Powerpoint file used to create this here.
Powerpoint file used to create this here.
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:
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!
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.
To investigate rounding more deeply, here is a structure for some questions that hopefully prompt the question, why does this happen?