Genuine Engagement and Genuine Learning: never the twain shall meet?

As I have stated previously, I loathe – with alarming fierceness – gimmicky lessons. I am very interested in developing lesson ideas that inspire and engage pupils and make use of their genuine desire to learn and to better understand the world. However, this often means working hard to avoid suggestions of ‘tricks’ that make pupils appear busy and engaged, but where little (or no) underlying learning is taking place; I’d much rather teach a ‘boring’ lesson (i.e. no bells or whistles, just teach, question, practice, check) than waste their time on a gimmick. I know I’m not alone in having this particular phobia; happily, I’m also not alone in trying to develop and share genuine ‘hooks’ which lead to meaningful, challenging and (hopefully) memorable learning. The ‘inspiration’ behind the lesson I’m about to describe came almost entirely from other people who generously shared their practice on blogs and twitter. I’ve lost track of one, but will endeavor to give credit where it’s due.

The focus of the lesson was spheres, particularly calculating radius from circumference, calculating surface area and volume, and analysing the linear and non-linear relationships between a spheres’ radii and other measures (including interpreting graphs of rates of change). It was a double lesson; hence the extended nature of some tasks.

We started with a question on the board: “The world’s population recently reached an all-time high of 7billion people. How much space is there per person?” (inspired by Don Steward, although we used a shorter version). I’m glad that the question was so ‘sparse’, as pupils had to identify what information they wanted/needed (e.g. they had to give some thought to whether they needed to calculate the earth’s surface area or its volume, and what sort of measurements are easily available). It quickly emerged – as predicted – that they needed to know how to calculate the surface area of a sphere.

This led to the first main activity: groups of 3-4 pupils were given a satsuma and a tape measure (the IKEA ones… surreptitiously taken!). They were instructed to draw several circles with the same radius as the satsuma, then see how many circles could be filled with the peel. I neglected to photograph this part, but a similar example can be found here.


Although completing the task in this way took a lot longer than simply giving the formula (i.e. 10 minutes instead of 3), there was a lot of ownership over the resulting conclusions and pupils were asking their own questions to extend their thinking (e.g. what is the total surface area of a hemisphere? What about the volume?). It had the benefit of being ‘naturally’ differentiated, in that weaker groups had plenty of time to think about and test methods to find the radius (a critical part of understanding and manipulating circle formulae), whilst other groups quickly moved onto generalising.

Finishing off that part of the lesson, pupils were told the length of the equator (as they’d asked for this to help them find the surface area of the earth) and the proportion of the earth that is ocean (as most wanted to know the land available, not just the space) and then set to work on planning their calculations.

As at the start, they seemed to be highly engaged by this question; it was exciting to see how keen they were to solve it, and the sort of questions they were asking of themselves as they worked (e.g. making decisions about appropriate levels of accuracy, asking themselves if their answers seemed sensible, deciding if using standard form made answers more or less comprehensible). Two boys demonstrated their working and conclusions on the board, giving a good opportunity to look at how their work – which was of a very good standard – could have been simplified and made more manageable and elegant (e.g. by using multipliers more effectively and using significant figures instead of decimal places to round more accurately).

The main part of the lesson was inspired by Sameer at samjshah’s efforts to make related rates interesting and memorable for a high school calculus class. They were shown 5 balloons, each with increasing amounts of ‘puffs’ in them (this obviously got lots of inane giggles…such is the world of 15-year-old boys) and asked to list the changing measurements (I erased ‘mass’ and ‘density’ as we didn’t have any equipment for measuring weight, which is regrettable).


I then set out what I expected them to produce (i.e. what to do – a table of results, graphs plotting the dependent variables against the radius, written analysis) and what learning they should be gaining from the activities (i.e. if that learning was escaping, they should be asking for help asap). We also looked at an example of what a ‘good graph’ and good piece of written analysis should look like (as their written commentary on mathematical work is a definite weakness in the class, despite typically producing super-clear calculations and a lot of strong oral explanation and demonstrations at the board). Here is some work mid-way through the lesson:



Some really interesting hypotheses were chosen; star-boy (above) chose an amazingly challenging one with his partner: “the ratio of volume to surface area decreases as the radius increases” (apparently he’d read something about why cats don’t break their legs when they fall, because they have a low surface area to volume ratio…..???? I was baffled, but loved what it led to).

I wish I’d photographed their final pieces and the presentation at the end, as some were really impressive, or represented real progress towards being able to given concise, technical explanations (not yet mastered, I’m afraid). All groups managed to calculate surface area and volume for 5 balloons (a grade A objective), all bar one produced at least one graph and decided if it represented a linear or non-linear relationship, and three groups got as far as deciding that surface area-radius relationship was quadratic, and explaining why the graph only showed ‘half a quadratic’ (they completed a similar analysis for the cubic relationship with volume).

Unfortunately I won’t see them again until next Thursday to see how much has been retained (an important test for any task…!), but the feedback from pupils was overwhelmingly positive and the level of effort, application and motivation during the task was some of the best I’ve ever seen.


Filed under Intrinsic motivation, Perceptions of Maths

7 responses to “Genuine Engagement and Genuine Learning: never the twain shall meet?

  1. I really enjoyed reading about your lesson, and now can’t wait until I get onto this topic! Out of interest, how long did the whole sequence of activities take? It seems more than an hour’s work including the orange peel activity too… And did they present their work to the rest of the class or just write it up for you to see? Thanks for sharing your ideas and pictures, am feeling very inspired!

    • Hi Dee, thanks for your comments. It was all in a double lesson (2 x 50 mins, which was *just* enough for the strongest/most organised pupils to get through all the tasks and work on a write-up/presentation). Some presented to the class, some wrote up their results, some didn’t get that far (as they were still busy with the calculating elements, or getting to grips with using the language of ‘linear’, ‘dependent’ etc) so I was satisfied with their oral explanations to me during the lesson. Some pairs were really frustrated at having to stop before they were ‘ready’, but some had completed the main ‘bulk’ of learning during the session and wouldn’t have learned much more with more time (I think???).

  2. This is super cool! I used the balloon idea from Sam in my Calc class and it sparked some great discussion. I find it amazing that the same activity can create discussion in classrooms across the world at very different grade levels. Thanks for sharing

    • Thanks:) I’m hoping I can try something similar(ish) with a much younger and (on paper) weaker group – trying to think about how to let those pupils access ‘higher order’ activities….it’s a challenge! I’ve had some limited successes in that respect – a post for another day.

  3. What group and set was this? Given the nature of what they were asked to do, it sounds like they have a remarkable amount of knowledge already, so much so that I’m surprised they hadn’t already encountered the formulae for surface area and volume of a sphere…

    • It was Year 10, set 1, 33 pupils, grades at the time ranging from C- to A+, with the bulk on B-. Some of them ‘knew the formulae’ (in the sense that they knew to look at the front of the exam booklet if such a question had come up) but most had never used or seen the formulae, and none of them had ever been formally ‘taught’ about them or engaged with them on a non-superficial level (i.e. plugging numbers in). We’ve spent a lot of time on algebra skills (e.g. substituting, rearranging, graphing), and ‘setting up the problem’, so it was possible to include the more challenging tasks of testing hypotheses, grappling with the language of ‘types of relationship’, and explaining why graphs took on certain shapes. That said, some were pretty busy with the rearranging/substituting/rounding/choosing axis labels aspects…

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