- Do you have an agreed departmental structure?
- Which method is slowest? Which method is quickest?
- Should we teach more than one method? i.e. On first learning, teach cell 3, but then transfer to cell 2?
- Which method extends most naturally to three or more brackets?
- Should cell 2 and cell 4 come with a discussion of the distributive property, or an intermediate step showing this? i.e. (x + 3)(x – 5) = x(x – 5) + 3(x – 5)?
- Additional thinking:
- Are all of these misconceptions directly addressed through explicit curriculum objectives, or is it luck as to whether they are addressed whilst doing algebra? i.e. Is there a specific lesson or sequence of lessons about the difference between (a+b)/a, and a/(a+b)?
- For each of the prompts, how exactly would you set up a pedagogical sequence to address and teach students the correct manipulations?
- Are there any other common misconceptions you can think of?
Additional common misconceptions:
- Is it helpful for the department to all use the same approach, or can different approaches be utilised depending on whether it is being introduced or consolidated?
- Cell 1: Why don’t we teach partial factorisations with the same importance? In proofs partial factorisations are often required.
- Which methods leads most seamlessly to area scale factors and volume scale factors?
- Is there an argument for method 2 being helpful to practice algebraic manipulation?
- Imagine you had a scenario in which there was a missing length on both similar shapes. Which method might be least confusing in this case?
- Which method connects most effectively to scientific concepts learnt in school regarding scaling things up or down – i.e. Antman is made 100 times smaller in length. How will this affect the strength of his muscles?
*Johnathan Hall’s Site (@StudyMaths): Link for cell 2
This is an obvious concept to me now, but I’m not sure that it was wholly obvious to me when I first started teaching. Finding common units is a necessary element to adding and subtracting all things.
This is a task I gave to a Year 8 mixed attainment class which I think would be a nice pedagogy prompt given the question set below:
- How do you show balance operations between steps? Is this consistent between department members? Should all department members use a consistent approach?
- Do you explicitly reference all variants of linear equations in curriculum documentation? i.e. I can solve a linear equation with unknowns on both sides when one or both unknown terms are negative, or, I can create a linear equation given the solution.
- Do you teach only one method for an equation like this? How do you approach different methods? Is this student led or do you explicitly instruct on a range of methods?
Which approach do you utilise to help students understand the order of operations?
Some interesting blog posts to back up this post:
- Cell 4: Dani Quinn (@danicquinn) – Tried and Tested: GEMS
- Cell 2: Colin Foster (@colinfoster77) – Higher Priorities Article in favour of BIDMSA
- Cell 5: David Butler (@DavidKButlerUofA) The Operation Tower
- Cell 6: Chris McGrane (@ChrisMcGrane84) – Order of Operations Area Model (A task which highlights that BIDMAS is not required).
Some of these are amended from the work of others – but unfortunately I don’t know which ones I made up, which I amended and which I found. It’s likely that one of them comes from Don Stewards blog.