Should all maths teachers in a department teach the same methods?

Revisiting an age-old argument

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A year ago, I wrote a post entitled "How prescriptive should a head of department be?". I argued that department heads should be more prescriptive regarding resources, pedagogy, and methods, and that doing so would improve students' learning.

Last week, I chaired a meeting with the heads of maths from a trust, and the question of methods was raised. Several department heads were planning on dedicating time post-exams to thinking about methods, and I figured that might be true of other readers. So, I thought I would write some more about it.

What do I mean by methods?

By methods, I mean the procedure used to solve a routine problem. Take expanding double brackets. Here, we have three common methods:

1. FOIL

2. Grid method

3. Using the distributive property:

All readers - assuming they are secondary school maths teachers - will have a preference, and many will be prepared to die on the hill of their chosen approach.

The same is true for lots of other methods.

• When teaching adding and subtracting negative numbers, do you use a number line (horizontal or vertical?), double-sided counters, or fire and ice cubes?

• When teaching decimal multiplication, do you teach students to estimate, convert to fractions, make place value adjustments, or count the decimal places?

• When teaching reverse percentages, do you teach proportional reasoning, bar model, set up an equation to solve, use a flow chart, use a ratio table, or use formula triangles?

If you want a sense of just how many different methods are out there, then look no further than Jo Morgan’s incredible A Compendium Of Mathematical Methods. As an example, Jo shares no fewer than 8 methods for finding the highest common factor of two numbers.

But shouldn’t we teach students several methods?

The dream scenario is that our students develop the capability to select the most appropriate (which often means efficient) method for the particular problem they face. But exposing students to too many methods at once, before they have had the opportunity to develop confidence and proficiency with any of them, often leads to confusion. It also makes that initial checking for understanding (I like to do Step-by-Step on mini-whiteboards) so much more difficult if students are doing several different approaches and setting their work out in different ways.

So, in this post, I am talking about the initial method we show students—the one we first get them to practice before we consider exposing them to alternative methods for comparison.

What I see during my school visits

In approximately 90% of the school visits I see each week, students in different classrooms are taught different methods for the same topic.

Would I see the same thing in your school?

Would it be a problem if I did?

The arguments against all teachers teaching the same methods

When I speak to Heads of Department about this, many are not concerned. They tell me:

1. Teachers must use the method they are most secure with so they can explain it better

2. Who is to say that one method is better than another?

3. Students in the top sets need different methods from students in the bottom sets

The arguments for all teachers teaching the same methods

1. If a student moves class or changes teacher at any point in their schoollife, they are essentially screwed unless they can quickly adapt to the new method they are being asked to use.

2. Some methods are more mathematically valid than others. A classic example is using “change side, change sign” to solve equations. I have seen Year 7 students happily whizzing terms from one side of the equals sign to the other, switching their signs with joyous abandon, with absolutely no clue as to why they are doing what they are doing. Of course, their whole world falls apart in Year 8 when the variable appears on both sides of the equation or squared terms come into play.

3. Some methods link better with other concepts. Let’s return to expanding double brackets. An argument made for using the grid method is that students are used to it, as that is how they multiply large integers (although it is worth noting that the grid method is no longer taught at primary school). An argument for using the distributive property for expanding double brackets is that it better prepares students to factorise quadratic expressions.

4. Teachers can collaborate more effectively, particularly when breaking down procedures into Atoms. The method chosen often determines the prerequisite knowledge that needs to be assessed and the new atoms that need to be taught. If everyone uses the same method, this complex task can be shared and discussed.

I believe the arguments for alignment far outweigh the arguments for teachers doing their own thing. I don’t think teacher preference should ever be the determining factor; it should be what is best for the students. Some methods are better than others regarding their validity and how they set students up for future success. Once you take teacher preference off the table, I can’t think of too many examples of methods that are appropriate for one class due to their mathematical achievement level compared to another class.

What doesn’t work

So, if you agree with me that more method alignment across a department is needed, then the next question is: How do we achieve it? I have an answer, but to get there, let’s start with some approaches I have seen fail.

1. Decision by committee

I sat in one departmental meeting, where the plan was to get everyone’s opinion on the best method to teach solving linear equations in Year 7. Twenty minutes later, no decision had been made, and no one was talking to each other. There are few things teachers are more passionate (and stubborn) about than the methods they use. Open forums rarely result in an outcome everyone is happy with, which is undoubtedly true regarding methods.

2. The dictatorship

The opposite of decision by committee is equally as problematic. If the head of department suddenly announces, say, that the grid method for expanding double brackets is banned and from now on everyone has to teach the distributive method, then they are likely to find themselves with a very unhappy group of colleagues.

3. The back-up method

One school I worked with thought they had devised the perfect solution. They had an initial method that every teacher had to start with, and then the option to use an alternate method if they found their students weren’t quite getting it. What could possibly go wrong?

I'll tell you exactly what went wrong because I watched it happen in a lesson. The teacher introduced the method they had been told to use with all the enthusiasm of a a prisoner on death row, pretty much coaxed their students into saying they didn’t get it, and then launched into what she described as “a much better method” with a swell of enthusiasm.

4. PowerPoint support

One head of department wanted a common approach to how his team taught completing the square. Now, this is a relatively tricky topic, so asking teachers to switch methods is much more challenging than for something mathematically simpler. So, to support his colleagues, the head of the department kindly animated the steps needed to do all the worked examples in the shared PowerPoint that teachers used.

I am sure you can predict what happened next. The teachers unfamiliar with this method simply clicked their way through the PowerPoint, missing out on all the benefits of modelling live.

What can work

What follows comes with no guarantees. I find it much easier to convince colleagues to change, say, the way they do their We Do, or to use a certain type of resource, than I do to change methods. Methods are deeply personal to teachers. Perhaps it’s how they learned as a student, and for some, it is how they have taught for many, many years. It is a habit few are willing to give up without good reason.

So, with that in mind, here is what can work.

1. Decide how many concepts you want to prescribe methods for. Some concepts are more important to get alignment with than others because of how they interact with other concepts, so start with those. Solving linear equations is a big one as it has implications for every type of equation students will meet, as well as anything that involves rearranging formulas, such as trigonometry and percentages.

2. Decide in advance on the method you want to use for those topics. Do some research. Look at the methods used by some of the big curricula that are freely available, such as Oak National and LUMEN. Think about the long-term implications of choosing a method. If you have one or two other department members to bounce ideas off, bring them in at this stage.

3. Present the rationale to the rest of the stream. Explain the need for alignment and the research you have done.

4. Dedicate time for teachers to get familiar with the methods. Pair them up with colleagues who use that method to plan together, or ideally see them teach.

5. Tackle just one concept at a time.

It’s not perfect, but it can work.

I am really interested in what you do…

What do you agree with, and what have I missed?

Let me know in the comments below!

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Thanks so much for reading and have a great week!

Craig