Saturday, December 19, 2015

Desmosification: Building Custom Parabolas

After an emoji was named 2015 Oxford Dictionary word of the year, I am holding out hope for next years' candidate:

1.   to transform the condition, nature, or character of a classroom activity using Desmos.

Starting with a Dan Meyer post, the art of infusing dynamic software into student activities changes the ways that students encounter abstract, functional relationships in mathematics. Desmos' activity builder gives teachers an extremely user friendly platform to create tasks that move students through semi-structured lines of inquiry. 

I decided to start with a task that I already liked.


I like to spend a few days at the beginning of a unit on vertex-graphing form of a quadratic function to focus on the roles of "a", "p" & "q". We do not run any calculations during this time; we do not input or output anything. We simply play with the parameters and decide their effects on the parabola's attributes. 

I encourage discussion about how we can predict what the parabola will do without any technology. We have "sketch-offs" where they have 10 seconds to draw an extremely rough sketch of the parabola and then discuss in partners their thoughts and possible revisions. (Currently begging Desmos people to make a "sketch" screen option in Activity Builder) 

This particular activity is not timed. It gives 10 separate sets of requirements, and asks students to build an equation that satisfies them all. I structured it around the ideas of what MUST be true about "a", "p" & "q", what MIGHT be true, and what CANNOT be true. Again, I love this activity. I essentially give it to them on day three, and it requires a full hour of profitable talk. The original handout is posted here for download and possible revision. 


In the initial activity, students would sketch parabolas before making their final decisions. I noticed that they were mainly accurate for their purposes, but some were lost because they were forced to make both decisions: how to change the parameter (i.e. make "p" larger positive) and the effect on the parabola (i.e. shift to the right). The result was some were not pairing the parameter change and geometric results correctly.

I created two screens for each challenge. The first gave the challenge criteria and a vertex-graphing form mother function. Here, the student played around with the sliders until the criteria were met. The next screen is a question screen where they type what MUST be true for the criteria to be met. (I omitted the MIGHT and CANNOT for sake of brevity, but plan on having these conversations in the moment). Each question screen also includes a screenshot of a possible solution. Odds are they have landed on a different parabola, and providing an example is designed to encourage comparison and contrast (again, around themes of MUST and MIGHT). I also omitted the last challenge and added a simpler one as Challenge #1. 

Click to test my new Desmos activity


All my favourite attributes of the activity remain intact, but the dynamics of the exploration are much improved. Students now have a visual aid to show why a particular parameter MUST be true, or provide a counter example as to why it only MIGHT be true.

Some clear benefits:

  • it provided me the opportunity to reflect on why I loved this activity in the first place.
  • it increased my fluency with this professional tool. 
  • it created a resource I can easily share.
  • it gives the students immediate feedback regarding how their decisions affect the mathematics of the situation.
  • it adds to the accuracy of the geometric representations.
I have some students who crave a piece of paper to document a task's learning. For that, I am spitballing a Des-Notes structure where students fill out their wonderings, distill further from discussions, and eventually highlight a key understanding from a task. This is still under development, but my first (very rudimentary) attempt at a recorder sheet for these guided activities is here. Feedback is more than welcome. 

If you have the infrastructure to use Desmos, do it! Take the time and become fluent in the basics. Use activities already developed (including this one), then branch out and desmosify your favourites. 


Wednesday, November 11, 2015

Counting Circles Brainstorm

Let it be known that Sadie Estrella is a Hawaiian treasure.

She made her way north for SUM2015 in Saskatoon and I got the opportunity to learn from her about counting circles (as well as share an eventful dinner). 

It is probably good to understand her work on counting circles before reading a couple of ideas I had during her session. 

I went to her blog and searched for #countingcircle, and the results can be read here

Use this time to read Sadie's work

A couple things struck me while she was talking:

  1. She is so honestly passionate. You can tell that she cares when she talks. I immediately felt comfortable as the "student". 
  2. She had such a precision of language. At one point, she noticed herself saying "anything else?" and purposefully corrected it to "other ideas?" Such a little detail that makes the idea generation flow. 

As she was talking, I started to think where this routine would fit in my room. I teach high school, so most of my students are past counting 10s, but still struggle with numerical flexibility. I had two ideas that I would love to try in my room. One is a structure, and the other is a conversation.


Sadie had us counting by 11s and dissecting our thinking. She modelled similar counting circles with decimals, fractions, qualitative sections of time (i.e. quarter of), etc. One thing that I thought would be interesting is attaching a function to the circle. So as the students were counting inputs (0, 1, 2, 3...) they were audibly giving the output of the function. 

If we started with linear, my hope is that students would see the pattern of outputs quickly. Then we could have them skip input numbers (possibly for functions like y = 1/2x +3). Why is it easier to skip? As the group gained fluency, it would lead into a profitable conversation...


Certain counting gaps are simpler than others, and that may result in an increase in speed around the circle. This analysis of speed is a fantastic conversation. Once students see the pattern, the counting will speed up. Stopping the flow and asking, "Why can we do this so fast?" might be an effective stem to get kids talking about the patterns they've noticed. 

Keeping track of the circle speed would be a great assessment tool for pattern recognition. The danger here is equating speed with standardization and getting away from the good conversations that occur around the circle. The goal shouldn't be to go fast, but going fast is a symptom of some great mathematics. 

If you cheated and didn't read Sadie's work, this makes no sense to you. I did that on purpose. Go read it now. 

Now don't you wish you had done that to begin with?

The routine is simple; it can embed into your room as a daily or weekly activity. It makes thinking visible and connects learners. Win-Win. 


Sunday, November 8, 2015

Clothesline Series

I joined a middle years math community organized by my school division. I have a growing interest in the transition of students from middle school to high school because many of the tasks I use or create get at middle years content. I'm wondering what knowledge students come to my room with and what atmosphere it was learned in. Both have huge impacts on how students operate in my room.

I was surprised to hear that middle years teachers lamented that students could not use number lines. I use number lines as a support in my high school classes because I (ignorantly) assumed that this was an accessible tool from their elementary days. As it turns out, what I thought was making things easier for kids to conceptualize, probably was causing cold sweats and night terrors.

Typical day in the office for me.

When Andrew Stadel blogged about clotheslines I had to investigate further.

It wasn't long until my intern and I had retractable clotheslines attached to our walls.

We started with Grade 9 fractions and decimals. We wanted to include a plethora of reasoning situations:
Like denominators
Unlike Denominators
Reducible fractions
Negative and positive
Terminating and repeating decimals
Equivalent fractions and decimals

I refined the resource creation process by using PowerPoint and then saving as a .pdf with two slides per page. I then brainstormed several situations where number lines would facilitate useful mathematical reasoning and discourse. They range from early grades to pre-calculus (with the ordering of radians and degrees). 

The complete "series" of slide sets is below. They include a brief instruction slide. My hope is that they are ready to print, cut, and use. 


Algebraic Thinking
Domain and Range
Improper Fractions
Fractions and Decimals
Radians and Degrees

How I used them:

I installed clotheslines onto my walls, but writing the line on the board or tacking a string across a wall would work as well. Sometimes I started by placing a few referents, but as they got better, I allowed students to begin conversations by placing the referents. I then handed each student a number one-by-one and asked them to talk through their reasoning as they pinned it to the line. Students were free to interact, but not interrupt. 

The key is the attention to reasoning and stress on mathematical communication. If I give more than one student a value at a time, it becomes a collection of independent (and silent) reasoners. I want students to argue about issues like sequence, scale, inverses, distance, etc. 

I ran two clotheslines simultaneously with a teacher asking questions alongside each.


Print extra blank values, and have students pair up and write a value for their partner to place. 

As an exit ticket, give students a value and have them write to you where they would place it and why. (I use journalling and portfolio work regularly in class, but a stand alone activity would be great also).

Change the scale by moving a couple referents. Ask the students what would happen and to adjust the values.

Any other curricular topics, classroom structures, or extension ideas are welcome. I've found that the impermanent nature of the placement encourages sense making, mathematical talk, and risk taking. Any activity that checks those three boxes is great at any level. 


** Check out Mary Bourassa's Log Clothesline here **

Wednesday, October 14, 2015

WODB: Polynomial Functions

If you haven't experienced the conversation stemming from Which One Doesn't Belong? activities, you are missing out.  
As far as I can decipher (#MTBoS feel free to correct me), this all began with Christopher Danielson's Shape Book centered around this structure.
From there, a crew of tweeps (headed up by Mary Bourassa) established (YES! Canadian) to curate a collection of problems of this format.
My unit on polynomial functions (either in Foundations of Mathematics 30 or Pre-calculus 30) requires students to decipher attributes of polynomial functions from their graph and vice versa. These include end behaviour, sign of the leading coefficient, y-intercept, domain, range, degree, and possible number of x-intercepts.
Each question has four panels. Each panel needs to have an attribute that is unique. (i.e. that could be the reason that it doesn't belong). I sat down to build some polynomial WODB questions, and had a hard time isolating unique attributes with sets of four. I ended up with three examples like the one below.

If you open up space for rich conversation and critique, these three could last upwards of 30 minutes.
(One student insisted that colour was always the answer, which, in hindsight, was my bad). After we were done, I should have had students try and create their own. That activity would show a deep understanding of the parameters. Alas, I succomed to the age old barrier of time.

Download the .pdf with all three questions.

Steal, modify, and adapt. If you make more, link in the comments.


Central Tendency: 10 Burning Questions

My intern just started a unit on statistics with my favourite starter question of all time.

(First blogged near the end of this post in 2011...)

The question is simple: floor is very low, and ceiling is very high.

Create a data set with the following characteristics:
Mean = 3
Mode = 3
Median = 3

During the teacher rotation between groups, I picked up on some lines of reasoning. (Not being directly responsible for the teaching of the lesson, allows me to sit back, be inspired, and follow the lines of inspiration).

Student justifications for their data sets were very interesting. It inspired me to pen some burning questions I may have asked groups if I were "in charge".

  • If the mean of a data set is 3, all data points must be multiples of 3.
  • You cannot have a mode of 3 without at least 2 threes.
  • Adding a data point lower than the mean will always lower the mean.
  • Adding a negative and positive data point will never change the median of a data set.
  • If the mean and mode of a data set match, so must the median.
  • The median of a data set must appear as a data point in the set.
  • The mode of a data set will always be closer to the mean than the median.
  • Adding a data point lower than the median will always change the median.
  • Adding a group of points with the same mean as a set of data will not change the mean.
  • If we keep adding threes to a data set, the mean will eventually become 3.

This type of extreme agree or disagree questions are great for discussion. I believe some teachers have coined them as "talking points". (If you have a link to a great description of "talking point" comment below.)

Asking students to prove or disprove these statements (ranging in difficulty) would allow them to surpass the calculation of the statistics and work directly with the mechanism of central tendency. That is a huge win for me.

If you have more questions, comment them!


Friday, October 2, 2015

Navigating Collectivity: Grade 9 Fractions

"I hate fractions"
- Everyone
Today an amazing thing happened; students put aside the endemic disdain for rational numbers and had a conversation. I'd go further, they weren't discussing their views on fractions, they were collectively conjecturing--the moves of the room enacted each other. I don't think that a written document can capture the movement of the body of learners, but I have to try something. Think of it as less of a remembering and more of a re-membering, a reconstruction of a living learning event from the past.
My intern and I have worked at fostering a spirit of collectivity in our grade nine classrooms. This begins with a starter problem where they are asked to respond and fully explain their thinking in whichever modality they are most comfortable with. After a partner discussion, the teacher anticipates, orders, and debriefs different ideas that appeared. Today, my intern gave a starter on fractions as an introduction to a review day. The thinking was mind blowing. The purpose of this post is hence two-fold: 1) try to capture the spirit of collectivity and 2) an open job application to school divisions looking for a fantastic teacher-to-be.
The *question was as follows:
Place four different digits (2-9) and one operation ( +, -, *, / ) in the boxes to create an expression with the largest result possible. Explain your thinking.
There were three amazing pieces of thinking that came from the collective that would not have occurred if their thinking wouldn't have met perturbations.
1)   She began by collecting answers from the class and placing them on the board. They stood as artefacts of intelligence, but needed to be investigated. As she asked probing questions, the following conversation occurred (paraphrased, of course).
Teacher: How do you know this is the largest?
Student 1: Because I made the bottoms as small as I could.
Student 2: Denominators.
Teacher: Right, denominators. So you chose 2 and 3?
Student 1: Yes, because they were the smallest, the best.
Student 2: Because you said we couldn't use 1.
Student 1: We would have used 1, because it is the best.
Teacher: 1 is the best denominator?
Student 3: Yes, well no, zero would be the best. It is as small as possible.
Many Classmates: No! Can't divide by zero! Zero doesn't work! etc.
Student 3: I  know you can't, but if you could, it would make the largest number.
The interesting (and very mathematical) idea of "best" comes out of the students' method of making a claim and supporting further claims based on their classmates. Student 3 is following a pattern established by the other two. When he takes it past the area where they were comfortable, the collective self-corrected. I'm not sure it gets to this moment with teacher questioning; the teacher constantly deflects back to the collective.
2)    Quickly after this discussion on division by zero subsided, another student made a conjecture and the collective employed rules to decide whether the move was mathematically legal.
Student 1: We could make the fraction bigger by adding a number.
Teacher: How do you mean?
Student 1: Like if we took a digit, 4, and put it out in front of the fraction. (motioning to create a mixed number)
Student 2: But that wasn't the question.
Student 1: I know, but it would always make it bigger.
Teacher: Creating a mixed fraction would always make it bigger?
Student 2: Yes, they are wholes.
Student 3: But that can't work.
Student 1: Why?
Student 3: Because you can't add a whole to an improper fraction? It already has wholes?
Student 1: But this would just add wholes?
Teacher: Adding wholes makes it bigger?
Here, we see the conceptual understanding of fractions (wholes make things bigger) and legalistic understanding of mathematics (can't be improper and mixed) at war. The students do not shy away from making a suggestion even though they are aware that the question forbade it. They are operating on proscriptive barriers and the question doesn't restrict their function. Notice how little the teacher says; the students form conclusions with each other.
3)   At the tail end of the discussion, my intern asked if anyone used addition. One student raised their hand and offered their solution. In a familiar line of questioning, she tried to explain why 2 and 3 would be the ideal denominators.
Student 1: Because they are the smallest.
Student 2: But, wait, they don't create the smallest number.
Student 1: Fewer pieces, larger.
Teacher: What do you mean?
Student 2: 2 and 3 make 6, but 2 and 4 make 4.
Teacher: When we add?
Student 2: Yes. Common denominator.
This initially blew my mind. What an astute comment to make. The chosen numbers would not create the smallest common denominator, so would they still create the largest fraction? Student 2 did not entertain an additive strategy until Student 1 suggested it; it was her explanation that spurred Student 2 to wonder about the effect of common factors in creating the largest possible fractions. While it may seem like the collectivity is one-sided, the thinking formed through communal action recursively challenges the entire class.
What stands out to me is three things:
1)  A simple question that allows for student action is the first step toward collectivity.
2)  The depth of thought the students achieved.
3)  The collective interplay of their thoughts when given the arena to collide.
I hear you:
"But I don't have time"
"My students are too weak"
In the end, it took the provision of a space, curation of conjectures, and presentation of a perturbation to open up collective space.

* Source of Problem:

Saturday, September 5, 2015

Integer War

Math 9 poses the specific challenge of pre-assessment. The wave of administrative details (lockers, fees, photos, textbooks, tryouts, etc.) creates a logistical whirlwind for teachers. On top of that, you have no clue who (most) of these new students are, or what their mathematical history is. Our department gives a short pre-skills exam to help with this process, but I like to use the first week to work on integer tasks to really see how the newbies move mathematically--beyond a number on an exam.
One such task was stolen (read: borrowed) from Timon Piccini. I decided to place focus on the idea of zero pair and the movement away from and toward zero.
What follows is just another set of (very simplistic) resources for the class to add to the excellent work that Timon has already created followed by a brief synopsis of the questions that worked well for me and my students.
The structure:
My grade nine classes curate portfolios for classroom starters and short pieces of assessment I need along the way. The day began with a start-up question:
2, 5, 9, 15, 4
Using addition and subtraction, create an expression (using all five numbers) that is as close as possible to zero.
Be prepared to explain your strategy.
This came after the previous day the class played a modified version of the game Zip, Zilch, Zero.
Both focused on the balancing of results around a central target by moving along the number line using addition and subtraction.
After the starter, the day began by grouping students into threes, and having them grab a large whiteboard. Each group got a marker, eraser, and handful of green (positive) men and red/white zombie (negative) men.
[Disclaimer: I couldn't find any colour except green so I spray painted half white and coloured red zombie effects. Truth: kids love army men. Corollary Truth: Kids love zombie army men more.]
The boards were horizontal (cue gasp...) and some even decided to create battlefield effects.
The task:
I began by explaining that each man had the same power despite many of them missing legs, arms, and wielding broken guns. Like Timon, I created a IWB file with an infinite cloner to mimic student battles. I used these two army men to create my digital version.
The idea of "0" battles or stalemates became important. The first slide focused on what would need to happen if the battle were to be a draw. Some students placed no men, most place one from each side in a man-v-man scenario. I stressed that a stalemate could happen with any number of soldiers on each side as long as it was the same amount on each side.

I used this .pdf file to take students through possible battles I wanted them to create. Each time, I asked students if they could create another scenario that represented the same battle or model it on the front board. The focus was on the conversations and animations. The slides include many examples, and I ended up skipping through once they became very proficient.

The result:

Students were engaged in the task, and it elicited some very important talking points about integer addition and subtraction. First and foremost, it built a strong connection between adding positives and subtracting negatives as well as subtracting positives and adding negatives. The last two questions on the slide show sparked great knowings.

The army men did more than wrap a math topic in a whimsical motif, they built the metaphor of retreat. This is one that I will continually return to (I'm sure) throughout the semester.


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.