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 some posts that outline my journey!
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.
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!
So I started my first Precalc 12 class (offered for adult upgrading students). The class had 17 people on the waitlist. It’s also a condensed course, so it’s basically two terms squished into one. FUN! I knew I had to minimize numbers a bit. So I decided to start off with a reality check. Here’s what it’s going to be like. Here’s an assessment for you to try in order to get a sense of the prerequisite knowledge that you will need in this course. etc. It worked . . . I think. A couple people gave up seats, and I let more into the class than I normally would because the energy in the room seemed to be working. Maybe it’s because my room is so NICE this term! It’s so big and spacious, and the best part is that THERE ARE TWO LARGE WHITEBOARDS ON THREE OF THE FOUR WALLS!!! The other one is filled with windows! This is extremely exciting. I believe that room ambiance has an effect on learning, but that’s a personal belief. Normally, I get rooms with way less vertical whiteboard space. So I have been resorting to using these:
However, students end up using them on their desks, and many students often resort to using their paper after a while. I don’t blame them, there doesn’t seem to be much difference other than the fun colors that can be used, and the fact that they are easily erasable. But also, they are easily erasable.
So students sometimes rather writing on something that will leave a record for them of their thinking. Anyway, this has worked well for the most part, but I really noticed the difference between flat non-permanent surfaces and vertical non-permanent surfaces! It’s true!
The engagement levels were much higher, and knowledge was more quickly mobilized around the room. This means that students quickly realized that I was encouraging them to look at what other groups were doing to guide them. One student expressed excitement that he is going to be able to look at other students’ work this term in this way. There’s something intriguing about that. Maybe it’s because for so many years, these students have been trained that it is socially unacceptable to “cheat” . . . but they have not always been encouraged to do this in a collaborative way.
So after giving these people a reality check on how demanding the course is going to be and how much they have to dedicate themselves, I gave them the following activity. It was inspired by a session I attended last August, where Dan Meyer introduced this similar tactic, but with coordinate points. Since my first topic is a review of sets, I decided to integrate it into a “let’s get to know people in the room” session. It seemed to really set the stage, and hopefully helps in creating that essential community aspect of classroom life. It also led us to some discussion around logical statements and key set related terminology. I actually tried this activity last term, and what I changed this time around was that I gave some prompts beforehand so that students had some examples of categories they could use for their set graphs.
So here it is.
First, I give my students slips of paper with these questions for them to answer. I gave them a couple minutes or so for this. The questions were rather obscure, and they probably wondered what this has to do with math class.
Next, I explained that I am going to group them, and that once I group them, they will be working on creating a Venn Diagram like this:
They will then be deciding on how to name each set so that they can then place themselves on this graph according to these categories.
I grouped them randomly using playing cards, and asking them to find the same number as them to form groups of 4. They were then to find some whiteboard space, and create their graphs.
Some groups were still confused. They drew two circles, but had no idea how to continue.
I gave them an example:
Say one category is a collection of those who hate cats. Then anyone who hates cats has to be in that category. But if someone hates cats AND bananas, then they would be in the intersection of the hates cats and hates bananas categories, and so on.
Here are some examples of what my students came up with:
And one group even created a way to make TWO sets of these on one graph :-0
Things that came out of our discussions included:
- how do you label the categories so that the points make logical sense?
- what is the “middle area” called? (intersection)
- why would a point be outside of the two sets?
- what’s a universe?
- if a point is ON THE BORDER, does it mean that the person represented by the point is unsure? (lol, this was a cute idea)
- what is meant by union and intersection?
This set the stage for more set related problems, and students got to present their categories to the rest of the group.
The one thing that I missed from this activity, that I would have liked to have in a student introduction activity, is the various reasons for why they have decided to take this class, and what they feel about math.
I am looking forward to working with this group, and only hope that I will be able to keep up with coming up with neat creative ideas for developing their engagement in mathematical thinking.
So often I have found myself marvel at how some teachers find the best activities. I always ask them, where did that idea come from!?! Recently, I discovered that there are SO MANY SOURCES OUT THERE! . . . and this is only because teachers take the time to share their ideas with the global mathematics teacher community online. I would like to see more connection between these seemingly single bloggers. Wouldn’t it be nice if we had a cohesive list of all the math bloggers? Well, it looks like someone’s already done that! I was so excited to find this link where there is a comprehensive list of mathematics teacher blogs! This means that at this point, there are 347 blogs chock full of potential teaching ideas. So why is this not visible to the average teacher, and why is it that teachers (like me) don’t have the ‘time’ to go through them? It’s overwhelming for me to think that I may never absorb all of those ideas that are out there.
However . . .
The time is now . . .
It’s time to be inspired.
In addition to the above link to blogs, here are some of my current favorites for teaching ideas:
Peter Liljedahl’s Teacher Resources
Dan Meyer Blog
Andrew Stadel’s Estimation180
Robert Kaplinsky’s Problem-Based Lesson Search Engine
Feel free to post your favorites in the comments bar:
This term, I have been giving group problems on my tests, and it has been fruitful.
Giving group problems as part of an assessment helps students treat group work during class time more seriously. In the past, I have tried to get students to work on problems in groups during class time, but they most often gravitated to working individually in the groups. This was because they knew that in the end, they were going to be tested individually.
This term, I noticed a point in time when there developed a resistance against group work. A colleague suggested that I try putting a group problem on a test to show that I value group work. I tried it, and it worked! No longer do I need to convince students to work in groups. Sure, there are still a few who just want to work on their own, but the majority have come to value working together and learning from each other.
I am still tinkering with the logistics. I have tried giving a problem for them to work on and then getting them to come up and ask for the rest of the test on which they will then write up the problem they solved together on their own. I have also tried giving a similar problem as the test for them to work on right before the test so that they will be ready for it when they complete their test. Finally, I included a group problem on the last midterm as described below:
Students were to in groups build a three dimensional object out of parts that I provided. They would then each describe and draw the object on their papers and give the object to another group to work on. The other group would then describe and draw the shape they received, and then work together to find the volume and surface area of the shape. Each student would then hand in a write up of the activity and then complete the rest of the midterm. The activity was graded as one of the midterm questions.
In giving such a task, I worried that students would just come up with simplistic objects in order to take the easiest possible route. However, the drive for most groups was quite the opposite. They challenged themselves with the most complicated shapes they could come up with! They even surpassed the outcomes of the unit in creating shapes such as:
Because they had created their own shapes, they had this drive to engage themselves in figuring out the volume and surface area regardless of how complicated it got! Many fruitful ideas and questions came out of their discussions.
If the stage is set right, group work can be very productive. However, I am constantly searching for new ways to switch up my group work strategies because too much of a good thing can take a hit. I have found that if I organize my class in the exact same way too many times in a row, students lose interest and it is no longer a novelty.
“Keep it fresh” is the advice I like to follow and remind myself of every time I plan a lesson.
Since I have much more class time available for students to engage in learning material, I have been playing around with various types of tasks. One of which is getting students to generate examples of questions they may be asked. Any teacher who has developed their own questions knows that doing this is more difficult that it seems. A lot of knowledge is required to make a question to which a solution may be found using prerequisite knowledge. I was teaching a unit on using properties of exponents to simplify exponential expressions. I asked students to generate a question in groups and give the question to another group to solve. I also asked them to create a potential unit test in order to build study skills. I was afraid that students would choose to create extremely simple examples and that they wouldn’t learn enough from the task. However, I didn’t realize how difficult it can be for a student to create a “simple” example. In order to know that it will be simple to solve, they need to be fluent with their knowledge. In my class, all the students decided to try to make the most difficult problem they could think of! I was shocked. It led to a lot of good discussion and I was able to clarify several key points to them through analyzing how they would solve the problem. For example, a lot of them threw in addition signs but didn’t realize how much more difficult it is to simplify rational exponential expressions with several terms in the numerator and the denominator! This stemmed from the fact that they were still unsure of the difference between multiplying exponential variables and adding exponential variables. Other nuggets came out as well. This task was so useful that I ended up putting a question on their unit test to create an example and solve it. The fact that they had to solve it themselves forced them to try to stick to using properties they understood.
I’d like to share my favorite student generated example from the group work activity:
This is truly brilliant. Is it not?