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. 


Saturday, April 11, 2015

The Review Day: Unit Analysis and Scale Factor

There seems to be three sacred cows in mathematics education:

  • the worksheet / exercise set
  • the review day
  • the exam

It is not surprising that these three feed off one another, and make up the bulk of assessment in the typical mathematics classroom (including my own). 

Here's my disclaimer:

While I have been known to slaughter a few of the sacred cows of the instructional process, I have lagged severely behind in my attention to assessment. I value the complexities of learning that occur when student ideas encounter perturbations, curiosities, and other conceptualizations. The type of assessment that comes out of these mathematical encounters is rich, connected, and constantly evolving. 

Assessment compacts this complexity into tiny chunks. 

My mind wanders to this scene from The Brave Little Toaster (a childhood fav) where the cars are compacted into tiny cubes by a vindictive magnet and junkyard trash compactor. 

Exams are not worthless, but are impersonal and static. I've employed the review day as a sort of buffer between the two. Class tasks are used to enact mathematics in a connected ecology, and the exam is designed to separate and operate on a static set of knowledge. The review day is a transition where we call upon the anchor activities of the first stage to perform well on the exam. Students typically complete questions from a textbook. The room loses life, and I hate it. 

The following are two activities I recently used in a unit with Grade 11s on Unit Analysis and Scale Factor. The goal was to discover a hybridity between the (en)active nature of knowledge formation and procedural fluency often needed for the exam. While I'm sure they are developed from the influence of others, I can say for sure that the second activity is adapted directly from Kate Nowak's "Speed Dating". She has many other promising review activities on her blog

1)     Problem Gallery

Before students came into the class, I printed four typical questions from the unit and taped them to the whiteboards around my room. (I have a front, side, and back wall boards). The students were told to solve each problem with as many people as they wanted. I did not assign groups, or even mandate them. I wanted it to be free-flowing. 

At each station, I placed a simple table with three gradations of difficulty. I gave each student a marker, and asked to them to self-assess the difficulty of the problem when they completed it. The result was a quick tally of which types of questions are giving the class the most trouble. 

The nice part about this activity was the free-flowing nature. If a student was stuck on a particular station, two or three new waves of help would arrive and share their perspectives and guidance. Students used the whiteboard space around the problems to solve them, but I asked them to erase solutions before leaving so everyone could have a blank slate. 

The conversation was natural, and I was able to spend time at stations or with individuals that needed more attention. I was also able to instigate discussion with the class afterwards around the prompt: "What makes one of these questions difficult?"

This was simple to set up, and allowed for a large amount of time to de-brief, discuss solution paths, and ask follow-up questions. It enhanced repetitive practice with peer conversations and problem analysis. 

2)     Speed Dating

I wrote 12 ridiculous questions regarding scale factor and unit analysis, and labeled them all with an astrological sign. (We always joke about pseudo context in my room, so the questions are designed to purposefully poke fun at the textbook's attempt to be "real world" and "relevant"). Each student was given a problem to complete and become an expert on. I gave 15 minutes at the end of class to make sure each student was confident in their own solution for the next day. They were told that they are responsible for explaining their process on a series of speed dates the following day. 

I arranged tables in a row and set a timer for 4 minutes. Students then had two minutes each to read and explain their problem to their date. As the activity progressed, they got so fluent with the explanations, that the timer was reduced to 2.5 minutes. It wasn't long before every student had seen the 11 other problems. 

I sat in on certain dates and helped students who were a little more reserved find partners. As the activity wore on, some claimed they had "dated" everyone necessary, and I set them up as a third wheel on a date involving two problems I considered particularly important. 

This, too, took about 35 minutes and we had plenty of time to 1) rank the problems from hardest to easiest (and justify our choices) and 2) hear some "elevator pitch" solutions from some of our favourite dates. One young man started with the line: "Do you want to date me? I'll tell you about my lump". (He was given a problem about the scale factor of an exotic skin disease.)

I feel that while I am still looking for a way to honour the complexities of an (inter)active community of learners on summative assessments, a better use of the sacred review day is a start. 


Tuesday, March 17, 2015

Math Challenge Activity

I built this activity for a group of 120 students from grades 7-10 at a provincial math contest. The problems themselves are a mixture of created, adapted, and stolen. I chose them because they fit fairly nicely into a multiple choice format while still eliciting deep thinking.

The puzzle moves forward as follows:

There are 10 stations, and 10 problems. Each problem is responsible for giving a unique letter for the final word scramble. Some of the letters are repeated more than once in the final answer (i.e. have a frequency more than one), but no problem leads to the same letter.

Each station contains a single problem and four multiple choice options. Each option is paired with a cipher key to apply to a letter also given on the problem card for that station. The students move from station to station answering problems, getting correct Caesar cipher keys, decoding the letters, and building a bank of letters to later unscramble to get the secret message and complete the activity.

I included a space on each station card for Station Number and Room Number. I had 10 separate rooms to work with, and wanted to get the kids moving around the building; you could set up 10 stations in the same room. Closer confines leads to more debate.

An example is given on the student recorder sheet:

If a student feels that the correct answer to the problem is the second one, then they would receive the Caesar cipher key of 5. That would move the letter "R" (pictured on the page) 5 places through the alphabet landing on "W". Because the frequency of this station is x3, three of those letters ("W") would go into the pool of letters for the final word scramble.

I've uploaded a .pdf of the problem sheets (one for each station) as well as the student recorder sheet I gave them to house their answers and guide their search. Word documents had insane formatting issues when uploaded.

Problem Sheets

Recorder Sheet

The final message should unscramble as:  THREE POINT ONE FOUR

A couple other anagrams were deciphered by students:


The problems could be changed to test a specific topic as a review or work period. If you do change the questions, be sure the correct answers still align with the correct cipher keys.


Tuesday, March 3, 2015

Large Whiteboard Project

Group whiteboarding has changed how I teach mathematics. It has also changed how students operate as a community of mathematicians. 

Since ordering my first set of large whiteboards, our department has ordered four times again, and given workshops to the division's mathematics teachers. (For a tour through my whiteboarding history, start here: mini whiteboards)

My running motto has become, 

"Whiteboards give me more than eight-and-a-half by eleven ideas"

This, coupled with the assertion that you can't expect limitless ideas with limited innovation space, caused me to think bigger. This is the result. 

Whiteboard paint from the HomeDepot coupled with ebay'd Washi Tape creates a new innovation space across my back wall. The new ultra-large whiteboard opens up opportunities for larger groups and impromptu collaboration as one group comments on the work now presented in plain sight. 

It also accommodates students who need some movement while thinking. 

Three pictures of the completed project:

For some reason, it makes the room feel larger. It encourages collaboration, curiosity, and conjecture. Next stop, find a classroom made entirely of dry erase surface. (probably a harder sell).


Sunday, February 8, 2015

What High School is (Often) Missing: A Conversation with a Kindergartener

Sometime after pyjama time and before bedtime, a math conversation broke out. My wife and I were visiting some good friends, when the topic of a recently purchased board game came up. It was bought at a teaching specialty store and designed to teach addition and subtraction of twos. After examination, I didn't like the overly symbolic structure, and asked their 5-year old if she wanted to play a math game. She ran and got a piece of paper. When she finally got called up to bed (much later than expected) I took the page and folded it into my back pocket.

Here it is:

Like most educational artifacts, they enliven with the story of their creation. In this case, our "game" made me think of my own classroom and the characteristics that are so often missing. They came so natural to her--a five-year old whose parents just finished telling me that they don't do much math at home. When my high school students think like kindergarteners, I leave knowing I've done my job participating in the action of creating mathematics. 

For the sake of clarity, I've dissected the image above. Each section carries a mathematical story, one of profound, and innocent, brilliance. 

I started by telling her the rules. I gave her a number, and she had to tell me how many more she would need to make five. We went through a few examples (my choices are on the left and she countered with the digits on the right). She started by holding up her hand with all fingers down, and raising the amount that I 'gave' her. Then her answer was the number of fingers still down. She counted the raised fingers one by one, but could recognize the 'down' fingers almost instantly. During this whole process, I was analyzing her thoughts. So much so that I missed her say, "There are some sames". 

She answered the last question without fingers; she answered it with 'sames'. She explained (later connecting the arrows) the pattern she saw. Her solution to "0" was a simple mirroring of the solution to "5". 

We then changed the rules. I asked if we could play the same game with 10. She obliged (eagerly) and we began. I decided to start with a large number to see how she would answer. She counted all five on her left hand, and then counted four on her right. Without announcing it, she scribed her answer. She answered all the questions correctly, but later asked if we could change the "7" to an "8". Then she announced that the answer would be "2". 

Another interesting process was her writing of the numeral "6". She rotated the page and examined the "9" already on the page. In the end, her second try was the charm. I downplayed the incident; the last thing I wanted was for notation to interfere with the great mathematics going on (which it so often does in high school). 

This was the most interesting case in our game of "10"s. I gave her "0", and she silently thought for a few seconds. Without counting fingers, she gave me a wry smile and told me she was going to use two numbers for the answer. Notice, she didn't ask, she told. I (naively) assumed she was going to write "10" and was calling that two numbers. Instead, she wrote "5 5". 

She was absolutely correct. The whole point of the game was to decompose 10, and she did just that. She had out-mathed me. I then asked her what the two fives make, and she nonchalantly replied with "10". I wrote the "10" and also an "=" sign. This was a mistake; I didn't want any notation. She didn't notice and we moved on. 

This is the coolest part of the page. She wasn't satisfied once we had finished with our game. She asked, "Is there any more sames?" 

When I wrote this on the page, she suggested that she write her name on it. Obviously, credit should be given where credit is due. Then a cool thing happened. She asked if we could make sames of a different number; she wanted to do 12s. I drew two extra fingers on the page (because she didn't think using toes was a good idea), but she got called to bed before we began. 

As she left, she turned and said, "We will do 12s the next time to find the sames". 

Something about this episode is incredibly desirable--especially to a high school math teacher. The strategies used are so free-form and the persistence contains such an honest agency. To her, the math was playful and pliable. In recognizing and extending the pattern of sames, she showed the curiosity to wonder; in justifying and owning her solutions, she showed the industry to get results. 

These two attributes are a potent example for any of my high school students. 


Sunday, January 4, 2015

Bloggable Distributions: Reading #MTBoS Blogs in 2015

Twenty-fifteen will be the fifth year that my little corner of the blogosphere has been dedicated to digitally curating my own thoughts and experiences regarding the teaching and learning of mathematics. It represents a wide array of posts regarding a wide array of topics. Much has changed from new teacher status to graduate student, and the posts reflect that. Still, the heart of its posts and pages is pragmatic: I write about classroom events that seem to matter (for some reason or another, they catch my attention) in hopes that other teachers might find the same phenomenon.

I am going to call these episodes: bloggable moments.

This post is inspired by Geoff Krall's, Math Blogging Retrospectus in which he (non-exhaustively) curates his favourite posts from the year. (This would be a great space to start if you are interested in starting somewhere in the online math-ed mess). While we are on this nostalgic walk down memory lane, Geoff was one of the first to offer support to me--a new teacher--when I sought it through the #MTBoS (Math Twitter Blog-o-Sphere). I was lucky to happen upon him, and I hope to offer that same support to others. 

When I sat to compile my own list of posts, I couldn't erase this thought from my head. I was lucky to gain "access" to the online world of math educators. There is no membership (so to speak) but there is a prevailing understanding that resources are powerful currency. You are measured by the tasks you present, good questions you offer, extensions you have, comments you leave, and day-to-day novelties you provide. It is unspoken, yet existent. I imagine it creates much the same effect as the "closed-door phenomenon" (where teachers crave solitude out of insecurity) or the "only/just phenomenon" (when teachers constantly qualify their lessons as only lectures or just drills). 

This post is not about the best posts of 2014, it is about how to read the best posts in 2015. A sort of meta-experience. 

This skill is growing in importance as teaching blogs become mainstream. I think it is safe to say that this is well established in 2015 by the incessant focus on educational technologies as well as the emergence of figureheads (none larger than Dan Meyer). We tend to place blogs outside our normal human experiences. They exist as edifices of perfect practice--of master teaching. The author has full creative control, and this robs you (the reader) of the opportunity to see the full picture. 

Posts (deemed bloggable) are only instances in a constant distribution of interactions we call teaching. Without the frame, it gives the impression that there exists a teacher that operates in a bloggable state at all times. This is discouraging and delusional. 

By way of framework, I've illustrated a number of possible distributions of day-to-day teaching and their relation to blogs. It is my hope to re-cast your impressions of the nature of bloggable moments before you go read the "best of the best" from 2014, and hopefully create your own posts.

The Uniform Distribution

This is probably the most damaging perception of teaching on the blogosphere. Here, the horizontal axis represents a passage of time or flow of semester. The vertical axis is the strength of classroom event. The red rectangles illustrate the instances that a teacher would deem bloggable and present to readers. (If teaching was indeed uniform, any random instance could be labeled bloggable; if such uniformity was ever reached, I would worry about either the teacher's impression of success or tendency toward pathological lying). In this case, the reader is given the impression that because all they see is top-notch teaching episodes, then this must always occur in that particular blogger's classroom. 

The Normal Distribution
A favourite of mathematics teachers. Now, the horizontal axis represents the strength of classroom event (maybe coining the term bloggability is too much for now). The vertical axis counts occurrences. If we take blogs to be representative of the blogger's entire practice, then the reader (if they subscribe to this distribution) may be led to believe that the posts are indicative of the most common lessons, tasks, or prompts. An event is bloggable because it (or similar ones) occur often. This takes an important step away from the uniform distribution, but still sends the impression that in order to be a good teacher, events like the ones detailing master teaching must be the norm in your class.

The Power Law Distribution

Probably the most accurate description of blog posts if we are going to work from the assumption that classroom events occur in relative isolation (as many--if not most--blog entries do). Here the axes retain the same representations as the normal distribution, but the bloggable moments no longer represent the most frequent (read: average) events, but those deemed to be the most potent. It is comforting to understand that those events with the most potency occur with the least frequency. Such a phenomenon is not unique to the classroom but lends itself to earthquakes, market fluctuations, and epidemics (Davis, Sumera, & Luce-Kapler, 2008). If you settle with this distribution to frame your reading, it wouldn't be the worst thing in the world. (It also wouldn't be the best...IMO)

An Alternative

I have two problems with the above distributions.

First, as mentioned above, these assume teaching happens in segments each loosely tied to a block of time. While many blog posts detail bloggable moments that carry projects, orientations, or themes intended to transcend these boundaries, teachers still think in this unit of currency: the lesson

Second, any teaching event occurs in a context--an ecology. Most of the time, bloggable moments occur in classrooms, but many posts present situations from home life, theorized classrooms filled with theoretical students, or everyday life (intended, of course, to bridge into a classroom). Some just provide contextless problems, tasks, or prompts--possibilities to be moved into a context. This means that bloggable moments don't represent a ready-made parcel of teaching, but a rich culmination of interacting forces. Try as we might, it is impossible to entirely relate the context to the reader. In order to be documented, a bloggable moment must be ripped out of context--disconnected from experience.

The Fractal Distribution

In the fractal image above, there is no scale (in fact, it is self-similar and scale independent). The simple governing rules create a rich pattern of interactions. Several of these patterns grow darker as they begin to attract more activity. This is a visualization of a complex classroom. The circled areas are the moments that may stand out as bloggable. Notice how they must literally be pruned out of the tree-like structure of associations, connections, and knowings. 

Good teaching is less about the judgement of "ineffective vs effective" and more about being connective. Don't read blogs thinking that what the blogger has deemed successful was pre-planned in that exact form. Teaching is an action; allow it to be such. 

The fractal image reminds us that a classroom is a complex system and judging task/lesson/prompt/learning success is more about attuning oneself to the clustering of connections rather than attaining chunks of currency in the form of bloggable lessons. The moments that you see strewn around the #MTBoS are those that caught our eyes or, better put, our imaginations. They occur within our classroom ecologies and with our students. Every system is unique. We can talk about how we attempt to encourage such clusterings--and I do believe there are attributes that encourage emergence of these attractors--but in the end, it is your class.

The influence of online collaboration in mathematics education increases with each passing year. In my opinion, the volume of material has exceeded the ability of situated consumption. These distributions are one way to frame our consumption, and hopefully encourage more to try their hand at creation. 


Davis, B., Sumera, D., & Luce-Kapler, R. (2008) Engaging minds: Changing teaching in complex times, 2nd ed. New York: Routledge.

Sunday, December 7, 2014

Visualizing Linear Systems

My Grade 9 students don't see an equation for the first two weeks of their unit of solving linear equations. That is because I think students get all bogged down in the notation, and lose their problem solving intuition. 

Instead, I play around with a key metaphor for solving linear equations--the balance scale

I've used the metaphor before, but only verbally alongside an algebraic representation. I would say things like, "What you do to one side, you must do to the other", or "What would we have to do to keep the scales balanced?" The whole time, I only referenced the metaphor; students were never required to work with it. In hindsight, this probably did little to help students with the abstract nature of variable quantities. 

Now, I get students to encounter a balance problem, and talk me through their solution. They begin to talk about things like "splitting weight evenly" when two circles are balanced against a value of twenty. They tell me that I can ignore a shape that appears on both sides of a balanced set of scales. The explanations come from them, and they encounter them through the lens of their everyday. 

The abstract will come. Until then, students encounter various principles of equations as different sets of scales are presented. 

Below are nine sets of scales to introduce the notion of systems of equations. Teachers--and textbooks--talk about "real world" as a way to tether topics to students, but I have always had trouble finding a task that can make the conceptual leap from the situation (phone plans, car trips at different speeds, etc.) to the algebraic notation. 

Instead of appealing to the students' surroundings, these scales are "real" in the sense that they appeal to their intuitions--their unavoidable tendency to organize, categorize, and achieve balance. 

Each question consists of two sets of scales, and students must find the weights of the circle and square. The nine questions are divided into three sets with small variations included. The three are designed to be used as a single object (Watson & Mason, 2006). This is done with the hope that students will use these small shifts to build a better understanding of how the variables act on one another. 

They are relatively simple to create. Obvious extensions include fractions of circles and squares as well as more "variables" (shapes). Any typical textbook exercise can be converted quite easily. Having these pairings (or having students create them) can be a powerful tool. 

Set 1

Set 2

 Set 3

Give them to students. Ask them to explain their thinking. All involved may be surprised by dormant algebraic thinking that just needs an intuitive trigger.