A few mornings ago, Rufus William got in touch with an interesting request:
@LearningSpy fancy doing a quick maths activity? You just need something to write with some paper
— Rufus (@RufusWilliam) June 14, 2016
I’ll admit to being a little anxious, but in the spirit of enquiry, I agreed.
This was the activity: A domino is made up of 2 squares. A pentomino is made up of 5. How many different pentominoes are there?
I got out my paper and pen and duly set to work:
I quickly realised the answer was ‘loads’. At that point I gave up and told Rufus the answer was “more than 50”. He then asked me what I thought of these:
My first answer was that if you cut out the shapes you could make a box without a lid out of each of them. Although Rufus liked this answer it clearly wasn’t the right one. I dredged up some memories of transformation from GCSE maths and hazarded that maybe they could all be rotated. Rufus nodded approvingly – as much as you can on Twitter – and agreed they could be either be rotated or “reflected on to each other. Mathematically, we don’t consider them different. They’re congruent shapes.”
Ok. Makes sense. Rufus asked whether I thought this was a good way of teaching congruence. I said I thought it was interesting, but possibly inefficient.
Here’s why. At the start of the activity I had no idea what I was doing. It took me a few minutes to work out that there were loads of different ways to arrange 5 squares, but that wasn’t really the point of the activity.
When I was asked what I thought of the patterns which turned out to be congruent my first thought was imaginative perhaps, but off topic. I was remembering making something similar with my daughters. When it was clear another answer was expected I though back to what I already knew. Luckily there were enough vestiges of the maths I’d learned in school to be able to supply a broadly correct answer. I still didn’t know what the point was until the concept of congruence was explained.
What would have happened if I couldn’t remember anything – or hadn’t been taught anything – useful? I would have floundered about with guesses until I lucked on, or was told the answer.
Now, I’m no mathematician – as should be pretty clear by now, but an alternative way of teaching congruence would have been to turn the activity around:
- Explain the concept of congruence.
- Show examples and non-examples to demonstrate.
- Ask me to draw some congruent shapes to make sure I’d got it.
The question is, which method is best? I think it would be quicker, and therefore more efficient, to use my alternative. Less time would be wasted guessing and working out that there were an awful lot of ways to arrange squares. But, would students have a better understanding if they had to puzzle out the concept for themselves? Maybe, maybe not. They might have ended up discovering something which turned out to be wrong or inappropriate. Because of the way our minds work, this might embed misconceptions, which are hard to get rid of.
It would be fascinating to run a mini test where a class was split with one half having the discovery method and the other having worked examples and seeing which group did better in a test next lesson, then again in another test at the end of the term. There have been a number of studies which indicate which approach would get the better results, but who knows – I’ve been wrong before and will certainly be wrong again.
The other consideration is that one method might be more ‘engaging’ than the other. Maybe students would prefer playing the pentoninoes game first? Maybe this would result in a more positive view of maths? That’s certainly what we as teachers often intuitively believe. And it might be true. The alternative view is that students become more motivated the more proficient they become. Being asked to play games might feel fun at the time, but is, perhaps, unlikely to intrinsically motivate students. But getting good enough at something that the basics become easier and you can have a go a more challenging stuff might actually be the best way to motivate students.
This is certainly borne out by my experience of nagging my eldest daughter to practice her piano scales. She’s now got to the point where she can play hands together and bang some of her favourite songs and we can’t get her off the bloody thing!
Rufus has written the following in response:
I encountered the idea on my PGCE, I’d like to discuss this experience as well as my observations of its effectiveness with my students; it’s quite a bit different doing it over Twitter than doing it in the classroom.
The activity is simply for the students to find out how many different arrangements of 5 squares they can find. There is an element of setting it up as a challenge to see if there are more than ten different ways, but I wouldn’t consider my students to be ‘playing the pentominoes game’. Also, whether the students consider this to be a fun activity or not is not relevant to me beyond my desire for them to consider my lessons to be both serious and inviting, and most importantly, lessons in which they learn maths.
Inevitably, when the class work on the activity, at least a few students will start to ask whether they can consider some of the shapes the same or different, and this will lead to a class discussion. For example, these two shapes generate a lot of discussion:
It is now that I go through David’s stages:
- Explain the concept of congruence.
- Show examples and non-examples to demonstrate.
- Ask the students to draw some congruent shapes to make sure they’d got it.
Therefore I am not saying that I am not giving clear definitions and examples, it is just that I think giving the students the activity first helps them to understand and embed the concept in a stronger way than if I had started the lesson with the explanation.
The reason I feel like this is that I was given the activity to do on my PGCE course. I have an excellent degree in maths, as well as excellent school grades, however, at that time I was not familiar with the concept of congruence. When I started to have a go, I too was apprehensive, and in trying it I was confused. However, in the course of a class discussion, I experienced the desire to want to clarify whether certain shapes could be considered the same or different, and I found that the concept of congruence made perfect sense to me when it was explained.
I am aware that this does not determine that my students will feel the same way, and I wonder how I can ever know if there’s a ‘best’ way to teach something. At the moment, my judgement is that it’s a good way to teach the concept because it always instigates a class discussion which effectively asks a question that I want to answer: “do we consider these shapes to be the same or different?”
I thin there are two different question to ask concerning the benefit (or not) of the ‘pentominoes game’, and it might be worthwhile distinguishing them. One (which seems to be David’s main concern) is the age-old debate of discovery type learning vs. direct instruction. But a second question is how important it is to give a motivation for why we want a particular concept. I take it that this is the main aim of the ‘pentominoes game’ — to give one reason for why we would want a concept of congruence. I notice that David’s sketch of a plan doesn’t mention this (although it obviously doesn’t rule it out).
The question of whether this motivation should be given directly, or through some kind of activity is then a further one. (And I’m not sure how much the research on DI vs DBL addresses this. It’s pretty clear that DI is better for teaching a concept, but perhaps there’s room for DBL to be a better way for getting students to understand the purpose of a concept? But I’m not sufficiently familiar with the literature to know if there’s work on this.)
There’s lots of research on this. The main finding is that novices and experts think in qualitatively different ways. Novices learn best when given clear structure and worked examples. As we become more expert in a particular subject this fades away until we get the ‘reversal effect’. Experts have internalised their own mental representations of how to do a task and additional guidance produces a negative effect.
If you’re interested, here are 3 studies which detail this:
Kalyuga, S. (2009). Knowledge elaboration: A cognitive load perspective. Learning and Instruction, 19, 402-410
Kalyuga, S., Chandler, P., & Sweller, J. (1998). Levels of expertise and instructional design. Human Factors, 40, 1-17.
Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38, 23-31.
Thanks for the references. I have seen some of these but not others. But it’s my understanding that they’re not quite addressing the issue that I was thinking of, which I perhaps wasn’t clear about.
To stay with the congruence example, the studies clearly show that direct instruction (and similar) is superior to discovery learning (and similar) for *learning what congruence is* (for novices at least, anyway). What I’m wondering is whether the same can be said for *appreciating why the concept of congruence is needed*. The latter of these is presumably very hard (if not impossible) to measure. For the former, things like performance on tests will serve as a pretty good proxy, but I can’t think of anything which would serve as a good proxy for the latter.
Of course, someone may think that the latter aim is not a worthwhile goal of education, or is only insofar as it leads to a better achievement of the former aim. In that case, the question of how best to achieve it is moot.
Anyway, I’ll have a look at the references to see if they can help me.
“the studies clearly show that direct instruction (and similar) is superior to discovery learning (and similar) for *learning what congruence is* (for novices at least, anyway). What I’m wondering is whether the same can be said for *appreciating why the concept of congruence is needed*. The latter of these is presumably very hard (if not impossible) to measure. For the former, things like performance on tests will serve as a pretty good proxy, but I can’t think of anything which would serve as a good proxy for the latter.”
This sums up so much of how I feel and find hard to articulate. Thank you.
This kind of argument is a bit of a slippery slope. I’ve written more about this view here: https://www.learningspy.co.uk/psychology/the-closed-circle-why-falsifiability-is-useful/
“I have an excellent degree in maths, as well as excellent school grades, however, at that time I was not familiar with the concept of congruence.” This seems to be pretty significant and very different from the experience of the children you teach. Also I can’t believe you have not used the concept of congruent triangles in your previous maths learning. You will already know about rotation and translation so many of the ideas are already present in your brain. The move to the different pentominos is a very small move for you but a very large move for your children. Perhaps the use of #ItWorksForMe is not the best way to decide on a method of teaching to naive maths learners.
Good point, thank you
It would also be interesting to see if the investigative learning approach described by Rufus is repeated with other maths problems what the learning effects would be. Confusion and a growing lack of confidence in students or a willingness to explore and go beyond the curriculum? I think it all depends on what type of learners we want and how they are supported in getting there. The ‘efficient approach’ may be quicker, and may help students pass exams. It may give confidence to those craving control and security in their own understanding. In my experience the ‘efficient approach’ only works with engaged students already interested in the subject and keen to do well. Unfortunately many kids would switch off after 30 seconds of 1. Explain the concept of congruence….
My point in the blog was that motivation seems to come from proficiency: when you can do a thing you are motivated to do more of it, not the other way around.
I liked Rufus’s example and think it has merit. That was why I chose it to argued against. Don’t tell me teacher explanations are boring. We should never select poor examples to explain why something is rubbish: https://www.learningspy.co.uk/featured/seven-tools-thinking-6-dont-waste-time-rubbish/ Let’s change #1 to “Explain congruence in a really interesting way that rivets the attention of all students.”
The point about this as a starter was to generate a reason for being concerned about congruence. It is fine for people to say teach the concept first, but there is a prior move which is that the audience has to ‘buy in’ to what is being offered by the author. The intensity of this requirement may vary from class to class, person to person, so there are no fixed rules here. The question offering an incorrect answer: So-and-so says there are 8 pentominoes… leads on to the question, what constitutes difference? If, as Rufus showed, the two linear pentominoes led to some considerable discussion about whether they were different, or not, is precisely what the activity was set up to achieve. The discussion can be tied down in a number of ways. If the linear pentominoes count as different in any orientation, then there is an infinity of possibilities. If each arrangement is taken as a ‘primitive’ then there are two further clear paths (there may be others): one is, how do you know you have found them all? This would lead to some form of proof by exhaustion. Two is, can we be certain when two shapes are congruent? This leads to an introduction of 2-D isometries and the inclusion of a glide-reflection required to compete the group. (Unfortunately this is omitted in the national curriculum, but perhaps it isn’t actually too concerned with mathematics). Pupils will not ‘discover’ the group, or a tight formulation of 2-D isometries, so further teaching interventions are needed. There is a nice little article on this by Wesslén and Fernández(2005) in Mathematics Teaching 191, p27-29.
The problem with ‘direct instruction’, or teaching by telling, is that empirically it does not always work; not least because whoever is ‘being told’ has to engage and make some sense of it themselves. If an audience is receptive to direct instruction then that might be an appropriate strategy. If not, then something else is required. The Pentominoes problem in this form has proved successful in many different trials. That does not mean it is the ‘best’ way, or ‘only’ way, but merely a strategy. If a mathematics teacher can deploy it successfully, then it is only the ‘success’ that counts.
Finally, it has always seemed to me that when introducing a new topic, or new idea, addressing the ‘why is anybody bothered about this?’ or ‘who cares?’ questions might be worthwhile. A current series of textbooks does precisely this through offering some apparent application of the topic or procedure being taught. Unfortunately looking outside of mathematics for answering these questions can lead to ridiculous suggestions. For transformations the response to the ‘Why do this?’ question given in the textbook is: ‘Transformations are useful for producing designs for tiles, mosaics, wallpaper and rugs’. This is to muddle decorative design and mathematical modelling. A rug weaver might look at repeating patterns, but might not say, ‘Oh, I’ll do glide reflection here’. The inclusion of mathematics language for a non-mathematical practice is neither one thing nor the other. The pentominoes problem is only a mathematics problem. The congruence of shapes does not lie outside of mathematics and the probelm has nothing much to do with non-mathematical contexts.
And that’s about it. It’s a pedagogical strategy to generate a question about same/different with respect to 2-D shapes. Pupils will come up with something, and the pedagogical ‘journey’ from there depends largely on what the kids say.
https://jemmaths.wordpress.com/2016/05/21/relevance-is-not-the-goal/ This might help explain things.
With respect to your point about glide reflections, I think this is very good:
https://clioetcetera.com/2016/06/18/genericism-and-the-crisis-of-curriculum/
“What was the point of this task?” is the first question which springs to mind here. I am not asking flippantly, it might be important that when you plan to use an activity, you consider the pedagogic aim of a task.
Here, it seems that the aim is for your students to increase their competence in identifying congruent shapes,
(and by “congruent shapes” here you mean pentominoes which have the same shape and size or are the same shape and size and are a mirror image of another pentomino (with a view to addressing the notion of congruent triangles, which appear in exams, or not?)).
There also is an aim of creating a class discussion.
What I would like to hear about, is the following:
1. To what extent did you consider what prior introduction was needed to facilitate your students’ engagement with the task? (Did you think they just need to know what a pentomino is, or did they need to know that they are going to find some which look the same, but may or may not be?)
2. How did you plan to respond to those students who asked whether “are these two shapes different, cause they look the same?” If you did not plan how, then how did you respond to these students anyway? Your response might matter, do you remain reserved, with a “what do you think?”, or a “what if they are the same, what if they are not the same?”. Or do you turn ostensive and give an explanation, “yes these two shapes are the same, they are the same shape and size and one is a reflection of another”.
3. If you could introduce congruency to some different students, what would you do differently, and why?
Regarding David’s comment: “Explain congruence in a really interesting way that rivets the attention of all students.” I feel you are suggesting that this method works always, are you? Do you think there would be an affect on students who just received these interesting explanations of all topics? (On their enjoyment of a subject, the amount of authority they have in their practice of a subject, and how they view the subject (here, the subject of mathematics).)
I note that Weinberg and Weisner (2011) have found that textbooks have an ideal reader in mind, but the empirical reader may not conform to expectations, surely this is true with a teacher explanation? Burke (2015) posits that the “effect” on learning mathematics of a demonstration/drill strategy might be different from its effect on a wider social difference in other settings, although he goes no further here.
My thoughts are influenced largely by my time at King’s College as a PGCE student, by the conversations I am currently having regarding teaching at BSix College (see http://www.squeaktime.com/) and also by this piece:
Being told or finding out or not: A sociological analysis of pedagogic tasks. https://hal.archives-ouvertes.fr/CERME9/public/CERME9_NEW.pdf
Other reference: Weinberg, A., & Wiesner, E. (2011). Understanding mathematics
textbooks through reader-oriented theory. Educational Studies in Mathematics, 76(1), 49–63.
All interesting questions, I’ll have a think
I would have thought that a more useful purpose in this exercise would be to develop systematic approaches to an enquiry, rather than teach the concept of congruence (which is a bit arbitrary as it merely relies upon the definition of a word invented to mean a 2D shape that may be translated reflected or rotated, but not changed in shape or size in any other way)
If the intention is to teach the concept of congruence why are we not showing children examples of shapes that are congruent and examples of shapes that are not congruent? Why are we limiting shape congruence to the pentominoes?
The point here was that it was ‘a way in’ to a discussion about congruence, in which what you state could then be explained.
Grouping all possible pentominoes as “the same if rotated” or “the same of reflected” or “the same if rotated AND reflected” or “different even if reflected and rotated and translated” may help to embed the criteria for congruence!
There are lots of things to do with pentominoes. e.g. https://illuminations.nctm.org/Lesson.aspx?id=4105
I think we need to ask ourselves what is the point of the kind of maths that doesn’t involve calculation; what is the point of investigation and problem solving in any subject area?
I also would be very slow to publicise that I think there are more than 50 pentominoes, if I wanted to be a leader of wo(men) and education policy. Your own education has been sadly neglected, my Dad would have said. Have a play and see what you come up with, please.
I’m still thinking about this, and two points arise for me. I’m happy to be proved wrong about both of them, and apologies if you’ve already addressed the issues
1. Misconceptions: my students do not come to class as blank slates or empty vessels. With the pentominoes, students do not have to know about the concept of congruence, they just have to respond to a mathematical activity. It might be that some students know the concept of congruence already, that is not a problem, they can help to clarify things for other students. If might be that that students don’t consider the possibility that two shapes may be considered the same if one is a slight rotation of the other, but it is certainly something all students can have an opinion on. To me, this is not embedding misconceptions but exploring what students think before explaining clearly what a concept is, so that they know for the future. To me, this seems like a more effective method than explaining the concept at the beginning of the lesson, which could lead to students ‘masking’ their previous thoughts, or holding two conflicting views at the same time.
2. The context of the learning: is a mathematics one, in particular they are learning about congruence in the context of a geometric activity. It seems to me that this context is important. I am not mythologising congruence as being about a ‘real-life’ situation, I am talking about it in a mathematical domain. Also, I am requiring the students to think about a mathematical situation in which knowledge of congruence will be useful, therefore setting up the students to learn about the concept in situ. I can contrast this with me telling the students about the concept at the beginning of the lesson, in which it seems to me they will be learning it in the context of relying on me as a teacher knowing what I’m doing.
[…] 3. Back to David Didau. He has written brilliantly on the difference between novices and experts. One of the reasons I realised I had been getting things wrong was that I noticed on Twitter that he was saying things that didn’t make sense to me. I argued my case with him and it rapidly became clear that he knew far more than me. He even wrote a blog about one idea that I put to him. […]
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