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
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 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.
This post is corresponding with the start of a new school year. For a few months now I have been watching the #observeme hashtag on twitter. The trend began with a teacher called Robert Kaplinsky (@robertkaplinsky) from the US and he has written a blog post about it and presented a 5 min talk on it. When I saw this begin to appear on twitter I was really excited by the idea.
I have been an instructional coach focused on mathematics for a number of years now as well as the mathematics coordinator at my school and have become very used to working with teachers and providing them with regular feedback. I have been fortunate enough that those teachers have felt that the feedback they have been given has helped the to move forward as a teacher, they found it to be a rewarding experience. But over that time I have also realised that I have not received the same level of feedback on my own teaching, any feedback I have received has been limited, and very general in nature, I have not had the same opportunities to grow in my teaching by using the feedback of others to improve my craft.
So over the last few days I have been slowly chipping away at my own sign. I was really excited to work though the process. However I came to a realisation near the end of completing the sign. This #observeme movement is much different to the normal process of teacher observation and although the normal process is more formal I think #observeme is much more daunting, more than I initially realised. This is not yet based on putting it into practice, the school year hasn't officially started yet, but it is based on what I feel it has the potential to be.
With more formal observation processes the visits are arranged in advance, your observer and you decide on a time, a place and in many instances a focus. Early on in my career I put a lot of extra work into making sure these formal observations were top notch lessons, they pulled out lots of bells and whistles, I wanted to really nail them. Over time I came to the realisation that this didn't help me as it was not my normal way of teaching at the time, if I tried to teach that way all the time at that stage of my career I would have burned out. Any feedback I was given at that stage was not based on my normal teaching but based on my inflated teaching, I saw these observations as a threat rather than an opportunity, and therefore it did not help me to move forward. Eventually I came to the realisation that I just needed to teach my normal lessons, but looking back on it over the course of writing my own sign I realised that I was still subconsciously teaching to what I thought the observer may have wanted to see, but this is only because I knew the exact time this person would be visiting, it primes it in your mind.
This is where the difference was in making up my #observeme sign. I came to the realisation that this observation and this feedback could occur at any time and without prior warning, I could not predict when it might occur and therefore I had no way of specifically planning for it, consciously or subconsciously. Anything those teachers see, and any feedback that I receive will most likely be based on the truest representation of my practice as a teacher, it will give me the greatest insights into the quality of education I am providing for the kids that I teach. Those goals that I have committed myself to on that sign cannot be covered by isolated activities once a fortnight or once a week (not that I am aiming to do that anyway), they need to be a strong component of most, if not all of my lessons. Someone should be able to drop in at any time and provide me with feedback on any or all of those three goals. Therefore I needed to make sure that I was really comfortable with these as goals.
lemsHowever at times I will want them all working on the on task. These tasks are normally the ones that will be super important for establishing the conceptual base. In these cases I would not want different students working on different tasks, instead the task will be differentiated through the the questioning that I use to support, probe and extend their thinking. The use of enabling and extending prompts will be important in achieving this appropriate level of challenge.
2 - Learning Conversations
I put a lot of work into designing activities that aim to build an understanding of mathematics on a conceptual level rather than just a procedural one. However the key to extracting the greatest amount of learning from these tasks is the conversations that go on in class both between students and between myself and the students. It is important that their thinking is represented strongly in these conversations, not my own. I want them to be able to develop their own reasoned arguments and then use the discussions with others to test the veracity of that argument. I realise that in the past may of my conversations in class have been getting students to share and talk about alternative pathways to solving prob, which is important and a good start. However I also recognise that I have not spent as much, or enough, time getting kids to critique reasoning. In the past I have had some great success and some spectacular failure with this, most of which hinged on the classroom environment. Ensuring that I develop a classroom environment where critiquing is seen as a opportunity rather than a threat will be vital in moving this forward.
3 - Feedback
High quality feedback is vital in helping to move kids forward in their learning and improving the overall quality of their work. Early on I probably gave a lot of 'autopsy feedback', by this I mean the bulk of feedback I gave was on the final piece of work, at that stage the feedback was not of much use because it was too late to do anything about it, although it could help them in completing the next task. When I started giving more feedback during the completion of the task it was only then that I began to see the quality of work improve. But I still have room to move on improving both the quality and the quantity of my feedback by trying to give much more feedback during the learning, at the time it is occuring rather than just looking as the samples of work after hours.,
As an addition to my #observeme sign I have also created the capability for students to fill out the feedback sheets. I definitely welcome the feedback of the teachers in how they feel I am progressing against those goals. However I am also intensely curious about how students will fill out my feedback sheet. I really like that idea that these signs are not just up there for teachers to see, but also for students to see, when they walk into that class each day from day 1 they will know what I am aiming at and therefore their feedback, is equally if not more important than the staff that visit. They are the recipients of the education I am providing for them and they need to be able to provide feedback about how I am going against these goals and whether they feel these goals are working for them.
In speaking to some teachers about mathematics learning there is sometimes a bit of misunderstanding about what constitutes mathematical understanding. The most common one is that if they can get it right, and they can regularly get it right then that shows understanding. But it is my argument that being able to apply a procedure is very different to understanding the procedure, it is easy to follow a procedure, but not so easy to know why it exists and why it works.
In workshops I run with teachers from pre-school through to secondary I like to use the following problem to explain the difference.
When three numbers are added in pairs the sums are 22, 39 and 45. What are the three numbers?
What I like about the problem is that it is simple enough that anyone from any level of schooling can understand it, but it is also complex for all those people. It also lends itself to a range of different ways to attack it.
When I present the task to teachers in workshops I split the room in half, one half have a go at it themselves with no prompts, hints or scaffolds, whilst the other half are given a very clear proceedure to follow. This proceedure is as follows
Typically I give each group about 10 mins to work on the problem. When I give teachers the method described I get responses as shown below. They normally work though a few of these in the 10 mins.
When I talk to the other group of teachers they have made some progress on the problem, but don't have an answer by the end of the time they are given. So if we were only looking at understanding as being able to get the answer then the group with the procedure would be seen to be more proficient. I give both groups the opportunity to talk about how their group approached the problem and after this I ask them one more question which is
Why do we add the sums together and divide by 2 in the first step?
This question throws them, initially they think it is an average until they realise that there are three numbers and we are dividing by 2. But a really interesting thing happens, the group that didn't have that procedure seems to pick up on why that first step exists and the group who did have the procedure doesn't pick up on it at all. The group with procedure where so used to following it without having to think about it that they could not see the reason for that step. The group who didn't have the procedure, and also didn't get the answer had to think much more deeply about the question, and in struggling with it they were able to develop a deeper understanding of the underlying framework.
In looking at the procedure I chose to answer the problem that first step is not intuitive and I choose that method particularly for that purpose. It is a very simple way of solving it and it takes some shortcuts, both of which are common practices for teaching mathematics in many classrooms. But in my experience, the easiest method and taking shortcuts also leads to a lack of understanding. Realising that you add the sums and divided by two because in adding the sums you have added each number twice, so dividing by 2 gives you the sum of the three numbers is an important pattern that they could have made for themselves had they not been first given the procedure.
A way of solving this problem I have found online and in textbooks is in the way below
Although this does not take any shortcuts, I also think that it tries to make it to "mathsy", it makes it more complicated than it really needs to be, it complicates the thinking to a point which the method of solving it does not closely resemble the original problem. Although some students really understand and can make sense of this method, for many in the past this has just become like the first way, a method to follow rather than something to understand.
For those who approach the question more intuitively and without the procedure I at times give them the following diagram to help them make sense of the question, normally though without the blue dot.
with this image it is clearer why the dividing by 2 is important, but when given this image most do not solve it that way. They look at of the boxes and notice that it is common to 2 sums, 45 and 39. They reason that the difference between the sums 45-39 = 6 is also the difference between the other two boxes. and the method they follow generally follows one of the two ways below.
Both of these methods make more sense intuitively of how to solve the problem and they don't bypass seeing the connections in the problem and developing a deep understanding of how it works. Hopefully as they progress though solving one or more extra of these problems they will be able to progress from the method on the left to the one on the right. Although the endpoint is the same the difference is in the connections they have made along the way and this gets to the heart of the difference between fluency and understanding.
I am mid way though a few other posts on the EC conference but I started this on on the plane ride home. The speakers from the stage at EC16 were people working on a diverse range of things on a range of different levels. Even though each talk was only for 10 mins the stories they told in that time were so incredibly powerful.
I had all intentions of writing some notes as I went though listening to the talks so I wouldn't forget, but I found myself captivated by what they were saying. Not only did I forget to write things down at the time, but I think it would have almost felt rude to do so. Therefore I am writing most of this from memory and trying to fill out the details some two weeks later, so I don't feel I am giving some of them justice..
Hayley and Liz
Most schools have avenues for student voice but how authentically are we actually seeking and listening to that voice. They are who we are go to work for, they are who our efforts are directed towards so their voice needs to be heard clearly and loudly. Many student voice models are run by teachers based on agendas organised by teachers that align with the priorities that the school wants them to work on. But if we go large and truly give students a voice then big things and much better outcomes are possible. Their student leadership model is fantastic. I really loved the ideas of students organising and running their own meetings and agendas and also loved how they research and run professional development sessions for teachers, I also loved the idea that student engagement/learning proformas were designed by the students themselves and used as part of performance development process for teachers
Schools can and should be places that we are not just nourishing the mind, but also the body and soul, they are also places where we can do some good for our planet. Their sustainability program is a great example of what can be achieved in this area. Setting them up with a range of life skills that will serve them well moving into their futures. From her talk I also took away that we really need to invest in what we value. If we promote healthly lifestyles in our education programs then we should also be promoting them though all aspects of our schools, including our canteens, this is hard since good food isn't cheap and cheap food isn't good, but they make it work.
Brett's message was a powerful reminder of the impacts of kids lives outside of school. Some kids that we work with are coping with a lot outside of the class, Whether they are acting as carers for siblings are having to work a lot to support their families who may be unemployed, they are dealing with a whole lot of stuff that at most times we may have little or no idea about. As much as we want their focus to be on their schooling during the day, for some kids they have much bigger things going on and their behaviour at school is probably a result of that. Care and compassion goes a long way to supporting kids.
As a student it was interesting to listening to Amy talk. What became obvious really early on is that she was very aware about what works with her in relation to learning and what doesn't work with her in her own learning and I suspect this is the same for a lot of kids. But are we listening to them? I also picked up from this talk ideas around curiosity and creativity. These are things that we have a lot of when we are very young but it manages to get driven out of us by the time we get out of the schooling system. However the future needs creativity and it needs curiosity, it may even be the key to getting more girls involved in STEM fields
There were a few messages from Jason that I picked up on. Firstly when you empower kids with the same sorts of entrepreneurial skills as we have developed though our involvement with EC then they can show you some really amazing things, they develop a whole lot of skills from that type of thinking, even from a very young age. Secondly was his message around working with and within systems and showing that sometimes you just need to do it and prove it gets results instead of asking for permission all the time, their response may be based on the perceived likelihood of success rather than how successful it could actually be.
Companies can be commercially successful and socially responsible, it doesn't need to be one or the other and we need to be encouraging much more of that. Unlike individual schools or individual teachers the have the resources and the connections to tackle some really big problems. However in deciding what problems they can or should tackle they need to talk to teachers like Yoobi did. Even though most teachers do not mind paying out of their own pocket for school stuff, they identified that teachers shouldn;t have to and that they can help. I loved hearing about their business model, for everything you buy, they donate one themselves to a classroom in need
We all we open up our schools and invite our communities in to learn from what we have to offer, we offer a whole range of things. We offer these opportunities based on what we think it is important for them to know, but is that what they want to be learning? Is that something that they really want to know? Are they getting anything out of it they will find useful. If we want to promote learning as a life long pursuit then surely our school environments should play their part in that. However in educating our communities we are not dealing with a set curriculum, if they are going to come in, we need t offer what they want to learn. When we truly embrace our communities as learners and as partners in their own and their families education, then the school community that builds is powerful.
Tristan's talk was one of the power of community, our local communities that we live and work in, and our shared communities such as the EC tribe. What we are doing in our own communites is important and needs to be shared with our wider communities like the EC16 conference, people need to hear our story, our journey, our success. It is through sharing this that others can learn from us and hopefully have the same success. However we also need to listen to the stories of the EC community to learning what we can from them t really hear what they did and how they did it so that we can go back to our own communities armed with this new learning to strive for better outcomes.
I learnt a lot from this talk, it was a really strong example of being able to be amazing things, in really tough environments against some significant odds. Her drive and her passion for what she is doing was, I think, key to her success. On message I really took away from her talk was a conversation she said she had with her father about understanding the reasons for doing things. It went something along the lines of if you don't understand why you are doing it and I don't understand it either, then I can't make you do it. It made me think about the nature of schooling, how many things do schools still do because they have always been done that way, do the teachers and do the students understand why it is done that way, and if not, why are we still doing them?
At times our kids with disabilities are being denied things that we take for granted, like the ability to communicate. Schools are places that can help kids with communication difficulties to have a voice, but how do we make sure our communities are listening and helping to support these young people to use this voice outside of school. It was also really great to hear her ideas or scale with developing apps to replace physical resources, stickers in cafe windows to identify which places are 'friendly' towards the change she is leading, etc. She had a very clear idea of where she wanted it to go and is a strong advocate for her students.
Jessica and Kerryn
Through EC we know the power of storytelling in sharing what we have learnt with others. Learning stories that I have started to become accustomed to because my daughter is at kindy. I know the power there is in being able to see her in the moment of learning something new, or working with others to complete a task, to have that photo or that video as well as the short story to go with it. For many kids I think that that grade at the end of the term and even the comment is too abstract. Kids need to see what we see in them when they are learning, it is incredibly exciting for us to watch but most of the time they may not be aware that it is happening. Made me think a lot about the nature and role of feedback in class.
Thanks for your contribution to the event it was very much appreciated, I would love to see how your projects are progressing in the future.
Mathematics Coach and Coordinator in Regional South Australia. Current driving the Empowering Local Learners project as a numeracy strategy from pre-school to senior secondary.
Opinions in this blog are my own and do not necessarily represent the views of my employer.