Mind the gap

December 1, 2017
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(This article was originally published at Learn and Teach Mathematics and Statistics, and syndicated at StatsBlogs.)

Teach the students you have

Our job as teachers at any level is to teach the students we have. I embrace this idea from Dr Kevin Maxwell:

“Our job is to teach the students we have.
Not the ones we would like to have.
Not the ones we used to have.
Those we have right now.
All of them.”

I believe Maxwell’s focus was on the diverse learning needs we have in our classes. I would like to take another angle on this. If students do not have the needed skills to learn what we are teaching, then we need to teach those skills.

In many subjects, content and the skills are largely uncoupled. For example in history, a skill might be to integrate material from two conflicting sources. You can learn this in multiple contexts, and you do not need to know the history of the world up until 1939 in order to study World War II.

In mathematics, there are clear progressions. It is very difficult to learn about trigonometry if you do not have a good working knowledge of the Pythagorean theory. And learning Pythagoras is built on applying formulas, which is built on basic algebra. I admit, that as I write this I can see other approaches, but the point is that later learning in maths is built on earlier knowledge, understanding and skills. Learning in maths is also built on earlier feelings – a post for another day.

The Gaps

There are two gaps we need to mind. The gap between levels of schooling, and the gap between what the preparation the students need, and what they have. I taught at the University of Canterbury for twenty years, and often heard colleagues complain about the level of preparation in our students. I am ashamed to say that it took me several years to realise that if our students do not have the foundation they need to learn what we are teaching, then we need to do something about it. As a result I created a course that started with making sure students knew how to use < and >, and which is bigger out of 0.04 and  0.2. These are necessary in order to make decisions about rejecting a null hypothesis.

Recently at a workshop I asked a group of about forty teachers how many of them have students starting high school who do not have the necessary knowledge of number skills – basic facts and multiplication tables. Every hand went up. There is a gap. I asked them what they are doing about it. Some suggested working in “Communities of Learning” to help primary schools to prepare the students better. This is fine, but what are they doing now! There was some discussion that if we are teaching lower curriculum levels at high school, they may never cover the materials at higher levels.

For that I have two responses. The first relates to the Maxwell quote I started with. “Our job is to teach the students we have.” Our job is to teach the students we have, the things they need to learn. If our students start high school without a good enough grasp of basic facts, then we need to help them to develop them. And we need to work out good ways to do this. I suspect part of the problem is that secondary maths teachers do not have training or knowledge in teacher beginning maths. Do we believe it is not part of our job?

The second response is that there is no point in moving on to later maths if the students’ foundation is weak. Now I say this with some trepidation as I can picture students being held back until they become fluent in their tables. This is not what I mean. One of the participants in the workshop asked me how I would go about setting up a programme to help such students. Obviously this is not a question I could answer on the spot, but here are some ideas and principles.

Ideas and principles for building foundation skills

Summary:

  • Do not under any circumstances give these students tests with time pressure.
  • Expect the students to be able to learn what is needed more quickly than they would have done when younger.
  • Engage students in deciding what they need to learn and how.
  • Integrate the skills into other activities
  • Make it fun

Explained:

No time pressure

Read Fluency without fear by Jo Boaler. Read this about Maths trauma. Do not add to the students’ feelings of inadequacy. One possibility if you wish to give a diagnostic test, and want to have some idea of how long they take, tell them they have as long as they need, but after a certain amount of time get them to change to a different coloured pen.

Learn quickly

My experience with teaching adults and teens is that once they realise they can learn, they learn quickly. Believe it. I don’t mean that they can answer questions quickly, but that they will be able to progress more quickly as they have better metacognitive skills, literacy, maturity.

Student agency

This is their learning. Make sure the students know why they need the skills and how they will help them. Talk to students about how they would like to learn them, and let them choose their own reward system if appropriate. Different students will have different areas of weakness, and different ways to improve.

Integrate the skills into other activities

I can’t imagine much worse than an entire maths lesson on basic facts. If we are working on multiplication, this fits well with area calculations. We also need to keep revisiting.

There is a place for well made and used flash cards to improve retrieval. There are multiple posts on using flashcards well. I would recommend them for some students for the last sticky facts, like 6 x 7, 6 x 8, 7 x 8 etc. Those were the ones that got me stumped. However, most knowledge is better gained in context. Create or find rich, open-ended tasks that help develop the skills the students need.

Make it fun

Maths lends itself to games and fun. If you can’t think up a way – find it on line. But if you don’t think it’s fun the students aren’t going to. (Not sure the converse is true, but…)

Mind the Gap

Our aim at Statistics Learning Centre is a world of mathematicians. My dream is for math trauma to be a thing of the past, and for all citizens to embrace mathematical thinking similarly to literacy. As maths and statistics educators we can work towards this. The most important student you have in your maths class is the one who becomes a primary school maths teacher. Make sure she loves maths!




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