Twitter has been full of intense debate over the past week about the methods that teachers use for different topics. While I’m sure a lot of it is very much old news to a vast majority of teachers, I always enjoy seeing the debate spark up. This post contains my reflections on one of the strands – prime factor decomposition. I’m not going to profess I have it sussed, or that these are unequivocally the best opinions, but I enjoyed thinking about the issue and would like to share those thoughts.
It’s important to keep in mind that, when talking about methods, it can feel very personal – teachers often choose a method largely on the basis of what they feel their students would do best with. I do feel that some methods have more longevity and are easier to generalise with than others, and therefore I prefer them, but at the end of the day only you know your students and context. That said, during a recent livestream maths chat Dani Quinn (who is amazing) asked (paraphrasing!) “why would it make sense that the best way to teach adding fractions changes depending on whether you’re in London or in Edinburgh?” I don’t think she actually said Edinburgh, but you get the gist.
I think I align with that argument to a great deal, although I think there is a lot of nuance in there that could be added. There isn’t really a single method or representation that teachers should be resigned to using it; Mark McCourt talks about instruction needing to be varied in metaphor and that teachers should be able to draw upon different representations depending on the students. The idea that if a student does not understand a method, or it is not ‘clicking’, then a teacher can draw upon their expertise and offer a different lens rather than doing what I’ve been known to do – repeat yourself, slower, and hope that at some point they’ll just give in to your pleading eyes and go ‘yup, got it now sir’. Maybe the idea is there needs to be a selection of battle-tested methods that are embedded into the maths canon, with a few being shown to be demonstrably better for deep mathematical understanding. Maybe that’s robbing teachers of autonomy and creativity. Such a debate is beyond the scope of this post, so I’ll just light the fuse and walk away innocently.
The overarching point here is that, please do not be offended if I am not favourable with your favourite method – these are just my thoughts based on my experiences and your mileage may well vary.
Prime factor decomposition
There are three methods that came to the fore here: the factor tree method, the ‘pure’ method and what I’ve referred to as the ‘ladder’ method. I’ve shown all of them below along with a ‘hybrid’ method that I’ll explain in a minute.
The factor tree method will be familiar to a lot of people. The way it works is you choose any factor pair of a number and write them underneath. If one is a prime number, then you circle it and leave that ‘branch’ as is. If it’s a composite number, then repeat the process of splitting it into a factor pair, circling any primes. Rinse and repeat until you only have prime factors. Your prime factorisation is the product of your prime factors.
The pure method (again, just naming it for convenience) is, inevitably, the same fundamental idea. It’s interesting to read Peter Mattock’s blog where he eloquently talks about the need to draw out the underlying mathematical principles explicitly, and that the method used to get to the maths is somewhat immaterial beyond that. He makes it very clear that students should be able to see that prime factor decomposition is a form of factorisation, rather than a discrete area of mathematics in of itself. It’s a very interesting point, and having students be able to draw links between appearances of factoring in the curriculum is not something I’ve dwelled on deeply before, so it’s worth exploring a bit further.
I absolutely agree with Peter’s argument, and I think it is important that prime factor decomposition is linked to the idea of factorisation and any factorisation that’s already been encountered by students (e.g. factors of numbers and factorising algebraic expressions). This will help long-term retention and also the development of a coherent and cogent understanding of the wider mathematical principle.
Peter is critical of the ‘tree’ method for a number of reasons, but one criticism that comes through quite strongly is that he feels that the method divorces the mathematical principle of prime factor decomposition with other forms of factorisation. We do not use the tree method when looking at other types of factorisation, so the danger is that students cannot see past the method and at the actual maths.
My own gut instinct is that I think it, by and large, depends on the actual teaching. What is the teacher drawing attention to as they go through their examples? What sort of questions being asked? What are the tasks that have been selected? All of these variables may have a larger effect in placing the concept within a student’s schema than the method itself. If one were to strip away the context of the classroom entirely and just provide a textbook (with an appropriately grim title, along the lines of “Mathematics for the youth”) with the algorithm, then I think the argument of the method being overly fussy is likely to be true. When we add in all the other factors of classroom teaching, however, it’s definitely not clear cut.
All that said, I will hold my hands up and say that I have lost count of the number of times that I have asked students to find the prime factor decomposition of a number and they’ve stared at me gormlessly for a little while before going “what, the tree thing?” Yes, the tree thing. So there is empirical evidence from my own practice that students place a high weight on the surface method – although, once they recall the method, they are generally able to do relatively sophisticated tasks involving factors, so it’s still not clear if the method is posing too much of a problem.
“I have lost count of the number of times that I have asked students to find the prime factor decomposition of a number and they’ve stared at me gormlessly for a little while before going “what, the tree thing?”“
The pure method is delightfully no-frills. It almost screams “MATHS” in its absolute single-minded intensity and there are no distracting features for students to get hung up on, potentially reducing the level of extraneous cognitive load. That said, a potential fault is that it could be difficult to keep track of which products have been reduced, and where those products came from. A mistake lower down in the working might be difficult for students to follow back up and spot, and I can see students overlooking factors that are not yet reduced to primes. A nice facet of the tree method is that it does organise thinking in a structured and clear manner, which makes systematic working and checking a lot easier in my opinion.
What about the next method – the ladder method? I’m sure there are a lot of variations of layout and specifics, but the way that I’m thinking of basically involves repeated division by prime factors, starting with the smallest prime factor and going up until you get a result of a prime factor. Conceptually, it is clear, and there is little fuss getting in the way.
I wouldn’t choose to use it personally for a couple of reasons, however. The first is that I wonder if some students will find it difficult to associate the fact that because an integer is divisible by 2 then 2 must be a factor of the integer. Of course, this should be solid understanding by the time students come to this process, but I am sure that I am not alone in understanding that this is not always the case. This is clearly not insurmountable, nor is it necessarily a bad thing to find (it is important that students are comfortable with the relationship), but it does drastically alter the sequence of lessons leading up to the idea. The second is that there is potentially the risk of it being too procedural – again, taking my caveat of needing to understand the delivery of the material into account. I could see students churning through the method without pausing to really understand it. There’s also the question that it may foster the misconception that you have to always divide by the smallest prime first, or that you couldn’t get the prime factorisation by dividing by any factor. Of course, a similar argument could be made for the factor tree method, although I think it is slightly more flexible.
Trying to draw it together
So what’s left? The factor tree method might have students thinking about oak dieback, the pure method might leave students with an incomplete decomposition and the ladder method might be too procedural. That said, all these methods, wielded by someone who has given it very careful thought, will undoubtedly be effective and they all have something to add. What’s my opinion? Get off the fence!
I like the down-the-line approach of the ‘pure’ method the best. I think it’s nice that it never deviates from explicitly referring to the factors of an integer as a product, and so I think I would find it easier to build connections with other areas of maths, or develop understanding for deeper connections (this is one of the reasons I shifted away from the Venn diagram method for HCF/LCM). That said, I vastly prefer the tree diagram method for the clarity and scaffolding it gives students.
My hybrid method isn’t really particularly revolutionary and you may just go it’s the pure method with circles in it, which is exactly what it is. I think it’s important that students record the prime factors as they come upon them, so they are still actively thinking about primes, and recalling that this is the motivation in the first place. It will also help students visually decode potentially long strings of numbers.
Layout is important and I have neglected it somewhat in my example. Perhaps there is an iteration where you take a great deal of space and clearly write the factor pairs underneath each integer in successive lines of working – taking the general idea of the tree method but altering it somewhat.
Of course, this is all theoretical. One could write all day about it, but it’s not worth much until we get the chance to get it in front of a good many classes and see where it falls down and where it works. Nonetheless, it has been fun thinking about such things, in a topic that I had previously assumed to be done and dusted and not really given a second thought. If you’ve enjoyed this post, I would strongly recommend picking up a copy of Jo Morgan’s book A compendium of Mathematical methods, which I have enjoyed reading immensely.