I compiled tweets related to my recent expedition to Ottawa to check out thinking classrooms! Each teacher implements the framework in unique ways that are often tethered to their characters, resulting in wonderful examples of places where students can engage in thinking. It was an incredible experience to be a part of. Follow the link below to see pictures and comments that outline my journey!
You know the classic question:
You have 832 ft of fencing and you want to build a rectangular pen along side a river with maximum area.
Well, as we know, the textbook approach to this is generally to create a length and width equation, use it to create an area function, and find the maximum value of this function.
Depending on how you go about it, it gets a bit thick on the calculations, and most students require a calculator for it. They can learn the procedure though and it works for them.
HOWEVER, I had a student come up to me after a class where I asked them to find the dimensions of a patio we could build with a given number of feet of fencing outside our classroom so that we have the maximum area possible. He told me that he thinks he has noticed a pattern within these problems where there is a 2:1 ratio between the sides for a maximum area.
My gut instinct was to introduce doubt, but encourage him to continue pursuing this idea. I told him about how mathematicians will notice things they think may always be true, and then find ways to prove or disprove their conjectures. I also told him that one way to easily disprove it is by finding a counterexample. So I set him on his way telling him to try many examples and see if his idea ever fails.
He had shared a few attempts at explaining his rationale with me, but they were not convincing. I wasn’t sure if he’d pursue it further or not, but I left it at that.
About two weeks later, which was last night, this question was on a quiz. This student, along with another student, were chatting after class . . . because they just can’t get enough of math class . . . and were discussing the farmer’s-pen-by-the-river problem. They are both very engaged and creative students who are both very strong thinkers. But the contrast in approaches was interesting to observe.
One student asked the other about how they approached this problem, and the student who had spent all this time thinking about his conjecture didn’t remember what he got on the problem. So the other student reminded him of what the problem was, and went on to explain the classic solution approach of using the vertex of the area function to find the maximum value for the length of the pen. The other student smiled, and said, “why do all that work when you can do this . . .” and he showed the following:
The part that blew me away was his reasoning:
HE EXTENDED THE RECTANGLE INTO A SQUARE!
Way to think out of the “box” . . . pun intended – lol.
By claiming that the area is maximized when something is a square, and that the pen problem is really always half a square, the proportions can be used to quickly deduce the length and width.
And even at this point, I continued my doubt. I told him that this is a very elegant and creative idea, and it clearly gives the right answer in this case, but that he should further work on how to convince someone that this is always the case.
What impressed me the most is that the doubt that I provided early on somehow motivated the student to refine his thinking.
I know for sure that without all of the work on building a classroom culture of thinking and communicating, this would not have happened in this way. Without random groups, these students would not be working together, commuting to class together, pursuing the next math class together, and discussing Greek Philosophy together after class (true story, they actually were)!
Opening up communication and visibility of thinking in class is so important. I am finding this idea being supported within the work of my colleagues. The other day, one of my colleagues told me she notices a big difference in student thinking when she has a classroom where students can work on whiteboards around the room. Another colleague also recently shared with me that she notices a huge difference in student participation with each other in a lecture based university level math course when she takes the time to implement a social activity during the first class. We really need more SOCIAL in the university learning context.
I just LOVE the MTBoS Search Engine! I want to send out a public thank-you to all who contributed to making this happen.
I was planning my lesson on systems of linear equations, and within an hour, I had a nice progression . . .
I started with Andrew Stadel‘s stacking cups Act 1 video. I proceeded a la Dan Meyer by asking – what questions do you have? This resulted in all sorts of answers, but soon, students began asking about the measurements . . . I decided to feed the hungry and tell them these cups were pre-measured for them!
This was great! I got them into their #vnps formation with a deck of cards, and off they went at their collective question of – how many cups until they are the same height?
Some groups expressed some frustration with understanding the problem so I brought in some stacked cups and suggested they try to find their heights. This helped.
Over the next 20 minutes, interesting discussions were generated within the groups:
- One was around rate of change as the steady increment of cup height.
- The other was around defining variables.
My favorite question was when a student asked:
“Why is it that at 0 we have a height? Shouldn’t 0 cups have no height?”
My response was:
“What does your x mean?”
Somehow, this question brought light into the situation, and the group revised their equation.
Being inspired by this group’s realization, I began asking other groups about what their variables mean. Some explained their x refers to the number of cups in the stack, while others explained their x refers to the number of cups in the stack after the first cup.
I chose to bring attention to this to the class as a whole. Once they sat down, I asked them what the equation should be. I had mixed answers. Then I asked what x means for each. As a whole class, we noted that y = 1.3 x + 9.2 is an equation for height y of x cups after the first cup, and y = 1.3 (x-1) + 9.2 is an equation for height y of x cups total. We also noticed that 1.3 was the rate of change in both cases, and it was the slope of the graph of that line.
Because of this discrepancy, the class as a whole has now established that defining variables is VERY IMPORTANT! This has become part of the classroom culture and discourse.
If I had told them to begin with, “you must define your variables,” it would have not made it into their practice – BUT, since it really mattered in this problem, it stuck! More importantly, a general focus on meaning was introduced.
This was followed by a general discussion about systems of equations, possible results to a system of equations, and approaches to solving.
Although this is the dry part, I find that the time we took to systematically have them work through solving the following systems was time well spent because it brought them into their comfort zone of algebra:
After this formality, I sensed the desire for more problems! Time to get them back out of their comfort zone and into ambiguity again!
The scenarios we explored included the following #MTBoS inspired ideas:
Double Stuff! (from Christopher Danielson’s “My Oreo Manifesto“)
Glue Sticks! (from Kyle Pearce’s “Piling Up Systems of Linear Equations“)
If we had more time, I would ask them to create their own scenarios.
The other day in class, I asked my students, “What do you want to get out of this class?”
I meant – what do you want to focus on today and what do you want to leave with today, but it just didn’t come out quite like that.
One hand went up, and the response was, “I would really like to get an A in this class.”
I chuckled softly and asked if anyone had anything else to add.
Another hand went up, and the response was, “I’d really be happy with a B in this class, even just a C+.”
And shortly after that, another student told me the grade they want “to get.”
I was baffled how no one seemed to think these were ridiculous answers. The norms of our culture, and the upgrading setting in particular, were prevalent to me more than ever. These people, in the end, want to get their grade, and move on. They want to forget about anything they learned, and move on with the interesting things in their lives. Fair enough, I get that.
But what baffles me more is that HOURS go into this! HOURS go into attaining a goal of a letter grade that doesn’t reflect in any specific way what was learned, and how that person as a whole has been enriched in some particular way as a citizen of society.
This is the system.
The system places value on this letter grade. It does not place value on:
- the social skills developed within collaborative problem solving activities
- the processes generated within the act of solving a problem
- the new found ability to explain a mathematical concept to a peer
- the insights garnered into why society has engaged in mathematics
- the time and dedication committed to understanding complicated mathematics
I have seen evidence of all of the above in my students while they have worked together on problems in groups, on whiteboards and off whiteboards. There is a community in the classroom that I see and feel when I enter it. A community where students feel comfortable walking across the room to check in on someone they care about and see if they are understanding the content, and where students listen to their peers and have the patience to hear them out before explaining their perspective on a problem.
How do these things get valued more?
It is my hope that throughout the term, the focus on the grade will become overshadowed by the focus on learning – as a process rather than as a product.
I fear, however, that it is very possible the pressures and demands of the system may ultimately prevail.
. . . unless we chip away at the system . . . ?
As teachers, we face many tensions about how to conduct our classes every time we teach! This has been even more prevalent than ever to me this term. Every time I have made a lesson plan this term, it has never gone as planned . . . actually, I don’t think it ever does, but it’s just been way more obvious to me this term than in the past.
First day, I walk in, and although I wanted to start them off with a problem, I felt the anxiety in the room. So I asked them what questions they had about the course, and we were able to cover many of the key aspects of the course outline without even looking at it! Then, I was about to give them a problem, but I thought, hmm, this is the first class, let’s see what their perceptions about math are . . .
So I asked, “What comes to mind when you hear the word ‘math’?”
Here’s what came out of this:
(I provided them with the ‘fake world math’ terminology after they described it to me!)
This was the best start I ever had, and it was because I created an opportunity to listen to them and to listen to what I sensed the room needed at the time in that context.
I have a very lively group this term, and am very happy with the energy in the room every time we have class. From the moment I entered the room on the first day, there was a classroom norm of asking questions set right from the start. I was blown away by the questions they were asking, and how collaborative this group was. It may be something to do with the size of the class. I haven’t taught a class of 30 in a while! Last term, my class was around 10 at one point, making it difficult to get random groups and vertical non-permanent surfaces work very well. But with 30, there seems to be enough density of ideas in the room to foster a ‘keep thinking’ atmosphere!
Now my greatest fear is – how do I keep this fire alive . . . all term???
So far, my favorite lesson was when we were reviewing order of operations. After getting them to remind me about what order of operations they know about, I asked them to (in random groups on whiteboards around the room) find as many ways as they possibly can to make the digits 1 through 9 to be 100. Digits need to be in order, any operations may be used, and numbers can be clustered if needed. (Source Martin Garnder’s “Further Mathematical Diversions” – see http://www.worldofnumbers.com/ninedig1.htm).
The room was buzzing!
Some groups had answers right away, but most didn’t. Everyone struggled, but most persevered. This group worked very hard, and by the end of it, they were SO PROUD of it!
This individual asked, “can it be super complicated?”
And this group, found that by identifying a way to make 50, they were able to get to this:
For those groups who after a while of persevering started to evidence a feeling of struggle, I told them to try some of the similar problems from Fawn Nguyen’s Foxy Fives. I came back, and they told me that was too easy, and they wanted to continue trying the 100 problem!!! I was in awe of their perseverance, and told them that this was what mathematics was all about!
We then could have a discussion about what they noticed themselves doing in this problem solving exercise, and how the process was more valuable than the product because in the process of trying to find a solution, they were using their order of operations abilities inadvertently.
Now, my tension as a teacher is that I don’t have such magical entry points to every topic in my course, and some topics are so dry that really the only way to ‘problematize’ it is to give them the tasks to figure out in their groups without me introducing any particular ideal way to approach the problem.
I’ve been playing around with this by giving them a more difficult problem from the section to solve before they have built the tools for that problem.
For instance, I asked them to find possible solutions to:
You have won the lottery and are about to invest $20,000, part at 3% and the rest at 4%. What is the most that you can invest at 3% and still be guaranteed at least $650 in interest per year?
Because they hadn’t yet been jaded by textbook approaches to this sort of problem, many of them began with trial and error, which is what I expected. This was great! During our reflective discussion, I guided them to seeing that if you create a trial and error process, you can eventually develop an algebraic statement that encapsulates that process. My aim was to show them that algebra is a generalization of a process.
Now, where my tension came in, was during the next class, when I was trying to introduce solving inequalities and absolute value inequalities. I started by asking them what the possible solutions were to:
However, I feel that we moved too quickly into questions that were too complex, and in this particular case, I think it would have maybe been better to start them off with simpler inequality problems, and have them work towards more complicated absolute value problems within their groups.
A lot of learning took place, I’m sure, but I left the class feeling as though there was still confusion about this whole topic, and when students were trying to solve the problems, they were making many mistakes. Mistakes are a place for learning, but I feel that this is a topic that I should have found a more interesting approach for.
So to identify the tensions that I have gone through these past couple weeks:
- Deciding which problem to give students next based on my interpretation of what the class needs at a given moment, regardless of any preexisting plan I have made.
- Deciding when to stop the problem solving process and move into a discussion.
- Deciding from which end of a topic to begin (complex to simple, or simple to complex)
- Feeling as though a topic may have been possible to make more interesting with all the aspects I theoretically know about such as creating a need, opening the middle, low floor/high ceiling, keeping students in flow, etc.
I have been learning about what being an “effective” mathematics teacher entails lately as part of my graduate work. Interestingly, there is no “theory of teaching,” there are only “theories of learning.” Not a big surprise given that teaching IS learning! Research also points to the fact that being an effective mathematics teacher is MUCH more than being good at mathematics. There are different explanations that we as mathematics teachers must develop and rework constantly. We also need to adapt continuously to ever changing conditions. In this sense, I’ve been drawn to theories of enactivism, which are based on the notion that we co-evolve with our environment in a continuous, complex, and adaptive way. We restructure ourselves to be able to remain viable. In a classroom, this means that we adapt to what students need in the moment. In many ways, we are always doing this. However, we also seem to be tied to our “lesson plans” and external curricular structures. These constraints can sometimes make us forget what teaching is really about.
Since learning more about this ecological perspective, and being inspired in particular by Brent Davis’ work, I’ve made a slight shift in mindset when teaching. This past term, I had quite possibly the best conditions for teaching in this manner. My room had two whiteboards on each of three walls, meaning I could finally try out a full version of Peter Liljedahl’s Vertical Non-Permanent Surfaces. I wasn’t able to blog about it regularly because of an extremely heavy term (teaching 4 courses, when 3 is a full time load, taking one graduate course, volunteering as part of a visioning committee for my institution that met weekly and had many assigned readings, and trying to work on my dissertation work). However, I resorted to a different form of tracking what I did in my classes. The purpose was more for my students, but it ended up being very helpful for a colleague, who used it to teach the course this term.
A typical class would consist of giving students a problem (by the end of it, I could really choose any problem because they were already in the ‘problematizing’ mindset). Students would form random groups, and would work on the vertical whiteboards. The benefit I experienced of vertical whiteboards vs horizontal ones, was that the ideas could spread across the room. I could also call a class meeting around a particular whiteboard to emphasize something someone had done. Here’s sort of what a typical class looked like for much of the period:
During each class, I would take pictures of every group’s work, and then plunk them into a document that I would share with the class. This way, students could take a look back through what all the other groups were doing during the class. The double edge of this is that now I have these “photo lesson plans” that my colleague is using to guide her problem based version of the course! This practice has become generative because she plays off my ideas, and now she’s doing the same thing, so I’ll be able to play off her ideas! For example, she came up with the idea to bring in a kids bike to class to pose the problem: How far will this bike travel with a quarter turn of the pedal gear?
(this is sort of what the bike looked like . . . I didn’t take a picture, I just saw my colleague walking around with a kids bike, and had to ask – just imagine a pretty lady in a dress with flats, walking around with a little kids bike! Ah, math teachers)
BRILLIANT! I hear it kept students engaged all period!
Notice how no one is using books – they are talking, and thinking. There were so many times last term that I found myself at the back of the class, watching everyone, in awe. However, what this meant, was that I had to be EXTRA observant. AT any moment, I needed to be ready to take them somewhere further, to a different realization. The vertical whiteboards helped me adapt to what they needed.
I only hope to further enhance my adaptive abilities next term.
Maybe I’ll learn something really neat at TMC15! . . . heck, I’m sure I will learn a lot of cool things that I will be able to adaptively incorporate into my lessons next term!
This term for me was quite a ride. I was teaching a full course load, taking my second to last PhD course, working on dissertation stuff, travelling to conferences, and trying to have a bit of a life too. It was my first term teaching a Math 12 equivalent upgrading course at my institution. The first round of teaching a course always has it’s challenges. Luckily I had some material to work from. However, I like getting creative with my teaching, and I always want to make changes. It never ends. Now that this crazy term is over, I have some time for reflection.
One thing in particular that I have been reflecting on recently, is my use of groups. I always start my term off with the use of random grouping, whiteboards, and problems (Hat tip to my advisor Peter Liljedahl). Lots of engagement is created with this, and it establishes the tone for a thinking classroom. I was impressed with how my students pushed me and questioned why everything is the way it is when they came across something. There were a lot of debates and opportunities for starting up mathematical arguments.
Fine and dandy.
However, there was one interesting occurrence. My students went through a bit of an emotional experience while writing a test while I was away (at a conference . . . and stuck in another country) and this changed the dynamics of my classroom completely.
I got the evil stare from many when I arrived back, and I worked really hard to recreate a good learning atmosphere.
One thing that came up was the use of random groups. Students really wanted to stick to working with students they found helpful to their learning. Fair enough. So I let this emerge naturally. Groups settled into these sort of learning pods, and people felt comfortable moving around to talk with others from different groups. Problems were tackled, questions were formed, justifications were made . . . great! I don’t think the groups would have worked quite this well had they not been placed in random groups for the first four weeks.
In retrospect, I notice that the groups that formed seemed aligned in their attitudes towards the course, their goals of learning, their mathematical abilities, and in the end, groups tended to receive the same grades. Now some of these things may have caused them to be drawn to each other, while other may have developed out of each group’s subculture. It’s interesting to me in what conditions a teacher should pursue random grouping, and in what conditions, a teacher should let them emerge (as I felt I needed to in this term). Either way, this indicates how strong the social is in learning.