Where I live in my part of Australia we are down to our last two weeks of school for the year before their long break over Christmas. Over the year I have been really trying to build their confidence in their own understanding and in the answer they are presenting as being a correct solution. A favorite phrase of both myself and Angela, the teacher I am team teaching with, is "convince me". When they come to us to check an answer, we don't want them to come to us for a tick or a cross. They know we are trying to build a culture where we examine not the answer, but the thinking that led to that answer. It is not their job to get the right answer, but to convince us that their answer makes sense. If the logic leading to the answer is sound and makes sense, then it also follows that the answer should also make sense. What I have found through this process is often is pretty easy to derail them. This is not something I do on purpose but often happens as I critique their reasoning. When I ask them a question about their solution they automatically assume that the answer is wrong rather than seeing it as me wanting to know more about their solution. Early on in the year I found myself needing to tell them it is correct but also let them know that I was not clear on how they got there, that there were gaps in the reasoning, knowing it is correct gave them the confidence to justify it further. As the year has progressed I have stopped telling them whether it is correct or in correct and have had them determine that through their justification. Their confidence with this is still developing, it is hard to really push your position firmly on whether it is correct if you have the uncertainty as to whether that really is the case. But I want that for them, I want them to get to the point of being confident in their mathematics and being confident in their reasoning even if they are not sure about whether they are correct. Today this culture was really strong in class, the confidence of our students in their answers and in their reasoning was firm, there was no way of derailing it, not today.
They first needed to use this information to figure out as much as they could about the missing numbers. This first part of Act 2 is not enough to figure out all the mystery numbers so it was only then that I showed them the video in the middle above (ACT 2B) which was how the mean, median and mode changed as the mystery numbers were added. The many ways that my students rocked this taskAfter only Act 2A
After Act 2B This one stumped them a bit at first, they didn't initially see how it could be useful, but in drawing them together and getting them to determine that they could use the mean effectively in that situation to help, many were away and gave it a good shot. The story I want to tell here is of two students who were able to use this Act 2B to find the 5 mystery numbers. They were really excited about it and went to check their reasoning with Angela. Angela had found a flaw in this Act 2B in relation to how Excel handles these calculations. I am not going to give away too much on this as I didn't pick it up and it is a nice one to think about, but I would be keen to hear about it in the comments. So when these students went to Angela to check and she asked for them to convince her, she said that she was not convinced as there was a part that did not seem to follow the pattern. When they noticed the inconsistency they thought they were wrong and went back and checked it all. As they began check it something interesting happened, rather than thinking there was something they were missing they started to get more and more convinced that they were correct and they said No, we are right, the data is wrong. This was amazing to me. They had developed so much confidence in their answer that they were happy to say that it was not them that was wrong, it was the way that the Excel was calculating the answer that was wrong, they were doubting the calculator instead of their own thinking. They were right, it was wrong, they had found a flaw that I knew Excel had, but I had failed to consider in designing this task, and I now owe them a chocolate which I am more than happy to do. I couldn't be prouder of my class today and it makes me really sad that my time with them this year is coming to an end, but it also makes me really clear of how far they have come this year.
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This post is one I have been meaning to write for a while, at least 2 years in fact, I have wanted to share it as it has had a dramatic impact on my own classes and also on that of other classes in my school, I guess I just never have gotten around to writing it down. Teachers at our school have made the decision to not to use textbooks to teach mathematics in the middle school. I think that this move has been a very positive one for our school. It is not that I feel that textbooks are evil, in fact I feel their quality has improved in recent years. However I feel there are still some parts of their design that impact on how people with engage with them, staff and students alike. When making the decision to no longer use a textbook for maths class I needed to find an appropriate replacement. Textbooks are good at providing lots of questions for students to practice the skills they have learnt, and this practice is important. However I needed to find an option that did not have the excessive scaffolding and the extensive quantity of questions that I saw in most textbooks. Many of the worksheets I found online has the same characteristics at the text books, so I began to think about how I would design problem sets myself. Designing the problem setsWhen I sat down to think about how I would design my problem sets I wanted to keep some design criteria in mind these criteria were:
One of the question sets I came up with are shown below. Underneath that image I will explain the design of them.
Impact of the problem setsGreater levels of metacognition One of the most interesting observations I have seen from students engaging with these problem sets is that they seemed to become much more metacognitive. This is evident in where they choose to start with the question sets. Some have talked about believing they can do the level 2 questions, but really want to work on a few level 1 questions first to make sure they have it. Others have attempted the level 2 or three questions, given it a go for a while, haven't made progress and have moved back to try the level 1 questions. There have also been students who have looked at the level 1 and 2 questions and have made the determination that they know how to do them and have spent their time only working on the level 3 questions. This process of having three different levels of question to choose from has made them much more aware of themselves as learners and of what they need to do to move their learning forward. All students have had the time they have needed to work on their questions of choice In looking at the problem sets with my class now, I feel that they all feel as if they have enough time to work on the questions they have chosen and feel comfortable attacking. If I take the level 1 questions for example, I know that some could be through those questions very quickly and there are some that will take much longer. If I look at my classroom about 5 years ago I would say that the time I gave them to do the questions was aimed at the the middle, the ones who had it finished early and got bored and the ones who were struggling never had enough time to finish them. With students working on different difficulties of questions they all seem now to have the time they need to finish the questions. They are attempting much more challenging work The comment I get a lot when students are working on these problem sets is "why do you have to make it so hard". This is normally from students who are working on the level 3 questions. My response in this situation is always the same "Doing the level 3 questions is your choice not mine, so you are making it hard on yourself" to which they normally reply something along the lines of "yeah well those other ones are too easy". What is clear through this is that they they are no longer just happy to do the easy ones, the could do that and finish really quickly, but they don't. Students seem to be really working on questions that they feel are just beyond their current level of understanding and they are striving to understand them. I can now see the slow, deep mathematical thinkers Having done the Jo Boaler courses one point that is emphasised a lot is not to make maths about speed. It talks at length about how mathematicians do maths, how they are deep slow thinkers. Reflecting on previous practice I realised that for those deep slow mathematical thinkers, there was a time that my classes did not offer anything to those students, I didn't even know they were there. I didn't see them because their diagnostic data backed up what I saw in class, but that was because I was asking them to work in class in similar ways to which the test was administered. What I am seeing now is that there are quite a number of students who do not perform well under the time and pressure of diagnostic testing, but have flourished with these problem sets as they have the time to sit with the problem and think about it rather than being pushed through endless problems. They are showing much greater levels of mathematical thinking than some others who score much more highly on those high stakes tests. The quality of their work is much better I have seen a noticable improvement in the quality of the work I recieve since using these problem sets. Much of it I think can be attributed to giving them problem types that allow them to demonstrate their understanding and then the time to work on them. What I find challenging about these problem setsDespite the massive, dramatic change these problem sets have made to my class, I still struggle with some aspects of them. Most of these are still the tug of war I have inside myself about getting the balance of mathematics right in my classroom
A lot of the activities I have talked about on this blog, or the ideas I talk about, are generally ones I have put significant time into. The activity I am talking about in this post has been a very successful one for me over the last few days. I would like to say that it was because of the careful and deliberate planning I did on the task prior to the lesson, but that would be a lie. Sometimes as the lesson is unfolding you see an opportunity present itself, and by following it through, sometimes planning the next step on the run, you can have a really good lesson. Planning the next step on the run was not a result of being disorganised, but a result of identifying an emerging need and recognising the need to follow up on it before moving any further.
They then had a conversation with their table groups about what they thought would be the colour of the other 5 blocks. Not surprisingly all the responses were some combination of red and blue blocks, however despite knowing that there were lots of other colours of blocks in the storage tub, they had not considered they could be part of the final five.
Next I wanted to introduce the idea of using probabilities to describe exactly what is in the box so I gave them the information listed below and then gave them the time in groups to have the discussions required to figure it out I have filled the box now with blocks according to the criteria below, can you tell me how many blocks of each colour are in the box? What was exciting about seeing them work on this was that I finally saw the classroom culture that I have been trying to build all year with them. In working together on the problem they were talking about the problem, they were critiquing each other's reasoning, they were asking questions of each other, they were willing to tell the group when the explanation still didn't make sense to them, forcing the person giving the explanation to justify their thinking more strongly. I think one of the most important aspects of their work though was their confidence with their answer. With these sorts of questions when they tell me they have the answer I try to ask a few questions to head their thinking down a line that creates some doubt that they have found the answer. This isn't done to trip them up, but is more designed to see if they have got to a point where they feel the have considered everything and have come to the only answer that works, to gauge their confidence in their own thinking. Normally when I ask students a question about their answer they take this as an indication that their answer is wrong, but this time, no matter what question I asked them about their answer they had confidence with it as they had determined that 12 was the only possible number of blocks as you could not have parts of blocks. The next day the aim was to move towards students being able to determine how to calculate the probability of pulling a block of a certain colour out of the box, to help facilitate this i made up a simulation of the box compostion from the previous day using excel. The excel file and a screen shot is shown below.
Like the day before i got students drawing the blocks from the box and this time we tracked the probabilities. when the first green block was drawing I asked them to predict what would happen to the probabilities of each colour.
Overall what started as a sidetrack, developed into some really great thinking, on some really important concepts and I couldn't be prouder of that lot today.
This lesson was one of last lessons on a week long problem on circumference, a video of the set up for the lesson is here and a blog post explaining more about it can be found here. If you haven't done so already take the time to watch the video below, but more important than that I would like you to listen to the video below... This video brought a massive smile to my face, in fact it is still there. Teaching can be tough work, but it is these sorts of moments that reaffirm why I have chosen it as a profession, it brings nothing but good stuff to your soul.
I didn't realise the reaction was this strong when we were doing it, I was wrapped up in ensuring we finished off the task before the lesson ended. So when I went back and watched it I thought "wow, the engagement here is really high"... but then I listened again, and it is more than that. The reaction those students are giving to this task goes beyond engagement, they are invested in it, they care deeply about it. Activities that will have this level of "buy in" are hard to cultivate, but when they work, when the invest themselves in the task, the learning is exponentially more powerful. The other day I did something that I have been meaning to do for quite a while now, I deleted every maths test from my hard drive. I did not do this selectively, I searched for the word test in my maths folder and deleted everything. I haven't done a test in any of my maths classes for the last 3 years (other than the compulsory 1 or 2 all maths teachers must do statewide) and can't see myself wanting to do any in the future, it is not a practice I place any value on any more, so this step is me wiping the slate clean. I have come to this decision by watching my students engage with tests over the years, and in more recent years through what I have learnt about the 'science of learning'. By the 'science of learning' I am referring to the interplay between the neuroscience and the psychology and how this impacts on learning. In particular in recent months my focus has been on developing a stronger understanding of the role of maths anxiety. To be honest I knew that maths anxiety existed, however it wasn't until I saw the video below that I realised how prevalent it was, nor had I taken the time to understand the neuroscience that underpins it.
.The anxiety felt is a result of increased activity in the amygdala, this part of the brain is responsible for processing our emotions, including fear, and for determining what memories are stored in our brains. This consequently reduces the capacity of our working memory which we use to make sense of, and solve, mathematical problems. Because our working memory is reduced we then do not perform as well as we could on that test and this further raises our level of anxiety and the cycle continues. All of this is symptomatic of maths anxiety regardless of the maths being worked on, however the effects of maths anxiety are further emphasised if a time limit is put on the piece of work like what occurs in tests.
I guess what prompted me to think a little bit more about why I hadn't just deleted all my tests was in watching the documentary "Race to Nowhere". In that documentary there were numerous occasions where kids spoke about cramming the information into their head for just long enough to put it down on the page but only a week or two later they could no remember the work at all. This is not the type of learning I want for my students, they deserve better than that To make that final move of pressing the delete key I needed to ask myself a few questions about tests, these were
However to finish this post off I want to address something you may have thought as you have read this which is, "they are going to have to do tests/exams eventually". I am not naive, I know that eventually, the kids I am working with be required to do tests and exams in their final years of schooling, I wish they didn't have to, but that is the reality of it at the moment. However I also believe that the putting them through timed tests regularly for many years before that is not the best way for me to prepare them for tests later on in their schooling. Kids need the time, the space and the opportunity to develop robust and flexible mathematical knowledge by understanding the connections between key mathematical ideas, this is where I feel the secret lies for success in tests many years down the track, and developing this understanding is where I feel that my role lies. This focus on understanding rather than speed may help them in their tests much later down the path, but more importantly it helps them out a lot with what they are doing right now. I am in my 14th year of teaching, when I look at how my own teaching it is sometimes hard to know just how much it has changed. But sometimes some small moments that bring that into real clarity, for me this moment was when I was clearing off my hard drive to make room for new stuff as it was getting too full. As I was going through my hard drive I sorted it by age, I figured that if stuff was going to go, it was probably the oldest stuff that hasn't been accessed or edited in a long time. I realised that I still had on there all the stuff I developed in my first year of teaching over 14 years ago. I was curious about the work I gave to my students back then, I was curious about how far I had come since then, so I had a look at it. When I looked at it I noticed a few things
When I reflect on that work now, I realise that at that time I didn't know any better, I didn't intended to do this to my students, I didn't think i was doing a bad job at the time as it was how I learned maths, it was what I learned at university in my preservice teaching program and it was what was expected when I was a student teacher. All of the models and examples around me for maths teaching that I was using to establish myself as a teacher were all saying the same thing. So out of interest I printed out just the practice problems that i am going to give to my students in the coming term of 10 weeks and then printed out all of the practice problems that I gave to my students on the same topic all the way back in my first year of teaching and the gif at the top of this post is the result of that. 14 years ago, for my 10 week unit of work, I was giving them 70 pages of practice problems. When I then looked at what I am planning on getting them to do the page count came out at 8 pages of practice problems across the 10 weeks. Just this little bit of data has made me reflect on what has changed in relation what I value now that I didn't value then. I am not expecting any less learning from them, so how am I spending the time that those extra 62 pages would have taken up. This additional 62 pages morphed into...
So 2 stacks of paper and 14 years later I have realised I am now asking them to do less busy work, so they can engage in a lot more, and a lot deeper thinking.
The last 2 weeks of my life has been monopolised by one question, a question for which I am still not confident in any answer, a question asked by my 5 year old daughter who just started school this year. One day just after I had got home from work and I was beginning to make dinner she asked me "Daddy, how do you count to half?". The question immediately intrigued me, because as a maths teacher I felt I should be able to answer that question confidently, after all it is just about counting, but I found my self unsure of how to answer her. I really wasn't sure if the answer was " you can't count to half, one is the first counting number" or whether to tell her "you can count to half by counting by numbers smaller than one", I just wasn't sure if a number less than one could be denoted as a count. So I didn't answer her at that time, I told her what a great question it was, it was such a good question that she was able to trick me, even though I teach maths. But I promised her that I would also speak to some other maths teachers to see what they thought about the question.
I went to school the next day and asked every maths teacher at my school the same question, and they all gave me the same look, kinda puzzled, but also very intrigued, it was clear that, like me, they had never thought about counting by anything less than 1's. Most of them were willing to give an answer, but they did not give the same answer, and when I spoke to them about the other thinking I had on the question they also became unsure and reconsidered their initial position. I also asked a lot of primary colleagues in different schools as I figured that they ultimately had the responsibility around teaching counting so I felt they would have a better understanding than me. But again, they could see both sides of it,. I went to my resources on how concepts develop and could not find anything there, I even went to Google and could not find the answer anywhere on there. When I told my daughter all of this she was really amused that all of these teachers could not answer it and even Google and Siri couldn't answer it. So as a final ditch effort I took to twitter and tried to tap into the collective knowledge of the MTBOS, a global tribe of mathematics teachers.
But again I had no luck, lots were intrigued by the question, and lots told me how I could talk to my daughter about it, but that is not what I was after, I know how to talk to her about these things, I wanted to know whether they felt that half was a number you could count to.
The discussions have continued for quite a while now, and are still going, the more I think about it the more I think i have settled on an answer, but as one of the teachers I have been working on this closely with put it "even if 90% of the people I talked to disagreed with my position, this is one question where I don't think I would be convinced by that enough to change my mind".
I still feel like this question only has one answer, I feel that if I was asked to count what was on the plate and asked 100 other people then we should all get the same answer, but that has just not been my experience with this one innocent question from my daughter. I think if she asked most people, it would be so easy for them to give an answer without giving the answer much thought, they wouldn't even think twice about the complexity of the question, so I am really glad she asked me first. It made it really clear to me about about how closely we need to listen to these questions, even from learners who are just embarking on this learning journey. What was clear from this question was that she had been thinking a lot about counting and also the number half, she had figured out that half was a number, but had never used it in counting. She had identified what was a gap in the concept for her, but interestingly enough it also made me discover that it was a conceptual gap for me to as I had never considered it. I have been teaching mathematics now for the last 14 years, and have had several roles over that time including classroom teaching, curriculum leadership and mathematics coaching on a school and regional level. I feel I have grown a lot in those 14 years, to the point that I do not recognise that teacher that I was all those years ago at the start of my teaching career. But I also like to think that I do not even recognise the teacher I was 4 years ago. There have been two groups that I have learnt an incredible amount from over the last four years, one is is the MTBOS, an online tribe of mathematics teachers who I have been following for over a year now. The other is The Flinders Centre for Science Education in the 21st Century, who are responsible for the success of the of the Empowering Local Learners Project in the region in which I work. I have been writing this blog now for a while, but only a little more seriously in the last year or two. I don't think I ever intended anyone to read this. In fact I probably hoped they wouldn't. Writing this stuff down has helped me to make sense of my teaching, my professional growth and how the thinking of my student's develops over time. I could have done that offline in a notebook or document, but in many respects putting it online in my own mind meant that I was making myself accountable to the work I was doing in my class as anybody who stumbled upon it could read it. However I haven't taken the plunge into trying to share it more widely as I still don't think it is as good as I would like it to be. However with some of the talk on MTBOS at the moment it seems that many are in the same boat as me. They have been silently watching the MTBOS for a long time, and are learning a lot from it, but have never posted anything. I know it is an incredibly supportive group who want are keen to tap into as many ideas from Maths teachers as possible, but the fear has been around how much I respect this group and in hoping that I am honestly applying their ideas to my own practice within my context. So this post is about me being brave and jumping in. The aim of this post is to hopefully share how much I have learnt through the lens of just one task on algebra, in fact it was my first task that I did on algebra with my year 8's. I am not sharing this task because I think it is a fantastic feat of mathematical teaching brilliance, I am sharing this task because how I approached it has become a normal part of my practice, it is how i try to approach all lessons. Therefore it represents how I have fundamentally changed my views on a lot of things from just a few years ago, growth I definitley wouldn't have seen without the aforementioned networks. I want to first give credit to where some of the ideas for this lesson have come from and then describe the lesson and the student thinking The Credit I know I will forget to mention people in this section, you won't know it, but I have taken so many ideas and lessons form so many people that I find it hard to keep track of what I have taken from who. What is represented below is not all the things I have learnt from these people, but rather just some of the things that are pertinent to this lesson Angela, Kristin, and Deb I really appreciate, more than you will ever realise, the frequent conversations we have. They constantly challenging my thinking, keep me honest, and they consistently keep me reflecting on what it is exactly that I am trying to achieve. They have helped me to bring clarity to my teaching and to iron out many of the bugs. Your expertise in task design and the effective use of questioning in class discussions, through the lens of executive function has help me to ensure that it is the kids thinking that is most strongly represented and valued, it is their voice, not mine that is heard most often. I have also appreciated the strong evidence base to our work. Thinking about the research that sits behind the work has helped me to think about my practice in some very deep ways, on my own I probably wouldn't have had to the time to come across this research myself. Jo Boaler (Stanford University) Your online course for teachers has changed my practice in a lot of ways, but probably for algebra more than any other. The clear importance of visual representations for building algebraic understanding through the examination of patterns is not something that I had given enough thought to until I did the course. This one change has done more for improving kids understanding of algebra than anything I had done in the past. Dan Meyer (MTBOS) Your work has influenced me for a long time but in recent times your talk on "beyond relevance and real world" has had the most profound impact. The idea of a maths dial that you start turned low approaching the content from a very intuitive base has allowed many more or my students to have an entry point and feel comfortable sharing their mathematical thinking, it has given a voice to those who didn't feel they had a voice in a maths class previously Fawn Nguyen (MTBOS) Your visual patterns website has been a great resource and an effective framework for my algebra work and hopefully this is represented here. But much more than that your passion for the students you teach and your drive to authentically and honestly represent the learning that goes on always keeps me accountable to trying to do the same for my students. Andrew Stadel (MTBOS) Your talk on the classroom clock has helped me to bring much greater clarity for how I choose to spend my time in my lessons and how I can further prioritise more time towards those more effective practices. It has helped me to regain the time I needed to give my students more time to think. Robert Kaplinsky (MTBOS) The #observeme process has really helped me to open up my class to others and to get the feedback that has helped me to grow as a teacher. Although I have not had many visitors, I am inviting many more people in and simply the process of writing the sign has helped me to articulate a vision for what I want my class to look like and what I am striving for. The Lesson The kids arrived at my lesson today on the first day of term, I stood outside to class in the common area to greet them. They asked if they can go into class, I told the "no we are staying out here", they asked what we are doing and when I told them algebra the groans went up almost in unison. It always happens, most kids seem to hate and fear algebra, so my first step is to attempt to take that fear of it away by showing them that it is not really as daunting as they think it will be. I can understand why they fear it, in my experience algebra has always been a topic that teachers feel the most need to teach from a proceedural base, they find it harder to develop conceptually. Algebra becomes about moving numbers and symbols around a page rather than really getting into what it is about, patterns and relationships. I started by isolating one table in the common area and asking them how many chairs they feel would fit comfortably around it to which they replied 4, I then put two tables together and asked again and then three, creating the pattern below. When I moved these tables I intentionally moved the chair on the right out of the way, joined the next table on and then moved that same chair back to the right to help develop the idea of a constant (left and right chairs) I then brought two additional chairs in to put on the top and the bottom to help develop the idea of the rate of change and also how the pattern is growing. As a starting point they were then asked to predict how many chairs would be around 4 tables and the quick answers provided were 10 and 11. When there was not agreement, rather than telling them what was correct, giving them a few extra minutes to discuss which was correct was enough to bring them all to the same thinking. Just giving them the time to stop and think, and to refine their original quick thinking has been valuable, they often answer quickly looking for me to give them the answer rather than taking the time to think more carefully. After settling on 10 as the answer for the 4th in the sequence their work for the task was to answer two questions.
The questions are very closed, but intentionally so, this was their first lesson on algebra and they had not seen questions like this before to the aim was to build my starting point, to see how they approached it, to see how I could build their intuitive understanding of the pattern into a deeper understanding of the concepts that sit behind the pattern. This is a task I could hook all of the learning on for the rest of the topic. With limited or no input from me they were away and the approaches varied, but the three that were most prevalent were the ones that I have shown below Some had attempted to draw the 54 tables and tried to count the chairs, and that is fine it worked for them. They hadn't noticed a pattern from the first three, but what drawing them all out allowed them to do is to recognise that they could then simplify their picture to one of the ones below it. The processing of drawing them all was the light bulb moment they needed. The thinking needed to be captured and shared, but it also needed to be refined, I wanted to take their thinking, which they were happy with but to them did not represent the algebra they were used to seeing and to transform their thinking to that more pure algebraic form by helping them to strip aspects away. The image below shows the process it took to do this It started with their visual models and their calculations. As they talked about how they did it we recorded their words. We started a process of removing some of the words that they felt were not necessary in describing the pattern. By also substituting some words for symbols and then substituting other words for pronumerals they were able to refine their inital thinking down to an equation . The equation made sense to them as they could see with much more clarity how their thinking was represented in that equation, that equation was no longer a daunting and unfamiliar thing. This process was repeated with the second of the two questions, but unlike the first question, there was a great deal of disagreement about the answer. In order to arrange 238 tables we had a roughly even split between the answers of 117, 118 and 119. We have had this at many stages over the course of this year and my response is always the same, first is to ask them whether this is a question that can have multiple answers, if it is then those three answers might be fine, but if not we need to agree on just one. They talked about it and came to the view that it was a question with only one answer, so the second part of this process is to have each of the groups try to convince the other that their process is correct. Whilst there was two or more answers on the table then as a group, we hadn't developed the understanding that we needed to in order to move forward. They were able to lead that conversation and the 118 group was able to successfully convince the other two groups that they had got it correct, there is a lot more power in them owning that process. Those so at the end of this we had four equations that looked very different. They were able to articulate that the operations use the same numbers but were the opposite operation ( instead of +) when you are trying to find tables rather than the chairs. This insight will serve us well as we move into solving linear equations at a later stage in the topic. However in looking at those equations I was not yet convinced that they were in fact different representations of the same thing so I asked them "if I graphed each of these" what would I see, all different graphs, 2 different graphs, 1 repeated graph?" Some thought they would all be different because all the equations are different and others thought that the ones on the left would look the same and the ones on the right would look the same. plugging these into Desmos it became clear to them that they were all equivalent statement and they were able to articulate that they had to be the same graph because they were all representing the same problem, the same pattern.
I was happy with how this went, they did really well, there was a lot of thinking here and a lot I can tap into as we build this understanding as this topic progresses. I have been thinking a lot lately about the best way to address student misconceptions on the unit of work that they are currently studying. Ideally I hoped these misconceptions wouldn't develop in the first place, we spend a lot of time discussing and developing the understanding of the ideas before introducing any more formal procedure, and when we do introduce a procedure, I try to make sure it is theirs, not mine. The aim of doing this is to ensure that we have thrashed out all of the ideas and tested their veracity so that the misconceptions can be brought to the surface and addressed as the new content is introduced . However in looking at the work I receive from students from time to time it is clear that misconceptions creep in, despite my best efforts, I may not have covered all the bases. The other day I tried to address misconceptions in a different way to had previously. The misconceptions became clear to me as I was looking through their books I picked up some misconceptions that were common across multiple student's work, but instead of talking about these in class and trying to correct it myself, I wanted them to notice and to correct the errors. The hope in doing this was to get them to think more deeply about the errors. To do this I wrote up some solutions to questions like they were legitimate solutions to the problem, but in reality these solutions incorporated the errors that I saw in their books, these samples are shown in the images below. Each table was given a different proposed solution, these were put in the centre of the table and students were encouraged to discuss the solutions in their table groups. It was interesting for me to see that when we had the discussion about the questions every group was happy that their solution was correct, none of the groups believed there was an error in the solution. To me this was very interesting, these errors were prevalent in a number of students work, but they definitely were not there in all of them. Even those students who had answered questions similar to this correctly in their books were not able to identify the error in these solutions. So this was quite a surprise to me.
I was caught a little off guard by this, but I also had to think carefully about the way forward. At this point rather than telling them what the errors were for each problem, instead I just said "what if I was to tell you that every question on the table is incorrect. Knowing this, what is the error in the problem that you have in front of you". This seemed to surprise them quite a bit, being utterly convinced the solution was correct and raising the possibility that it is incorrect created that conflict in their mind, and stimulated a lot of discussion as they tried to find the error. They did not find this easy but eventually were able to notice the mistake. What I hoped they would get of this process is exactly what was achieved, they were then able to successfully examine their own work, and their own thinking, and correct any of the mistakes that had been previously made. They were able to look at their own work, which they thought was correct, and find the changes that they needed to make. I feel that it was a much more powerful way of looking at the errors they made, as the fix was not handed to them, they still had to work for it, they still had to own the learning. I was reading a post on Dan Meyer's blog the other day titled You can't break math. There were a number of aspects to that post that resonated with me, but I came across the passage below that really made me think about the subject I teach in a slightly different way, this passage is.. One advantage of my recent sabbatical from classroom teaching is that I am more empathetic towards students who don’t understand what we’re doing here and who think adding 2x to both sides is some kind of magical incantation that only weird or privileged kids understand. I started thinking about my own experience with my own mathematics education and in many ways the approach I took to my first few years of teaching. I realised that when I was learning maths in school, it was presented as magic rather than logic, and that is the way I presented it in my first years of teaching. My teachers didn't intend to present it in this way, and neither did I.
Part of the beauty in mathematics is in the patterns that emerge and in the certainty we have in our conclusions. However in the highly formulaic way that many classrooms still operate, more time is spent practicing how to use the formula under the guise "trust me it works", rather than spending the time to get students to develop a sound line of reasoning where that formula is the only logical conclusion. The formula is simply the highly refined end point of a lot of thinking about a particular mathematical idea. That formula is the point at which all the uncertainty in their argument has been stripped away and what remains is the pattern that has emerged. When we present that formula without the thinking behind it, without developing that understanding prior to presenting the formula, this is when when it appears that we have 'pulled a rabbit out of a hat', it is something that has come from nowhere. However it is important to remember that even magic is not magic to everyone, this has become abundantly clear to me through watching a lot of the TV show Penn and Teller Fool Us. For those who do not know the show, a range of aspiring magicians come on to a show and try to fool a world famous pair of magicians. Most of the time, these magicians cannot fool them. Even though the trick has been previously unseen, these magicians strongly understand the principles of magic, concepts such as misdirection and sleight of hand, so even if the trick is unseen, they can unpack the thinking that may produce that result. Their knowledge of these concepts is so strong, and so flexible, that they can apply them to any trick and to create new tricks of their own. This is what I want for my students in relation to their mathematical knowledge. More than just understanding the individual tricks, I want them to understand the underlying framework of the mathematics they are studying so that they can apply it in unfamiliar and sophisticated ways. 
Mathematics Coach and Coordinator in Regional South Australia. Current driving the Empowering Local Learners project as a numeracy strategy from preschool to senior secondary.
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