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.

NCTM 2016 Take-Aways Summary

My head is still swirling with everything that my NCTM experience made me think about math education. However, from all of my notes, I wanted to synthesize my bite-size takeaways from this conference. 

Really Important Reminders:

• Select tasks that combine different content areas
• Students should only be working with other people when it’s actually beneficial to work with other people.
• We should be engaging students not just in how to do math but in why and when.
• Having a problem solving protocol(s) is more important than what the particular protocol is
• Students often don’t distinguish between the tick-marks (addresses) and the intervals (distances) on a number line
• Formulas: throw most of them away

Things I can implement tomorrow in my classroom:

• “Pencils down, brains on” One minute to think about the problem
• For Estimation 180: “If you say a million [as your too high] you get this conceptually but be brave”
• Visual Patterns: Fold paper into 4 parts, and reveal and have students draw 1 step at a time. After each step ask students “How many in the next one?”
• Pull a kid from each table for a small group huddle and then send them back to be experts

Resources to Look In to:

• NCTM research brief on problem solving
• Math Visions Project
• Graham Fletcher’s Progression Videos (obviously in conjunction with the actual standards)
• Geoff Krall’s posts on PBL and revisiting his curriculum maps

And finally, I’m thinking a lot about engagement lately. Here are some things that I want to remember from the conference:

• “If I don’t ask you to guess, you don’t have a stake in it”
•  Using a high motivation Do Now as leverage for the rest of the class—if kids start excited and engaged, this sets them up for success in the rest of the lesson
• Students will be curious about (SWBCA) instead of students will able to (SWBAT).
• “People can’t understand solutions to problems they don’t have”
• Find a problem you are interested in and kids will probably be interested in it as well.

NCTM 2016 Days 2 & 3

Ignite:

Peg Cagle – promoting teaching as a profession

Michael Fenton – What we can learn about mathematical sequels from movie sequels

Good - best sequels take advantage of what we know about context and characters and hit the ground running - balance between familiarity and innovation - shed new light on original work

Bad - know when to stop - don’t wing it. You have to do the prep work - do not abandon what works

Marilyn Strutchens – equitable classroom strategies --> positive math identities: math autobiographies, low threshold/high ceiling, math teaching practices, formative assessment, social justice activities, ask students what they want to study, students can contribute to the development of math, engage families in doing math

Andrew Stadel – allow time constraints to improve teaching: 20% of most effective functions should take up 80% of the time

Annie Fetter – students will be curious about (SWBCA) instead of students will able to (SWBAT). “People can’t understand solutions to problems they don’t have”

Max Ray-Reik – borrowed lessons are good because they come with a story, but it’s hard to build coherence with a hodge-podge of borrowed lesson. The ideal curriculum is not free and open-source, but open-story and crowd-engaged.

Tracy Johnston Zager – fluid and powerful collaboration comes from thinking partnerships (generative, supportive working side-by-side), cross pollination (credit for ideas becomes interesting), math disputes (representing own ideas, listening to others, publically changing your mind), and peer feedback (how do I make this better). We should only be working with other people when it’s beneficial to work with other people.

Lee Stiff – All students deserve high quality education. We have created the achievement gap and we can change it.

Jennifer Wilson – The slow math movement.

Matt Larson – We should be engaging students not just in how to do math but in why and when.

 

Fumbling Toward Inquiry: Starting Strong in PBL

Geoff Krall

Take-aways:

• 5 design principles
• notch some early wins where students are talking to one another about math and thinking of themselves as mathematicians (start short and successful)
• provide an iterative framework and protocol (having a problem solving protocol(s) is more important than what the particular protocol is)
• choose tasks that support targeted instruction and group-mate experience (example pull a kid from each table for a small group huddle and then send them back to be experts)
• start slowly and strategically
• don’t go it alone
• Ira Glass: "Nobody tells this to people who are beginners, I wish someone told me. All of us who do creative work, we get into it because we have good taste. But there is this gap. For the first couple years you make stuff, it’s just not that good. It’s trying to be good, it has potential, but it’s not. But your taste, the thing that got you into the game, is still killer. And your taste is why your work disappoints you. A lot of people never get past this phase, they quit. Most people I know who do interesting, creative work went through years of this. We know our work doesn’t have this special thing that we want it to have. We all go through this. And if you are just starting out or you are still in this phase, you gotta know its normal and the most important thing you can do is do a lot of work. Put yourself on a deadline so that every week you will finish one story. It is only by going through a volume of work that you will close that gap, and your work will be as good as your ambitions. And I took longer to figure out how to do this than anyone I’ve ever met. It’s gonna take awhile. It’s normal to take awhile. You’ve just gotta fight your way through."
• Select tasks that combine different content areas
• Find a problem you are interested in and kids will probably be interested in it as well.

What I am wondering now: I used to primarily use tasks at the end of unit for students to solidify and apply their learning. Now I am often using tasks to introduce a topic and draw upon students’ prior knowledge and reasoning. I want to move towards integrating problem solving with mini-lessons and other instructions. What structures do I need to make this happen?

My Journey from Worksheets to Rich Tasks

Michael Fenton

Take-aways:

	“Before”	Now
What Math Teachers Do	- Answer questions - Present “the notes” - Assign Practice	Facilitate opportunities for students to engage meaningfully with math
Role of Math Teachers	-Explanation-Giver - Answer-Provider	- Question-poser - Thought-provoker - Discussion-starter - Math-instigator

• Estimation 180:
• Have them struggle with a specific estimation before giving the tools of too low/too high
•  Too low/too high a confidence builder for students who think that wrong answers are “their game”
• “If you say a million [as your too high] you get this conceptually but be brave”
• Visual Patterns
• fold paper into 4 parts, and reveal and have students draw 1 step at a time.
• After each step ask students “How many in the next one?” (Acknowledge that this is unfair after step 1)
• Make a table, sketch a graph, write an equation. Check with Desmos.

What I am wondering now: I am familiar and excited about all of the activities that Michael Fenton brings up as ways for students to engage meaningfully with math—3-act problems, estimation 180, visual patterns, would you rather, my favorite no, open middle, which one doesn’t belong. I have used them in isolation in my classroom. But I am really wondering how to fit them into the larger structure. How can really intentionally I use them to build the  math practices while working toward specific content goals.

 

ShadowCon

Robert Kaplinsky – give power, gain influence

Gail Burrill – Listen to the voices in your head:

            - Think deeply about simple things (Ross)

            - Never say anything a kid can say (Reinhard)

- I know that my students understand when I see them in a place they have never been (Cuoco)

            - Let the students do the work (Wiliam)

Kaneka Turner – “When were you first invited to the math party and how did that feel?” Who will you invite to the math party?

Graham Fletcher – Really get to know your standards: be a wise consumer

- draw the standard or use tools to support your understanding

- be dumb… surround yourself with brilliance

- be vulnerable

Brian Bushart – explore, play, and find joy in doing math

Rochelle Guitérrez – These populations need access to math vs. We need this population’s contributions to mathematics

The Power of the Number Line: Understanding Fractions as Numbers

Ryan Casey

Take-aways:

• students often don’t distinguish between the tick-marks (addresses) and the intervals (distances) on a number line
• in the common core, the fraction a/b is defined as “the quantity formed by a copies of size 1/b”
• “mathematicians are convinced that if something is on a # line then it is a #”
• prerequisite skills for putting fractions on a number line
  
• principles of linear measurement
• measurement is iterative
• measurement involves partitioning
• measurement involves processes that combine partitioning and iterating

What I am wondering now:  I really like the framework of using whole number understanding and leveraging that to extend into rational (or irrational) number understanding. I am wondering what would be good activities to do with both rational and irrational numbers in order to build my eighth graders’ understanding of irrationals.

The Life-Changing Magic of Tidying the Math Curriculum

Jason Zimba

Take-aways: How to tidy: teaching vs. turning in to a topic

• Never ask about manipulatives on tests
• Forget about calculators K-5
• Formulas: throw most of them away
• Headings: only for distinct topics
• Methods: don’t teach too many (having too many around means less time on the challenging one that gets us over the next hurdle. Strategies go stale)
• “Do mental calculation with multi-digit numbers, but save it for cases where a readily apparent mental strategy is both faster and more reliable than the standard algorithm”
• Procedures: they are for procedural tasks
• “A concept is worth 1000 procedures”
• Terms that never appear in the common core
  
  
• Store topics inside of one another

What I am wondering now: What am I teaching as a topic that isn’t really a topic? Exponent laws/Simplifying exponents. Solving systems using elimination (save that for high school). Anything else? And what are the most important methods that will get us over the next hurdle? Am I focusing on those? I think I need to spend some time tidying, but doing it with the text of the standards and the input of at least the other people in my department.

NCTM 2016 Day 1

Motivating the Unmotivated: Access to Learning

Barbara Dougherty and Lisa Bendall

Take-aways:

• Motivation through games that are high interest, require collaboration or cooperation, and incorporate significant mathematical thinking (ex: Bowl-a-Fact and Find-a-Place)
• Using a high motivation Do Now as leverage for the rest of the class—if kids start excited and engaged, this sets them up for success in the rest of the lesson
• Public accountability for engagement—homework through expert groups or collaborative groups

What I am wondering now: What is the “just right” fit for skill level in games that require some sort of prerequisite knowledge? How do students get feedback to improve if they are consistently demonstrating misconceptions in these types of games? What are the characteristics of a game that is both engaging and builds mathematics?

Cuisenaire Rods and Number Lines: Multiplication and Division of Fractions

Adam Harbaugh, Kurt Killion, and Gay Ragan (Powerpoint here)

 Take-aways:

• Sequence of building understanding: word problem--> concrete model with Cuisinaire rods--> # line model --> matching pencil and paper algorithm
• They are only doing “sharing” division (how many 1/3’s in 3/4?) rather than “measurement” division (3/4 is 1/3 of what whole?)  because “measurement” contexts with two fractions are too contrived. Therefore, this leads to the development of the common denominator algorithm.

What I am wondering now: They require their students to make the model so that it is easily partitioned by both denominators. This makes the numbers come out nice and the model easier to use, but it does not promote sense-making. Where is the line between sense-making/flexibility and streamlining for success?

Choosing Tasks for Productive Struggle, Not Frustration

Jackie Murawska (Powerpoint here)

Take-aways:

•  Characteristics of problems that are good for productive struggle:
• the problem solver must decide what math to bring in
• the task uses real life (often messy) data
• the task requires mathematical modeling
• “If I don’t ask you to guess, you don’t have a stake in it”
• “They’re not making mistakes because they aren’t thinking. They are making mistakes because they are thinking.”

What I am wondering now: These characteristics define a subset of problems that are engaging and good for productive struggle. But I think that the type of modeling problem described is only one type. How would I categorize the other types the are highly compelling?

Linear or Quadratic? Engaging in Two Effective Math Teaching Practices

Amy Hillen and Jennifer Outzs

Take-aways: 

• “In this figure as the step changes, the _____ also changes.”
•  For assessing questions stay and listen, for advancing questions walk away

What I am wondering now: I started my year building linear relationships with a lot of visual patterns work. I want to revisit that structure now and use this much more open prompt in order to review linear vs. non-linear.

Supporting Productive Struggle in Secondary Classes

Michael Steele

Take-aways: 

• “Pencils down, brains on” One minute to think about the problem
• “[Students] eavesdropping [on other students] is a great sign because it means there’s something worth listening to”
• When students are starting to say “What if…” that is a good sign

What I am wondering now: What should help look like in my classroom?