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.


Tuesday, November 25, 2014

Polynomial Personal Ads

Every year, my students study the general characteristics of polynomial functions. We investigate the various shapes of various functions and slowly shift parameters to watch changes in the graphs. Eventually, we deduce the roles of the constant term, leading coefficient, and degree.

It should be noted that Desmos makes this process much easier than years previous. Just set up the generic polynomial, add sliders, set specific ones to play (depending on what you want to investigate), and have students discuss in groups.

See sample here. (Sliding "a" to "0" invites an excellent conversation; same with "b" etc.)

After we work with the transition from function to graph, we go the opposite direction. (makes sense, right?)

My favorite types of problems, however, ask students to play with parameters to influence results while leaving some characteristics consistent. For example, they might be asked to write a polynomial function that has an identical Range but different y-intercept. Or an identical end behavior but a different number of x-intercepts.

We play with these choices for a while. (I have them come up with lists of characteristics that are impossible...this is a great conversation)

The student work below comes right before the exam is written. They are asked to write a personal ad for a polynomial of their choice as if it were joining an online dating service. They cannot state their degree, leading coefficient, or constant explicitly. The result is an interesting exercise in encoding and decoding sets of possibly parameters in polynomial functions.

I am a polynomial function, super fun and curvy with two turning points. I am currently on a down slope in my life, but I don't want to sound negative. I am an infinite range of y-values and infinite domain of x-values. It will never be a dull moment. I am looking for a polynomial that is more calm than me. Someone who is basic but positive and going up in life. They need to have 1 y-intercept and 1 x-intercept. I don't want anyone who will throw o curve at me. I hope to have domain and range in common--be something special we share.

I am a very negative and odd function. I have been working my way from quadrant 2 to quadrant 4. I have no curvy parts and I like to rest right at the origin. I am looking for a function to put a little more life in me. Two turning points is a must. I'm, looking for a positive influence in my life.

I am mostly laid back and enjoy to stick close to home. Ever since I was born I have never changed. I prefer to not cross paths with my enemy, but am willing to take the chance to see my friends on the other side. I don't have much of a range. I can't tell which quadrant I start and end which makes me mysterious. I am looking for a soul mate which will help me take risk in life. Someone who leaves the x-axis regularly; three times would be the perfect number. I would like someone who picks me up regularly and doesn't mind hanging out at our common y-intercept. Be curvy and outgoing, but willing to stay close to home as well.

Once students have a grasp on the abstractions, they can begin to play.


Tuesday, November 11, 2014

CCSS: Support from the North

I can't--for the life of me--understand why someone would argue to eliminate high level mathematical reasoning in favour of memorized tricks, but that seems to be the case with those arguing against the Common Core State Standards. I cannot fathom how this can be the case except to chalk it up to a case of "he-said-she-said". Change (especially in something as resistant to it as mathematics education) breeds ignorance. And Ignorance breeds fear.

Let's face it: The public are scared of reform efforts and most teachers aren't far behind. 

There are multiple (legitimate) reasons.
  • Time
  • Mathematical Proficiency
  • Control / Power
  • Outside Testing Pressures
  • Belief about University Requirements
  • Personal Histories
the list goes on and on. 

All of these things considered, I still find it impossible to figure out why a teacher would want to condemn the methods aimed at deep understanding. (Other than, of course, they haven't taken the time to see what it is all about). 

Some would say that the methods do not actually achieve the deep understanding they claim to, but I would caution them (alongside Dewey, 1938) to tread lightly if they believe they can control what anyone learns at any time. Learning is a complex process that is not prescribed; claiming that it is optimal to internalize a 'canon' of 'truth' exactly as it is presented is one assertion, but assuming that such a feat is possible is something much larger. (In my mind, ignorant). 

With this incredibly provocative preamble behind us, let's take a look at what sparked this thought. A series of four responses (many more were worthy of exposure) to a typical question on surface area in my Grade 9 regular stream classroom. Each student "received" the same educational experiences, but came up with different ways of conceptualizing the task. People may dismiss this as fluke or ignore it all together, that is your prerogative. 

Just know:
  • This is real. It is not fabricated. It is not doctored. The scans were simply cropped by myself to ensure complete anonymity. 
  • This unit didn't take more time than usual. I didn't spend a thousand hours allowing for "aimless" exploration. 
  • I didn't prescribe what method to use, but encouraged discussions on efficiency. It wasn't an "everything goes" culture; strategies were analyzed. 
  • The students generated each one of these in consultation with their peers, histories, and classroom experiences. 
If you choose to ignore the possibilities (albeit from a very concrete, accessible topic), fine. But if you care to do so by intelligent and sophisticated means, please tell me why students using these strategies is a detriment to them, their future, and the education system. 

One last appeasement. Even if one believes that right answers and streamlined edges are the purpose of mathematics (**cough** which they are certainly not **cough**), each of these arrives at the correct answer. This is a nice precipitate come exam time, but certainly wasn't the case during concept development. The mistakes were culture forming; they encouraged reflection and recursion--not something to be shunned. Some of the strategies even looked somewhat "standard" **gasp**

Without further ado, the question and four responses:

Nat Banting


Dewey, J. (1938). Logic: A theory of inquiry. New York, NY: H. Holt & Co.    

Sunday, October 26, 2014

Egg Roulette

I find probability to be one of the most difficult topics for students to grasp. Beyond the simple experiments of spinners, coins, and dice, students have issues operating on uncertainty. This issue is compounded when multiple events each involve such a calculation as well as the relationship between them. Soon they find themselves neck-deep in notation and lose all rationality--they forget what they are solving to begin with. 

This past week we found ourselves mired in another battle with conditional probability. The initial questions were completed at a high level:

Sally draws two cards from a standard deck without replacement. What is the probability that she draws two spades?

Yawn... yet something is nice about the sharpness of the problem's boundaries. There is something objective--mathematical--about the task at hand. The students can count the sample space and favourable outcomes. 

The question then becomes:

Sally draws two cards from a standard deck without replacement. What is the probability that the second one is a spade?

Breaking down the question proved difficult for the class (as it always seems to do). I gave a narrative proof, we ran experiments, developed notation, and tethered to set theory. Still, at the end of the class, the exit slip data was not good. We were a mess. I needed a way to make the problem real again. 

That night I happened upon a YouTube video in a random playlist. The next morning I picked up a dozen eggs before school and spent the noon hour in the home economics lab. I had found the hook. 

When class came, I began with the video and then asked the class:

Sally draws two eggs from this carton of eggs. What is the probability that the second egg is raw?

I then asked if anyone wanted to play. When I had my volunteers, I asked one of them if they wanted to go first or second. His response--after a lengthly pause to think--was:

First, because then I know my chances of being hit. He won't know until I'm done

When I asked the other player if he was okay with that, he replied:

Sure. Because if he gets hit, my chances go way down. But if he doesn't get hit, mine aren't that much worse. 

This couldn't have gone more to plan. We quickly drew up the notation and explained the conditional probability involved in the two cases explained by the volunteers. 

We played the game. Eggs were smashed, videos were recorded, and snapchat stories went viral around the school. At the heart of it all was my hook--I had found the anchor problem.


For those of you interested, there is no advantage to going first or second. I don't do this often, but I've worked through the problem below. 

H1 - Hit on the First Egg
H2 - Hit on the Second Egg
M1 - Miss on the First Egg
M2 - Miss on the Second Egg

P(H2) = P(H1 AND H2) + P(M1 AND H2)
(These are the two cases, each of which is conditional. I spent quite a bit of time discussing this)

P(H1) = 4/12
P(M1) = 8/12
P(H2 | H1) = 3/11
P(H2 | M1) = 4/11

P(H2 | H1) = P(H1 AND H2) / P(H1)
P(H2 | H1)*P(H1) = P(H1 AND H2)
(3/11)*(4/12) = P(H1 AND H2)
1/11 = P(H1 AND H2)

P(H2 | M1) = P(M1 AND H2) / P(M1)
P(H2 | M1)*P(M1) = P(M1 AND H2)
(4/11)*(8/12) = P(M1 AND H2)
8/33 = P(M1 AND H2)

P(H2) = P(H1 AND H2) + P(M1 AND H2)
P(H2) = 1/11 + 8/33
P(H2) = 3/33 + 8/33
P(H2) = 11/33
P(H2) = 1/3

A simple calculation shows that P(H1) = 1/3 as well. 

Tuesday, August 26, 2014

Math Class Starters

I am very distractible. Students know this; I know this. For this and multiple other reasons (including insipid tardiness on the part of my students) the first few minutes of class is often filled with retrieving forgotten textbooks, quieting down the pockets of flirtation, and acknowledging the students who show up two minutes late with a coffee. 

Numerous factors have led me to the institution of class starters for grade 9s. I will do my best to summarize them here and introduce my framework, theory, and pedagogy behind them. 

Why Starters? (The multiple influences)

  • Success with my Enriched 9 class last semester
    • I used starters to establish a culture of curiosity and conjecturing with my set of "gifted" grade 9 students last year. The tone was beautiful. It taught me valuable lessons about how I need to approach and ameliorate the process into the class time, as well as important logistical tweaks that need to be made. 
  • Colleague suggestion
    • A teacher at my school uses quick start-up questions to encourage punctuality. His are graded, and essentially consist of textbook problems from the previous day. While his motives for using the system are different, the hope of greater efficiency does appeal to me.
  • Graduate studies
    • My interest in classrooms as complex systems (see Davis & Simmt, 2003 for a good summary) is growing as my studies continue. I believe that teachers need to base their practice in theory--a deep personalization of theory. This doesn't mean a citation justifies my actions, but my daily enactment of the theory with my students does. It takes a being open to reflection and reflexivity. 
    • I have found that providing students with a stage to enter into a topic best facilitates the climate of complexity. This means my starter questions are often roundabout ways at deriving the curricular content. They roughly align with the chapters in the textbook, but provide multiple avenues for interpretation and pockets of organization. 
    • In this way, starters are not meant to assess the students on whether they did their homework, but provide the spark for organized chaos. Some repeat to provide a chance for deeper analysis, and others lend themselves nicely to a "one-and-done" philosophy. It is important for me to root my classroom practice in theory; it enables me to make justifiable decisions on the fly. 
How are they used?

Class begins with a question projected on the board. Students get into the routine of having their starter portfolio out when the class starts. (I will explain this detail later). As I take attendance, students work on their own problem, but are encouraged to compare with others after they have prepared an argument for their own. This way, the class slowly gets noisier as the multiple strategies are discussed. I generally sit and allow students' ideas to make the rounds. I do not circulate (like I normally do during an open task). I want them to own the work. 

I simply ask for responses and give them autonomy over the room. Some grab the marker and go to the board. Others use the IWB controls. Some prefer to give a verbal explanation, and I scribe it for them. For the first few days/weeks I ask networking questions suck as, "Who else saw this the same way?" or "How is Suzy's answer like Donny's?"

It doesn't take long before students share, "That's what I did" as the question is being explained. After the student volunteers are exhausted, I usually do one of two things: 1) I highlight a particular student that never shared but I happened to overhear some genius thinking or 2) I ask a leading question about an interpretation that was missed or a connection to something previously encountered. 

In this way, the starters do not only serve as an efficiency piece, but a crucial way to start the process of community networking and establish the organizations of complexity. 

Just a side note: If you are wondering where to begin to amass these types of problems, visual and are great places to start. From there, be imaginative. Find connections to your units of study, anticipate student responses, or scour your existing sources for problems. 

What do students use to answer the problems?

Last year I had them bring in a three-ringed folder to put sheets into. This offered a couple problems. 1) those folders become disorganized quickly. 2) full-size sheets were daunting for many students. This year I decided to have them purchase a small exercise book with 1/2 blank and 1/2 ruled pages. In Canada, they are made by Hilroy; I do not know where you Americans can get them. Maybe someone can help me out in the comments section. You can buy these exercise books in Canada almost anywhere. (Walmart or Staples for example). 

The blank space allows them to copy down the problems which are often given in the form of a graphic. The ruled space provides space for their calculations and written explanation. Some students also use the blank space for solution. The two spaces facilitate many conceptualizations. The book is smaller than normal. This sets it aside as important to students and also provides a manageable response section. 

On the cover of each notebook, I have the students put a label that I print for them. My personal tech guru/genius @evandcole provided me with this great video on how to do a mail merge from a class list in Excel. I be sure to include the QR code for their class digital exit slip. When they are finished, they look something like:
This way, the exercise book becomes more valuable than their textbook. I like the feeling it creates when their own thinking is valued above everything else. 

I'm expecting to expand this practice to other classes once a) I can establish a set of good problems and b) become better at facilitating them. The point is to take a common practice, steep it in educational theory, and produce a dual edged sword of efficiency and complexity. 

As always, I'm open to failure. It is the best way for me to learn. 


Davis, B., & Simmt, E. (2003) Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education, 34(2), 137-167.