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. 

Tuesday, June 2, 2015

The Scale of Coffee Cups

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

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

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

The result is a simple task:

Are the coffee cups similar? 

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

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

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

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

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

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


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

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

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


Monday, June 1, 2015

Teacher Hack: iPads in Exams

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

How is it used in exams?

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

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

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

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

How to Enable Guided Access

               Guided Access

Turn it "On".
Set Passcode

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

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

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

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

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

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

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

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

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