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. It seems like a simple enough question, am I a good teacher? The question however is not so simple. Education is an area that is both measured and researched extensively so your impact as a teacher can be measured, compared and analysed in any number of ways, but what measures do we value most and which measures best define us as teachers.
I have been teaching now for 12 years. This is long enough for a student who started their schooling to now be at the stage where they are completing their schooling. I have taught a lot of students in a lot of subjects, some of these students have now been in the workforce for long enough to reflect on school at the time. I want to share some of the seemingly contradictory impacts of some of my current and my former students
Looking back at some of the examples above, which of the criteria in each example above do I use to measure my worth as an educator. Is it that I place more emphasis on the fact that some of my students are just not doing any work and therefore I am not doing a great job, or do I place greater emphasis on the idea that a student knew I would never give up on them. Do I place more emphasis on the fact that test scores have not improved, or do I place it on the fact that students are gaining a real love for mathematics as a result of my teaching. Do I place it on a huge increase in test scores despite no work being done in class and the student disliking the subject more than they had in previous years. How do I reconcile what these contradictory bits of evidence say about me as an educator? Do I see these contradictory bits of evidence in a different light to how others see them? Do I value only what can be measured or do I value more the stuff that cannot be measured? How do I truly know my impact as a teacher? I am so proud of the kids I teach today. Since the start of the year we have had a much stronger focus on patient problem solving than they might have been used to in the past and until today there has been a lot of resistance to attempt some of the tasks. But today they were superstars, it is the first time where almost all of them had persisted in solving a problem, even when they found it difficult. There had been pockets of this in the past, but to see most of the class doing it today was fantastic.
The lesson that we decided to look at was based on an ad produced by the motor accident commission in South Australia. The idea behind the lesson, which they came up with at the start, was to find out how long they were actually in the back of that taxi. For the sake of trying something a bit different we combined three maths classes with three maths teachers to see how it would go. Students were then asked to generate some ideas about what is the sort of information that they might need to know to answer the question and they came up with
From here they need to have some information in order to be able to progress forward firstly was how they calculate the taxi fares. Students were given the formula below, but were not shown how to use it at all. Cost of the fare = Distance cost + Waiting Time Cost + Flagfall It was great to see how students had made the decision that they needed to compile information from several sources realising that they needed to use the time of departure to find out what tariff they would use and then using the tariff information to relate back to the distance traveled. This continuous movement between different sources of information was really great to see. The path they took from this point was pretty different with each group. some were able to work through it themselves and some needed some further prompts, but very few actually gave up on the problem. Even when they did not have much success initially they really took on the prompt and tried to incorporate it into their thinking and that is why I am so proud of them today.
I have been wanting to talk about this for a few months but have only just got around to it now. It was a conversation I had with one of my students whilst he was attempting some difficult questions in class and having little success with them. He had asked for help and I had provided some assistance with the question, however the conversation quickly turned to him wanting me to show him how to do it. It would seem to be quite a reasonable request but I knew that if I did, I would be taking the learning away from him, it was just a question that he had tried and failed a few times and he didn't want to think about it any more. The assistance he was given was enough to get him started, but not enough to show him what he should be doing. This is where he said to me
"This class is pointless, you don't teach us anything, you just make us learn" It was said in a way that I think was supposed to make me feel somewhat bad about what I was doing, that by not showing exactly how to solve the question he believed I was being a bad teacher and I should feel bad about that. However the comment had quite the opposite effect on me. It made me more convinced that I was on the right path with them. As I said to him at the time, I believe that it is one of the most positive things anyone has ever said about my teaching. I think in that moment he had recognised that the schooling process was not about me as the teacher anymore, I was not the central person in the process. He discovered that he was the person central to his learning, it is only my role to be there to support that learning. I have been making a very conscious effort to support productive struggle in my class, to not jump in and save them at the first sign of struggle, to let them to continue to think about it and to try new things to work with others. This comment I believe was a clear indication that I am on the right track. In mathematics, I think the focus on getting the correct answer has permeated much of maths education, with a bit focus on tests, worksheets and textbooks in a lot of classrooms around the world it is easy to see maths as a range of questions that you need to get correct answers to. I can tell that students believe that this is the case as often all they write in their book is the answer thinking that it is that number that I am only interested in. This focus on trying to find one magical number amongst all of those infinite possible answers I believe is a major contributing factor to maths anxiety. I can see the nervousness on the faces of students as they ask me "is that right?". I feel that if I say that it is correct then they will feel a sense of accomplishment, but the level of anxiety will come back on the very next question. Conversely if the student gets the question wrong then their whole world tends to cave in, and you start to get comments such as "I can't do maths", "I'm stupid" and "this work is too hard" creeping in. Their whole selfconcept as a learner of mathematics tends to hinge on whether each question they come across is correct or incorrect, this selfconcept can change from question to question as they get questions correct and incorrect. With a quick google search I find the following definitions for the word answer. What I find really interesting about this is that when you look at it, especially in the first definition, it mentions nothing about correctness, it simply describes an act that you do in response to a question. It also points to the idea that an answer is more than a single statement response, it includes how you thought about it, your solution, how you thought about the problem becomes an important part of the answer. So in this way we can define the answer to a problem as the point where you felt you could not add any more to your thinking about the problem. The answer is not some magical number, it is where your thinking about the task stopped stopped.
Therefore I tell my students now that I don't care what the answer is, that is not important to me, what is important is your thinking. By looking at their thinking I can see how they thought about the problem, what mathematical thinking they may have used to solve it. By looking at where their thinking stopped I can see where they believed was the end point of their thinking, the point where they believed that there was no more to do. If they are multiplying fractions and have followed a correct line of thinking but have not simplified the answer it is not wrong, it just tells me that they did not recognise that they could do more thinking about the solution that they had come up with. So why is this so important? I have also pointed to some of the reasons such as the anxiety it causes. But I also think it is because students believe that if they don't get the correct answer, then they haven't learned anything, the time they spent getting a wrong answer they feel is useless. But wrong answers can teach us a lot. Often with more complex problems mistakes are expected and celebrated as they gives us more of an insight into the problem. If you are sitting there for an entire 90 min double lesson and working on one complex problem without giving up there is a lot of learning that has gone on there regardless of whether you have got a correct answer or not. The simple act of trying different approaches means that you are learning a lot and using a lot of mathematics in some highly flexible ways. On multiple occasions through my teaching career I have seen a number of students get the correct answer with some incorrect thinking, and an equally high number of students get the wrong answer with highly insightful and correct thinking. For me now it is really about changing the messages that I give to my students, it is no longer "what is your answer to the problem" it is now "can you tell me about some of the thinking you had in relation to this problem?" I think that most of the time it is not the highly positive praise that drives you to be better, it is the highly negative criticism that drives that growth, especially if it is criticism regarding something you really care about.
For me this moment came as a preservice teacher in one of my teaching practicums. In my final report for the teaching prac one of my supervising teachers described me as "dispassionate about teaching". To be honest at the time I was devastated with this comment, however I also thought the comment was unfair as given the teacher I was working with I know the comment was one somewhat based on me having a very different teaching style to them. I have always loved to teach, I taught my brothers from a young age and have known I want to be maths/science teacher since year 8. My passion for this profession started at a young age which is why the comment was so soul destroying to me. However, in some ways, even though I still hold a great deal of anger towards this teacher, I have to also thank her. It is a comment I think about regularly and in many ways it always keeps me honest, and it keeps me improving. With everything I do in my classroom and in my job, I think about whether it is showing enough passion, is it my best work, is it working for my students, how can I make it better next time, what went well and what didn't. The thing about praise is that it gives no incentive for change if you go about your job day to day and get praised for what you are doing, then it gives the impression that what you are doing is enough, or even more than enough. So if what you are doing is enough, why would you try harder, there is no indication that there is anything that needs to change. Criticism on the other hand has the opposite effect, by definition it suggests that there is something that needs to happen, some change that needs to be made. Criticism demands you to grow as a person or grow as an employee if you do not want to continue to be seen in that light. If I was currently thought of as a dispassionate teacher by students or colleagues I would be quite horrified as I don't believe it was me all the way back then, and I do not believe that is me now. However if I am to be sure that I am thought of as a person with a passion for teaching I need to set myself on a path of continuous improvement. Central to do that is to constantly seek feedback from my students and colleagues and to focus predominantly on the negative feedback. But it also requires me to be very selfreflective in my practice. By working to continuously fill the gaps in my own practice it will ensure that i will not become complacent in what is a very important job. With the new school year starting for me in just over a week I have been thinking about what I want my mantra for the year to be. What is it that will not only raise the achievement of my students, but also that of my own work and the teachers in my faculty. "Fail more often" is going to be my starting point for the year as I think it has the potential to be the path to greater success. This mantra is similar to what was said during my leadership training with Education Changemakers, quite often the program facilitators Aaron Tait and Dave Faulkner would say "Don't worry, be crappy". This idea really resonated with me and fit well with some current reading I have been doing on the Growth Mindset by Dr Carol Dweck.
The idea of failure, no matter how small, is debilitating to lots of people. They would rather not try and avoid failure, rather than give it a go and having the possibility of not being successful. Why is this... because generally as a society we are very intolerant of failure, it becomes a label on the person rather than a commentary on the situation. Instead of "I failed to get this right", the internal dialogue becomes "I'm a failure", instead of "that program didn't really have the intended outcomes" it becomes "They are bad at their job". This view has to change, students need to see that failure is a major component of success, it is not the opposite of success. They need to see that failure is an important part of the learning process and that it is vital to fail if you are going to identify and rectify those areas of weakness. So why do I think this idea of failure is so important, well I have a few main reasons. You will have to forgive me with this post, I am a bit of a fan of a nice quote and this post is going to contain a few
I am curious to see how this goes and how students and staff respond to my quest to fail more often, but I will comment on this as the year progresses. 
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
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