each time. And he straight away wrote down the correct result for his problem with the derivative of an integral and articulated a general rule he could use in the future.

I couldn’t have been prouder of him. Internally, I couldn’t have been prouder of myself for using this approach. I am led to believe that this sort of thing is called a “problem string”, and I want to notice more opportunities to use them because I have a feeling it will be way more empowering for students. I’ll certainly use problems strings next time I am called upon to help a student understand using the chain rule when a function is defined as an integral!

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I was talking to a student about his calculus last week. He was trying to differentiate xx. (Actually he was trying to differentiate x ln(x) and had decided the best place to start was to raise e to the power of it, thus producing xx.) At first he tried this:

I asked him what he thought […]

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It’s university holidays again (aka “non-lecture time”), which means I’m back on the blog trying to process everything that’s happened this term. Mostly this has been me spending time with students in the Drop-In Centre, since I made a commitment to do more of what I love, which is spending time with students in the […]

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Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff like this:

There were several small issues a lot of them had, which combined to stall their calculations. One of the more important problems was about how the calculator interprets what they type […]

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I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I’m not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind […]

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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, […]

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Converses
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”, […]

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This is the last (for now) in a series of posts about Where the Complex Points Are. To catch you up, I discovered a way of visualising where the complex points are in relation to the points of the real plane. All the complex points (p+ri,q+si) are in a plane attached to the real plane […]

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