16 February 2023

Don't forget the units?

Sometimes, the units (e.g., cm) that come with a quantity can really help to make sense of what's going on. But do we always need units?

Recently, I was with some teachers who were arguing about a question like this:

What is the area of this rectangle?

Some of the teachers were complaining that the question was ‘wrong’, because the question writer had apparently ‘forgotten the units’! This was seen as ironic, because we are always telling students, “Don’t forget to include the units”, and yet here was a situation where this error had apparently been made in the writing of the question.

“You can’t have an area of 8,” someone said – “it has to be 8 somethings, like 8 centimetres squared.” The whole question was completely unspecified – what on earth is a “4 by 2” rectangle – imagine going into a shop to buy a carpet that is “4 by 2” – without some units it is completely useless!

I didn’t agree. I am quite happy to have a line segment of length 4 or a rectangle of area of 8. In pure mathematics, these things are dimensionless numbers. When we calculate the area enclosed between the curve $y=x^2$ and the curve $y=x(2-x)$, the answer is $\frac{1}{3}$. It doesn’t have any units, even though it really is an area. It isn’t $\frac{1}{3}$ of anything in particular, although I suppose you could say that it is $\frac{1}{3}$ of a unit square, if you wanted to. It would certainly be absurd to write the answer as $\frac{1}{3}$ cm$^2$ unless you were in some applied context in which you’d stated that the scales along the $x$ and $y$ axes were marked off in centimetres. The same applies if we use a formula like $$\int_a^b \sqrt{1+\left( \frac{dy}{dx} \right)^2} dx$$ to calculate the length along a curve: the answer has no units.

Confusion about units with things like this leads to students thinking that they need to write the words ‘square units’ after a definite integral. Similar confusions sometimes lead students to want to write the word ‘radians’ after $$\int_0^1 \frac{1}{1+x^2} dx=\tan^{-1} ⁡1=\frac{\pi}{4},$$ or possibly even writing $$\int_0^1 \frac{1}{1+x^2} dx=45°,$$ which makes no sense at all! (How would you respond to the question: "When you do a definite trigonometric integral, should you give the answer in radians or degrees?")

The debate around units seems to be one where both sides think that the other side is demonstrating some kind of dangerous misconception. Contexts are very important, as are the applications of mathematics, but I am not convinced that everything is always made clearer by setting it in context. The abstract concept of area can be used to solve real-world problems, like painting walls and laying carpets, but there is also just the abstract notion of area, which is measured in dimensionless numbers. If you are happy that you can have a number like 8, all on its own, which isn't a measure of anything in any particular unit, then it ought to be OK to have a length of 8 or an area of 8 too.

A similar issue arises when people object to tasks like:

Write down 5 positive integers with a mean of 7.

Write down 5 positive integers with a mean of 7 and a median of 4.


Sometimes, teachers object that when you are calculating a mean of a real-life dataset the data points are very unlikely to be nice, neat positive integers. And would it really make sense to calculate summary statistics from data sets containing so few numbers? How meaningful is a median when there are just 5 data values altogether? People who object in this kind of way would be much happier if all of the data points had a couple of decimal places, and ideally would like us to have 500 data points, rather than 5, and handle them in a modern fashion using technology.

This all seems very valuable to me, and I am all for students grappling with realistic, messy datasets, with all the opportunities they present for data cleaning, examining outliers and using descriptive statistics to get a handle on what’s going on. In such a task, there is a purpose - something you want to find out from the data - and the focus becomes less on the nitty gritty of adding up and dividing and more on asking meaningful questions and using the mathematics to figure out meaningful answers (i.e., mathematical modelling).

But I don’t see that kind of work as an alternative to tasks like the ones above. The arithmetic and geometric means (as well as the harmonic mean and other kinds of mean) are all essentially (and certainly were originally) pure mathematics concepts. Certainly, they have important applications to statistics, and elsewhere, but if you are dealing with the AM-GM inequality, for instance, there is no reason to think that the quantities being averaged must constitute some kind of 'realistic data set'.

There are times when context really helps students to get a sense of the underlying mathematics, but there are also times when context can get in the way. It seems likely to me that learners might get a better understanding of what the mean ‘means’ by using - at least at first - simple, easy to apprehend numbers. Tasks, for instance, in which you add an additional small integer value to a small, simple data set, and notice if the mean increases or decreases, or combine two small, simple data sets of different sizes, and explore what happens to the mean, seem very valuable to me. It can be through this kind of work that learners build a sense that the mean is the 'equal shares' value that 'balances' all the values in the set. In such tasks, it would be impossible to notice anything amid the noise of vast quantities of awkward numbers. Later on, of course, when applying the concept of the mean to the real world, we can bring those insights to bear on larger, more realistic data sets, but having messiness from Day 1, as the default, seems undesirable to me. I think there is no reason to feel guilty about asking learners to find the mean of a few small positive integers.

This is not to say that it's always wrong to 'begin with complexity'. Often that can be motivating and lead to powerful mathematics. But paring back the complexity at times, so as to see the mathematical structure, can also be really insightful and can support powerful generalisations. When we say that a 2 by 4 rectangle has area 8, we are effectively making a general statement that subsumes all of these:

A 2 cm by 4 cm rectangle has an area of 8 cm$^2$. 

A 2 km by 4 km rectangle has an area of 8 km$^2$. 

A 2 mile by 4 mile rectangle has an area of 8 mile$^2$. 


For me, to say "a 2 by 4 rectangle has area 8" isn't wrong, even though it doesn't apply directly to any one specific real-world area.

Questions to reflect on

1. Do you feel that 'an area of 8' is wrong? Why / why not?

2. When do you think that contexts are helpful and when do they get in the way?

02 February 2023

Non-expository video clips

How can video clips be used effectively in the teaching of mathematics? And I don't mean clips of someone explaining something...

If I do a Google search for "maths videos", I discover a tonne (258m hits) of short clips of mostly people explaining various bits of mathematics - with varying degrees of clarity and accuracy. I'm sure that some of these may have their uses, but they're not what I'm interested in in this blogpost. I assume that clips like those are rarely used in the classroom, if you have a live teacher who can do the explaining themselves in a more interactive fashion.

What I'm focused on in this blogpost are what I call 'non-expository' video clips. These are not trying to tell you something or explain something mathematical. They might not even be created with mathematics education in mind, although sometimes they are. But they are intriguing and engaging in their own right, and have obvious potential for mathematical discussions - whether it's to introduce a new concept or to apply some recently-taught ideas. They are just a minute or two at the most in length.

Lots of these I first found via Twitter, and I'd like to acknowledge whoever it might have been (now long forgotten) who forwarded them to me.

You don't need sound for any of these.

Here's the first one:

Cookie cutter

I think you could use this clip with pretty much any age of learner. You could just play the clip and let discussion emerge, or you could ask:

What do you notice? What do you wonder?

Getting students to describe as precisely as they can in words what they have seen can be helpful in getting them talking about it, and mathematical terms might find their way into what they say quite naturally.

If the discussion doesn't take a mathematical turn by itself, you can always ask:

Where is the mathematics here?


What do you think is mathematical about this?

For me, this particular clip triggers thoughts about area and perimeter. If you wanted to be more directive, you could ask explicitly:

What does this have to do with perimeter?

There is still lots of room for different comments to be made, even with this level of direction. Someone could start by saying what they understand by the word 'perimeter'. Someone else might say that the perimeter is getting smaller, and then someone else might disagree with that and say that the perimeter is constant, but the area enclosed is getting smaller.

Another way to use the clip would be to introduce the idea of a cookie cutter first, so that everyone knows what one is, bearing in mind that not all children may have had experiences of home baking.

Or you could begin one step back from that, by showing a picture like this:

You could ask what kitchen equipment would be needed to make these, and then how they think cookie cutters themselves are made. Students might have interesting ideas about that - and then you could ask, "Would you like to see a video clip of how they are made?"

Drawing a freehand circle

Here's another example of a clip that never fails to engage students:

Students could come to the board and see if they can do it. If you have an electronic whiteboard, you could record snapshots of their attempts (they could sign their name in the middle of their circle, so that you can tell which one is which). If you have a traditional board, you would need to take a photograph of the board after they have walked away (so that they aren't in the shot), before the board is wiped. Each student gets only one go. Then you can print out some of the best ones (you could get six onto one sheet of A4 paper) and ask:

How can we fairly decide which one is the best circle?

This can go in broadly two directions.

(i) Students think about what measurements they would need to make on the drawn circles. There is a lot of mathematics to consider, and this is not straightforward, because, for instance, shapes with constant width are not necessarily circular. They might suggest covering the drawn circle with accurate circles - maybe sandwiching it between the largest circle that will fit completely inside it and the smallest circle it will fit completely inside, and then finding the difference between the diameters of those two circles, a smaller value representing a 'more circular' circle.

(ii) Students treat it as a statistical project and ask everyone in the class to rank the 6 circles from best to worst. Then they have to think about what to do with these rankings to come up with an overall answer (there are several options here). If they take this approach, to avoid bias they might prefer to obscure the names written in the middle of the circles. Ideally, they might wish to use participants from a different class, who wouldn't have a chance of remembering who had drawn which circle.

Work based on drawing freehand circles could test hypotheses such as that it is easier to draw a large circle accurately than a small one, or that people improve at this the more they do it, or that doing it faster is better than doing it more slowly.

I think that lessons that use activities such as these can be very memorable. The idea that the perimeter could remain constant while the area changes might be referenced by saying, “Remember the cookie cutters?”

If you browse around https://www.youtube.com/ or a site such as https://free-images.com/ you can find all sorts of interesting images and videos that could support this kind of activity. Many years ago, I collected some together myself at http://www.mathematicalbeginnings.com/.

Questions to reflect on

1. Do you have examples of non-expository video clips that you often use?

2. Would you use clips like the ones mentioned here? Why / why not?

3. Which class(es) would you use them with, and how would you use them?

Teaching specific tactics for problem solving

This is my final blogpost, as my year as President of the Mathematical Association draws to a close, so I've allowed myself to go on at ...