Like most secondary teachers across the country, I spent term one getting to know a new S1 class. I always find the initial weeks with a new class to be challenging. Trying to find out their prior knowledge, their outlook on the subject, their traits as learners (and indeed as people) whilst setting out expectations and showing them what I value can be tricky. This difficulty is magnified with S1 classes partly because they themselves are adjusting to such a significant change but also because my knowledge of their prior experience is even worse than it is for other year groups. This shouldn’t be the case but it is. One thing that didn’t take long for me to identify was the wide range in my class’ ability to divide a whole number by a single digit, a topic that seems to be a perennial headache for primary and secondary colleagues alike. This post is a summary of my attempts to remediate this important skill. This episode spanned three periods so when I use the word lesson I am not referring to a single period. I also use the word concrete to refer to digital manipulatives. I’m not looking to debate the use of physical over digital, the resources I used were based on what I had available. This post focuses entirely on the remediation component of my lesson but there was planned enrichment for the pupils who were already fluent with division by a single digit. The focus here was to help them push their understanding and connect division with some other aspects of their learning in S1. The resource I used to support this is available here: https://www.dropbox.com/s/ol2akfi6a6rkj08/Division%20Extension.docx?dl=0 Of a class of 26 I established that 13 were completely unable to divide by a single digit if there was a requirement for carrying a remainder into the next column. These 13 became my target group. Through starter activities, observations and discussions with pupils, I established:
E.g. 21/3, 48/6, etc. This fluency was not necessarily automatic recall, in many cases it involved skip-counting from zero or another known multiplication fact.
E.g. 21/5
I think the last bullet point is particularly interesting as the narrative I’ve encountered seems to be, “Concrete resources are used in primary schools but not in secondary.” This was a good example of concrete resources being used but without a plan as to how they would support progression to, and understanding of, the abstract method. Not having concrete resources available at the time of the lesson (an order for a good starting set is on its way at the time of writing), I decided to utilise online manipulatives using the Mathsbot website (https://mathsbot.com/#Manipulatives) since my class had access to laptops. This was a workaround rather than my ideal scenario. I was concerned that dragging lots of blocks using the trackpad on a laptop might be a very slow and frustrating process. For this reason, I decided to use the Place Value Counters rather than the Dienes Blocks because the former has a button that allows you to rapidly exchange one counter for ten of another. This was a decision I came to regret for reasons outlined later. Task 1– Creating Rectangular Arrays Show me 12/3 Typical Response: "I've made four groups of three so the answer is 4." This matched my expectations in that prior encounters with concrete materials involved making groups but not in an array. Me: Fantastic. The answer is indeed 4. I’m going to ask that you structure your groups a bit more. I want you to lay them out in vertical columns so they form a rectangle. Them: Blank I decide to show them exactly what I mean. We discuss that the rectangle has an area of 12, a breadth of 3 and a length of 4. I hope that some start to see the bones of the algorithm but do not make it explicit that that’s what the layout represents. Their reaction suggests that the connection is not made. I present them with a few more to try before we move on to problems involving remainders and exchanging. Using only the unit counters, show me: (a) 12/4 (b) 12/6 (c) 18/6 Common responses: Chaos! They’ve gone clicking “exchange counter” and “scatter” without my instruction to do so. I’m immediately annoyed and frustrated but don’t let it show. If I’m honest I’m probably a little hurt. A lot of thought has gone into trying to get this right and I know that I need more concentration from them in order to make a success of this. Then I remember something from a few months back. I was asked by Stuart Welsh (@maths180) to help out with his online tutorial, hosted using Adobe Connect software. I was given clear and explicit instructions not to click anything but the minute I was confronted with a new interface I couldn’t help myself and went exploring, causing damage to the setup, which Stuart then had to put right. The thing with Mathsbot is there’s no undo button so the pupils went exploring the interface erroneously believing they could quickly go back and get on with the task. My annoyance quickly dissipates and I talk to the class about what the buttons do and why it is important that they stick with the task in hand. The pupils quickly get on with the task and get to grips with the more-structured grouping I’ve asked for. One or two are creating arrays oriented in the opposite direction (making rows containing the divisor instead of columns). I’m in a dilemma as to whether or not I should insist that they don’t do this. I decide that I will allow them to continue like this until the connection between the array and the algorithm becomes more explicit. Task 2– Introducing the 10s. We repeat a similar task but instead of using only 1s we use the 10s and 1s to represent the dividend. Show me: (a) 39/3 (b) 69/3 (rookie mistake in my choice of dividend – some chuckling ensues) (c) 99/3 (d) 48/4 In retrospect I think I missed a trick here in not asking for some learner generated examples to follow on from these. There could have been a rich discussion about the restrictions imposed on us by the fact we are looking to have no remainders. There’s always the next time, I suppose. Making the arrays goes to plan but I begin to stumble as I see the issue with the place value counters and why Dienes blocks would’ve been the superior choice. I want the place value to be explicit so I supplement my array by writing the length and breadth of the rectangle as follows: I’m happy with the pictorial representation, but the concrete? What a mess. The fact that length isn’t proportional to value on the counters has left me with 3 being 3 times as long as 10. This would not had happened if I opted for the Dienes Blocks. Someone asks, “Would that not be 30 on the breadth then?” I don’t recall how I handled that excellent observation. I was probably a bit flustered because somehow I hadn’t seen this coming. We agree it would be much better to write a 1 above the 10 but I labour the point that the 1 is one group of 10s and is distinct from a group of 1s. We now have: I’m now happy with the diagram on the left but feel there’s a lack of synergy between the two. Pupils are comfortable with the task so we move on. Task 3– Carrying remainders Show me 56/4 Most of the class progress as follows: I ask what has happened. “We’ve got one left over.” “What should we do with it?” “Well now we’ve got 16 including the 1s so that is 4.” “4 what?” “4 columns.” “Show me 4 columns.” “I can’t because I only have 7 counters but I know that it’s really 16 counters.” “How could you make it 16 counters?” Someone else has a eureka moment and shouts, “Exchange it!” I click the exchange button to obtain the layout below and something interesting happens. One girl gasps. I hear her whisper to the person next to her, “I get it! I get it! I actually get it!.” I can tell that this has been a demon of hers. She asks if she can go and try some division problems on her own. I tell her that’s fine and give her a Corbett Maths worksheet but ask that she checks her answers after every three questions. Away she goes and she doesn't re-enter my target group. I’ve now dropped the written area model from the process. I was dis-satisfied with the relationship between it and the concrete model after I changed the notation in the 10s column in task 2. I worry this will create a missing link in the progression of the lesson but I’m confident that the act of exchanging with the digital manipulatives can explicitly be related to the “carrying over” in the algorithm. I also know that there’s no need for things to progress in a linear fashion from C to P to A. I ask them to carry out the following divisions: (a) 48/3 (b) 72/6 (c) 65/5 (d) 96/6 The first three questions are designed so that only one 10 needs to be exchanged. Question (d) requires three 10s to be exchanged for thirty 1s. This is an onerous process and the question is in there for exactly that reason. I am trying to generate some motivation to move away from the concrete method towards the algorithm. Of the 12 pupils I am now targeting, 10 can confidently and fluently use the counters to work out the answers to these questions. The other 2 are undoubtedly struggling. They need a prompt to count out the dividend, not the divisor. They then need a prompt to arrange the first column but can continue the process from there. I reason that they will need more practice than the questions I’ve provided so I write down some more for them to do while I move the rest on to the next task. Later on I get the chance to revisit both pupils and notice the key difference between them and the 10 who are getting it; they’re unable to predict what is unfolding in their model. When the others have twelve 1s that are to be divided by three they seem to be “predicting” (or perhaps just know) that this will generate four columns. The arrangement of the counters is serving to confirm their thinking. For the two stuck pupils the concrete model isn’t confirmation of the answer, it’s the mechanism by which the answer is generated. I reason (or perhaps hope) that the extended practice questions I’ve given will help them make progress since they can now tackle problems with much less support. Task 4– Towards the algorithm As I anticipated, the pupils found the process of dragging thirty-six 1s counters into place to be a chore. Drawing on this, I ask, “Wouldn’t it be great if we could move away from needing to do that with the counters?” I believe I’ve given them a sufficient enough headache for them to happily consume the aspirin I intend on dispensing. I am wrong. They don’t say anything but their faces are a mix of disappointment and frustration. I ask one boy directly the question I’ve just posed to the class. His response is honest and illuminating. “Sir, I get this. It’s really annoying when there’s loads of counters but I can get the right answer.” I pause. I’ve given them a strategy that allows them to do something they previously couldn’t. They’re enjoying being successful and I’m trying to take away the crutch that’s supported that success. I ask, “Can we try laying the working out next to the counters to see if it helps?” They grudgingly agree. As an example we use 54/3 I ask the class to progress until the point where they are about to exchange then stop. Firstly they count out 54 using 10s and 1s. They now begin grouping the 10s into columns of three. This is the point that the 10s will need to be exchanged so we stop and I construct the algorithm next to the model. I don’t write the entire dividend, only the 5 to represent the 10s. I ask if they can see any similarities or any differences. One boy tells me that the 5 isn’t obviously representing fifty. He knows it is but it doesn’t “feel like” fifty. Off the back of this, another boy asks why the concrete model shows three 10s but the written one shows 5 in the 10s column. Someone interrupts to point out that the other two are up above the 1 in our concrete model but I think there’s an excellent point here and I regret that I didn’t explore it more during the lesson. When we write our working the way I have above, it seems to suggest that 3 x 10 = 50. I wonder if it might be more useful when learning this algorithm if we scored that 5 out and replaced it with a 3 as well as carrying the remainder of 2. It’d look more like this: Perhaps some people already do this but it’s not a layout I’ve seen. Back to the lesson. I ask what our next steps are. “Exchange those two 10s.” I click exchange on my concrete model and ask how we could represent this in the written algorithm. “Put 24 in the units column.” We now have: I make explicit the connection between the two models and have to give no further direction for the majority of pupils to arrive at: I ask pupils to go back over the 4 questions they did in the previous task but to supplement what they did with the written algorithm. All but 2 are in a position to do this. My plan was then to model one involving hundreds but, feeling things had gone well, I just put it up on the board and ask them to go for it. Ten were able to do so, the majority of whom asked if they could bypass the counters and go straight to the algorithm. They then moved on an attempted a few questions from an exercise. I was surprised at how smooth the transition was but put it down to the investment in the previous tasks.
The next day I gave a few division problems as a starter and was heartened by the level of fluency across the room. Of the 26 people in the class, 20 were able to independently divide a three-digit number by a single digit. The others needed to log on to the laptop and create the model using counters. 4 of those were able to translate this model to a written algorithm and 2 were able to compute the quotients correctly but only using the counters. I realise at this point that the absence of a pictorial strategy is leaving the group of 6 learners a little hamstrung since they will not reasonably be able to access the concrete resources every time they need to carry out a division. Having reflected on this, I think that James Tanton’s “Exploding Dots” model would be the perfect accompaniment to the concrete model used in this lesson. The next time one of my S1 class ask for the counters I will show them this method so that they have a pencil and paper strategy for division. Although pleased with the outcome of this lesson, I am acutely aware that what I’m currently assessing could be performance, rather than learning. I know the retrieval is there because little time has passed, but will this stick? Only time will tell. I think I’ll get a good idea when we return from the October break and some forgetting has occurred. Even if pupils are now completely fluent with this process, there is no way I can say they’ve achieved mastery of division. I like Dr Helen Drury’s summary of mastery: “A mathematical concept or skill has been mastered when, through exploration, clarification, practice and application over time, a person can represent it in multiple ways, has the mathematical language to be able to communicate related ideas, and can think mathematically with the concept so that they can independently apply it to a totally new problem in an unfamiliar situation.” Whilst some of these conditions have been met, every problem has been routine and there has been no test of whether or not learners can select division as an appropriate strategy in a given context. As we move through S1 together, I will aim to develop their multiplicative reasoning and provide repeated opportunities for them to strengthen their understanding of this foundational concept. Some final reflections:
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AuthorMy name is Lee Gray and I am the Principal Teacher of Mathematics and Numeracy at Trinity High School in Renfrew. I tweet from the account @mrgraymath ArchivesCategories |