Percentage Composition

I wanted to make this lesson a bit more interesting as well as incorporate opportunities for connections with other maths topics.

I knew my year 8 class students loved music so I devised a very simple poll on Rob Thomas (one of the students was going to his concert): like him, don’t like him, don’t know him. This was a good opportunity to revise Frequency Distribution.

Just by fluke, our 3 fractions turned out percentages of recurring decimals (another connection). In that sense, it’s really not a very good first example! Anyway, someone asked if the percentages will total a hundred (I love it when they think!). So, we added and of course it didn’t. We got a result of 99.8%.  This opened up a discussion on the consequences of rounding/truncating decimals.

I made a statement that the most accurate way to represent not whole (fractional) values is via fractions. Nods all around until I asked, “‘Do you believe me?” The consensus was yes because I was the teacher. I was quick to point out that they should not always believe everything teachers say – they have to think if it’s reasonable; teachers are humans, after all, and can make mistakes. Besides, I really wanted to challenge my students to develop their thinking and reasoning.

So, I went back to the Frequency Distribution Table and 3 fractions. Adding the fractions naturally gave a total of 1 and nearly everyone said that’s 100%, as expected. Joy!

Were it a more able class (we stream our Maths classes), I could have pursued more connectionist opportunities but I did sense I’ve pushed enough today. I know I’ll refer back to this lesson when we actually do Data later in the school year. The class was engaged because they knew the meaning behind the numbers.

By the way, the statement is a fact I had truly learned in my previous career in IT. Particularly when I worked for a bank calculating interests, calculations in programs (software rather than curricular) were designed to stick to fractions as much as possible to minimise rounding errors.

In summary (more for my future reference),

Learning strategies: connectionism, motivation and engagement via meaningful and relevant examples, question/reason validity, compare/contrast

Fractions, Decimals and Percentages – provide different ways to represent fractional values, of which fractions are the most accurate

Reflection as part of learning

Our maths faculty head teacher set a ‘soft’ challenge a while ago to have some subjective assessments.

I took this on board and have devised an assessment task that incorporated self-reflection which was another skill I’d like my students to practice; 2 birds in one stone. Reflection is essential in developing sound mathematical thinking, apart from the fact that it is a good life skill itself. At its core, it necessitates looking at something from a different perspective such as looking back, looking deeper, looking from a different angle. From a Jesuit perspective, it also involves Action. Solving mathematical problems often involve all these.

Here’s what I’ve learned so far:

1. Reflections (or learning journals) are just as good for assessing understanding.

What a number is and what it stands for can be dramatically changed just by putting a tiny, little decimal point in front of it. For example, the numeral ‘1’ is a whole number and stands on its own. But as soon as you make it ‘0.1’ it now stands for 1 over 10.

This shows an understanding of the decimal point as the anchor that defines place values.

2. Aim for conceptual understanding; maths is more than just mechanics and shortcuts

I did not really understand the purpose of it and therefore was confused as to why we were doing it. It made me realize that just because you can do something and get the answers right, it does not mean that you don’t struggle with it.

Do I stop to check for conceptual understanding when the students have demonstrated they can do the mechanics?

3. Passion rubs off – case in point, Algebra

Many of my students expressed struggling with Algebra and yet have found it interesting, even fun. Algebra is my favourite maths strand and know that many are put off by it – often from what they’ve heard adults or older students say. With extra effort as well as creativity – and surely joy – many of my students have admitted to struggling with, but finding Algebra interesting; some even said useful and fun! No amount of objective testing will ever show this.

4. Students can be Teachers

I see myself in the reflections of my students. I can see where they’ve struggled and what they’ve enjoyed. I am learning new ways of perceiving topics in a way that is impossible to capture in ordinary class discussions – it is just impossible to hear everyone out every lesson. I can see that some students do appreciate the variety in delivery – modelling, direct instruction, investigation, problem-solving, questioning.

5. There can be some objectivity in the subjective

This sounds more like a philosophical discourse and, a lifetime ago, it was indeed (Philosophy of Man at uni). For now, all I mean is that there can be guidelines to help in marking subjective answers. I used Depth, Breadth and Details.

While I initially dreaded the prospect of marking these assessments, I was surprised to find myself enjoying reading my students’ comments especially the like the ones quoted above. These are just year 7 students. I look forward to Parts B and C of this assessment task.