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Let’s encourage student spread in mathematics units

Part 3 in a 3-part series
Richard Andrew
"Why on Pythagoras' Earth would we want to encourage student spread in mathematics lessons?
​Surely the aim of the game is to keep them all together?"
This is the final article in the 3-part series on the relationship between student-spread and a conceptually-based, student-centred approach to teaching and learning mathematics.

The first article, Conceptual approaches to teaching and learning mathematics need to be student-centred, proposes that a conceptual approach should, ideally, also be student-centred; that increased student spread is an inevitable byproduct of student-centred learning and that student spread poses a big problem to many high school teachers.

The second article, Catering for student spread within mathematics education, outlines three effective strategies which enable teachers to embrace student spread within a student-centred approach.

If your reaction to the title of this article was in line with the quote above ("Why on Pythagoras' Earth would we want to encourage student spread ... ") then this article will hopefully help. I’ll be arguing the case in favour of encouraging student spread, rather than seeing student spread as an inevitable, problematic byproduct of a student-centred approach. 

A conceptually-based, student-centred approach fosters engagement

To quote the first article – Many factors contribute to engagement. Some of the important ones are student ownership over learning, choice, inquiry, freedom to progress at one’s own pace and collaboration with fellow students to ‘work stuff out’. There are others, but the factors listed above are arguably the most critical when it comes to math education. These four factors are the domain of an effective, conceptually-based, student-centred approach to teaching and learning.

Engagement is key

The point I would like to expand upon is this – engagement is key to any form of authentic learning. If students are not engaged in their learning then, at best, any approach used to teach them anything will be significantly less effective compared to the the same approach applied to engaged students.  Engagement, in my view, should always precede 'curriculum' because, without engagement, teaching the curriculum is essentially a waste of time. The first task for any teacher with a new class should be to engage the students. Of course, rather than spending the first month playing games and telling jokes (not advised!) the most efficient approach to engaging students is to use inherently engaging, conceptually-based instructional methodologies which serve to establish an engaged learning environment.

It all comes down to learning

Humans have a way of learning. Yes, there may be different learning styles (?!?) but at our core, we have some common aspects to the way we learn. There are some important principles to the way I learn, to the way you learn, to the way any functional adult human learns. And I propose that these common principles of learning apply to all humans – toddlers, pre-school children, school students, young adults, adults and elders.

The common principles of human learning

I could leave a blank space here and suggest you write this paragraph because we all know what the common principles are.

What are the necessary ingredients we need in order to learn something new? What do we ideally need in order to learn how to cook a new meal, to paint with oils, to build a cupboard, to use a sewing machine, to become a better parent or to become an exceptional teacher?

In regard to learning anything, we all - I assume - understand that the following are key:
  • We need a reason to learn the “thing” in the first place. If we don’t have a reason to learn it, then how can we possibly be engaged? Our reason may be intrinsic (I want to learn how to play the guitar), it may be somewhat externally driven (I’d better learn to master GeoGebra so that I am better equipped at bringing understanding to my maths students; I'd better get my head around blogging so I can incorporate it with my humanities students), or it may be mostly external (I've enrolled in a course but there's one aspect that bores me. I'd better learn it anyway because I need to complete the course).
  • We need to own the learning. If this is not intuitively obvious to you then ask yourself this – when was the last time you learned something in which you were told exactly when and how to learn it? When were you last in a situation in which someone was telling you information you didn’t want to learn? When was the last time someone was trying to teach you something in which you had no ownership or sense of control over the learning? My guess is the last time for any of us was in an education institution. But today, in mostly informal contexts, we own our learning, I certainly do. Why shouldn't students have the same privilege?
  • We need to be free to learn in the way we learn best. Again let me ask – when was the last time there was an attempt made to have you learn something that was being delivered using a learning mode which does not suit you? Most will probably respond with a memory of some kind of poorly-delivered lecture format. But - today - when we need to learn something what do we do? Without even thinking about it we immediately go about the learning in the way that suits us, whether it be reading, whether it be computer-based, whether it be via a video. Many of us go straight to YouTube! We may take notes or we may simply practice over and over. We may work for 10 minutes at a time, or for hours in one sitting. We may sit or we may stand. Whatever it is, we almost always choose our own learning mode.
  • We need to be able to progress at a pace that we choose. Further to the above (choosing a preferred learning mode) we need to be able to choose the pace at which the learning occurs. We know when our brain has had enough, or in the case of the middle-aged guy learning to operate the new power tool, when the body has had enough (!) And if there’s a looming deadline, we set our own timeline. Or we choose to ‘wing it’, knowing there will be consequences at the final hour. But we choose. Always, we choose.

Choice drives engagement

An underlying factor in the above four common principles of learning – having a reason to learn, owning the learning, choosing our preferred mode of learning and choosing to progress at our own pace – is CHOICE. Choice is a major key to engagement in learning. That choice is a major key to engagement is a principle, which, I believe, is lost on many teachers because until we have witnessed this in action we are blind to it. 

The conclusion

Here it is folks ... we now have the argument:
Student spread results from a student-centred approach ...
... A student-centred approach breeds choice ...  
​... Choice drives engagement.
​
CONCLUSION: Encourage student spread!

OK, so this is really about embracing a student-centred approach!

Yes, OK, I'm being a little provocative here. The title could have been 'Embrace student-centred-ness' rather than 'Bring on student-spread'. But the fact is that articles on student-centred approaches are common yet I've never seen an article - other than mine - that addresses the issue of student spread in a mathematics context. And because most maths teachers avoid student spread by default, I chose the 'Encourage student spread' title.

So we could wind the article up here, but before we do let's address some ways of dealing with student spread. The tips below are additional to those covered in 'Catering for student spread in mathematics classes'.

Dealing with students who are ahead of the pack.

We need to avoid the race in which some math speedsters like to engage. Any student who treats maths lessons as a race to the finish line needs to be coaxed out of this mindset immediately. Talking it through with the student, setting in place some checks and balances, and generally being creative when solving this issue should do the trick. As for students who are naturally fast workers, but not engaged in a race, simply keeping an eye on their work, and ensuring they are being thorough – not rushing – will be critical checks to have in place.

Tips for dealing with the tail-enders

The main issues with the 'tail' is to ensure two things:
  • Firstly, that the slower students are working at an acceptable, authentic pace. You will need to manage this one as the need requires.
  • The second is to ensure the ‘slower’ students are feeling OK with being 'the tail', that they are not comparing themselves to those at the front. Mostly, this requires some regular dialogue, for example, by regularly asking rhetorical questions to the whole class: “Do we all work at the same pace?” … “Should we expect everyone to be up to the same point in the unit?” … etc., which lead to a re-agreement that it is normal for 'us all' to be spread out within the unit. This needs to be repeated regularly. And then to immediately pick up on any condescending attitude directed towards anyone in the tail. In addition, it is vital to give genuine and regular praise to the tail enders, especially when they work at their authentic pace, and to reiterating that its OK to be where they are at.​

But what about the early finishers?

This is a common source of concern for maths teachers new to dealing with student spread. They always ask the question "What do I do with those who finish early?" Of course there will be some students finishing several lessons earlier than others. But it’s simply a problem to solve. You will need some extension work planned ahead of time and on hand, available when required. Below are some ideas:
  • An engaging, quality problem solving series, well organised, so that students can continue to work from where they left off at the end of successive units of work. Or there needs to be a system in place allowing early finishers to easily commence a new task. Again, these need to be engaging tasks!
  • Have students make up a ‘chunky’, worded question, of a specified type and level (one to two levels easier than they can readily solve) where the end result is a neat, photocopy-able version of the question as well as a neat, fully-worked, photocopy-able solution. There is an incredible amount of mathematics involved in making up such a question. You could also ask them to write up (or create a video of) the process they went through to reach their final product. This activity is one of the highest quality and easy-to-run extension activities I have consistently used and with great success. It's a 1-3 lesson activity and the student-produced questions can be incorporated in the teaching unit for other students when they reach the relevant points in the unit.
  • Early finishers can also be used as peer teachers.  There is much research to back the fact that people learn something better when they teach it to someone else.
  • There’s three quality ideas. Please feel free to post any others you use below. 

In conclusion (again)…

  • We want our students engaged.
  • Well-scaffolded, student-centred, conceptually-based units of work are infused with choice.
  • Choice breeds engagement.
  • Student-centred units cause an increase in student spread.
  • So let's encourage student spread!​

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Topics:   Student engagement    Student-centred learning    Blended-flipped learning    Concept-based mathematics teaching    GeoGebra