# Estimation – an undervalued skill.

It was a great pleasure to host Rob Eastaway at the London branch ATM/MA meeting this morning.   His theme was Arithmetic, and how some techniques are almost becoming a lost art. There was so much energy in this session, the room was positively buzzing with pencils and pens scratching away!  We covered so much ground in two and a half hours, I’m not going to attempt to write about everything but I am going to pick the thing that resonated most with me.

A number of techniques we explored were about getting exact answers, but this section was as rough as you like! Rob introduced us to his idea of “Zequals“.  When teaching rounding, I always enjoy introducing my students to the “approximately equals” sign, ≈.  I hadn’t really considered how this symbol, on its own, doesn’t give the complete story.  All of these statements are true…

```7.3 ≈ 7
7.3 ≈ 10
and even 7.3 ≈ 7.4```

but they don’t give an explanation of what you have actually done to the 7.3 and in the last example here, it really would require quite a bit of explanation!

So Rob proposes “Zequals” which has a precise method.  It looks like this:

and it involves rounding the numbers you calculate with AND the result to 1 significant figure.

The Numberphile video explains in more detail here.  An interesting question to ask might be, what calculation would give the biggest discrepancy between the accurate calculation and the Zequals calculation? And what would the % error be?  The blog post explains this beautiful graphical representation of that % error, which turns out to be a fractal.

Now, to be honest, I would be hesitant to “teach” non-standard notation and methods as part of the regular timetable of maths.  There is already so much to learn and time is so precious, why would I take a lesson explaining something they are unlikely to ever encounter again in this form?  But dismissing it on that basis, misses the point, I feel.

Estimation as a topic features in a fairly minor way at GCSE but is a critically important skill in many jobs and life in general. There was some discussion amongst the attendees this morning that as students progress through KS3 to KS4 and A-level they become more and more reliant on their calculator.  With the demise of the C1 paper, there is no longer a requirement for a non-calc paper at A level  which is inevitably going to mean that our students will get weaker at this skill rather than stronger. This seems like a real mismatch between our education system and the requirements of employees and our broader society.

An idea which might help is to explicitly teach estimation as a technique to be employed when doing calculations with large numbers or decimals. Typically these types of calculations would involve some sort of “ignoring” the decimal point or the zeros, doing the calculation, and then “putting it back”

```3.23 × 3
323 × 3 = 949, so 3.23×3 = 9.49 (counting 2 d.p.)

23.1 × 0.31
231 × 31 = 7161, so  23.1 × 0.31 = 7.161 (counting 3 d.p.)

3200 × 40
32 × 4 = 128, so 3200 × 40 = 128000 (counting 3 zeros)

```

Maybe instead of, or as well as, “counting the decimal places” when doing these calculations we could do some rounding / estimation. So

```3.23 × 3 ≈ 3 × 3 = 9
323 × 3 = 949, so 3.23×3 = 9.49 (same order of magnitude)

23.1 × 0.31 ≈ 20 × 0.3 = 6
231 × 31 = 7161, so 0.23 × 0.031 = 7.161

3200 × 40 ≈ 3000 × 40 = 120000
32 × 4 = 128, so 3200 × 40 = 128000```

Now I am not claiming that this is a more efficient or reliable method. It does depend to a certain extent on the examples chosen and “counting the decimal places” is a method that will always work.  But I feel that the approximation step helps with number sense:  the idea that 20 × 0.3 is a bit less than half of 20 so must be 6 is really valuable for life beyond exams.  It provides an opportunity to practise these skills, practice which I believe our students currently have precious little of.

# One plus one equals …?

Powerpoint file used to create this here.

# Mathematical debates, and Bounds

It’s possible that what I am about to explain is already completely obvious to many a maths teacher.  But after 5 years of teaching, it’s only just dawned on me (a recurrent theme of this blog by the way!) so I thought I’d share.

I like teaching the concept of Bounds.  It highlights a fundamental difference between concepts that exist in Maths and the real world.  We measure things using numbers as continuous variables.  When we measure the length of a table, say, as being 120cm, we are rounding to the nearest centimetre.  We could be more accurate and say it is 120.4cm or even 120.4187234cm but this is still an approximation.   It reminds me of my engineering days, looking at milling machines that were capable of cutting to the nearest 0.1mm. I used to wonder about how those milling machines were made and the tolerances required in the dimensions of their components.  What machines were used to make the components? And what about the machines that were used to make those machines? Where does it stop?!

But one of the stumbling blocks with bounds is always with the Upper bound.

“But you told us that 5 rounds up!” is the usual complaint.  And then we get into a heated arguement about whether:

Which of course it is, for the same reason:

(these images from Don Steward can be useful for this one).
Sometimes that argument can be fun, but I think that as teachers we need to beware of going down rabbit holes. The more inquisitive students might find this philosophical debate stimulating but it can be a turn-off for others. And even the ones that do actively engage in that discussion may not be convinced at the end of it.  It’s just another one of those decisions that we make in the moment as we read the mood in the class.  In making that decision we should ensure that we are taking into account the feelings and needs of all students, not just the more vociferous ones.
So, to the point of this post, how to move this debate on.

Earlier this week, I was marking this GCSE question, which most of my students, including those sitting Foundation were answering no problem.

I’m pretty sure that I have previously suggested that students state the “Lower Bound” and the “Upper Bound” almost as though they are two separate answers.  But expressing the range of possible values in this way makes more sense. It is also entirely consistent with the well-understood rule that “5 rounds up”. In the question above, if the actual length was 13.5, it would be rounded up to 14, but a value below 13.5 would not.  Whereas a measured value of 12.5 would be recorded as 13.

All makes perfect sense now!

# Reflections on Pythagoras

I was with a Year 10 class doing Pythagoras’ Theorem recently.  This was a low attaining group who had encountered it before but were shaky.  They had recently been doing rounding to d.p. and s.f.  They had made progress with that but were getting a bit bored so the teacher decided to go over Pythagoras which I thought was a nice way of interleaving topics.  Calculator answers needed to be rounded (for those triangles which were not pythagorean triples).

It got me thinking about the knowledge and understanding required to apply Pythagoras’ Theorem and also about planning topics. In this lesson, they stuck to finding the hypotenuse and didn’t do any problems where they had to find the shorter side. This decision was made by the teacher because she knew the class and the context. By the end all students had practised it a few times and had built self-belief that they could do it. In other contexts, another teacher might decide to introduce both cases side-by-side.

Is there a “best” approach?  Has there been educational research looking into such a finer point of teaching this one topic? If there has, please point me to it because I wouldn’t have the time to look for it.  Nor the belief that if I did eventually find something, reading it would actually improve the outcome of my learners.  There is no single perfect way of teaching any topic. As teachers, we need to keep our eyes and minds open to approaches that we hadn’t thought of or used before. But we must not agonise over trying to find the holy grail, the golden nugget that will suddenly enlighten our students.

As I wasn’t actually teaching the lesson, just helping out, I had time to draw a quick mindmap in the lesson.  I was trying to put myself in one of the students shoes.

`What are the things I need to know, understand and be able to do to be successful in this topic?`

This, fundamentally is what they care about.

I’ve included a picture of my mindmap here, not because it is a stunningly useful breakdown of this topic, but just to show the complexity of what our students need to learn.  And this is without much page space taken up on the “And then…” topics of proofs, pythagorean triples, etc., some may say the “interesting bit”.

I am going to try to do this mindmapping exercise more often in my lesson planning. I found it quite easy to do when I was sitting in this lesson, but I find it harder when faced with a blank sheet and maybe some resources and ideas that I might have used before.  The point is, I think, not to agonise over the perfect mindmap showing the perfect sequencing of “learning nuggets”. It needs to be done whilst thinking about the class and the context. And if there is ever time for such luxuries, doing it with a colleague surely makes the process more satisfying and enjoyable.

# Rounding Errors

To investigate rounding more deeply, here is a structure for some questions that hopefully prompt the question, why does this happen?

Here are three examples as a pdf and as a Google doc.  It doesn’t happen in the first example, but it does in the second two.

Next, see if students can create their own to investigate why this happens.  I have created a blank sheet (3 on each page).  Again, as a pdf document and as a Google doc.