While I understand and appreciate the value of consistency across a department, I respectfully hold a different view. I believe that the choice of method is an integral part of a teacher’s personal methodology, and experienced teachers in particular should be trusted to select the methods they feel work best for their students. Of course, this doesn’t mean teachers should be resistant to trying different approaches; on the contrary, having more than one method at one’s disposal (within reason) is essential to adapt to the diverse ways students think and learn.
The concern that students may struggle when moving between classes due to different methods being used is, in my opinion, not a strong enough reason to enforce uniformity. A good teacher should be able to explain a concept in various ways and help students make connections, regardless of their previous experiences. Personally, I teach most topics using two different methods, and if a student comes with their own approach, I assess whether it’s effective and whether they’re confident with it. If so, I encourage them to continue using it. This promotes not only understanding but also flexibility and open-mindedness, skills that are invaluable beyond the classroom.
I am reading this and thinking about how boring would my teaching life be, if I was to use prescriptive methods. Also can you undo, I am guessing here, 30 years of teachers being told that students need to use different methods to understand?
We have a departmental policy to model and teach using a consistent method. We collaborate to try out different ideas and reach a shared agreement, ensuring that students receive a consistent learning experience, regardless of set changes.
This is a very thought-provoking article, thanks Craig.
There should definitely be methods that link to other or further concepts.
Multiplying brackets:
FOIL is a good starting point, but needs to lead onto "take the first term of the first bracket and multiply it by every term in the second bracket, then take the second term of the first bracket...". Within one lesson, I have taught expanding brackets and had students multiplying brackets with 3, 4, or more terms.
The grid method works well, as it essentially does this (probably a bit of explanation needed). But this would be good to lead onto lining up the "rows" of the grid so that the like terms are underneath each other. Then the addition become incredibly simple.
With a few small tweaks, I have taught addition, subtraction, multiplication and division of polynomials all using primary school methods that students learned with numbers.
Now, I also think that there should be professional development time devoted to teachers getting to grips with their subject better. If a teacher is faced with a method they have not seen before, they should very easily be able to assess whether the method is mathematically valid, or just a random fluke for the particular question used to demonstrate the method. After a few years, there should be very few methods that teachers have not seen... unfortunately, due to how little time is allocated for teachers researching their subjects, this is idealistic and does not happen.
But here's an idea for making a change:
At the next department meeting, the head of department could have a selection of topics that have multiple methods written on pieces of paper. Each teacher picks one, and researches the different methods for those topics. They each get 10 minutes to explain/demonstrate each of the methods they find, and explain which one is the most beneficial due to how it links to other areas of maths. That way, all teachers in the department have seen multiple methods (in case students bring it up), and they now have a clear reason as to why to teach a topic with a certain method. It also gives the individual teacher a sense of ownership over the chosen method, a feeling of being able to influence a student's education (as opposed to just being a cookie-cut teacher).
Now, I am also a firm believer that as a student gets older, they should have their education tailored to they chosen pathway in life. For example, not all students will need to be able to multiply numbers without a calculator. So why do we enforce it? Because the government says so? Well, maybe something needs to change there. Obviously, if a student has chosen to take A Level Maths, then they need to be taught those topics. I'm talking about students who want to take up a trade, or do something with very little maths.
"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."
More often then not those these few examples are quite foundational methods for Foundation students which can be the difference of a Grade 3 or 4.
For example, multiplying two 2/3 digit numbers. Most foundation students insist on using column method, despite the fact they struggle to keep track of carries after the first line of working. However, this 'one topic' bleeds into so many other topics which are vital for those students who struggle with maths to achieve a grade 4.
The way we align methods at our school is by using booklets instead of a textbook. This is initially very time-consuming for the person compiling the booklets but it is a game changer. All teachers have to teach the methods in the booklets and all learners have access to the same examples and methods. So changing classes is not problematic. If a teacher feel strongly about a certain method, then we have a meeting where we discuss the new method proposed and if the majority is an agreement, then the booklet is edited and the new method put in. There is only one topic that I've ever had to change- and that is factorising quadratic trinomials. And with that topic we changed more than once. But there has never been a situation where a teacher refuses to teach the method in the booklet. They have all seen the benefits of teaching the same method and as you say, Craig, it far outweighs Teacher autonomy on that aspect.
The grid method is my favourite method for expanding brackets and factorisation, leading nicely into dividing polynomials later on. Lovely forward facing methods the whole department can get behind... :)
I have brought in area model as the consistent multiplication (grid), however, rather than being a one method approach it is a string of an approach to align multiplication across the curriculum.
Sometimes the single method in isolation approach doesn’t work. So considering how you want ideas to be foreshadowed numerically so that students are more comfortable numerically I think is importantly.
We do use PowerPoints. But as default these are blank questions there are then hidden slides with annotated solutions. Teachers have to make a teaching choice to use the animation rather than the blank model.
These are also annotated to replay, so students still see the ink being drawn the way that the teacher drew it initially. Anecdotally these feels like a happier medium. It also means that even if teachers aren’t using these they still have a model example for what the questions will look like.
The back up idea comes into play December of year 11. If by that stage an idea isn’t work show em select students what they need to see to have something to use.
While I understand and appreciate the value of consistency across a department, I respectfully hold a different view. I believe that the choice of method is an integral part of a teacher’s personal methodology, and experienced teachers in particular should be trusted to select the methods they feel work best for their students. Of course, this doesn’t mean teachers should be resistant to trying different approaches; on the contrary, having more than one method at one’s disposal (within reason) is essential to adapt to the diverse ways students think and learn.
The concern that students may struggle when moving between classes due to different methods being used is, in my opinion, not a strong enough reason to enforce uniformity. A good teacher should be able to explain a concept in various ways and help students make connections, regardless of their previous experiences. Personally, I teach most topics using two different methods, and if a student comes with their own approach, I assess whether it’s effective and whether they’re confident with it. If so, I encourage them to continue using it. This promotes not only understanding but also flexibility and open-mindedness, skills that are invaluable beyond the classroom.
I am reading this and thinking about how boring would my teaching life be, if I was to use prescriptive methods. Also can you undo, I am guessing here, 30 years of teachers being told that students need to use different methods to understand?
We have a departmental policy to model and teach using a consistent method. We collaborate to try out different ideas and reach a shared agreement, ensuring that students receive a consistent learning experience, regardless of set changes.
Use the methods taught by Big Curricula as a default? Are we even sure those methods are research validated? My skepticism says no
This is a very thought-provoking article, thanks Craig.
There should definitely be methods that link to other or further concepts.
Multiplying brackets:
FOIL is a good starting point, but needs to lead onto "take the first term of the first bracket and multiply it by every term in the second bracket, then take the second term of the first bracket...". Within one lesson, I have taught expanding brackets and had students multiplying brackets with 3, 4, or more terms.
The grid method works well, as it essentially does this (probably a bit of explanation needed). But this would be good to lead onto lining up the "rows" of the grid so that the like terms are underneath each other. Then the addition become incredibly simple.
With a few small tweaks, I have taught addition, subtraction, multiplication and division of polynomials all using primary school methods that students learned with numbers.
Now, I also think that there should be professional development time devoted to teachers getting to grips with their subject better. If a teacher is faced with a method they have not seen before, they should very easily be able to assess whether the method is mathematically valid, or just a random fluke for the particular question used to demonstrate the method. After a few years, there should be very few methods that teachers have not seen... unfortunately, due to how little time is allocated for teachers researching their subjects, this is idealistic and does not happen.
But here's an idea for making a change:
At the next department meeting, the head of department could have a selection of topics that have multiple methods written on pieces of paper. Each teacher picks one, and researches the different methods for those topics. They each get 10 minutes to explain/demonstrate each of the methods they find, and explain which one is the most beneficial due to how it links to other areas of maths. That way, all teachers in the department have seen multiple methods (in case students bring it up), and they now have a clear reason as to why to teach a topic with a certain method. It also gives the individual teacher a sense of ownership over the chosen method, a feeling of being able to influence a student's education (as opposed to just being a cookie-cut teacher).
Now, I am also a firm believer that as a student gets older, they should have their education tailored to they chosen pathway in life. For example, not all students will need to be able to multiply numbers without a calculator. So why do we enforce it? Because the government says so? Well, maybe something needs to change there. Obviously, if a student has chosen to take A Level Maths, then they need to be taught those topics. I'm talking about students who want to take up a trade, or do something with very little maths.
It's incorrect to say that grid method is not taught in primary schools any more. Many of our primary schools use it.
"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."
More often then not those these few examples are quite foundational methods for Foundation students which can be the difference of a Grade 3 or 4.
For example, multiplying two 2/3 digit numbers. Most foundation students insist on using column method, despite the fact they struggle to keep track of carries after the first line of working. However, this 'one topic' bleeds into so many other topics which are vital for those students who struggle with maths to achieve a grade 4.
The way we align methods at our school is by using booklets instead of a textbook. This is initially very time-consuming for the person compiling the booklets but it is a game changer. All teachers have to teach the methods in the booklets and all learners have access to the same examples and methods. So changing classes is not problematic. If a teacher feel strongly about a certain method, then we have a meeting where we discuss the new method proposed and if the majority is an agreement, then the booklet is edited and the new method put in. There is only one topic that I've ever had to change- and that is factorising quadratic trinomials. And with that topic we changed more than once. But there has never been a situation where a teacher refuses to teach the method in the booklet. They have all seen the benefits of teaching the same method and as you say, Craig, it far outweighs Teacher autonomy on that aspect.
Why is the grid method not as effective when factorising quadratic equations?
The grid method is my favourite method for expanding brackets and factorisation, leading nicely into dividing polynomials later on. Lovely forward facing methods the whole department can get behind... :)
Love it Craig!
Would love your advice:
What would be the top 10 concepts that you most recommend a Maths team to cohere on in teaching a clear agreed-on method?
I have brought in area model as the consistent multiplication (grid), however, rather than being a one method approach it is a string of an approach to align multiplication across the curriculum.
Sometimes the single method in isolation approach doesn’t work. So considering how you want ideas to be foreshadowed numerically so that students are more comfortable numerically I think is importantly.
We do use PowerPoints. But as default these are blank questions there are then hidden slides with annotated solutions. Teachers have to make a teaching choice to use the animation rather than the blank model.
These are also annotated to replay, so students still see the ink being drawn the way that the teacher drew it initially. Anecdotally these feels like a happier medium. It also means that even if teachers aren’t using these they still have a model example for what the questions will look like.
The back up idea comes into play December of year 11. If by that stage an idea isn’t work show em select students what they need to see to have something to use.