ack in 2004, at the end of my first year of teaching, I went on a trip to New Zealand. During that trip I visited Queenstown and decided that I would have a go at bungee jumping. Given my experience on my first bungee jump, that first year of teaching could have been my last. The jump was quite a substantial one and before the jump the first thing they did was to weigh everyone, and to write the weight on your hand, in this way the jump operators could ensure that you got the right bungee cord for you, to make sure the jump is both safe and thrilling. This is where my issue arose, they weighed me, wrote my weight on my hand and I waited as they did the same with the others, as I sat there I looked down at my hand and saw the weight that was written there and saw that they had gotten it wrong, the weight they had written was about 20 kg below my actual weight at the time. If I had left it, and they had thought I was lighter than I actually was, then the rope would stretch much more than intended and my safe but thrilling ride, would no longer be safe. Luckily I noticed it and had them reweigh and change the weight on my hand.
At he start of 2019 a new STEM centre opened at my school. Early in the planning stages for the implementation of programs in this centre was that all students in the school needed the opportunity to be involved in STEM learning experiences. In years 8 and 9 was to take the form of a 5 week unit of work in their maths lessons and also 5 weeks in their science lessons. As the head of mathematics at my school I was tasked with developing those learning experiences for maths, learning experiences that all teachers would deliver at some point during the year.
In my initial thinking for the year 9 task I really wanted mathematical modelling to play a strong role. I really wanted students to see the power of mathematics in being able to accurately model a situation and to test and refine that model over time. I also wanted their model to be robust, to be able to handle anything we threw at it, so that it could deal with uncertainty. Finally I wanted them to be able to communicate that model clearly, not just to me as a mathematics teacher, but to any audience. I wanted them to take all the complexity of the situation being investigated and to synthesise it into something really clear to anyone, regardless of their level of mathematical confidence.
In the years I have been keeping an eye on the #mtbos, Barbie bungee has always seemed to be a popular activity, throughout the year there are numerous posts showing the excitement that students bring to the problem and it provides a good opportunity for some mathematical modelling. It was a task that I had always wanted to try, but never had the opportunity to since I did not teach in those year levels. However after reading a number of blog posts on the activity I could see that for many, this activity lasted a few lessons, or maybe a week or two, I was trying to make this into a 5 week task. This is where I started thinking back to my experience with the bungee jump. The other activities that I had seen, they had just considered the number of rubber bands as a variable that determines how far the Barbie drops, and that works brilliantly as a shorter unit of work. However in my own jump, weight was the variable that almost sealed my fate. I needed to bring more variables, not less, to the problem, I needed to introduce more uncertainty. As I thought about it more I knew I wanted to incorporate weight but I also thought the height of the jumper would also be a factor that I could get students to investigate.
As I reflected on my thinking I realised that I was asking students to consider how many rubber band sections would be needed for any jump height (up to 8 meters), any jumper weight (up to 200 g) and any jumper height whilst still maintaining the criteria of a safe but thrilling jump. I began to wonder if it was possible and whether I was asking too much. From looking at other posts I knew that the fall distance scaled approximately linearly with the number of rubber bands, the stretch of a rubber band being the slope, and the height of the Barbie being the y-intercept. However the addition of weight as a variable also meant that the slope of the model, the stretch of the rubber band, would also be a variable as less weight would mean less stretch and more weight would mean more stretch. In further research it was revealed that the stretch of a spring scaled linearly with the force applied (Hooke's Law) and therefore I hypothesised that a rubber band would behave in a similar way and that the stretch would be directly proportional to the weight, but to be honest, I wasn't sure.
I was amazed with what they came up with in this task. Some used different numbers of rubber bands and looked how far the Barbie fell and then repeated the same numbers of rubber bands with differing weights. Others decided to determine how many rubber bands would be required to fall to set distances and then looked at how weight would effect the number of bands needed.The third and probably most intriguing approach involved not dropping the doll to collect data at all. They researched the physics of bungee jumping and attempted to use all of the physics formulas to determine the best approach, they only dropped the Barbie enough times to verify some of their numbers.
It wasn't until the day of the final jump that they were told how high the jump was, hot tall their doll was and how heaving it was going to be. They had 15 mins to calculate the number of bands needed, and construct their bungee cord before the drop this is why the jump calculator tool became so important. They had three opportunities to drop their Barbie, the video below only shows the first attempt. It is clear from this that many of the drops are not as close to the ground as ones I have seen online, however given they are having to adjust according to more variables this is a great result. and as you can see some of them are quite close and did very well as a predictive model
Overall I feel that this activity has been very successful and as more classes complete this task I will have more information about what changes I might make. All the resources that I used to run this activity can be found here.
For 15 years now one comment on my final teaching report that I had as a pre-service teacher has driven me as a teacher and not necessarily in a good way, that comment was
Shane is a competent teacher but he is a little too dispassionate
That comment destroyed me at the time, and it has stuck with me ever since, no teacher wants to be known as just competent, nor do they want to be known as dispassionate. This comment comes into my head when I have a bad lesson, but it also comes into my head when I have a good lesson with the thoughts of what could have been done a bit better. It drives me to work beyond my level of exhaustion in the pursuit of becoming the best teacher I can become. I think teaching can really be one of those jobs that no matter how much time you spend on it, no matter how much you improve, and no matter how much recognition you may receive, you can be still left with the feeling that you are not doing enough and that you need to be doing more, and that you can always find reasons to justify that you need to work harder.
At one time this year when that feeling was particularly strong Twitter Math Camp 2018 (TMC) was running. Living in Australia my chance of ever attending it is highly unlikely, but I kept a close eye on the twitter feed over that time. It was at this time that I saw stickers appearing in the TMC feed saying
You are not an imposter, you are enough
After a bit of digging further I found that these stickers images were stickers by Julie Reulback @jreulbach. At that time that was the message that I needed to hear, and I have looked at that image a number of times over the last few months, however I think that needing to see it has been one thing, but believing it has been a whole other issue.
So tonight I did something I haven't done in the last 15 years. I opened that teaching report from my pre-service time and I read it from start to finish. It wasn't a comfortable read, and I am still not sure it was an entirely fair report (for reasons I will explain further down), but I needed to read that report now, it was the right time, I needed put that report behind me.
It was directly after reading it that I went and watched Julie's talk from TMC 18. I had seen the sticker but needed to see the talk that went with it. Watching the video has helped me to start the process of believing the sticker, not just needing it.
Some of the things that really resonated for me as part of this talk were
1. No teacher goes to school trying to do a bad job
I guess in reading the report again I am not sure if that report was true of me then, but even if it was I know I was doing the best that I could with what I knew at the time. I also know that what was true of me then is unlikely to be true of me now, I have developed and improved a lot in the last 15 years but my approach is still the same, I am doing the best that I possibly can with all that I now know, and the same could be said of any teacher I have come across. The difference is now I not only have a lot more experience, but I have also benefited from reading much more widely and connecting with online communities like the MtBOS, my practice is now based on a much more well informed base. However I have, and will continue to, stuff up from time to time, not because I am not prepared, and not because I don't care, but because teaching is hard and complex work.
2. There is no one way to be a good teacher
In looking over the report, this became very clear to me. My teaching style at that time did not align very well to the teaching styles of my supervising teachers. At the same time as reading that report I also went back and read my pre-service teaching report from the previous year and it was a very positive report. However my teaching style tended to match more closely with their way of teaching. The reports did identify some common areas for improvement, but overall to look at the reports they seemed to reflect how well I matched with how they thought I should teach, not how well I taught in line with what I was trying to achieve, they wanted to see themselves reflected in my teaching rather than looking at the effectiveness of what was there.
I am very aware that when others see my class, whether that is students, staff or parents, they may think it is too loud, that there are off-task conversations and that it is not helped by students sitting in groups. But I want them to know that it is a deliberate choice for me, if my class is quiet, they are not discussing their maths, if they are sitting in rows and not groups then they are not bouncing ideas off of each other. I want to create a class environment where there is a free sharing of ideas. I want them to know that I know that there are off-task conversations occurring, and that I am prepared to tolerate some of this if it means that students feel safe to have conversations about their learning.
I am also aware that when others see my class they see a lot of problem solving work, they see kids struggling with the work, they see them getting frustrated and they are not seeing much teacher directed instruction. But I want them to know that teacher directed instruction is still a strong part of what I do in the classroom, I do value it, but I use it as it is needed. I want them to know know that I don't feel they are struggling with the work, I feel they are grappling with it, that they are working at that point where they are trying to make sense of it and it is just outside of their grasp. I want them to know that my students have chosen to be in that state and that they could have chosen something easier, but they chose to do the harder questions. I want them to see how I interact with, and question my students, I want them to see how it is building their ability to work much more independently and to think for themselves.
Most of all I want people (students, parents, teachers) to take the time to understand what it is I am trying to achieve, to have a conversation with me about it, to let me know what is not working for them with it so we can figure out a way to move forward, and I want to make sure that I hold myself to the same standard.
3. I need to share more
Following on from the previous one part of letting people know what I am trying to achieving and why is sharing it more widely. I am proud of, and believe in, the work I am doing and I am proud of the work my students are doing. But I think a big impact of this imposter syndrome is that you never feel you are quite happy enough with your own work to share it more widely, you are concerned that someone will find a hole or flaw in it. But I need to get it out there, the work my students are doing is amazing and they deserve me to share that with others, to share what it is that they have done and what it is that they are capable of.
A strong part of this need to share more is sharing with teachers what I have really enjoyed about something else someone has done in their classes. Whether that is in my own school or whether it is online through the MtBOS, sharing with others about something you have really liked or got a lot out of helps to lift that belief that you are an imposter. It lets them know that you have seen value in what they have been doing. If as a profession we are to shake off this imposter syndrome we need support each other in whatever way we can.A quick email, conversation or tweet really doesn't take a lot of time but can make all the difference.
I am lucky enough to have people around me who do not believe that I am imposter, and they feel that I am doing more than enough, and for now that works for me until I completely believe it for myself.
I was stumped, where was I going wrong? There I was standing in front of my class and I couldn't get the maths to work. There was nothing wrong with my thinking, there was nothing wrong with my calculation, but yet it still wasn't working. It was then that it struck me, I looked at the question and realised that the maths wasn't working because the question I had put up wasn't a triangle. If the longest side was 9, then there was no way that sides of 3 and 4 would form a point above that side, they wouldn't even come close to joining. I felt like an idiot, I wanted to crawl under the desk and hide.
On that day, I had ran through my carefully prepared examples and they wanted more, so I made the questions up on the spot, I drew a triangle, put some numbers against them and tried to work it out. This is something I had done on other occasions without incident, but this time I picked wrong. It was a valuable learning experience for me as a young teacher and this moment still sticks with me, even 12 years later.
Late last year I was doing some work on mathematical reasoning at the school, I gave the maths teachers the question above and gave them the time to figure it out. Not surprisingly they were all able to figure out that B and C were definitely not right angled, however they all used Pythagoras' Theorem to determine the answer, which was fine because I put no restrictions on it.
So my next question to them was "If you knew nothing about Pythagoras' Theorem, how would you approach this question, how could you eliminate at least some of the possibilities?" This was an interesting process, to look at at how they now approached the problem. It wasn't until I asked them to draw triangles B and C that they realised that they were impossible, they didn't need to draw them, it was just that stimulus that got them thinking about it in that way.
I began to realise that knowing that the sum of the two smaller sides must be bigger than the largest side is not a frequently discussed property of triangles, at least from our high school perspective of shape, I really don't think I knew much about it until I was standing in front of that Year 10 class and trying to figure out why the maths wasn't working. We are often told that triangles have got three angles and three straight sides, we also are told sometimes that the sides must be connected, however I never remember exploring the conditions under which those sides would be connected and how this property extends to polygons with more than three sides. We are told about angle sum in polygons, but don't explore ideas around the sum of sides.
This experience has taught me a few things
For a number of years now a colleague and I have talked about teaching together. Over the last three years we have both taught year 8 maths, but in different years, never in the same year. Over that time we have worked together extensively in developing, refining and adjusting our teaching program to make it more effective. We bounce ideas off of each other all of the time so it was exciting to find out that this year everything aligned so that we could teach together we also have another teacher and their class who is joining with us.
The double class works across 3A:09 and 3A:10, and the triple class works across all of the spaces shown with tables, we did start off with the triple class in only 3A:09, 3A:10, and 3A:11 but found that they needed a bit more space. When they are working on questions or when we are working on more open problem solving tasks then we always work in this way, one big open space. The are able to sit with anyone from any of the classes in that space, as long as they are working productively. The two or three teachers in that space work as a team to address student questions as they arise, we are not three separate classes, we are one big class.
However having one big class does have it's limitations for some activities, when we are doing some specific teaching of key ideas it requires a smaller class. Often with this specific teaching you need to respond to what you see in the room, you can get a feel for the class in relation to whether it making sense to them and you need to respond to the questions raised this is too hard to do effectively with a class of 75 and therefore at those times we shut the walls and teach smaller groups. However those smaller groups are not necessarily the people on the class list of that teacher. Because each student in the large class has generally received help from each of us, they have come to understand which of us is able to help them best, so they are free to go into any of the three classes to get this specific teaching from the teacher they feel they will be able to understand the most. The examples we use in each class are always the same, we do that preparation in advance, but they just hear it from the teacher of their choice.
Why the change?
At the start of this post I talked about how we had wanted to do this for a number of years, part of this is because we get on really well, and we teach in similar ways, but from my perspective at least there were some really important reasons for students in why we trying this approach.
How is it working...so far
It all seems to be working fairly well. It has been a very new experience for us, and there were some teething issues such as how do you mark attendance when you have told 75 kids to sit with whoever, but we sorted that out.
we have learnt a lot along the way, but it seems to be working better than it normally would if we had been separate classes. The structure seems to be working like we intended and seems to be addressing the things that we thought needed addressing. Kids are more comfortable in taking risks, they have greater agency, they are getting the help they need and they are producing some very good work. Is it perfect at the moment... no, but we need to be aware that this is still a very different for the adults and for the kids in the room and we have had enough success early on to see it is worth pursuing
Last week I attended a day of training for the Empowering Local Learners Project a project I have been involved heavily in for the last 6 years. The thought experiment above was one of the first things we did during that day and it has been an incredibly powerful thing for me to think about over the last week.
As I began to think about what pack I would choose, I immediately went with pack A as I knew exactly what I was dealing with. The odds were clearly defined and although I could not control what card came out first, I knew my odds of wining were the same as my odds of losing. Choosing pack B did not even enter my thinking for a good few minutes, there was just too much unknown, did they choose the 10 cards themselves or do it randomly, the person holding the decks of cards doesn't want to lose $1000 so they probably stacked the deck in their favour. But as I thought about it more, with with only about a minute of thinking time to go, I started to look at pack B a little differently, I knew that I probably should pick B, but I had to justify it in my own mind. When I went to share my thinking with the group I said that "I ultimately picked pack B because there are as many possibilities of a better outcome as there are of being a worse outcome". In my own head at that time I had still made it a 50/50 chance.
The next question about the packs was "Is there any difference in how pack A and pack B feel?". For me this really got to the heart of this thought experiment. I didn't feel anything towards pack A, there was nothing we didn't know about it except what the top card was. My heart may have raced as the card was picked, but picking that pack to bet on was easy. Pack B on the other hand didn't feel like that at all, I avoided thinking about it to start off with as there was too much that was unknown. When I did start thinking about that pack I got tense, I had to really force myself to think about it. In the end I made the choice of pack B, but it was not a comfortable choice, as soon as I said it I wanted to change my mind. This was really interesting given that none of my money was actually at stake.
Over the last week this has really made me think about my role. For the last 7 years I have been in the role of an instructional coach on both a school and a district level. I have been about working with teachers to implement more student-centred approaches that include a greater emphasis on students' problem solving and reasoning skills. I have spent the last 7 years looking at the idea of pedagogical shift.
When I began about the above thought experiment in relation to pedagogy, I began to think about how pack A and pack B might look different in a classroom. However as I began to go down that path and think in that way I realised that there is no such thing as pack A and pack B pedagogy. The difference between pack A and B is in relation to the perceived level of risk, the extent to which you are stepping into the unknown, it is a feeling more than an action. It is the way that the lesson feels before, during and after the lesson is delivered, something that would be mostly invisible if you looked for it in a classroom. Teachers could be teaching the same lesson but could be feeling very differently about it.
Teaching is a challenging and complex job and we learn a lot about what our students know, how well our students work, who they work best with, how they interact with others, and how long they can work for. These, amongst many others, are variables that can impact the learning on any given day. We often control for these variables in designing learning experiences to ensure these experiences have the greatest chance of success. However even if we have had great success with the task in the past on any given day we know that that lesson may work well, or it may fail. For me this is similar to picking pack A we have a pretty good idea of the odds of achieving the same outcome as we did previously.
So what about pack B? In my role as an instructional coach I have been asking people to consider picking pack B for a number of years now. I have been asking them to consider trying some approaches to teaching maths that they may not have used before without knowing for sure how comfortable they will be. I have asked them to try them without them knowing how their students may react to the tasks, without knowing if they will even attempt it or what questions they may ask. I also realise now that I have been doing this from the position of pack A, I am doing it from a position of relative comfort with the practices I am asking people to implement. People tend to see this comfort and assume that it is just the way I have always taught. However this level of comfort has not come from just being "that sort of teacher who has always taught that way" but through years of putting myself in that position of choosing pack B. These practices have transitioned from pack B to pack A over the years through taking a risk, by both failing and of succeeding and in turning the ambiguity of trying something for the first time into an experience that I now know more about.
Therefore in my role I feel it is important for me to continue to put myself in the position of pack B to keep trying things that I feel will work, but make me uncomfortable. This is important for my students in ensuring I am always working to improve the quality of their learning experiences. However it is also important for me to never lose sight of just how difficult making changes to your own teaching practice can be.
Here in Australia we are about to start our school year, with teachers back at work next week and the kids the week after that. This year marks the start of my 15th year in the classroom, and I consider myself very lucky to have the career I have had so far. I have had the chance to be involved in a lot of projects that have transformed learning for kids at my school but they have also transformed me as a teacher and transformed me as a leader.
Over the last 5 years I have been strongly involved in the Empowering Local Learners Project and for the last 3 of those years I have had the privilege of leading the implementation of this project across 16 schools and preschools from preschool to year 12. As a result of this work, kids from across all of these 16 sites have shown significant growth in their numeracy outcomes national testing (NAPLAN), but more importantly that that kids are now loving maths and enjoying challenge, this was not their relationship with maths in the past. For me professionally it has transformed every part of my teaching practice. I have had the opportunity to work with some amazing teachers locally as well as building strong relationships with Flinders University, particularly Deb Lasscock and Kristin Vonney, two phenomenal teachers from the Flinders Centre for Science Education in the 21st Century and that centre's leader Professor Martin Westwell. This work has given me the opportunity to present both nationally and internationally at both education conferences well as research conferences and has been recognised in our states teaching awards over a number of years.
However this year I am finding myself in a place where I need to let this project go. I will not be in a position that I am leading it, and there will be much few people involved than previously, the project will be on a much smaller scale than previous years. Therefore my level of involvement in the project will be limited at best. I have taken this hard, this hasn't been easy for me to reconcile in my own head, it really seems like it has been a grieving process. So much of my time and my energy over the last 5 years has been poured into this project, and so much of my identity as a teacher is wrapped up in the work, so the process of letting it go has been hard.
But as the saying goes as one door closes another opens. This year I find myself in a new position at our school, I have responsibly over leading pedagogical innovation across our school, in taking the work that I have done with the Empowering Local Learners Project across the town, as well with my own mathematics faculty in the school, and extending this work to other subject areas in my own school. Part of this position is the development of curriculum and pedagogy for our new STEM centre that will be built over the course of this year. This is a very exciting time for our school, in creating new opportunities for both our students and our staff. In embarking on this new job it is also an exciting time for me and I look forward to the challenges ahead.
I guess the point of this post is that it was started by what I perceived as a big loss to me professionally, the loss of something I had spent more than a third of my teaching career on, but in reality it has been a gain. Even though I am sad my involvement will be limited, this is not a project I need to let go, the work will continue in my own practice despite my level of involvement. As one teacher said "There is no way that you can take this project out of me", and it is true, no matter where I go with my career, this project has permanently changed me as a teacher. I know that with this project I have been given a lot of opportunities that some people never get in their teaching career so in my head I need to reframe it it is something I need to move forward with, instead of something I need to let go.
As with any professional learning experience that we have found particularly inspirational, it is about what we do with what we learned once it is all over. We can be sad that it is over, but we need to look for those open doors that we can use to help move it forward. This was part of my reason for choosing to do the #MTBoSblog18 challenge this year. Part of me helping to move the project forward desipite my lack of involvement is to keep myself accountable to that and this blogging challenge is one small way of keeping myself accountable to it. Much of the work with the project was inspired by MTBoS members so it is only fitting that I share what I have done with this work back with that community, to also let that community know how I am moving forward with some of their ideas.
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 task
After 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.
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 sets
When 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 sets
Greater 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 sets
Despite 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.
Senior Leader of Pedagogical Innovation and Mathematics Coordinator in Regional South Australia.
Opinions in this blog are my own and do not necessarily represent the views of my employer.