Could this dramatically improve some parts of mathematics education?

I love Multiple Choice Questions (MCQs), in principle, I almost never use them.  Intuitively them seem like a bad idea in contrast with Cued Response questions, and I cannot even begin to go into the theory behind MCQs here and now, but if you’d like a brief primer, this is a great place to start.  In short, they have a powerful role to play, one that we’ve been overlooking, rather than being a replacement to anything.

So I played around a bit with Plickers yesterday.  Not in a proper classroom setting, but enough to be very, VERY impressed by the technology.  Damned slick, simple, sexy; it’s fast.  Fast in reading pupils’ cards, fast in updating the live display on the board.  It’s got me thinking a lot more about using MCQs in the classroom.

There’s a kind of MCQ that I find particularly appealing: multiple correct answers, and you don’t know how many correct answers there are.

For example:

Which of these are considered wonders of the Ancient World, according to Herodotus and Callimachus?

  1. The Colosseum of Rome
  2. The Great Pyramid of Giza
  3. The Statue of Zeus, at Olympia
  4. The Library of Alexandria
  5. The Mausoleum at Halicarnassus
  6. The Great Wall of China
  7. The Sphinx

Give it a go, without looking them up.

Without knowing that three of these seven possible responses are correct, the probability of guessing correctly at random is a mere 2.9%.

There are other clever things going on here – this is not only an assessment, it’s a pedagogical tool operating on many levels:

  • Some of the distractors are considered wonders of the world in later lists, they’re just not a part of the original Seven
  • The Colosseum of Rome has been chosen deliberately because it sounds similar to The Colossus of Rhodes, which is a wonder of the ancient world
  • The Library of Alexandria has been chosen because it is an important feature of Alexandria’s history, and is often referred to even as simply The Great Library, however it is The Lighthouse at Alexandria that was considered one of the original Seven
  • The inclusion of the Sphinx forces the student to ask the question “Is it both the Sphinx and the Pyramid that were considered wonders… or only one… or neither?!”

For the pupil who really knows their list of the seven wonders off by heart, without any doubt, this is an easy question to answer, but for the student who doesn’t, a substantial amount of thought and differentiating between ‘bits of history’ they’ve heard before needs to take place.  Most pupils would likely fail in this question on first attempt, yet that’s from where in large part where its pedagogical benefit arises; now feedback can be issued, and in the process the pupil realises ‘Oh!  It’s the lighthouse of Alexandria, not the library, that was one of the seven wonders… but wait, the library is important, right, so why is it not one of the seven wonders…?’

Back to that pupil who really knew the list well, what do they get out of this?  Similar to the above in cases, except they approach is from a slightly different perspective.  It’s while answering the question that they might be wondering ‘Why aren’t the library and the sphinx a part of the seven wonders?’  You don’t get that kind of thinking if the question is: “What are the seven wonders of the Ancient World?” although that kind of Cued Response question does have its own merits.

This is all before we start getting into the clever increases to memory storage strength that a question like this can promote (albeit that has its caveats.)

It doesn’t even end there, I’m just going to end it there because this one question alone will sprawl into the thousands of words if I keep going.

I promised you maths.  Here’s how that could look in maths, any or even all of the questions below:

1 – Which of the following are examples of pyramids?

2 – Which of the following are Platonic Solids?

3 – Which of the following are examples of prisms?

Solid Shapes

Both of the MCQs above, from history and maths, are examples that deal with Categorical Concepts.  You start by learning what fits into a category, then explore in ever greater depth why those categories exist, and why its boundaries are set where they have been set.

Obviously vast swathes of mathematics do not fit into this grouping… and here’s where I actually get excited, and what prompted me to write this now.  What can this look like when dealing with Comparative Concepts?  On the surface, it seems so much more boring.

Which has the higher gradient?

  1. A
  2. B

Gradients 1

Bleh.  I mean, useful, no doubt, but no clever probability tricks here.  Straight up, 50/50 guesswork.  If you know that ‘higher gradient’ means ‘steeper’, it’s obviously A.  The Plickers graph would (certainly should!) look like this:

Plickers Graph 2

We can do some interesting things here.  For example asking the question before explaining what gradient is, especially if you show that image, then this one and ask again:

Gradients 2

Do they just select the longer line, rather than the steeper line?

You can eliminate that with a question like this one:

Gradients 3

But anyway, enough of all that, let’s make this really interesting now.

Try this one yourself, now:

Which has the higher gradient?

  1. A
  2. B
  3. Don’t know

Gradients 4

Difficult, right?!  Yet, one of them doesdefinitely, have a higher gradient than the other.

The nice thing about combining a question with this with a software solution like Plickers is that we should be able to see a graph something like this:

Plickers Graph 1

The whole group can see that no-one really knows…

So now we get to an interesting question: What additional information might make it possible for us to figure out the answer to the question, without giving the game away?

Here’s one that’ll make it super easy for you, check this out:

Which has the higher gradient?

  1. A
  2. B
  3. Don’t know

Gradients 5

And now by looking at the points that both lines leave the grid, you can see clearly that line A is just a little bit steeper than line B, and therefore has a higher gradient.

Next question: What if I didn’t give you a grid, but I’d told you that line A went along to the right by 8 units, and up by 6.2 units, while line B went to the right by 8 units, and up by 6 units.  With those numbers how would you know that A was steeper?

What if we try different numbers, and I tell that you that Line A goes to the right 5 units, and up 3 units, while Line B goes to the right 5 units, and up 2.8 units?

Oh you think Line A is steeper?  Because it goes up more?  Fine, correct, but what if I now tell you that Line A goes to the right 7 units, and up 5 units, while Line B goes to the right 11 units, and up 7.5 units?  Why is this now more difficult to answer?

Would you answer it by trying to ‘scale up’ Line A’s 7 units to become 11 (which relates it equivalent fractions and finding common denominators.)  Would a child look for the functional multipliers in the system, where child A gets:

Line A: 7/5 -> 1.4

Line B: 11/7.5 -> 1.47

And concludes: Line B has the higher gradient

While child B gets:

Line A: 5/7 -> 0.714

Line B: 7.5/11 -> 0.682

And concludes: Line A has the higher gradient

Given that Line A does indeed have the higher gradient, we now finally have an explanation as to why we calculate gradient as ‘rise over run’ or ‘Change in Y over Change in x’ or ‘dy/dx,’ if you prefer mathematical symbols, rather than dx/dy, which does feel more intuitive: ‘We use x all the time.  X is more important or comes before y in the alphabet; why not have dx on top?’  Well, now we know.  If you do, your smaller number will suggest a bigger gradient, and that’s ugly.

And so on, and so forth.

Dan Meyer has a series of posts that look at mathematics as the ‘aspirin,’ and ask ‘so what is the headache?’  He argues that we tend to force feed pupils aspirin when they don’t even have a headache – in other words we teach the ideas that have been developed to solve problems or answer questions people had way back in time, without looking at what those problems or questions might have been.  We give answers to questions no child ever asked…

I do have some sympathy for his views here, and I wonder if what I’ve explained above overlaps.  I also wonder how kids who’ve never been taught anything about gradient might interact with the sequence of questions differently from those who have been taught something about it.  If I run it with my Year 11s in a few weeks’ time, will they ask me for coordinates of two points on the line?

I didn’t teach gradient this way.  I wish I had.  Instead I just jumped in almost immediately with ‘Here’s how to calculate what I’m calling the gradient.’  I really want to try this out with Year 11 and see whether it deepens their appreciation for the process of calculating gradient numerically.

It doesn’t end here; I’m sure something like this could be attempted with any comparative concept.  Area’s an obvious one.  Assuming the concept of area has been effectively communicated, try this:

Which shape has the bigger area:

  1. Rectangle
  2. Circle
  3. Triangle
  4. Don’t Know


…guess we better learn how to put a number on the area of each shape!

About Kris Boulton

Teach First 2011 maths teacher, focussed on curriculum design.
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3 Responses to Could this dramatically improve some parts of mathematics education?

  1. youwhatnow says:

    I use MCQs in English, but I don’t use the more than one correct answer idea because there’s always some smart-alec kid who says “What if I just choose all of them?” Without telling them how many are correct, how do I get around that idea? If I say that choosing them all with get a zero, then they say “Well, I’ll choose all of them but one.” What would you (or your readers) suggest as a response to avoid this?

    • Kris Boulton says:

      Interesting question, I guess for me it comes down to culture – I almost wouldn’t even expect that to happen! Because… what’s the point? I suppose it depends upon the context of the question. If you’re thinking of it as a test, then yeah, perhaps I’d go for 1 point deducted for each incorrect option selected, or something similar – in fact you’d have to, otherwise the correct ‘exam strategy’ would just be to select them all.

      If it’s a hinge question during a lesson, though, I would simply expect a person to justify selecting all 7. If they genuinely thought they all sounded reasonable… though they perhaps weren’t sure (which is most likely) well then fair enough, that’s why we’re hear, to learn why they aren’t all examples of Wonders or Shape Type. If they were just opting out of the question or attention seeking, then I would deal with that at some point. If someone overtly stated out loud ‘I’m just going to choose them all then,’ as in, they overtly stated they were opting out of thinking, to be honest I would just point out how ridiculous that choice was. We’re here to learn, the question is there to help that learning happen, if you were to do as you suggested, then you won’t learn… so why on Earth would you do that?

      I wouldn’t really expect anyone in my classes to belligerently continue with that line of reasoning after that initial gambit. If they did it might be a demerit, for lack of effort, or it might be a phone call home.

  2. dodiscimus says:

    MCQ are the defining example of what’s missing, in terms of effective collaboration, within the teaching profession. ExamPro and other exam board resource banks have some, TES resources have lots, Socrative a bunch, etc. but searching through that lot isn’t a rewarding task. Obviously having multiple platforms is a big problem but also the effectiveness of the search tools and the very mixed quality of the questions. If half of the maths teachers in this country contributed one question per week, there would be the best part of a million within a year so quantity isn’t really the problem. For those of us old enough to remember the rise of Google over Yahoo! there’s hope that in a few years we might be looking back on this era and wondering how we managed without a single, fast, MCQ search tool that automatically took account of question popularity. I keep thinking that project has TF Ambassador written all over it. It’s the sort of thing a university with strong education and computer science depts. ought to be able to do but it’s really too risky (might neither generate income nor have research impact) unless the there was a pot of money to bid for.

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