Explaining Significant Figures - why are they significant?

I have always been somewhat dissatisfied with how myself and other teachers explain significant figures to pupils, in particular why certain figures are significant and others aren't, and why certain bits of logic only apply in certain situations. The only really acceptable link I have found up to now is to that of standard form, in that when represented in standard form the digits multiplying the power of 10 are only the significant one. However I am still not very happy with this, as we often teach significant figures after standard form - I feel like a proper understanding of significant figures should inform a pupil's understanding form, not the other way around. Thankfully I think I have found a way of describing significant figures to pupils, which should allow them to begin to understand...

"In any number, figures are significant if they are needed to describe a number using separate columns without skipping any columns."

Now this seems a little bit of overkill I must admit, but given that the next most satisfactory definition I could find was to have four separate rules to cover situations without and without decimal points.

Lets examine some numbers to see how this can be applied:

456 - this would be described as 4 hundreds, 5 tens and 6 units, so all 3 figures are significant.

4057 - this would be described as 4 thousands, 0 hundreds, 5 tens and 7 units, so all 4 figures are significant (note we need 0 hundreds as we cannot skip the hundreds column)

300000 - this would be 3 hundred-thousands and so would have a single significant figure.

000456 - this would still be 4 hundreds, 5 tens and 6 units, so again has 3 significant figures.

0.00042 - this would be 4 ten-thousandths and 2 hundred-thousandths, so would have 2 significant figures

0.0506 - this would be 5 hundredths, 0 thousandths and 6 ten-thousandths, so would have 3 significant figures (again, the thousandths is needed so as to not skip columns).

For me the only place where this definition is a stretch is with decimal numbers that have trailing zeroes, as it would be tempting not to include the trailing 0s, such as in the number 300000 above. To counter this one would need to ensure that pupils had a proper understanding of place-holding. The zeroes in 300000 are place holders, they are there only to ensure the 3 lands in the correct column. The zeroes to the right in 0.4600 are not place holders (or shouldn't be) as they do nothing to ensure numbers find their way into correct columns. These digits must have a separate purpose (and of course when it comes to measurement they do - they tell us the accuracy applied, which is a natural point to eventually take significant figure understanding to) and it is this purpose which makes them significant. This allows us to show that the number 0.4600 will have 4 significant figures, and indeed that while the number 300000 only has one significant figure, the number 300000.0 will have 7 significant figures.

So if you are struggling to justify to pupils why certain digits are significant, and others are not, try first of all giving them a proper understanding of place-holding zeroes, and then applying the logic above - it may not be perfect, but I haven't seen anything better in 10 years of teaching maths.

Secret Teacher - a response.

I rarely write political or commentary blogs, preferring to stick to talking about what I know (i.e. teaching maths and leading a maths department). However after becoming involved in a discussion with @siobhanorb and @andylutwyche on twitter about the latest Secret Teacher article I felt the need to expand on the problems some people are finding with the articles and also the responses to them.

First off let me say I at least partially agree with @andylutwyche in particular; the secret teacher is giving us one view of education based on their personal experience and we shouldn't try to invalidate their experience just because we don't agree with it. The problem that I, and many others, have with the articles is that they always seem to support a negative view of the profession and there is a real danger that the public will take these individual and sometimes isolated experiences as indicative of the profession itself. Unfortunately too many of the responses, particularly online, are negative themselves, attacking the writer for their view even when they do only speak from their own experience. I would much rather see us as a profession acknowledge these experiences, and use our response to paint the different picture of the profession that many of us see. So here is my response to the Secret Teacher.

Dear Secret Teacher,

I am really sorry to hear that you are not looking forward to the start of the new term; and I can understand why that may be the case if you have felt under-supported in the last year. I do hope you choose to stick with it for a bit longer; you will find these things easier as time goes on and hopefully it will mean that in a few years time you will be in a position to use your negative experience to ensure that new entrants to the profession get a more positive one, by supporting them in the ways that you know you needed the support yourself.

It was really nice to see that you didn't blame others for not supporting you, but rather that you recognised that people had a huge amount on their own agenda and that was limiting their opportunities to help you. I know it can seem that everybody around you is too busy to help, and speaking as a Head of Maths of a few years experience I can tell you that you can approach your department lead and experienced team members at any time. We know that the success of our pupils and our teams is dependent on you developing well and feeling confident in your work (as well as retaining your expertise to create a consistent experience for our pupils). I have 3 NQTs joining my department next year and will make sure that I am holding sessions to support marking, report writing etc but my new staff will be clear that they should approach me or their assigned department mentor at any time if things are difficult. If you find that you are asking for support and not getting it, can I then suggest you try another school before giving it all up; you may find that a new environment will suit you better.

As a new teacher you may not have a wider awareness yet, but can I suggest you take care over comments that can paint a profession that many people love and are heavily invested in a negative light. In recent years the profession has taken a lot of stick from the government and the media in particular, and for teachers to validate this view doesn't help us change things for the positive. For example you talk about people you trained with leaving the profession, but don't speak about all those hundreds of people who would have been in your overall cohort (taking all subjects training into account) that will be continuing in the profession and have had a very positive experience. My suggestion is that if you have (completely valid I might add) points to make that your experience has not been ideal, that you try and give the balanced view and also that you reflect a little more and suggest ways you might have improved your own experience. This way your writing will become more about advising other people that may be struggling rather than simply expounding your own woes to the world.

If you do choose to stay this year then I hope that you find the experience better and that you find yourself more supported and more confident. If you continue to struggle I hope you will reflect on your experiences of the first year and use any insights to support yourself and others through the difficulties.

Multiplicative counting - the different types

In preparing for teaching the new GCSE, one of the new topics is the explicit use of multiplicative counting. While preparing problems I came to a realisation that there are 3 different types of counting here (at least before we get into the whole permutations and combinations area), and I thought I would take this opportunity to share my insight.

1) Powers for counting

Students and teachers of A-Level stats modules will likely be intimately familiar with this idea through the binomial theorem. This occurs when the same choice or outcome is independently repeated. The typical example might be flipping coins; flipping 1 coin has 2 outcomes, flipping 2 coins has 4 outcomes, flipping 3 coins has 8 outcomes etc, leading to 2ⁿ for the possible different outcomes of n coins. These are probably the easiest type to make up as there just needs to be a repeat that isn't affected by the outcomes before. Tweaking independent event probability questions to focus on the outcomes can be a rich source of ideas here.

2) Multiplying for counting

This one is probably the most familiar to GCSE teachers, as it is a reasonably well used stretch for pupils on systematic listing. Similar to above, this occurs when choices are independent, but when there are different number of choices for each outcome. Probably the most typical here is the menu problem: choose a three course meal from 3 starters, 4 main courses and 3 desserts and there are
3 x 4 x 3 = 36 different meals you can choose from. Take any systematic listing problem from the current GCSE and it can be tweaked to encourage multiplicative counting; just change the question from "list all the possible outcomes" to "work out how many possible outcomes there are" and maybe add a few more options to make the multiplicative reasoning justifiable over just creating a list in the first place.

3) Factorial counting.

This is perhaps the one that will be least familiar, although still familiar to the A-Level aficionados amongst us; this occurs when an outcome is removed once used and the process repeated until no outcomes are left. A nice example of this is given here:

Anne, Barry, Colin and Damien book 4 seats next to each other in the cinema as shown.

Seat 1

Seat 2

Seat 3

Seat 4

One way that they could sit with each other is like this:

Seat 1	Seat 2	Seat 3	Seat 4
Anne	Barry	Colin	Damien

(a) How many different ways can they sit?

The result here is 4 x 3 x 2 x 1 = 4! = 24, because there are 4 places the first person could sit, then once settled there are 3 places for the second person, then 2 for the third, then only 1 for the fourth (or alternatively, 4 options for the first seat, then 3 options for the second seat etc...). A nice place to go from here is to ask 

(b) What is the probability that Anne and Barry sit next to each other?.

This is a fairly simple adaptation to the problem, which could be asked directly instead of breaking down the problem into part (a) and part (b), but importantly it leads to a further question:

(c) How is your answer to part (b) affected if the four seats are not next to each other in a cinema, but around a table on a train as shown?

Seat 1

Seat 2

Seat 4

Seat 3

This sort of variation isn't really available to the first two problems, but is a lovely adjustment in these sort of factorial counting problems. Although they are harder to come up with, modifications of problems with conditional probability can be fruitful here, often with the reduction of options (i.e. going from choosing 2 or 3 from a bag of 20 sweets to eating 4 or 5 sweets in a certain order).

I guess I have two points from this blog. The first would be to make sure that pupils are aware of these different approaches to counting, and the second is to suggest that you don't need to come up with lots of new contexts for this new topic, just tweak things that are already there.