Differentiation

The first thing to note is that Maths is usually taught in sets. However, that does not mean that we do not have to differentiate. Even in a set class, there is a (sometimes large) range of abilities. That being said, we are not dealing with the top of the year student with the lowest achievers in the same group. This also means that a large part of the differentiation is achieved with a whole class by the teacher. You would not teach the same lesson to a bottom set and a top set. This includes both the presentation of new material and the exercises used to practice.

Most differentiation within class takes one of two forms. Either by providing more structure or more time. By providing more structure I mean breaking concepts down into smaller chunks, or providing scaffolds where students start by filling in gaps, before progressing to doing it completely independently. In my opinion it is important that all students are moved on to this point though, as we want them to be able to do it independently eventually. Scaffolding is a temporary measure.

Differentiation by time involves allowing all students the time to complete the task, and this means having something more challenging to push those who finish (successfully) earlier. For me this will usually be linked to the topic, though in the past I did also use mathematical puzzles to develop mathematical thinking.

One way to achieve this is to use activities that are "low floor, high ceiling", which all students can access and succeed with quickly, but that have depth allowing them to dig in.

For example, a simple question: Half the pages in a book have a page number that starts with the digit 1. How many pages could the book have? This is easy to access, and almost everybody can find an answer very quickly. But then more possibilities open up (can you find the next answer after 2 pages?). With a simple twist, you can change the task. What if we were interested in the pages starting with 2? Or 3? What about the last digit?

I would be reluctant to differentiate by instruction. Even those who know more going in to a lesson will gain something from a good demonstration of a worked example, even if it is just clarifying their own thoughts. The risk of misconceptions being developed by not instructing clearly is high, and I work hard on creating clear and concise explanations that all students can access.

Another common approach I am suspicious of is getting the high achievers to teach the low achievers. This is not beneficial to the low achievers (an explanation from a student can not be as good as one by a teacher, as the teacher will always have more expertise) and some argue it is not beneficial for the high achiever either (they would be better served being challenged by something that really makes them think, rather than repeating the same boring thing).

A common strategy used in maths classrooms (indeed I have used it in the past) is to have multiple tasks of varying level, and allowing students to move on when they believe they have mastered one. My problem with this way of differentiating is that students are not always effective at assessing how they are doing on something new to them. I do still sometimes use this strategy, but I would only allow students to move on if they have demonstrated to me that they have a good understanding. This does not address the issue of those students who are getting it right, but not confident enough to try something more challenging.

For me the most important thing is to have high expectations of all students. Help them develop their confidence by making them successful, but then challenge them to think hard.