The Association of Teachers of Mathematics currently runs Twitter events every second Thursday between 8.00 and 9.00pm. (@ATMMathematics, # beingmathematical). I find these are a great source of CPD with a difference and would recommend you get involved! The discussion prompt for the first one of this academic year was the following set of questions.

- Write down an improper fraction.
- Write down an improper fraction which lies between 3 and 4.
- Write down an improper fraction which lies between 3 and 4 and is closer to 3 than 4.
- Write down an improper fraction which lies between 3 and 4 and is closer to 3 than 4 and whose numerator differs from its denominator by a multiple of 3.
- (Make sure that each answer is a different improper fraction.)

Notice that each question builds upon its predecessor by preserving the same wording then adding a constraint. Thus having started with an extremely general (and generally extremely easy) question, we arrive at a significantly more specific (and, specifically, more challenging) one. Note also the requirement for four different answers, which draws our attention to the fact there is a range of correct answers – albeit a narrowing range – at each stage.

Participating in this event made such an impression on me that I suggested a face-to-face version with my departmental colleagues, none of whom had been aware of the Twitter session. Five of us got together during a non-contact period and had a very productive discussion. We set ourselves the task of each devising a set of questions to use with our S1 classes who had just started the topic “decimals”.

Our brief was to come up with a short activity that could be used as a starter or plenary task or, perhaps, as the basis of a mid-lesson discussion. We all duly compiled our question-sets, which have now been added to our resource bank.

At a personal level, I felt I could get more out of this “increasing constraints” format – perhaps an activity that would last a whole lesson, or maybe more. We are currently working with mixed-ability sets in S1, having delayed the decision to set which would normally have taken place by this time. From our decimals diagnostic test the most striking finding for me was that only one of the 27 pupils present in my class managed to answer the following question correctly:

*Calculate: 0.9 – 3 hundredths *

On top of this, during teaching time I judged that, overall, the concept of place-value, especially when expressed explicitly as in the question above, needed further investigation and development. I devised a set of increasing constraints questions that I felt would help address this issue. I decided to start with whole numbers, with a view to repeating the process with decimals.

The first question-set I used is offered here for your consideration – give it a try, I hope you’ll find it both challenging and interesting. Even better, maybe you’d like to try it out with your learners and let me know how it goes. I’m particularly interested in the range of strategies folk come up with. From a more experiential and phenomenological perspective, I’d also be interested in knowing – as in the original ATM activity – *“What happened when you worked on this set of questions?”* Please use the comments facility below or send me an email or a tweet (perhaps with a picture or two?).

*Answer the following four questions in order; each answer should be different from the others.*

- Write down a whole number between 1000 and 2000 inclusive.
- Write down a whole number between 1000 and 2000 inclusive that gives an answer greater than 2000 when you add 4 Hundreds.
- Write down a whole number between 1000 and 2000 inclusive that gives an answer greater than 2000 when you add 4 Hundreds, and an answer less than 2000 when you add 4 Hundreds then subtract 4 tens.
- Write down a whole number between 1000 and 2000 inclusive that gives an answer greater than 2000 when you add 4 Hundreds; an answer less than 2000 when you add 4 Hundreds then subtract 4 tens, and an answer greater than 2000 when you add 4 Hundreds then subtract 4 tens then add 4 units.

Here are some observations from my lesson. Around one-third of my S1 pupils managed four correct answers. Interestingly, one of my most confident and vocal learners quickly became very frustrated and declared it was “impossible”, while another – who is currently receiving remediation intervention – was beaming to be numbered among the successful group. One boy – who had up to this point given me much cause for concern regarding his motivation and behaviour – was totally absorbed and came up with all the possible solutions to the final question. (Interestingly, he has maintained noticeably improved performance and behaviour in class since). Contrast this with the response of another boy – normally both able and enthusiastic and whom, on the basis of his performance during the first month of term, I nominated to participate in the S1 extension programme. He lacked the perseverance to complete the task, saying it was “boring”. More generally, I found that a class discussion with pupils giving answers and me modelling a written check whether they met each and all of the constraints helped learners manage both the linguistic and mathematical demands of the task.

I personally found it a challenging task initially. Furthermore, compiling and checking a collection of similar question-sets to incorporate tenths and hundredths, as well as varying other details and structural aspects of the question proved particularly difficult and time-consuming. (A link to a selection of these, as a Microsoft Word document, appears at the bottom of this post.) With subsequent practice, both compiling and answering these question-sets has become much easier. To allow you to find a suitable strategy I shall not reveal my own; however I will say that I’ve tried, unsuccessfully as yet, to come up with an effective strategy that uses a pictorial representation. If you find one, please share it with us!

I would encourage readers to explore this “increasing constraints” model across a range of topics and for different purposes (starters, plenaries, homework or more centrally within a lesson.) I believe this model, with appropriate planning, could form the basis of low-floor-high-ceiling activities with more able learners generating and investigating their own questions and conjectures. Supplementing this with *“what happens ..?”* questions to develop awareness is, I feel, a worthwhile exercise when learning, teaching and doing maths. Once again, I’d love to hear from anyone who tries this and is willing to share their thoughts and experiences.

Thanks for your time – hope it proves worthwhile!

Increasing Constraints Place-Value Germinal

Thanks for taking the time to write this. I’m interested in how you got students to work on this task (which I think it’s a great set of questions BTW!) Did you present all together or one at a time? Working in books or MWB? And what went before it in terms of explicit teaching, exercises, or assessment? Sorry for the barrage questions but I think the context is important. I’ve tried tasks like this in the past and bailed because the ground was not fertile! Or was it because I didn’t let them struggle for long enough?

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Hi Mark – no need to apologise for the questions, I’m happy to do my best to answer them. Being honest, I was so focused on the task itself, I didn’t put much thought into how I’d deliver and manage it. I used a projector slide containing all four questions together; the type size was such that I had to scroll down gradually to revealing two or three of the questions at a time. Pupils worked individually in their jotters, although I encourage on-task discussion with neighbours (kids sit in pairs) in general. Prior to this we had been working on the four operations with decimals. Questions involving adding or subtracting “unlike” pairs such as 4 .5 and 3.75 caused the expected problems for some learners. I found myself insisting they set up and carried out each addition, subtraction or multiplication question in headed columns (H – T – U – . – t, etc), which despite some protests, definitely increased success levels. Back to the task under discussion, the first question was easy, as expected and just about everyone managed the second without difficulty. This meant I could display the third and fourth together, allowing pupils to work at their own rate. I don’t know about you, but I think the language demands ramp up significantly by Q3. That’s when the girl claimed it was “impossible”, and “didn’t make any sense”. I found that within 10 or 15 minutes on these last two questions (I wasn’t really keeping track of the time: I tend to teach responsively and intuitively – although I sometimes end up wishing I’d planned more carefully), pupils had either succeeded (the approximate 1/3 I quoted in the post) or given up. It was at this point I went to the board. This was more productive in terms of allowing most of the other 2/3 to contribute to some whole-class solutions. Taking question 4, I asked for a suggested answer (ie starting number) and worked through each of the add or subtract steps, asking for the correct sum or difference and writing it on the board. At each stage, we checked if it met the constraint of being greater or less than 2000. Having rejected incorrect starting numbers and found one that worked, we were able to establish the boundaries of the full solution set. The boy I mentioned who found the full solution set on his own was actually working away obliviously during the discussion – I was aware of this but did mind that he was “opting out”, in fact I was pleased about it. Of the others who had found a single solution on their own, most seemed happy to engage in the discussion and checking process. Some pupils copied the boardwork into their jotters, others were happy for me to do the writing. Again, I’m comfortable with these choices, as long as learners are engaging and attending to what’s happening. Our classes are due to be set next week, so I might try another version with my new pupils. The thing I would change would be to begin with the check-modelling process, the whole class working together, perhaps as far as the second operation of Q3. They could then attempt to complete independently. I’m now thinking pairs and mwbs would be good. I’d hope to get through at least a couple of question-sets (little or no modelling by set 2?) using this approach. Thank you so much for posing these questions – they’ve actually helped me move forward myself with this task.

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Thank you so much. Really helpful. I think I will teach a similar lesson next week as my year 7 are at a similar point with adding and subtracting decimals. I’ll let you know how it goes if I do teach it!

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