In any classroom and any learning situation it is important to have an idea of what students are bringing to the learning. Having a sense for what they already solidly grasp, what they find difficult, and what their misconceptions are all help you to better target your teaching. In looking at what students bring to the learning it is often the diagnostic testing or a pretest on the particular content that may form the basis of knowing what they bring. However I think that this only looks at a very narrow band of understanding, what formal mathematical tools they have or remember. This is approaching it from a basis of procedural fluency rather than conceptual understanding. In looking at what they bring to the learning it is very important to me to know if they have a feel for the concept, do they have a sense of what it is before I start to apply too much formal mathematics to it. Can they intuitively determine a way to do the problem without knowing the formal mathematics and if they can, can I use this intuitive understanding as a base for building further understanding. An activity that we did in class other day really brought that into clear focus for me as it does most years. The problem and the specifications for the task are shown below. Obviously with this task the food helps, there is some competition here to help maximise the number. It is clear that this becomes a problem on volume, but this was not something students were told, a few came to that conclusion as they completed the task and it was good for me to know that some had a sense of volume, but it was not necessary to complete the task. The intention behind the task was to see if they could connect their idea of area to that of volume of shapes with uniform cross sections. How that intention was set up was by limiting the number of cheese balls they were given. By limiting the materials you force them into thinking about the problem in more depth they need to think flexibly about how they can use the resources they have to solve the problem they had been given. The most common approach to solving this problem was the one shown below Using this method students created a layer of cheese balls on the bottom and multiplied it by how many of those layers could fit in the container, they determined this by measuring the side in cheese balls. In this case the student would have found that they could fill it with 80 balls (10x8). What is interesting about this approach is that it is the exact method they would be taught to calculate the volume of these shapes, multiplying the area of the end by the length of the shape and they had got to the point where they had generalised the solution. Many had started with rectangular prisms and determined the area of the base by multiplying the length in cheese balls by the width in cheese balls but then ran into difficulty when looking at a cylinder, they had to come up with a way to approach that and realised that area was the link that generalised for any shape.
It was a situation where they did not need to be taught it, they had a sense for it already, my role was then to formalise the thinking at the end of the task or
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Sometimes ideas for lessons get away from me and take me to some unexpected places, and sometimes the reaction of classes to those lessons are also unexpected. I was thinking about a lesson idea for circumference and I thought about a lesson I saw once from Andrew Stadel (lesson link here). Where he wanted students to determine how many rolls of tyre it would be for a tyre to hit a target, I liked to lesson but was not sure how my students would "buy in" to the task. I wanted to redesign the task a little to give students a greater reason for getting into the maths. Like with many of my ideas I ran it past a teacher at the school that I often collaborate with on tasks such as this and we came up with what I think is an interesting alternative to that original task. The Task Students had to determine where on the ground they were going to put a chocolate frog so that it met two criteria:
The twist with this task was that they were only allowed to use 1 item from their bag to measure both the tyre and the distance on the ground. In relation to this task I think there are three incredibly important aspects to it. Firstly is that they really have to think about the positives and negatives of anything that might be in there bag in relation to it's use as a measuring device, there was nothing in anyone's bag that could accurately measure the circumference of the circle in one measurement so they had to think about aspects such length, flexibility and having actual measurements on the device. Secondly was the idea that the criteria and the restrictions I provided made the task of achieving it quite difficult, Using something that was not ideal as a measuring device meant that there was going to be inherent errors in the way they could possibly measure the tyre. They had to be very accurate in what they used as the measuring errors could compound quickly as they measured both the tyre and the ground.Finally there was a high degree of ownership over the task as they seemed to really want to be able to not have a squashed frog. That little element of competition was very valuable in this task, That element of competition I have found to be very useful in a range of tasks I have done previously. The Approaches and Challenges The range of measuring devices people used to measure the tyre was quite interesting. Most chose their ruler from their bag to measure it. However since the ruler was only 30 cm and the tyre was significantly bigger than that both in circumference and diameter they needed to make lots of measurements and therefore compounding any measurement errors. Most were also using a rigid wooden ruler to measure a curved surface which made the measuring more difficult again, some had a more flexible ruler that they could wrap around, but some of the compounding errors remained. Many of these students decided to use a ruler as they could then stick within their comfort area of standard units of measurement. Others chose to use headphone cables as measuring devices, they found them to be much more flexible and longer than the rulers, therefore they could accurately follow the contours of the tyre and it was long enough that the measurement errors were not as significant. However they seemed to encounter difficulty when dealing with lengths that were not quite a full headphone cable length. They were not sure how to figure out what fraction or proportion of the headphone cable was left over. When Introduced to the idea of folding the headphone cable to figure out how many parts were used they found this area. For example in the image below if they measured the tyre and got a total diameter of the blue part of the line below, they could fold it as shown to give an approximate length of just over 3/4. The other part of this that students seemed to find challenging is using headphone cables as a unit of measurement in a mathematical formula. The seemed to have this impression that the formula for circumference of a circle would only work if you were working in standard units of measurement. They didn't realise that If you said that the diameter was 3/4 headphone cables then the diameter would be 3.14 x 3/4 = 2.36 headphone cables. This however was not unexpected, in mathematics we get so used to dealing with measuring devices with standard units of measurement that we do not often realise that sometimes the nonstandard units of measurement can be just as valid and just as accurate.
Some of the students who engaged in the task a little more strategically came up with multiple ways to measure the tyre using the same device. The measured the circumference directly, but then they also measured the diameter and used to formula to determine the circumference, where there was significant differences between the two values they took measurements again to verify their results and if they were accurate then they had some discussion about which of their measurements they felt was more accurate. They realised that a discrepancy of 3 cm was not much but when you multiply it over 10 rolls you have just potentially lost a third of your target area. The final challenge most of them faced was where to place the frog because that question is not as simple as it seems. If you put it too close to where you think the tyre will finish and your measurements are off then your frog will get squashed. Conversely if you put it close to 1 m away from where you think the tyre will finish and your measurements are off then you could be outside the target range. Most people put it closer to the tyre and a few put it closer to the end of the 1 m, very few considered putting it where they thought of the middle of the range may be which in my humble opinion is where I would be aiming. The Results Overall when we rolled the tyre the level of excitement was high, they were all keen to see how their calculations faired. Of the 8 groups that placed frogs down 1 group was more than 10 rolls away but outside the 1 m target range, four groups had their frog squashed (but by less than 15 cm) and 3 groups met both criteria. They seemed to really enjoy the activity and everyone got to eat the frogs at the end. Below is a video of a lesson I have done for the last few years that is in relation to area. Students are required to determine both the number of post it notes that will fit on the board and the time it will take to complete the task. In getting students to complete the task I first deprive them of the measurements I start with a set of estimation exercises. First they use the video of the first 20 as shown above as a way to estimate the time and the number for the whole board. They are then shown an image of 100 postits on the board (shown above) and are asked to adjust their estimate and give reason for what change they made and why. This process is repeated with 200 and 300.
What was interesting about the process of this activity this year is the strong desire of the students for the measurements initially. The just wanted to apply a formula they knew to the problem, they placed absolutely no value on the process of estimating both the number of post its and the time, they did not see it as a valid way to approach the problem. However in reality estimation is an incredibly important skill for students to become comfortable as we use it all the time, in fact we use it a lot more than we use an exact calculation. In an article I read a few years ago (I can't remember the source of the article now) it stated that of all the maths we doing in our lives 80% of it is estimation, only 20% requires an exact calculation. If this is the case then the estimates I was getting them to do is arguably a more important process than getting them to calculate the exact numbers. However I also feel that little emphasis is place on estimation within mathematics classes, our desire as teachers often is to apply the formula and to deal with exactness of the solution, we don want to look at the messy and imprecise nature of an estimate. Students also seem to have a fundamental level of misunderstanding of how to estimate. In some of the diagnostic testing we do this misunderstanding of estimation also becomes evident. When asked how they would estimate the value of 18 x 79 the most common response is to calculate 18 x 79 and then give the answer. Other common answers have students performing the calculation and then rounding. The number of students who calculate 20 x 80 instead as a good estimate is very small but is the most appropriate way to approach the task. It makes me think that estimation needs to be a much larger section of my program than it currently is and I know that one very good source of questions requiring estimation is Estimation 180. It is a great website that potentially has tho ability to provide students with 1 estimation question for each lessonn of the year as a warm activity.I am also thinking more about how I can get students to estimate before they calculate more often over the course of their year. 
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
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