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.

Wednesday, July 23, 2014

"__BL" : Education's Obsession With Labels

Last week there was an interesting twitter discussion on the nature of projects versus the nature of problems.
It occurred with specific reference between the differences of PBL (project-based learning) and PrBL (problem-based learning). If you follow this blog or scan the provided tags you will find PBL does occupy some space here. There is also a large amount of posts detailing "tasks". This is a rather artificial term I use to refer to a piece of mathematical work to be done or undertaken.

To me, the potency of all of these ideas is lost on many teachers. Not just my work, but the work of math educators worldwide. (and yes... even some textbook writers. Shock). Teachers love to label different approaches and then subsequently develop (or collect) resources that carry that specific label. Some labels just repackage old ideas while others aim to describe a pedagogy as much as their content. I think such is the aim of both PBL and PrBL. The only problem is that the label shifts the focus away from what is important: the pedagogy.

It is probably easiest to lay out the harmful effects of labels in a bulleted fashion.

  • Labels create commercialization.
Once there is an established "type" of resource, the entire educational machine revs up to churn out books, memberships, sample materials, presentations, and professional development. It isn't long until the original inputs for the label is diluted to suit mass production.
  • Labels attract connotations.
It doesn't take long for the rich understanding (and well meaning) behind a label to become simplified and vilified by those (as ignorant as they may be) opposed to the idea. Just today a chart was tweeted that labeled PBL mathematics as "fuzzy" and opposed to memorization. Ignorant over-generalizations take on lives of their own, and labels create easy targets. 
  • Labels put the focus on one stakeholder.
Whether it is PBL, PrBL, student-centered, or teacher-centered (to name a few), labels highlight but one piece of the educative puzzle. You can't honestly say that a certain type of room is teacher-centered? Or student-centered? An isolation of one of these players renders the entire process null. It never begins. All the players (including the content and culture) are co-implicated in an educational setting. Lecturing is not teacher-centered, if anything the teacher is just a passive mouthpiece void of any initiative. Their role is then to pass on pre-conceived knowledge. That doesn't sound teacher-centered, it sounds more like teacher-proof. 

People sit in the middle and say that their classrooms are "learning-centered". Well... duh. What well-meaning teacher doesn't want (and even think) this to be the case. A "deceit-centered" or "ignorance-centered" classroom is either non-existent or pathological. Why even label that?

Finally, and most importantly:
  • Labels often highlight the resources at the expense of pedagogy.

In the specific case of PBL and PrBL, we are debating what attributes make a educational artefact a problem rather than a project (or vice versa). What that does is pull the focus away from the pedagogy behind the label (the spirit in which they are to be encountered by the students), and place it on the specific instance of content. This means that teachers attempt to collect these artefacts, and once their repertoire is robust enough, they can then execute the "type" of classroom under that label. (i.e. If I can only find enough good problems, I could run PrBL).

This generates a mindset of attainment in teachers. We see it during curriculum renewal; we see it during internships. Teachers scrambling to attain the "stuff" needed to keep up. Labels pull us away from an attunement to the pedagogy behind the resources. Some do a better job at embedding the two mindsets. Recently, the phenomenon of 3Acts has generated a whole new label. The inherently great thing about these problems is the pedagogy was built within the content. It created a potent mix that could take relatively humdrum things like stacking cups and printing paper and create engaging classrooms. It wasn't the content (like the label might insinuate), it was the pedagogy behind it. 

I continue attempt to throw off labels; I try and describe to others how I teach and run my classroom (or at least attempt to). Maybe "Occasion-based learning" (OBL) or "Discourse-based learning" (DBL). Something that resists definition and implies that learning happens in the places where content, pedagogy, and curiosity meet. 


Wednesday, June 25, 2014

Problem Posing with Pills

My class always welcomes conjectures. This is made explicit on the very first day of the semester. This goes for everything from grade nine to grade twelve. As the grades advance, the topics have us venturing into increasingly abstract concepts, but conjectures are always honoured. 

Certain class structures promote conjecturing more than others. Students offer questions during lectures, but they are often of a surface variety. They notice a pattern that has occurred in three straight examples, or think they have discovered a short-cut. I don't like using tricks, but if they are "discovered" or "re-invented" (to borrow a term from Piaget and genetic epistemology), then we use them. 

Several posts on this blog have been born out of conjectures offered in class. Many task ideas come from shifts in tasks that we were working on. By far the most profitable, conjecture-rich structure has been that of large whiteboards. There is something about the organization mixed with the rich problems and communication obligations that opens student minds. Regardless of class structure for that day, if a conjecture arises, I get everyone's attention:

"I need your attention please. [Insert name here] has a conjecture to make."

After the conjecture is made, we have a conversation around its feasibility and even vote on its validity. Some groups (or individuals) devote themselves to refutation, while others remain in whole-hearted affirmation. It creates an interesting (yet non-competitive) dynamic. Some students even like having their conjectures disproven and get right back to modifying them to make them stand. 

When I feel like I have loosened the curricular pressure, I take my students through explicit problem posing (conjecturing) exercises. We use the process documented explicitly in Brown and Walter's book. I complete this exercise with a task that is solvable in 5-10 minutes and has many discrete attributes that can be changed. That leaves plenty of time to pose new problems, exchange, attempt solutions, and discuss. 

Side note: The Art of Problem Posing is a must read for any teacher of mathematics at any level. It's algorithmic encapsulation of the fluid process of posing problems jives well with beginners and is extended easily for experts. Like seriously, stop reading this post and get that book. L-I-F-E C-H-A-N-G-E-R.

On this particular day, I gave my grade 9s the "Poison Pills" problem from Stella's Stunners. (The initial solution offers avenues into factors and multiples)
The problem has nice places for students to create conjectures. For example, students quickly realize that two poison pills can ever be adjacent. They can use these certainties to build on their solution. 

After we solved the problem and discussed, I asked them to change attributes and exchange new problems. These are the new problems they created:

  • We split pills in half and put in two containers. Take pill from #1, place in top of #2 and then eat from bottom of #2.
  • Two players. We used three containers. Take one pill from bottom of all three. Ingest one and place the other two back in whichever column you like. Goal is to be last one alive.
  • Switch the bottom and top pill. Then take two from the bottom and put them on top. Eat the next pill.
  • Instead of taking two pills, you take three. Eat the third and place the first two back. 
  • Take three pills and eat the second. Return the other pills if you are alive to do so. Don't take third pill if you die when you eat second. 
  • Split the poison pills into six half-pills of poison. You need to eat two of them to die. 
  • Have 16 pills (12 good, 4 poison) and 12 prisoners. Look if pattern still exists.
  • Odd prisoners eat first pill and pass the second one back to the top. Even prisoners pass the first pill and eat the second. 
  • Add one antedate pill that can make one prisoner invincible to poison.
We managed to solve a few of them, but had to leave some for them to work on in their problem journals. 

There are three large benefits to encouraging problem posing in class:

Mathematical Intuition
Students were able to recognize and reason why new problems were too easy to even spend time on them. This is a great thing to see as a teacher. Four months ago, these same kids would have happily accepted work that resulted in no cognitive struggle, now they ditch their new problems because they see quick solutions. 

Mathematical Complexity
Students quickly discover that some of the hardest problems come from a simple switch. Their instinct was to change almost everything about the problem until the original was a distant cousin and fleeting memory. Some groups found that shifting one small attribute can create a difficult problem. The beauty and intricacy of mathematics shows itself. 

Mathematical Ownership
Students would much rather work on problems they authored. It makes me think of a quote from a recent presentation: "You know what students are interested in? Their own thinking". Students took these problems home and created new ones from them. I even had one student bring back an idea for a card game based on his problem. This ownership can be authored into problems in subtle or extreme ways. 

I remind my students that a good mathematician will try and keep problems alive. We are so used to math being about killing the problem--problem solving. While this is a noble pursuit, I am more interested in resuscitating problems and extending their mathematical lives--problem posing. What can we do to revive or extend this curiosity. A good class, in my books, leaves with more problems than it started.