Bar Models 1

For me, one of the most important continua in learning Maths is the following:

Concrete-Diagrammatic-Symbolic

Too often, I feel that Math teaching rushes to the symbolic representation of the problem and downplays the importance of the diagrammatic and concrete representations. Which is strange really: if I get stuck on a problem, one of the first things I do, is draw a picture to help me better understand what is going on. And if this doesn't help, then I'll get some hands-on materials to help me figure out what to do.

When solving problems, probably the most common question that students ask their math teachers is "What do I have to do?" Often they want us to tell them what they have to add, subtract, multiply, divide (or any combination thereof). Over the years, I have become more convinced that this is because students haven't developed a schema of what the operations look like in real-life. Instead they are reduced to looking for keywords that might (or might not) be in the question and that might (or might not) actually mean what they are supposed to mean.
This is why I have been recently singing the praises of bar models to the educators I work with. These are sometimes referred to as 'Singapore Bar Models' due to their extensive use by students in that country to solve problems. However, I saw bar models when I began teaching in 1990; they were used to illustrate how to visualise percentages in the SMP Red series texts. And, I suppose, my first concrete experience of bar models as a student, was when I worked with Cuisenaire rods as shown in this earlier post.
For those unfamiliar with bar models, see how this Grade 7 student used one for the very first time in solving the following problem:

A crowd of 2400 go to see the local hockey team play but as they are doing so poorly, three-quarters of the fans leave at the end of the first period. A further third of the remaining fans leave at the end of the second period. How many fans watch the third period?

What I think is great about this solution is that by drawing the bars, the student can see what operations need to be done (and then does these in his head). If he was struggling before about what operations to use, now he can see them and, if necessary, he can write these symbolically.
Much as I love bar models, I don't think it is a method that students will come up with themselves independently. It will require a lot of co-ordinated effort and consistent modelling from teachers all the way from Kindergarten up. I will post some ideas on how we can develop students' efficiency in using bar models in future posts, but in the meantime, it is worth checking out these summaries here and here.

Developing Formulas (3)

A few years ago, I was in a room of 100 or so educators who were asked to draw a picture to represent the Pythagorean Rule. Maybe 5 or 6 people were actually successful. I would say half didn't even know the Pythagorean Rule. Others drew a triangle (sometime right-angled, sometimes not) a bit like this:

But this doesn't illustrate what the Pythagorean Rule is, namely that the sum of the areas of the squares on the two shorter sides are identical to the area of the square on the longer side:

At the time I made the point that if we don't understand this, then we don't truly understand the Pythagorean Rule. I reckon that this is the result of teaching the Pythagorean Rule purely from an algebraic point of view. I was once shocked to see a textbook from the UK that told students to memorise these three formulas for the Pythagorean Rule:

a²+b²=c²

c²-a²=b²

c²-b²=a²

But this is not how the rule was discovered. It was discovered geometrically long before algebra was invented. Here in snowy, frigid Ontario, the curriculum has got it right: in Grade 8 the focus is on a geometric understanding which is followed up in Grade 9 by connecting this to an algebraic understanding. If this is the way it was discovered, then it should be the way we develop it with our students.
So let's not just give students the abstract formulae.
Instead, have them try some decomposition activities like the following (notice how one of the triangles is not right-angled: I always like to offer a counter-example!)

This can be followed by this video which has been doing the rounds on social media (I don't know who first created it so apologies for not giving this person credit):

Now this isn't a proof as such but merely one example. Using a dynamic geometry software such as Geometer's Sketchpad or Geogebra

helps us generalise that it doesn't matter what the original right-angled triangle looks like, the sum of the two smaller squares adds up to the area of the larger square. This is the point where we should bring in algebra to PROVE that it works for all right-angled triangles

But what happens if we draw other shapes on the sides of the triangle such as semi-circles?

It appears to hold true for semi-circles too. Can we prove this algebraically? Spoiler alert:  Yes we can, and it is a lovely wee proof too.
If we get our students to develop a geometric understanding of the Pythagorean Rule, they will be in a better position to use the algebraic representation to tackle problems like the one above or the one below (which is in my Top 10 favourite math problems of all time).