Tuesday, June 23, 2015

Classroom Clean-Up

No more students for this year. I've spent a full day cleaning up and re-arranging my space for my incoming intern (for whom I'm very excited for).
Amidst the broken calculators and stray linking cubes, I found a note that a student wrote me from my first year of teaching. It served as a brief reminder of why I attempt to curate a community of mathematical action with my students. It isn't the easiest way to teach, but has a limitless ceiling.
Dear Mr. Banting,
You called me a "genius" in math class once, and that night I went on the web to search up the word and its definition, and I came across this quote.
People are in the misconception that geniuses are born, rather than made.
After reading the quote about 5 times, I realized its meaning. I remember in grade nine I used to be impatient for math class to start. I would treat every question as a fun challenge, and along the way, if I didn't get the answer, at least I learned something new. But for some reason, I don't know, I lost my passion in the following years. And then you came along and challenged me with rich questions and I realized that somewhere deep down, it still excites me to be challenged with a math question.
In short, students like me need genius teachers like you to make a difference... Thank you for making a difference for me.
 This, to me, cuts through the year-end mess.


Sunday, June 14, 2015

Fraction Talks

Discussion is one of the organic ways through which human interaction occurs, but not all discussion is created equal in the math classroom. The tone of discussion relies on the mode of listening (Davis, 1996). Most classroom talk focuses on an evaluative mode of listening. Students are expected to share, compare, and contrast solutions to problems.

I do think that justification of their solutions gets at some important points regarding mathematical reasoning, but would like to move the discussion to center around that exact feature--the reasoning.

Rather than piecing together the pieces of isolated reasoning (which I still think has value), I want to see a collective reasoning emerge through the discussion. Students don't "think-pair-share" their solutions; students bring their conceptualizations and reason to a collective understanding. There is something about the power of this collectivity that makes the learning unique. (And I am insanely interested in how it differs, why it differs, and if it is viable in a classroom).

To try and harness this type of reasoning, I created (with my students) a series of "Fraction Squares" inspired by a keynote given by Ilana Horn and SUM2015. They all involve a dissection of a square using straight lines.

I project a Fraction Square, shade a section, and ask, "What fraction of the square is shaded?"

I moderate the discussion and entertain all solutions. It isn't long until the discussion no longer runs through me. The students turn, pull out paper, run to the board, and justify their thinking.

As they talk about slicing the sections into pieces, I do my best in interject precise vocabulary. For instance, a student might say, "A half of a half is a quarter", and I may probe this to elicit which operation they are really talking about. I don't push the standard algorithm of multiplying fractions, but write number sentences beside their quotations.

This way, the vocabulary and notation is slowly imbibed into their collective action. Students start to talk about the four operations (instances of division by fractions are rare), and the number sentences act as a bridge to conceptualize the standard fraction algorithms. It provides an active visualization of processes like creating a common denominator. The picture is not presented and explained; the picture is presented and then acted out through collective discussion. In a sense, it becomes personal.

Some groups may focus on certain lines of argument, and others find them completely disinteresting. It is the job of the teacher to remain fluid, keep end goals in mind, and network the collective.

Some other prompts to use:

Which section is the easiest to find?
How many different sections exist and what are their values?
If the total area was "x", what is the area of the piece?
Can you find all pieces that are exactly "1/x" of the total?
Can you find a piece that is double / half of the section?

Above are a few more examples of fraction squares my students created this year. After a few fraction talks, have your students do the same and then exchange them. These seem to introduce enough perplexity to facilitate a collective discussion where students think out loud to arrive at a strong conceptualization of part-whole relationships. 


Davis, B. (1996) Teaching mathematics: Toward a sound alternative. New York, NY: Garland. 

Wednesday, June 3, 2015

Math Wars North

O Canada!

The debate about best practice in Canadian math education has exploded once again. This time attracting high profile combatants.

This post is not meant to resolve deep-seated values, but rather provide a perspective that gets lost in the partisan arguments. It wouldn't take a long time to place me in a camp, but that would be assuming that there are two camps that want drastically different things.

Now there are certainly battle lines drawn, but would it be possible to distill the highbrow mudslinging into a succinct cause--a common denominator?

The recent tweet conversation between Dan Meyer and Robert Craigen (storied by @MathewMaddux here) serves as an excellent example of the arguments entertained 140 characters at a time. It is dizzying to follow, but if you do, you just may find the essence of the entire debate:

It just may be that Mr. Craigen summarized the problem with his argument: students don't operate like streams. The idea of "standard" and "empirical" only begin to describe the complex process of education. Herein exists the disconnect.

It is a slap in the face of mathematics educators to reduce our craft to the transmission of facts and administration of discipline. Mathematicians view math teachers as deficient mathematicians, and math teachers view mathematicians as deficient educators--islands of incommunicable knowledge. Teachers view it as pretentious when mathematicians imply that the way to be mathematical is to become like them, and mathematicians scoff at the gambits teachers necessarily entertain to allow students to experience productive struggle.

In short, the war is personal and the conversations often go there. Rather than having a conversation about what is best for something both parties value (mathematics), it becomes a dance of rhetoric. Both sides take astute stabs at one another. Rather than trying to understand the opposite lens, we (yes, we) shroud petty insults in deft wordplay. It is a self-righteous battle waged on high horses.

Below are plenty of examples (from both discussants).

Mudslinging aside. I'd like to suggest summary statements from what I've read from both sides. (Admitting fully that I am a math teacher and also have an ego to protect).

Mathematicians: Math doesn't operate solely on memorized algorithms, but they form an important foundation for the interconnected and fluid process of mathematics creation.

Mathematics Educators: Teaching math effectively is not simply a summation of predetermined teaching scripts but a dynamic process that involves the interrogation of foundational facts.

It could be that mathematicians and mathematics educators have one common thread: we want to control our own volition. There is more to the respective crafts than the other is aware of. Math education cannot be brushed aside as a make-work project where the memorization of routine facts would suffice--even if they are artifacts of our heritage. Mathematics--the work of mathematicians--cannot be dismissed as stoic or inhuman.

Until this becomes an actual conversation, it will remain, in fact, a war. And that helps a total of zero students.


Quick Update:
I tried to engage Dr. Craigen in productive Twitter conversation, but he told me he wasn't interested in pedagogy or the classroom--only educational change. 
I offered a space to offer longer responses, because I was losing control of my tweets as they filled with a one-sided, non-responsive rant every time. (I'd call it a response, but I don't think he listened.)
I asked him to return to my questions once he cared about the teaching and learning of mathematics and not just the process of doing research mathematics. That is, after all, what the conversation needs to be about. 
Two of the last tweets he sent me said, "the goal of procedure mastery is less thought, more power" and "students don't need detail". Until these things change, I don't think conversation can take root. 

Tuesday, June 2, 2015

The Scale of Coffee Cups

A colleague is a religious McDonalds' coffee drinker. One day she showed up with a medium coffee and a cream on the side. It was in two separate cups:

I asked her for her cups when she was done. (She is also a math teacher so understands that this is not a creepy request. It is no weirder than the time I bought 400 ping pong balls, or 1500 bendy straws). I then made her a request to buy a large and small coffee in the future and save me the cups.

The result was a family of coffee cups stored in the bottom cabinet behind my desk. I had a student in a grade 10 photography class take pictures of the cups in various arrangements. 

The result is a simple task:

Are the coffee cups similar? 

I'm hoping that various lines of reasoning emerge. Personally, I love that the diameter of the two largest cups (Large and Medium) is the same. This means that a student could pick up on the fact that a scale factor of one cannot possibly exist between their heights so they cannot be similar.

I assume that some type of prerequisite knowledge would lead some to create the (scaled / original) ratios to determine that they are not similar.

I am hoping that the fact that the two middle cups (sizes Medium and Small) have exactly a 1 cm difference in diameter sparks a conversation. I often have students dealing with scale factor as an additive relationship. (i.e. Diameter is 1 more, and height is 2.3 more, so scale factor is 2.3). If this comes up organically, it is a huge win.

After initial conversations, I am planning on giving groups of students extension tasks:

Design an XL cup that is 1.3 times the Medium cup.
Design a cup that is similar to the size Small.
Design a cup similar to the Medium cup, but no taller than the Large cup.
What would the height of the Large cup need to be to make it similar to the XSmall cup?
How much taller would the Small cup need to be to fit a Large lid?

Students finish at various paces, and with varying degrees of work. I love it when students are not working and have a fantastic reason for doing as such. I asked a group if the Medium and XSmall cups were similar, and one student told me that he didn't need to do the calculation. He reasoned that he could tell the height was increased by a scale factor grater than 2 (12.9 / 6) and the diameter of the Medium cup was not close to double the diameter of the XSmall cup (8.9 / 6.3). Therefore, the cups couldn't possibly be similar. 


If you wanted to direct the class toward scale factor in 2D & 3D, you could assume the cups are cylindrical (or pieces of cones) and ask:

Which cup offers the best bang for your buck?
How many XSmall cups  of coffee will fit into a Large cup?
Design a cup that has exactly twice the coffee as the Medium cup.

I suppose the point here is: tasks don't have to be amazingly unique contexts. Great mathematical opportunity can be found in regular places, teased out with curiosity, heightened with interesting questions, and curated through collective action.


Monday, June 1, 2015

Teacher Hack: iPads in Exams

My department has a set of 10 iPads for mathematics instruction. I use them primarily for the powers of Desmos. When I introduce teachers to the program, they get excited about the possibilities, but are immediately worried about one thing:

How is it used in exams?

While this may be a tad short-sighted, it is a legitimate concern. Teachers simply don't have the resources to constantly be monitoring a class of students to be sure that they are not accessing the internet or communicating with each other (which is fairly easily fixed in settings).

The greatest part of iPad technology is the connectivity. They have the potential to facilitate inter-actions between students and their thoughts.

A friend of mine introduced me to the solution: Guided Access.

Guided access is a feature where the teacher can lock the iPad into a specific app. It requires a passcode to get out. It even has the ability to disable certain portions of the screen. (I'm not sure how this would come in handy for Desmos, but may work for other apps).

How to Enable Guided Access

               Guided Access

Turn it "On".
Set Passcode

Also, turn on "Accessibility Shortcut". This will enable you to triple click the home screen to change settings.

Close Settings and open up the app you want and triple click the home button.

The menu will appear, and click Start to lock the app.

If a student tries to leave the app, they will be prompted to enter the passcode. If they enter an incorrect one, it makes them wait 10 seconds to try again, then 60 seconds, etc.

On top of that, triple-clicking and the entering passcode allows you to draw on the screen where you would like it to be disabled. In the same screen, selecting Options allows you to disable touch and motion altogether if desired.

If you want to end Guided Access, simply click End.

This allows you to open Desmos, engage Guided Access, and (temporarily) limit the connectivity of the devices.

See this post for more details about Desmos and standardized test use.

This setting--also great for occupying young children--alleviates the worry of academic dishonesty and arms our students with powerful learning tools.


Friday, May 29, 2015

Connecting Quadratic Representations

I always introduce linear functions with the idea of a growing pattern. Students are asked to describe growth in patterns of coloured squares, predict the values of future stages, and design their own patterns that grow linearly. Fawn's VisualPatterns is a perfect tool for this.

While stumbling around Visual Patterns with my Grade 9s, we happened upon a pattern that was quadratic. The students asked to give it a try, but we couldn't quite find a rule that worked at every stage. While I knew this would happen, the students showed a large amount of staying power with the task. The pattern growth was an engaging hook. After a conversation about what made this pattern ugly (the non-constant growth), we looked at the growing square.

It was then that I decided to try and insert patterns into the introductory unit on quadratics--a notoriously dry and abstract topic (at least in my class). I developed a series of quadratic patterns and animated their growth on Vine. They can be found on the Vine page on this blog. (Look for Quadratic Patterns). Each one is represented as a yellow set of blocks, but then as coloured blocks to illustrate possible ways to visualize the growth.

The plan was to play with patterns, develop quadratic functions, and then use the blocks to find the shape of the new, non-linear graph.

How I Structured Class:

I randomly grouped the students into groups of three, and started them off with a linear pattern from my Vine page. Once we discussed how to generate a relation, I asked them to copy down the first three stages of this quadratic pattern:

I then gave them each a handful of tiles, and asked them to model stages one through six. This took a little bit of trading as some groups became obsessed with having the colours match in all the squares. This (while slightly OCD) came in handy in visualizing the graph later.

Once all their stages were built, I collected residual blocks, and then ask them to do the unthinkable--destroy their patterns. But keep the stages separate. I drew a Cartesian Plane on the board and told them to use the tiles to build a sort of bar chart on their tables.

The results were the shapes of the graphs of parabolas.
Here, the +2 is represented by one red block on the bottom and one on the top of each stack. 
After stage seven, the pattern had outgrown the table
Some decided that, to save space, they would stack their blocks.
This stack preserves the +2 with the use of two blue foundation blocks on each stack.
This quickly became the method of choice throughout the class.

OCD colour matching comes in handy for pattern visualization
The cool thing was the conversations that emerged from the blending of the representations. Students began to use the metaphors of scaffolding that was holding up the graph, and roller coaster supports while the graph was the track. Some began to comment that it is good they didn't go past six, because it would have got out of hand. It provided an excellent image for them about the increasing slope of a parabola.

We talked about the cause of the rapid growth, and settled on the square. If we wanted it to grow slower, we would need to minimize the effect of the square. This brought the idea of width into play. We asked what would happen if negative stages could be represented. We used the function we built to help us conceptualize a "stage -1". It didn't take long until they had the concepts of vertex and axis of symmetry. (Although formal verbiage only became established when it became necessary for efficient communication).

I found myself referring to the activity when we were talking about the concept of range. We recalled how the patterns quickly out grew the tables, and would have continued on if we would have added stages. This helped them anchor the idea of an infinite range.

Quadratics occupy a large chunk of our grade 11 math curriculum. This was a nice way to connect a representation commonly used for linear relations and functions in the previous grades (that of the growing pattern) to the graphical and algebraic representations of quadratics.

Dan Allen captures the activity in 5 seconds. (The lesson now loops full circle)


Thursday, May 7, 2015

On Collective Consciousness and Individual Epiphanies

I would like to begin with a conjecture:

The amount of collective action in a learning system is inversely related to the possible degree of curricular specificity. 

The mathematical action of a group of learners centred on a particular task gives rise to a unique way of being with the problem, but also reinvents the problem.

In short, what emerges from collectivity is not tidy. 

How can I justify curating a collective of learners, when school is so interested in individuals?

Learners commerce on a central path of mathematical learning while acting on a problem, but each take away personal, enacted knowings from the process as well. Collective consciousness grows as agents interact, but we live in a system that values individual learning--often in a very narrow sense. Although I cannot be sure where the problem will go, students will become more mathematical by acting on it. 

This wondering has been pushed to the forefront of my thought by two events today. First, a Skype call with a graduate supervisor regarding the nature of collective consciousness and its relation to the outcomes-based school system we teach in. Second, a moment of personal significance from a Grade 11 class on quadratics. 

Here is what happened and why I think it illustrates the essence of education:

I have a group of grade 11 students who have never seen quadratic functions. In a effort to tether the idea to linear functions, I organized them into random groups of three, gave each group a large whiteboard, and asked them to graph the function using ordered pairs:

I anticipated students beginning their table with "x= -3" because the values -3 through 3 were commonly used in our study of linear functions. I purposely gave them a quadratic that returned large numbers for the first few inputs. 

I watched as the groups began organizing themselves around the task. It wasn't long until each group developed a personality. Some groups divided inputs among themselves, and built a joint table of values. Others worked through the arithmetic together. Conversations around input choice and error correction began as their pattern-finding skills took over. 

Why is that so big?
Make it smaller!
It won't fit, so change the scale.
The square is making everything too large.

The large output values perturbed the groups' thinking. Some handled it by changing the scale while others chose inputs that made the square as small as possible. I wanted groups to do the latter, but some resolved the problem as a matter of scale. They changed the problem, and I gave them the licence to do so

I wanted them to get at the idea of a vertex, a lowest point, or a turning point. I asked them why the graph was turning around, and because they had been given the opportunity to experiment with choice, they knew that the exponent was creating positive outputs from negative values.

This was perfect. It was exactly what I wanted them to get out of it, and I had harnessed collective action to get there. 

I would have been more than happy to distill these experiences into grade-level competencies, but then I took one more stroll around to discuss students' work with them. One group had a table of values on their board:

I commented on the growth of the y-values. One student said the following:

"We noticed that the growth wasn't constant, but it did grow constantly"

It was this moment that I pulled out my phone and wrote this quote down, because it clicked. She had described differential calculus. 

I took the time to act collectively with my students and it couldn't have paid off more... for me! They didn't know it, but I pulled an extremely valuable individual knowing from our collective knowing. They centred their group work on the idea of change. From there, they looked at the symmetry of change and how it created a parabola. This is valuable work. Each had encountered the math and created personal coherence from the task as well. 

Where these personal knowings landed, I could only assume based on our interactions.

For me, the personal knowing was centred around the connections between tables of values and calculus. The pattern they showed me occurred because the rate of change of a quadratic function is not constant, but does grow constantly--linearly. I had pulled the idea of a first derivative from a lesson on introductory quadratics. 

Circling back to the point of the post: I saw great learning from the groups. It wasn't all identical, and that enriched the fabric of the lesson (and the intended aims). From the space opened for collectivity, I pulled out a personal meaning--one deeper than I ever would have anticipated. 

This is the very essence of education. In a system obsessed with individual scores on specific competencies, we lose sight of the fact that deep meaning is pulled out of collectivity. It isn't one or the other. Curating a collective consciousness in the classroom allows students to build understanding in context as they change their problems with their actions on them, but that doesn't preclude them from creating powerful, personal meanings. 

The episode provides an illustration that even though collective student action softens control on what students digest mathematically, it doesn't mean that the classroom events provide only group knowing and lack personal meaning.