Why Too Much Compartmentalising Is A Bad Idea When Teaching Mathematics
Richard Andrew
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Most skills are best taught compartmentalised - i.e., in progressive parts.
For example, an effective way to teach someone to throw a ball - for example, an outfield throw in baseball or cricket - is to have the person:
The point is, a learner's confidence is built when the skill is broken into easily achievable parts. You wouldn’t teach a complex skill - a tennis serve as another example - by expecting the learner to execute the entire skill from scratch.
A similar principle applies when teaching students to write stories. There is no point asking a child to write a ten-page story if they haven’t progressed from writing words to sentences to paragraphs and then to one-to-two-page stories. And, of course, along the way, learning the grammar.
So what about school mathematics?
School mathematics can be divided into hundreds of components that build upon each other. Therefore, it is logical to assume that if we want students to experience success, we must compartmentalise mathematics when teaching it.?
And clearly, compartmentalising has been integral to mathematics education since we started teaching the subject en masse hundreds of years ago.
Compartmentalised maths teaching in action ...
I’m sure we all understand what compartmentalised mathematics teaching looks like. My twelve year’s worth of mathematics lessons supplied me with 100% compartmentalised mathematics!?
But to help drive the point home, consider a fractions, decimals and percentages example ...?
And on and on … and on. (Whew!)
I suggest that a similar level of compartmentalising occurs in most maths classrooms for most topics.
In the article, Too many students find fractions, decimals and percentages confusing - you’ll find two short videos where I demonstrate what compartmentalised fractions-decimals-percentages look like in action, followed by some short video examples of teaching fraction division with a focus on understanding.
Over-compartmentalising in mathematics lessons
As we have established, compartmentalising is ideal for learning physical skills. However, the level of compartmentalising described above makes learning mathematics more difficult, not less.?
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The problem with compartmentalising
There is a significant downside to the compartmentalised learning of mathematics. It turns out that learning trigonometry, straight-line graphing, fractions-decimals-percentages?- and indeed most mathematics topics - via a compartmentalised approach is MUCH MORE difficult for students than when learning via a more unified, less compartmentalised approach.?
Compartmentalising has its place …?
There is a place for compartmentalising. Students need to explore basic trig before tackling complicated, real-world, trig-based problems. They need to deal with gradients and ordered pairs before determining the equations of more difficult straight lines.?
However, mathematics teaching tends to be over-compartmentalised. Heck, I way-over-compartmentalised my teaching for years until I realised that compartmentalising was largely responsible for the disengagement that permeated my students.?
WHY over-compartmentalising makes maths more challenging to learn …
It is difficult to convincingly explain WHY compartmentalising makes learning mathematics harder, not easier - especially to sceptics.
Ironically, once we EXPERIENCE presenting mathematics in a more unified, less compartmentalised manner, we don’t need the explanation. We see improved engagement and understanding in the students, and the effect of reducing the level of compartmentalising becomes obvious.
When we analyse highly compartmentalised maths teaching, it is obvious that it tends to be heavily focused on teaching procedures - in the hope that understanding will follow. (We refer to this as Procedures-1st, Understanding-2nd teaching.)
Conversely, If we were to analyse a less compartmentalised, more unified approach, we would see a much stronger focus on having students understand the related concepts of the unit as a priority with a strong but secondary focus on procedures. (Understanding-1st, Procedures-2nd teaching.)
A highly compartmentalised approach is very step-by-step, requiring students to replicate examples (low cognitive load) and tends to foster passive, non-thinking learners.
On the other hand, a less compartmentalised, more unified approach has students thinking and collaborating and gives students a sense of control over activities. This translates to deeper understanding, engagement and agency in students.
To watch a teacher explain her success when implementing a non-compartmentalised, straight-line graphs unit with the ‘bottom’ year nine class (a highly-structured, student-centred. conceptually-based approach) watch from the 25-minute mark to 27-minute mark?of this video.?
Your turn
What about you? Do you suspect you compartmentalise too much? Do you strive to present mathematics in a more unified, less compartmentalized way??Have you experienced using less compartmentalising for a topic and seen improved engagement and understanding? Are you a leader wanting your teachers to compartmentalise less?
Whatever your thoughts, I'd love to hear them.
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2 年Fantastic read, Richard! On this topic, it actually makes a solid argument. When several related topics are discussed, over-compartmentalizing can occasionally result in problems and misunderstandings. It is preferable to teach kids the relevant ideas first to ensure that they grasp them.
Founder | CEO | Teacher | EdTech Specialist | Learner | Lover of Mathematics
2 年Agree on many points here Richard Andrew! Adding a little to your 'why'... Math(s) is beautiful/engaging/relevant when it makes sense and it has purpose. In order for it to make sense and have purpose, we need to see how it is connected the other concepts around it. Compartmentalising completely limits these connections. Moreover, it makes it feel isolated, abstract and arbitrary... not to mention boring! Part of the problem is that we often measure success on how well students can answer compartmentalised questions that require rote learned methods. A 'successful' student is too often one that can repeat a learned method shortly after being demonstrated it. If we change our definition of a 'successful' student to be one who thinks deeply, who is creative, who asks good questions, who takes risks, who reflects, who communicates well, who can patiently solve real problems (the list goes on), then we will be forced to teach in a more connected and less compartmentalised way.
CEO at Linked VA
2 年Excellent read, Richard! It actually has a good point on this topic. Over-compartmentalising can sometimes cause complications and confusion when one specific topic is just related. Better teach students first to establish their understanding of the related concepts.
CEO at The Expert Project
2 年I agree, Richard. Over-compartmentalising makes mathematics more difficult to learn because students can't focus on one method. It is best to understand the basics first before moving on to the more complicated methods.
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2 年Great article, Richard! I totally agree that teachers should teach from the basics before going to the complex to prevent students from being overwhelmed.