Lesley University Days 2 & 3

Using Multiple Representations to Think About Middle School Math

Karen Gartland

- Doing this task on ratios using manipulatives gives access to kids who don’t know the procedures for comparing ratios

- Make bar diagrams online with Thinking Blocks

- Patternary game (from her book Well Played 6-8):

            - Given the first two steps of a visual pattern

            - The first team draws a possible next step

            - The second team guesses a possible next step

- If the second team matches the first team, the round is over. If the second team does not match the first team, the first team shows what they drew for the next step in the pattern. Then they repeat the process for the next step until the two teams match/are thinking of the same pattern.

Multiple Representations on  the Number Line through the K-8 Standards

Kate Marin

- In order to emphasize understanding of measurement and the unit on a number line, have students construct a number line on a piece of adding machine tape using the cuisinare rod unit block (or some other length measure) to measure and mark the intervals

- With a number line, students need to understand…

• what is the unit
• the numbers on the line increase to the right and decrease to the left
• the number line represents distance
• distance between 0 and 1 is “unit distance”
•  unit distance can be used to locate additional points both inside and outside the unit interval
• once two numbers are marked on a # line, all other locations are fixed

- open number line tasks as formative assessment

• Example 1 (Goal: how do kids do with relative size of intervals, are they inclined toward -#s and fractions): Label an open # line with a tickmark of 2, put some tickmarks to the left and right and label
• Example 2 (Goal: what is the comfort level with fractions, do they understand the concept of inbetween on a # line) Label -1, 0, and 2 on the number line. Where would you place \(\frac{1}{2}, \frac{3}{2}, \frac{-1}{2} \frac{7}{8}\)

- Construct right triangles in order to exactly place irrational numbers on a number line (See this article from NCTM’s Mathematics Teacher )

What If…

Steve Yurek

- Towers of Hanoi

• Traditionally (see this geogebraapplet) there are 3 posts, and you are trying to predict the minimum number of moves for any number of discs
• What if there are 4 posts?

- Red and yellow chips: You have a pile of chips that are red on one side and yellow on the other. Some number of chips, r, have the red side facing up. Blindfolded, split them into to groups. Then make it so that the number of red chips is equal in the two groups. What is the strategy that will make this work every time?

• Solution: Count out r chips that will be in one group. Flip over all the chips in that group. You will now have an equal number of red chips in each group.
• At this point, my question was WHY? I have figured it out since then, but I want to figure out more ways to set my students up to have that burning need to figure out why.

Multiple Representations Make Learning Come Alive in Elementary and Middle School

Anne Collins

- Conjecture board: Record student conceptions before introducing a new topic, then have students see if they can disprove the conjectures through a counter-example (it only takes one)

• Example: What do you get when you multiply any two numbers?

- Which elementary models build into algebra?

- Partial products (and partial sums) extend better into algebra than the standard algorithm

- Using the algeblocks quadrant map for multiplication and the algeblocks basic model map for integer addition and subtraction—directionality shows the sign (just like the # line) rather than the color of the chip

- Ratios on the Cartesian plane: comparing \(\frac{2}{3}\) and \(\frac{3}{5}\)

• \(\frac{2}{3}\) is steeper than \(\frac{3}{5}\)so by inspection \(\frac{2}{3}>\frac{3}{5}\)
• Can convert these fractions into percepts by letting each interval on the y-axis equal 10%, then look at the points (10, ?)
• Anything under y=x is going to be a proper fraction or ratio
• Vertical lines show common denominator
• Horizontal lines show common numerator
• Subtract the fractions by finding the distance between the lines (it’s not constant on the graph because the x value determines the denominator)
• For division: \(\frac{2}{3} \div \frac{3}{5}\), can think about how many \(\frac{3}{5}\) fit into \(\frac{2}{3}\). If you look at x=15, you can see that \(1\frac{1}{9}\) \(\frac{3}{5}\)s fit into \(\frac{2}{3}\)

Structure Through Skip Counting: Seasonal Activities to Keep Kids Counting

Susie Schneider and Sarah Clark

- Skip Counting Progression: Put finished projects in a triangle with 1 in the first row, 2 in the second row, 3 in the third row, etc.

•  Start with chart paper and recording it all together
• Students do the skip counting on the own paper, but can still go up to the triangle to count

- Other skills (using glue, tape, scissors, working on handwriting, planning, sharing) progression

• Step-by-step as a whole class
• Show the whole process, then students do it themselves
• Directions written on the board, only show highlights

- Book suggestions

• “Mathematics Their Way” Mary Baratt-Lorton
• Number Wonders: 171 Activities toMeet Math Standards & Inspire Students, Grades K-2 by Catherine Jones Kuhn

Preparing Students for High School Algebra: Critical Foundations for Developing Function Sense

Judy Curran Buck

- Function sense indicators

• Students can translate between different representations
• Appreciation for and ability to apply the function concept and procedures to a real world setting

- Foundations for developing function sense

• Number sense: flexibility with numbers, and recognizing characteristics and relationships between numbers
• Operation sense: meaning of operations, relationships of operations, when to use operations
• Symbol/variable sense: generalize a pattern, use an unknown, joint variation
• Expression sense: being able to translate between English and math, equivalent forms, properties, equal sign
• Graph sense: coordinate plane, describe a relationship between two variables and make predictions

- Data collection ideas for lines of fit: jumping jacks (x = time), the wave or wrist squeeze (x = # of people), # of pennies in a cup to break the noodles (x = number of noodles)

No More Rainbows, Butterflies, Or…

- Main point: “Get away from the cutesy and empower the students to use appropriate language”

• We need high expectations for all kids around appropriate math language and representations. It is an equity issue that students with IEPs and ELLs are most often taught these tricks

- Needs in math education:

• Move away from computations without understanding
• Shift from getting the answer to understanding the problem
• Use errors to explore student misconceptions

- A focus on key words leads to kids not recognizing that this problem is not possible to solve

- Visual approximation of square roots. Ex: \(\sqrt{31}\)

- Have students construct a circle: Start with a point, draw line segments of equal length through it so that the point splits the segment in half.

Lesley University Day 1

My take-aways from Day 1 of Lesley University's Summer Math Institute: Mathematical Representation: Looking at and Making Use of Structure  

Successfully Structuring Math Lessons, Courses, and Programs

Jim Matthews

- In the 180 days you teach, how many lessons are traditional vs. one (or more) of these formats?

• Guided discovery
• Reasoning and proof
• Data collection and analysis
• Interesting application
• Interdisciplinary connection
• (don’t immediately know what to do) Problem Solving
•  Counterintuitive Phenomenon

- Two problems, that among other things, fall under the counterintuitive phenomenon:

• Train-track problem for Pythagorean Theorem. Question: How far off the ground will the tracks be where they meet?
• Rip a sheet of newspaper in half and place the pieces on top of each other. Repeat for a total of 52 rips. How tall will the stack be?
• Key: have students make a prediction before solving. Ex: Would a telephone poll fit underneath the traintracks? An algebra textbook? A piece of paper?

- What do mathematicians do? They do verbs (in red) much more than nouns.

How do you teach structural thinking to students?

Amy Lucenta and Grace Kelemanik

- Structural thinking sits in the noticing not in the answer. This is why we are always going back to the noticing stage.

• In sharing stage: “We noticed _____, so we ______”
• In reflection stage: “Noticing ____ helped count/calculate quickly because _____.” And “Knowing _____ comes in handy when quick counting/calculating because _____.”

- Contemplate then Calculate specifically designed to push structural thinking, whereas number talks are about multiple strategies and highlight something about a particular operation. The first three examples here all have one structural shortcut that makes them easier to solve. The last example has many structural shortcuts that make calculate easier.

- The most productive Contemplate then Calculate tasks have multiple shortcuts that leverage structure

- To choose tasks, first think about the structural thinking that is helpful in current grade-level content, then think about where that type of structural thinking can be used in earlier content. Those are the tasks that you want to start with. (My next steps are to do this thinking with eighth grade content)

Algebraic Formulas Make Sense!

Natalya Vinogradova

- algebra is generalized arithmetic

- algebra is a beautiful and efficient expression of an idea, but you have to develop the idea firstà visual representation described in words.

To illustrate these ideas: factoring the difference between two squares:

- The product of numbers that are two units apart is one less than the square of the number in the middle: \((a-1)(a+1)=a^2-1\)

- The product of numbers that are four units apart is four less than the square of the number in the middle: \((a-2)(a+2)=a^2-4\)

- The product of numbers that are six units apart is nine less than the square of the number in the middle: \((a-3)(a+3)=a^2-9\)

- The product of numbers that are 2b units apart is \(b^2\) units less than the square of the number in the middle. And so we can generalize: \((a-b)(a+b)=a^2-b^2\)

- You can then use this idea for quick mental math computation:

• Forward: \(79^2=80*78 + 1=6241\)
• Backward: \(125^2-123^2=(125+123)(125-123)=248*2=496\)

What is Mathematical Structure and What Does Attention to Structure Afford Problem Solvers in the 7-12 Classroom?

Roser Giné

- From Mason, Stephens, and Watson:

• Mathematical Structure: “the identification of general properties which are instantiated in particular situations as relationships” 
• Structural Thinking: “a disposition to use, explicate, and connect properties in one’s math thinking”

So an example of structure and structural thinking with quadratics:

1. First, we can identify some of the properties of quadratics by completing the following table for the general equation \(p(x)=ax^2+bx+c\)

2. Next, write a closed-form equation for the quadratic function given by the following table of values:

Based on the properties of quadratics I saw in the first table, I can tell that a=7 based on the second differences and that c=-2 based on the y-intercept. My inclination to find b then is to just choose a random point and solve for b.

However, Roser suggested a different way to determine b.

Once we know that a=7, we have the equation \(f(x)=7x^2+bx+c\), so therefore \(f(x)-7x^2=bx+c\). In other words, the difference between our unknown quadratic function f(x) and the function \(g(x)=7x^2\) is the linear function h(x)=bx+c. So let’s actually look at input-output pairs of that function:

x	\(h(x)=f(x)-7x^2\)
0	-2
1	7
2	-12
3	-17

We can see that the slope in this table is -5 and the y-intercept is -2. Therefore, b=-5 and c=-2.

Finally, we can see this linear relationship of the difference between the functions f(x) and g(x) in the graph below. We know that the two quadratics will be translations of each other because they have the same quadratic term, which determines how skinny/wide the “u” is.

Structural thinking that happened here:

- When we take away the leading terms we are left with a linear relationship

- comparing the two graphs

- properties used:

• that a determines how open the parabola is
• linear functions have constant 2^nd differences

Morning Math 2016 - Part II

Continuing from my previous post about what I learned from the PCMI morning math problem sets, this post will focus on various proofs of the geometric series formula. Variations on the geometric series came up many times as we were calculating expected value. 

To calculate the sum of a geometric series the following is true:

 \(1+\frac {1}{b} + \frac {1}{b^2}+ \frac {1}{b^3}... = \sum_{n=0}^{\infty}r^n = \frac{1}{1-r}\) where \(\frac{1}{b}=r\) and \(|r|<1\).

a) Paper splitting: Say that there are b people and one of the people, the “dealer”, starts with b papers. The dealer proceeds to give each of the other people a paper and keep one for herself. She then splits her paper into b parts, and gives each of the other people one part and keeps one part for herself. She repeats the process until she has no more paper and therefore, her piece of paper is evenly split between the other \( b-1\) people. Each other person gets  \(1+\frac {1}{b} + \frac {1}{b^2}+ \frac {1}{b^3}...\) pieces of paper, which has to equal the original piece they got plus the \(\frac{1}{b-1}\)they were given of the dealer’s paper. Then, if you don’t believe intuitively that \(1+\frac{1}{b-1}=\frac{1}{1-\frac{1}{b}}\), here’s algebraic proof: \(1+\frac{1}{b-1}=\frac{b-1}{b-1}+ \frac{1}{b-1}=\frac{b}{b-1}=\frac{1}{\frac{b-1}{b}}=\frac{1}{\frac{b}{b}-\frac{1}{b}}=\frac{1}{1-\frac{1}{b}}\).

And here’s a beautiful visual proof that my table group created for when b=4.

b) Algebraic substitution:

\(S=1+r+r^2+r^3+r^4…\)

\(S=1+ r(1+r+r^2+r^3…)\)

\(S=1+rS\) (Substitute S into the equation above)

\(S-rS=1\)

\(S(1-r)=1\)

\(S=\frac{1}{1-r}\)

c) Algebraic elimination:

\(S=1+r+r^2+r^3+r^4…\)

\(rS=r+r^2+r^3+r^4…\) (Scale the original equation by r)

\(S-rS=1\)

\(S(1-r)=1\)

\(S=\frac{1}{1-r}\)

d) Recursion:

A dog’s bowl starts filled with 1 liter of water. By the end of each day, the dog has \(\frac{1}{b}\) of the water it started the day with and the owner adds another liter of water. If this process continues forever, how much water will the bowl eventually always start with?

Days n	Amount of water W(n)
0	1
1	\(\frac{1}{b} + 1\)
2	\(\frac{1}{b^2} +\frac{1}{b}+1\)
3	\(\frac{1}{b^3} +\frac{1}{b^2} +\frac{1}{b}+1\)

Recursive formula: \(W(n) = \frac{1}{b}*W(n-1)+1\)

Since we know that this will lead to a steady state, eventually…

\(W = \frac{1}{b}*W+1\)

\(W-\frac{1}{b}*W=1\)

\(W(1-\frac{1}{b})=1\)

\(W=\frac{1}{1-\frac{1}{b}}\)

Learning an Instructional Routine

While I was at Twitter Math Camp, I went to the morning session (three two-hour sessions) on Contemplate then Calculate. I am extremely excited about this routine and plan to use it in my classroom this coming year. If you want to know more about Contemplate than Calculate specifically, you can read David Wees’s “Getting Started with Contemplate then Calculate”, access all the materials from the morning session, and/or read this blog post from Dylan Kane, who was also in this session at TMC. However, this post is going to focus on the process that David Wees, Kaitlin Ruggiero, and Jasper DeAntonio used to teach us this routine. 

As I am thinking about next year, one thing that will really help to build vertical and horizontal alignment between K-8 math teachers at my school is for PD (weekly department meetings and once-a-month meetings with all math teachers) to focus around a specific content strand as well as an instructional routine.

To quote directly from the materials that we were given at TMC:

Instructional Routines are “designs for interaction that organize classroom instruction” (Lampert & Graziani, 2009). This distinguishes them from classroom procedures, which organize behaviour, or routines for handing out supplies, which organize distribution of supporting materials.

Instructional routines are both flexible (the math always changes) and consistent (the format remains the same). The consistency of the format reduces the number of decisions teachers need to make, allowing them to focus entirely on the parts of the lesson that are most important for student learning. The intention is that the teaching within a routine responds more directly to what students do as they engage in problem solving.

I am not sure yet what instructional routine will be the best fit for the teachers who I will be working with. However, I think that focusing on an instructional routine will both streamline parts of our planning and teaching and give us a common framework where we can really dig into the differences in what we are doing. No matter what routine we end up focusing on, I plan to draw a lot from the process that we went through at TMC to get to know Contemplate then Calculate.

Part 0: Start with goals, schedule, agenda and norms.

With the norms, we talked at our tables and then shared out about which one would be the easiest to do and which one would be the hardest. I particularly appreciated “say the thing,” which is a norm that I hadn’t heard before.

Part I: Experience the routine

We were then participants in the routine three times—David, Caitlin, and Jasper each ran the routine with a different problem. Before we started, they told us that the purpose of us seeing multiple examples was to see what was fixed and what is flexible in the routine.

Part II: Make sense of each of the five sections of the routine through an idea carousel

Step 1: People made groups based on which step they were most interested in starting at. With one person writing, they identified the steps, rationale for the steps, and any questions/wonderings.

Step 2: Rotate through each of the other posters and annotate them using the following symbols:

Step 3: Go back to your group’s original chart and put an ! next to anything surprising.

Here is what my group’s chart looked like at that point:

Step 4: Then, as a group, we discussed things that were still on our minds at that point. There were some questions about where one section started and the next began. David emphasized that this was extremely worth discussing in order for us to have common language, not because there was a right answer.

Part III: Planning the Task

We split into groups of 2 or 3 and planned an iteration of this routine that we knew we were going to rehearse with the group. Planning was extremely streamlined because we were given a bank of tasks (elementary and high school), a powerpoint that we could modify to suit our needs, and a planning template (there's a version here if you click on "prepare") with the think-throughs required.

Part IV: Rehearsals

Several groups then got to tag-team the rehearsal of the routine that they had planned. Before we started, David clarified the purpose of the rehearsal: “In our work, the objective is not for individual teachers to practice teaching (although this happens) but to develop a shared understanding of pedagogy amongst all participants in the rehearsal.” This shared understanding allows teachers to more easily discuss what is happening in their classroom and focus in on the small variations.

In the rehearsals, the partners traded in and out of the teacher role. The teacher and the facilitators also had the opportunity to call a time out at any moment. Teachers were encouraged to call a time out if they weren’t sure what to do next or if they wanted to analyze the efficacy of a decision they had just made. Facilitators often called time outs to highlight something that a teacher had just done or offer additional insight based on their greater experience.

Also, not all groups did all sections of the routine. The facilitators had groups start and stop in different places which allowed us to focus in on different things in different rehearsals.

Morning Math 2016 - Part I

I had the privilege of doing another summer of morning math at PCMI. This year Darryl and Bowen wrote and facilitated problem sets focused on "Probability and Big Data." You can find the problem sets here. Spoiler alert for the rest of this post if you are thinking about doing the math yourself (which will be way more fun than reading a summary of my thinking!).

I'm having a hard time tidily describing what I learned and what questions I still have. So instead, I want to map out the evolution of my thinking around one problem. One of the questions that we followed over the course of the whole three weeks was if you are flipping a coin, how many times do you need to flip the coin in order to get heads twice in a row?

A warm-up: What is the average number of flips until you get heads?

Looking at this tree, we can create the following series:

\(1*\frac{1}{2}+2*\frac{1}{4}+3*\frac{1}{8}+…\)

Here the whole numbers represent the number of flips and the fraction is the probability of getting your first H at that flip. Adding about 10 numbers in this sequence we determined that it sums to 2.

What is the average number(or expected value) of flips until you get two heads in a row?

So now our tree diagram gets a whole lot more complicated.

Here are the observations based on the tree diagram:

# of flips	Probability of having gotten HH	Probability of not getting HH
2	\(\frac{1}{4}\)	\(\frac{3}{4}\)
3	\(\frac{1}{4}+ \frac{1}{8}= \frac{3}{8}\)	\(\frac{5}{8}\)
4	\(\frac{1}{4}+\frac{1}{8}+\frac{2}{16}=\frac{8}{16}\)	\(\frac{8}{16}\)
5	\(\frac{1}{4}+\frac{1}{8}+\frac{2}{16}+\frac{3}{32}=\frac{19}{32}\)	\(\frac{13}{32}\)
6	\(\frac{1}{4}+\frac{1}{8}+\frac{2}{16}+\frac{3}{32}+\frac{5}{64}=\frac{43}{64}\)	\(\frac{21}{64}\)

It’s interesting to note that the numerators in the “new” component of the HH and in the total for the not HH both follow the Fibonnaci sequence. I’m not sure why this is yet.

Based on this pattern, I can create the following series, but I still don’t know how to find the sum. And the pattern of the probabilities is harder to describe than when we were just trying to get one H.

\(2*\frac{1}{4}+3*\frac{1}{8}+4*\frac{2}{16}+…\)

Next a re-framing of the question and a new representation of a steady-state diagram (this is the language from Day 12, but it first shows up in Day 2. I’ve added the arrows between states.)

Let a(n) be the probability of being in the 0 state, b(n) be the probability of being in the 1 state, and c(n) be the probability of being in the 2 state, where n is the number of flips.

We can write a recursive formula as follows to determine the distribution after any number of flips based on the distribution before the flip.

\(a(n) = .5*a(n-1)+.5b(n-1)+0*c(n-1)\)

\(b(n) = .5*a(n-1)+0*b(n-1)+0*c(n-1)\)

\(c(n) = 0*a(n-1)+.5b(n-1)+1*c(n-1)\)

We can then translate this into a matrix and do some multiplication to tell us what the distribution between states is after a certain number of flips. All of these decimals here can either be interpreted as the probability that you’ll be in each state, or the % of people who are in each state if you were running n simultaneous games. I found situations where each conception was more helpful to my thinking.

The decimal that is the bottom entry of each of the vectors is equivalent to the HH fractions found in the tree diagram above.  But we still don’t have a way to find the value of the sum. Matrix multiplication can primarily be used to show us that if we flip forever, eventually there is a 100% of having gotten HH. (Duh.)

So let’s look at a modified version of the steady state diagram (also from the Day 12 problem set, with additions by me).

Now we can rewrite our sequence from above:

\(EV=2*.25+(2+EV)*.25+(1+EV)*.5\)

So this is showing us that you can get HH in 2 flips ¼ of the time, ¼ of the time after 2 flips you have to start over, so that’s 2 flips plus whatever the expected value is, and ½ of the time you have to start over after 1 flips, so that will take 1 flip plus whatever the expected value is. Now you just have to solve for the expected value!

\(EV = .5 +.5 + .25EV + .5 + .5EV\)

\(EV = 1.5 + .75EV\)

\(EV-.75EV = 1.5\)

\(.25EV = 1.5\)

\(EV = 6\)

On average, it will take 6 flips to get HH.

Finally, we found that one way to think about the expected value of the number of trials it will take for a specific occurrence in the inverse of the probability of that occurrence. For example, with 1/2 chance of flipping a heads, on average it will take 2 flips to get heads. There are 4 possible combinations when you flip twice, so you could say that the probability of getting HH is 1/4 and thus the # of flips it would take is 4. But this would only make sense if you were repeatedly flipping two coins and recording what you got. I think the fact that the sequence of flips is continuous explains why this reasoning does not work and the expected value is not 4.

Silence is an Act of Complicity

Over the last three weeks, I have had the privilege of being part of two amazing math education communities—Park City Math Institute (PCMI) and Twitter Math Camp (TMC). At PCMI, we spent two hours a day in a class focused on doing math, an hour and a quarter a day in a class focused on reflecting on our teaching practice, and two hours a day collaboratively developing PD. At TMC, our days were similarly full with one two-hour session, two one-hour sessions, a keynote speaker, and two sets of 30 minutes of announcements/my favorites per day. In the unscheduled time in both programs, there were optional sessions, continued conversations, side projects, and an endless stream of interesting people to talk to who shared my passion for math education.

There were so many things I was excited about that I wished I had more hours in the day to get to them all. As a result, I put much of the rest of my life on hold. I spoke with very few people who I didn’t see in person, I put off responding to emails unless I absolutely had to, and I only briefly skimmed through facebook. I knew that this was an opportunity to immerse myself in thinking about math teaching in a way that was impossible for me to do at any other time during the year.  I was in my happy place.

Then, one morning at PCMI, I noticed that my facebook feed had blown up with news stories about the murder of Alton Sterling. And then the murder of Philando Castile. To my embarrassment, my first reaction was to just get off the internet. I was tired of the heartbreak, the anger, the helplessness, and all of the other emotions that I have come to associate with the racial injustice and violence in this country. I wanted to stay in my happy place. And at first, PCMI enabled me in the privilege of being able to do so. I kept doing math, I kept thinking and talking about teaching math, I kept admiring the beautiful mountains, and no one brought up what was happening outside of our bubble.

But as time went by the feelings that I had shoved off into a corner started to break out of their pen. I couldn’t keep them separate from everything else that I was thinking and doing. I noticed that other teachers at PCMI were posting things on facebook, though I had still not heard anyone bring it up in person. And I started to feel betrayed by PCMI leadership.  In their silence, in my silence, we were implying that the teaching of math can and should be separated from the social and cultural context in which it is done. We were saying that the lack of value placed on black lives had nothing to do with our focus on math and education here. So emboldened by the facebook posts of my fellow teachers and the status and relative power that I held in the PCMI community, I approached one of the program directors and asked if there could be a PCMI-sanctioned space for a conversation about the recent police brutality and what it meant for us as people and as teachers.

I was relieved when the response was yes, that this was important, that I wasn’t the only person who had brought this up, and that they were creating a space and time to start this conversation. That night, instead of hanging out with other teachers or reading a math education book from the stack I had accumulated, I connected back into the world outside of PCMI and tried to educate myself as much as possible about these most recent occurrences of institutional racism in our country. I was overwhelmed by all of the feelings that I had been trying to avoid and felt alone. But then PCMI gave me a gift—they sent out the email saying that for all who were interested, there would be a safe, facilitated space to talk about recent current events. This gave me the push that I needed to start a conversation with the people around me. It was easier for me to say “Hey, what do you think about that email?” as a conversation starter than “Hey, I’m feeling overwhelmed and confused and angry and alone.”

When two days later the larger group met to talk and grieve and share in these feelings with each other, I did a lot of listening. People shared what they had heard and felt about recent events, their own experiences with institutional racism in their lives both outside of and within the math and education communities, and their questions and strategies for addressing social justice in their school contexts. People authentically said what was on their mind and were vulnerable, with the goal not of coming out all on the same page, but of being able to talk in a room where people were willing to listen. This was only the very start of a conversation, with so much of the work left to be done. But I left with the knowledge that, at least for these people who had showed up to talk and listen, not only were they passionate and interested in deep conceptual understanding of matrices, multiple solution strategies, building on prior knowledge, and student engagement, but they also deeply cared about the lives of black and brown people and dismantling the systems of oppression that we are complicit in as teachers. These were the people who I could follow up with to continue the conversations we had started and whose ideas and support I could draw upon when I needed it.

A little bit less than a week later, I showed up at TMC. As primarily a lurker in the MTBoS, this was a community of people who I deeply respected and idolized, but with whom, for the most part, I did not have personal relationships with. In the morning session on the first day, I once again settled into my happy place. There were great norms, including “say the thing [that everyone is thinking but no one is saying or that only you are thinking]”, which led to a supportive working environment. I dug deep into structure and examining a routine to surface and leverage it, surrounded by people who were passionate and thoughtful about the work we were doing.

And then in the afternoon, Jose Luis Vilson, gave the keynote speech for the day. He pointed out how race was a relevant conversation that we weren’t having, shared the overlaps in the habits of mind for mathematical thinking and the ones needed to have tough conversations, and drew attention to the fact that he was one of the only black men in the room, which was one of the reasons why his wife had concerns about him coming to speak here. By the end of his talk, I was pretty emotional. I was so relieved that he had “said the thing” and that the organizers of TMC had intentionally made space for him to do so. Because this conversation had been started at the keynote, it was easier for me and other people to jump into follow-up conversations and reactions afterward.

Like PCMI, I knew that the TMC community was deeply passionate about math and pedagogy. But I didn’t know until that point that they understood that issues of race and oppression couldn’t be separated from that passion and that they cared enough to place value on those tough conversations. Now obviously one conversation and a couple of follow-ups are not the end of this work. They are only the very start. But this post actually isn’t to address what my next steps are, or the next steps for either of these communities.

Instead, I want to think about parallels between my feelings at PCMI and TMC and those of my students. How often is it that I am asking my students to check their concerns and feelings at the door so that they can “focus on the math”? What is it that I am expecting them to keep in a separate part of their brain, but is spilling over into everything that we do at school? What is it that I am telling kids that I don’t care about because I’m not talking about it?

Although it was only a small step, it meant so much to me that both TMC and PCMI said this is important and we’re going to talk about it. It demonstrated that as institutions they cared and it gave me the courage to be more connected and vulnerable than I otherwise would have been. As a teacher, by definition, I hold power in my school community. It is my responsibility, and my school’s responsibility, to make sure there is time and space for students to talk about and act upon the issues that they care about. Silence is an act of complicity.

Math Practices to Provide Clearer Focus for Tasks

Last year, I totally revamped my first unit on linear relationships. My goal was to have the students develop procedural fluency from conceptual understanding and also to work from concrete to abstract problem situations. I designed the unit so that it was extremely heavy on tasks, and had students notice patterns/generalize rather than a lot of direct instruction. I anticipated that students would build their conceptual understanding at different paces and tried to choose low-floor, high-ceiling tasks where all students could make progress wherever they were in their understanding. I also anticipated that it might be hard to measure and track the development of understanding. I had seen students experience learning trajectories  where it seemed like nothing was building in exposure after exposure and then all of the sudden something clicked about the concept.

In many ways, my revamp of this unit totally failed. While there were probably many factors in this failure, I think that one of the biggest ones was lack of student buy-in. I did not do a good job of communicating my vision to students. I didn’t tell them where we were headed in the long run because I felt like that would give away the end and defeat the purpose of their own building and discovery of knowledge. Because my main goal was for them to slowly make connections and I expected them to do it at different paces, I didn’t have a neat, tidy goal for every task. This required my students to put their trust in a vision they couldn’t understand or see, which was an unfair expectation for them. Although students did engage in many of the problems and start to get at the big ideas, the overall result was that students who came in confident in math felt as though the class lacked rigor and students who came in less confident in math felt like they still weren’t making any progress.

So I am going to revise again. I still believe in the premise of using a series of tasks for repeated exposure to build conceptual understanding over time.  I think most of the tasks that I used do a good job with that. What I am going to change is my messaging around the tasks. I need to do a much better job in getting my students to believe that they are learning math through this process, even if it doesn’t seem like it. I also need to do a better job in building a common understanding of how students (and I) can tell if they have used their time in a worthwhile manner.

I want to join this idea with my desire to figure out a way to have students explicitly reference and use the math practices. When I explained my math team’s focus on tasks, one of my other colleagues asked, “If the goal isn’t to get the answer, what is the goal?” For me, the process is the goal and I think the math practices could help both my students and me assess their process of trying to solve a problem.

I am imagining having students take some time to reflect at the end of their work time (or even in the middle if work time extends more than one day) on each task. They would identify one of the math practices they had used and give an explanation or illustration of how they had used that practice in their work. I would introduce the practices one at a time and spend several weeks choosing tasks that particularly lent themselves to that practice, so that we could develop a common understanding of what that practice was and what it might look like as you are working on a problem. As the year went on, kids would be able to choose from an increasing number of math practices as the focus for their evidence. I would probably start with math practice 1 and then 3, as I see those as the most over-arching, and then sequence the rest based on which practices are the best fit for the tasks we are working on at the time.

It would be my hope that this focus and reflection time would give my students a much better illustration of what to value during the time we spend working on tasks. Whether or not this is something that I will grade (or have the students self-evaluate), I have not decided yet. I also hope that the explicit teaching and re-visiting of the math practices in this manner will help students develop those skills so they can really leverage them to access new content. It doesn’t address how I will share the end-goal and overall trajectory, but I will keep thinking on this.