If you don’t have anything growth mindset to say, don’t say anything at all

I had been going to a new gym for a couple of weeks. The gym emphasizes scaling to the appropriate weight for the individual (classroom teachers, think differentiated instruction). As a new gym-goer I exercised this scaling option generously. I would use the lightest weight possible so that I knew I would make it through the whole workout. That is until the head trainer dropped two twenty-five pound weights in front of me saying, “In this gym, we aren’t afraid to fail.”

I’ve repeated this phrase to myself daily–in the gym, in my classroom, on the running trail. It has completely changed my mindset. Twenty years ago my fixed mindset convinced the rest of me that I just wasn’t one of those “athlete kids” who could perform well in gym class. I would never be able to climb a rope or do a push-up, so I stopped trying. Two weeks ago I did 8 rope-climbs in a row, 3 pull-ups, 10 push-ups. This gym and the trainers, the other members have empowered me to believe in myself. For the last four months, I’ve looked failure in the face and forged ahead anyway.

At no point on this gym journey did I ever look in the mirror and say, “Damn, check out those biceps” or “How’s the 6-pack coming?” Joining this gym was never about how I looked. It was about proving to myself that I am far more capable than my mind lets me believe (in fact, we all are).  So here’s my issue. When people find out that I’ve been working out, they don’t comment on the hard work, the lessons learned, they comment on the superficial, the results–“You’re so skinny.” “You look awesome.” “I wish I had your arms.” I’ve started to focus more on the results. Instead of overcoming obstacles, I think about toned tummies. I feel shame when I don’t get to the gym.

Is this what happens to my students when they are praised for being smart instead of for the effort, the challenges, the mistakes and failures they endured along the way? They sink into despair that they won’t measure up to the external standards placed upon them? I want to believe people are just ignorant. They make fixed mindset comments like “You’re so smart” and “I wish I had your arms” because they don’t know any better. You’re at a cocktail party making small talk; it’s harmless. But it isn’t. These comments create a larger value system. We value toned tummies, right answers, defined arms over learning and empowerment.  We develop fixed beliefs about ourselves-I’ll never be as smart, as strong, as toned.

What if instead of commenting on what we see right in front of us, we commented on the deeper stuff? What if we asked people how they got so strong or so smart? Or what they learned or had to overcome to get to where they are today?

My sister is the best representation of a growth mindset. She has never been afraid to fail. She worked incredibly hard to get a D1 soccer scholarship. She walked onto a mountain in the middle of the Cascades in Washington and taught herself how to big mountain ski. She’s got a gnarly collarbone from a separated shoulder, eyeglasses from a couple of concussions. For her these are road bumps on the path to greater learning. Since I started going to this gym, she has never made a comment about how I looked. She asks the question that only someone who values empowerment could, “Isn’t it so fun to be so strong?”

 

 

Building Understanding of Linear Functions through Pattern Talks

I originally wrote this with my colleague, Bethany Chalmers, as a possible article for Teaching in the Middle magazine. But after reading Dan Meyer’s blog about the kinetic energy of blog posts over printed articles, I thought I would throw it on my blog. It’s a summary of what we learned from a semester of tying pattern talks to linearity in an 8th grade Algebra 1 classroom.

Why pattern talks?

“Slope isn’t just a number; it’s a unit rate of change.” “No, not change in x divided by change in y; it’s the other way around.” “No, you can’t just divide a y-value by an x-value; it has to be the change in y and the change in x.” “Okay, now how can you use the slope to find the y-intercept?”

These are the comments and questions we continuously find ourselves repeating seemingly to no avail as we push students towards the explicit, non-tabular reasoning necessary to write linear equations. Every year students grab onto the tabular recursive understanding of linearity with ease but then get lost in the formulas and procedures of calculating slope and y-intercept. Slope becomes just the answer to a formula and the y-intercept some abstract concept that only the “smart” kids can figure out. In an attempt to bridge students’ misconceptions with their understandings, we started making pattern talks a regular part of our math lessons.

Goals for our talks

After doing some reflecting and researching on our students’ common misconceptions about slope and y-intercept, we embarked on this math talk adventure, with two main premises in mind. First, students need a quantity intervention. Too many students saw slope as a difference instead of a ratio. In fact, when discussing similarities between calculating driving distances on a grid and slope during Connected Mathematics’ Looking for Pythagoras unit, students insisted that the driving distance of 12 was the slope (Figure 1). Even with substantial questioning and examples on our part, we could not get our students to budge from slope as a difference. According to researchers Lobato, Ellis, and Munoz, when students examine linear patterns with a recursive understanding, they see quantities independent of each other. As a result students tend to see slope as a difference instead of a ratio. Furthermore, the three researchers assert that while a recursive understanding is a necessary building block, teachers need to intervene to make sure students make the explicit connection needed to write formal linear equations (2003). Once students have seen tables and formulas, they need to have their attention drawn back to co-varying quantities and units (Elliis, 2009). Pattern talks were our attempt at a quantity intervention.

Figure 1: The misconception of slope as a difference.

 

Second, we needed students to attend to structure and the similarities between different strategies. Although writing the explicit linear equation was our ultimate goal, this could not be the focus of our talks. Boaler and Humphreys (2005) found that when finding the explicit equation was the goal, students (and teachers) lost sight of the underlying mathematics. They note, how it is “…less important that students be able to find the algebraic rule than that they recognize that the different visualizations of a pattern can be described symbolically in equivalent algebraic expressions.” For this reason, we were intentional about how we wrote student responses on the board so students could make connections between strategies, the picture, slope and y-intercept.

Structuring our talks

Before we were ready to start orchestrating talks in our classroom, we spent time planning how we wanted to structure the talks. We used Sherry Parrish’s “Five Small Steps Toward Teaching for Understanding” (2010) as a guide in this planning phase. She encourages teachers to:

1. start with a smaller problem;
2. be prepared with a strategy from a “previous student;”
3. be okay with putting a student’s strategy on the back burner;
4. limit talks to five to fifteen minutes;
5. be patient with yourself and your students (as you learn how to effectively orchestrate talks into your everyday classroom).

We decided to implement math talks into our warm-up routine, with students analyzing a figure or visual pattern about once per week. Many of the figures and patterns we used came from Fawn Nguyen’s websites, Visual Patterns and Math Talks. We showed students a figure or visual pattern with an accompanying question or questions (Figure 2). We gave students 3 or 4 minutes of quiet think time to develop an answer to the question. In the beginning, we had students do this thinking without writing, to encourage reliance on mental picturing in finding their answers. Later we encouraged them to draw figures and write down calculations or expressions to arrive at an answer. When students arrived at an answer to the question, we asked them to show us a thumbs up in front of their body. This subtle indication of being ready let us know when the majority of students were ready to start the discussion and didn’t put undue pressure on students who were still thinking. When the majority of students showed a thumbs up, we moved on to the selecting and sequencing phases and started the discussion as a class.

We decided that we would ask for answers only first and then student reasoning. We started the discussion by calling for answers only. We recorded all answers on the board with no indication of whether it was right or wrong. After we exhausted the different student answers, we solicited strategies by asking, “Is there anyone willing to defend their answer?” As the student spoke, we thoughtfully recorded their thinking at the board. Once a student finished explaining, we opened it up to the class by asking, “Are you convinced by this thinking? Does anyone have a monster to throw at this strategy?” We define a monster as any part of the thinking that didn’t make sense. If students seemed satisfied, we would call for any other strategies, recording each strategy on the board and labeling the strategy with the student’s name.

As teachers we guided the discussion towards our two goals of a quantity intervention and multiple representations. With that in mind, we had three main tasks. One, to record student thinking so that students could begin to make connections between their mental pictures and the accompanying algebraic structure. Two, to orchestrate a discussion where students were questioning, defending, and making connections between different students’ strategies. Lastly, to hold students accountable to the content of our talks by ending each talk with a connection or extension phase.

Pattern Talks: The 5 Practices in Action

Once we knew how we wanted the talks to play out in our classroom, we had to decide how to scaffold the talks to achieve our two main goals of a quantity intervention and multiple representations. We decided students needed to understand representations and structure before they would be able to reason more deeply with quantities. Following Parrish’s advice of starting with a smaller problem, we began with what we called a “figure talk.” Our goal here was merely to get students to start picturing mentally and seeing different representations of solving the same problem.

 

Figure 2: Figure Talk Students were shown this question: In choosing this figure, we spent time anticipating different solution strategies and how we would represent these strategies on the board. The arithmetic number sentences shown below were a conscious choice on our behalf since structure was a big part of getting students to write explicit equations.

 

 

Continuing our focus on structure and representation (with explicit rules for patterns as our eventual goal), we knew it was important for students to get in the habit of looking for structure in a figure. Before asking our students to extend a pattern or write a rule, we simply asked them to to describe what they saw. We gave them a figure like the one shown in Figure 3 and asked the question, “What makes this a step 2 figure?”

 

Figure 3: What makes this a step 2 figure?   Student Responses: Student 1: “This is a step 2 figure because it has 2 archways.” Student 2: “This is a step 2 figure because it has 2 breadsticks on the top level.”

 

In simply looking for what made a figure a “step 2 figure,” students looked for structural elements and then generalized to a larger step number. We then asked, “What would a step 10 figure look like?” Students were able to use the structural elements they identified in the step 2 figure to predict the number of breadsticks in the step 10 figure (Figure 4).

 

Figure 4: How many breadsticks are in step 10?   Mohamed’s response to Suhayb’s simplistic strategy of 10 groups of 4 plus one more, “Whoa, that’s so cool!”

 

Now that our students were familiar with structure and representation, we needed to start to push the quantity intervention by connecting recursive thinking to explicit thinking. We began by showing students consecutive steps of a pattern and asked them to predict for a later step. Seeing consecutive steps, led some students to think recursively, while other students still made predictions based on an explicit expression. When both types of thinking came out in our class discussion, we were able to help students see the connection between the two (Figure 5).

 

Figure 5: Recursive and Explicit Thinking     The choice of color and structure used to write each expression was very intentional on our part as the teacher. During the anticipation phase of planning this talk, we came up with an exhaustive list of strategies and planned what and how we would write each strategy on the board. We wanted to connect parts of the expression to the picture while also encouraging the linear structure y=mx+b.   To extend our students thinking and hold them accountable for explicit thinking, we followed up this number talk with a Pear Deck that asked students, “How would Ryan find the number of tiles in step 42? 95% of student responses used the structure xm+b=y: One student used the structure b+mx=y: Only one student reasoned incorrectly by forgetting the starting three tiles:

 

After spending a couple of weeks doing pattern talks with consecutive steps, we knew it was time to really push the quantity intervention. We needed to force students to have to reason with slope as a ratio and not just consecutive sums or differences. In our next set of pattern talks, we gave students non-consecutive values of the independent variable to begin to necessitate this understanding of slope as a ratio of two quantities changing together (Figure 6).

 

Figure 6: Encouraging Explicit Thinking   The resulting responses and discussion were everything we could have hoped for as a teacher.   This question resulted in five different student answers. Sam raised his hand first to defend his answer of 48 hexagons. (This alone showed students found these pattern talks to be a safe space for sharing and thinking as Sam is extremely shy and hesitant about his math capabilities.) Another student quickly threw a monster at his strategy.   Student: “At first I thought the rate of change was 4, too, but then I realized that only step 1 had four and then there were groups of 3, so I think it’s going up by 3.”   I circled the four hexagons at the beginning and then prompted the student to be more specific with her language.   Teacher: “What do you mean by going up by 3?   Student: Like every time you go up a step, you add 3 hexagons.   I circled the groups of three. This was sufficient for us to agree as a class that 48 did not make sense.   Isaac defended 39 next. He reasoned proportionally. Reasoning proportionally makes so much sense to students. Pictures were especially helpful in getting students to see the errors in this reasoning. Suhayb threw the first monster by showing Isaac that his strategy overcounts the hexagons.   Suhayb: In step four, there are four groups of three plus one more, so in step twelve there will be twelve groups of three, not thirteen. Then you have to add the one hexagon at the end for a total of 37.   Mohamed then said he agreed with Suhayb because using his method, he also counted 37 hexagons. With the connection between the numbers and pattern clearly visible, students came to the consensus that there were 37 hexagons in step twelve.

 

The last phase of our pattern talks needed to bridge students’ linear reasoning with a visual pattern to standard equation-writing situations from graphs, tables or coordinate pairs. In some of our later talks, like the hexagon one, we made sure to explicitly define the independent and dependent variables, slope, and y-intercept. This was important because in our last phase, the task was reversed. Students had to create visual patterns to align with a tabular representation of a linear relationship. We did this first by giving students a table showing consecutive values of the independent variable. Creating a visual pattern with the correct rate of change was easy for most students, while creating a pattern with a clear and meaningful y-intercept was more of a challenge (Figure 7).

 

Figure 7: Creating a visual pattern from a consecutive table.   Xavier’s pattern produces the correct values for the dependent variable and shows the correct rate of change, 2. The y-intercept, however, is not clearly shown in the visual pattern. The step 2 figure has more of a “step 3” quality due to the 3 groups of 2 circles. Katie’s visual pattern is a stronger representation of the information from the table, as both the rate of change and y-intercept are clearly represented in the figures. The step 2 figure has a “step 2 likeness” with 3 circles to start and 2 additional groups of 2.

As we knew before we embarked on this journey, students grasp rate of change with some ease when it is simply a consecutive sum or difference. To push them further, we gave students a nonconsecutive tabular representation to model with a visual pattern (Figure 8). We found that students did not have much difficulty identifying the rate of change for the table and embedding it into their visual patterns. This was great to see, as we have seen students struggle so much in the past with finding rate of change when given nonconsecutive coordinate pairs.

 

Figure 8: Creating a visual pattern from a nonconsecutive table.   As we reflected on Katie’s pattern as a class, students agreed that this pattern had the correct rate of change, 3, as well as the correct number of squares in step 1. One student expressed confusion over why the added squares for each step were place in the manner that they were. Katie wasn’t able to defend her reasoning and agreed that the squares could have been placed in a different manner. When asked how the y-intercept was represented in this visual pattern, students agreed that it wasn’t clearly represented. As a class, we modified the visual pattern to the version shown below. The y-intercept of 1 is clearly shown in the leftmost square, and the rate of change of 3 is shown in the new column of 3 squares added to each step.

 

 

Going further

 

With visual patterns, students never reasoned ,𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥–𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦.. It intuitively doesn’t make any sense. To strengthen students understanding of slope as a ratio, we encourage teachers to continue to build the bridge from visual pattern to linear situation by providing more situations like Figures 7 and 8, translating graphs, tables and situations into visual patterns. If students equated every linear situation to a visual pattern, by thinking of the dependent variable as “number of squares” and the independent variable as “step number,” students would be able to calculate slope and attach meaning to it.

 

To strengthen the ratio concept, we also encourage teachers to experiment with non-integer rates of change. For example, giving students a pattern like the one shown in Figure 9, where number of pizzas increases by three for every two steps.

 

Figure 9: Non-integer slope

 

We also encourage teachers to make visual patterns a regular part of their classroom routine. Students need immediate opportunities to use the strategies and thinking from a pattern talk before they forget.

 

Overall, we were pleased with the progress our students made around representing visual patterns with structure and beginning to see slope as ratio of two varying quantities. Students became comfortable with the routine of pattern talks and liked seeing their classmates strategies.

 

 

 

Resources:

 

Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: Middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann.

 

Ellis, A. (2009). Patterns, Quantities, and Linear Functions. Mathematics Teaching in the Middle School, 14(8), 482-491.

 

Lobato, Joanne, Ellis, Amy Burns and Munoz, Ricardo (2003). How “Focusing Phenomena” in the Instructional Environment Support Individual Students’ Generalizations. Mathematical Thinking and Learning, 5:1, 1 – 36.

 

Nguyen, Fawn. Math Talks. Retrieved from http://www.mathtalks.net/pt-1-4.html

 

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies, grades K-5. Sausalito, CA: Math Solutions.

 

 

 

 

 

 

Lesson: Area of a Circle (or How I Got Students Hungry for the Formula)

I teach CMP. I love discovering pi in Investigation 3.1 of Filling and Wrapping. It’s hands-on and engaging. Students love discovering that pi is a real thing not just a random number and excuse to eat pie on 3/14. If the lesson goes really well, I’ll even get a kid to ask, “Isn’t this a proportional relationship? Isn’t pi a constant of proportionality?” To which I want to use every happy emoji ever created in large, bold, capitalized letters with lots of exclamation points!!!

But then Investigation 3.2 comes along, where students discover the formula for area of a circle as pi groups of radius squares. I’ve never been able to get students to make sense of this lesson. There seem to be lots of understandings (what a radius square is, how to calculate how many times one number goes into another, understanding multiplication as groups of…) needed in order to get valid data that would allow students to conclude pi groups fit inside the circle. It just gets messy and doesn’t seem to make any sense to students, so this year I decided to take a different approach.

I decided to go with composing and decomposing. I would show them how to decompose the circle into a rectangle that is πr by r. Here’s how the lesson went (spoiler alert- a Domino’s pizza makes an appearance):

MCA Warm-up (The MCA’s are the big comprehensive state test in MN. We start each class by doing a couple of review questions similar to what they will see on the test. After 6 warm-ups, they take a quiz online to further mimic the testing experience and hold them accountable to remembering the material.)

Go over homework

• Discuss how you know proportional.
• Make proportional conclusions:
  • Constant of proportionality
  • Proportional equation: C = 3.14d

Notes

• add pi and Circumference formula to notes
• have them draw a circle with diameter, calculate circumference
• Area = ?? still need a way to calculate area of a circle…

Launch Inv 3.3:

• Hand out circle drawn on cm grid paper.
• Talk about diameter and radius of circle then ask, “If I asked you to find the area of the circle, what would I be asking you to do?”
• Work with partner to estimate the area of the circle using whatever strategy they thought would give them the most accurate estimate.
• Entered their data on a Google Form when done.

Explore

• As a class:
  • Look at Form data, talk about estimates and strategies
    • Area of 72 (area of rectangle)
    • Area of 144 (area of large square circle inscribed in) These two estimates give up good lower and upper bound
    • Show student strategies for estimates in between 72 and 144
    • If someone used formula, ask them why it works? are they sure that makes sense?
  • Say, “We have these estimates that seem like they’re probably somewhat accurate, but there must be a more exact method where we don’t have to count squares. I don’t know how to find area of a circle, so I’m going to see if I can turn my circle into a shape that I do know how to find the area of.” Here’s where it got fun!
    • Method 1 (this is what I did first and second hour):
      • have them cut out circle
      • fold circle in half – ask them for circumference of half of the circle: πr , outline this distance with a highlighter and label
      • then fold circle in half again, talk about length of straight edges, r
      • fold in half again, talk about length of straight edges again, still r
      • unfold and cut out along fold lines, should have 8 slices of pizza
      • Start arranging on desk to form rectangle, talk about dimensions of rectangle, area = πr•r
    • Method 2 (I had been talking about pizza all morning with my first 2 classes, so last hour I decided I should just order a pizza and use that to derive the area formula)
      • Order a pizza (Domino’s large cheese worked great!)
      • Reveal pizza to class, watch them go insane!
      • Have students gather around your front table
      • Slice pizza into 16 slices,
      • talk about circumference of 8 of the slices or half of the pizza: πr, record this on the pizza box
      • then start arranging pizza slices into a rectangle, listen to student “Ahas!” and “No ways!” when they see it is clearly starting to form a rectangle
      • Talk about dimensions of rectangle and then the area
  • Add area formula to notes
    • Find area of practice circle in notes
    • (let students eat pizza upon showing you their completed area)

Practice – labsheet

• in groups
• circumference and area practice WS

Summary

• Circle Song!!!!

Resources:

Circle notes:

Circle they estimated area of (wish I had saved some student work here. They did some really great reasoning. I had a couple of students who were only a hundredth off of the actual area):

The great pizza discussion:

 

What I liked:

• The hook: Having students estimate area first got them hooked. They wanted to make sense of formula so that they could see how close their estimate was to the actual area
• Composing and decomposing: Christopher Danielson tweets about this a lot and it’s the basis of all math talks, but seeing how visibly wowed students were by decomposing the circle into a rectangle was so fun to see. Before the lesson, I thought the circle as rectangle might be a hard sell since visually it’s more of a trapezoid, but students didn’t get hung up on this at all. They were the ones that told me it was rectangle.
• Ordering a pizza: my students have never been more engaged than they were when I pulled that pizza out.
• More students seemed to follow this logic than the pi groups of logic in previous years

What I would want to continue to improve:

• I did a lot of the leading, especially in the classes where ordering a pizza wasn’t possible. Is there a way to get students to do this or do they need to be lead here?
• More circumference practice before adding area of the mix. We talked a lot about the proportionality of circumference and diameter, but students needed practice just finding some circumferences before throwing in area.
• Still only had about half of the class following the area discussion and where the dimensions came from. This may be due to the fact that I went to straight to the abstract, I didn’t use numbers when talking about length and wdith of the rectangle. If I had students do a couple of specific examples first, would that help all students generalize better and come to their own conclusion about dimensions and area?

Better Questions

The best compliment I have ever received came from a student at the end of my first year of teaching. (Yes, my first year, when I was naive, clueless and a general mess of a human being and teacher.) I had a student named Emma. She was one of those rare students who enjoyed thinking deeply about mathematics. She was never satisfied with a procedure. She wanted to know why and how. She often stayed after class just to talk about a math problem we had done in class. At the end of our year together, she approached me in her quiet, always thoughtful manner and said, “Ms. Lee, I really liked having you as a teacher because you asked me really good questions. You asked me questions that really made me think about math.” Cue, Ms. Lee, standing still, mouth agape, speechless. She will never understand the compliment she paid me, the late nights I stayed up that first year just thinking about questions–what would I ask them, how would I ask them? If they came up with this strategy or got stuck here, what would I ask them to move them forward or connect back?

All of this is to say, I take questioning seriously. I think it’s vitally important to getting kids to reason, critique, and problem solve. Never say anything a kid can say. Ask them a question. Make them say it instead. I have two question moments to share from this school year:

1. The question I’m always trying to get better at asking:

What does it mean to be proportional?

Proportionality if the central concept of 7th grade math in Minnesota. Every year I teach proportional reasoning strategies-unit rates, scale factors, scaling ratios, setting ratios equal to each other, etc-but every year I feel like students never quite get a good clear grasp of what it means for two quantities to be proportional to each other. This year, I tried to be more deliberate in my attempt to tie proportionality to fairness. I came up with a unit question, “How can proportional relationships help us make decisions about how resources should be distributed?” This helped me in planning the questions/problems I asked students to work on and helped ground the complexities of proportionality in something 7th graders are always keen to argue about, fairness. One problem that finally got them to reason, think and question fairness while using their proportional reasoning strategies came when I asked this warm-up question:Some students were hell-bent on 20/20. Other students had a gut-feeling that this wasn’t fair because there were more girls. Students who found unit rates (questions per gender) were finally able to shed some viable proof on why the 20/20 plan wasn’t fair. They were using math to construct arguments and their arguments were grounded in the concept of proportionality. My only regret, this question came at the end of the unit. I need to give them more questions like this throughout the unit that make them wrestle with fairness by using their proportional reasoning strategies.

2. The question I’m still searching for:

My friend tells me he was born on Thanksgiving Day, November 28. Instantly my noticing/wondering wheels start spinning. I wanted to know how often since then his birthday has fallen on Thanksgiving.  At first, I thought maybe every 6 or 7 years, depending on leap year. The actual answer? So incredibly more fascinating than I could have ever hoped for. I’m left with more noticings and wonderings than when I started. In fact, I have so many questions now that I don’t know what exactly I would ask students.

What do I ask students?

What if he was born Nov. 27, 1990? Would his birthday repeat in the same pattern? How does the timing of Leap Year effect the pattern? What are all of the possible dates that Thanksgiving can fall on? Do they all repeat in the same pattern?

Any suggestions here would be greatly appreciated!

#MTBoS

My Favorite

I have so many “favorites” when it comes to why I teach and what happens in my classroom everyday. Here just a couple:

• My favorite way to stay organized: Notability. I hate paper. I lose paper within minutes of first being handed it. Notability helps keep all of my papers organized in a place where I will be able to find those papers weeks and even years later. I use Notability to store and organize notes that I take at workshops, articles I find, answer keys, and student worksheets. My students are 1:1 with iPads, so I also have them using Notability to do all their note-taking. They groan at first, but then realize how incredibly organized it keeps them since they don’t have to keep track of papers.
• My favorite part during a math lesson: The few times I get students to challenge and question each other. It’s the space between the right answer and the wrong answer. The space where a student either knows something doesn’t make sense and asks a question or he/she knows it does make sense but wants to know why.
• My favorite part about teaching junior high students: When you finally figure out what makes a kid tick. That kid that’s been challenging for me all year. He/she is unorganized, disruptive, and unreceptive to any help or feedback until that moment when I figure out how exactly he/she wants me to relate to him/her.

#MTBoS

Is this how my students feel?

A concept has been explained to them multiple times, from multiple different angles. At a very basic level, they get it. They practice it in low stakes situations where they get reinforcement and encouragement, but every time they get their test back, Fail. They fail over and over again. They try, and try again, but it never seems to pay off. Why can’t I get this? Frustration sinks in, hope fades, they give up.

I’ve always been a quick learner. If I don’t get something, I go home; I research; I practice until I get it right. I don’t like to make mistakes, but I’ve learned they are inevitable and invaluable, so I tell myself to embrace them–make them make lots of them, but make sure you learn from them. Don’t make the same mistake twice. Enter my fourth block class. For nearly 90 days, I’ve tried and tried and failed. Everyday I fail. The basics and fundamentals of managing student behavior have been explained to me multiple ways from multiple people from multiple different angles. I’ve read blogs, attended conferences, listened ardently to the advice of colleagues. I get it; fundamentally I get it–If I have to redirect a kid more than twice, it’s no longer the student who is an issue, it’s me; follow-through on expectations and consequences; replace 3 minutes of math time with non-math time; give them brain breaks. When it comes to game time, though, I fail every single day. I lay out expectations at the beginning of class and don’t follow through on them. I redirect a kid over and over and then give up. I let kids argue with me, “It wasn’t me talking, Ms. Lee. Why are you always singling me out?” Is this how my struggling students feel? They practice. They tell themselves this time is going to be different, but then it isn’t. I get it. Give up. For three years, I’ve been telling myself that this year, this time it’s going to be different, but it isn’t. I walk from my classroom to my office with my head hung low. I know that I have failed the 80% of the class who show up to learn,  who do what I expect every day.

I’m a mathematician, a problem-solver, but I don’t know how to move forward. For the first time in my teaching career, I don’t want to go to school tomorrow. I don’t want to face that fourth block, that block with whom I have earned neither integrity nor respect. How do I move forward? I’m starting to become that student I struggle to teach the most.

But Explore MTBoS asked me to take a crappy day and notice the good, so here are three shiny starts I can pull out of my Wednesday:

1. I hooked my 8th graders into thinking a boring stamp investigation was pretty interesting after talking about the rare 1856 British Guiana 1-cent stamp that sold for $9.5 million at auction in 2014. They came up with the growth factor/growth rate connection seamlessly.
2. Both of my 7th grade classes (including the naughty fourth block one) took full advantage of quiz-fixing time in class. They worked with their groups, asked great questions and really wanted to understand where they went wrong.
3. I worked with a student who had been struggling lately in class. She just needed a little one-on-one time to straighten out a couple of concepts, but she advocated for herself, and that’s awesome?

#MTBoS #ExploreMTBoS

Turtle Races – Understanding proportional relationships

To help my students understand what proportional relationships look like on a table, graph, equation and verbal description, I created the Turtle Races. I was inspired by an activity I did with Terry Wyberg at the University of Minnesota that used turtles races as a way to match different representations of different functions. Because I teach 7th grade and all of our standards center around the idea of proportionality, I adapted this turtle race idea to help my students understand what proportional relationships look like as a table, graph, equation or verbal description.

I tried this lesson for the first time last year. It was okay, showed potential but need some logistical changes in execution. This year it went really really well! All groups were able to match the pieces with little help from me, and most groups were able to completely fill in the missing pieces.

I treated the questions more as a post-game analysis activity. Students had to start them on their own at the their seat. They used their matching to start making some connections between time, distance and proportionality. Only after we talked about #1 as a class, did I let them find a partner to finish the questions.

Here are the materials. Post any questions and I’ll try to respond to them!

Turtle Races – Student version

Turtle Races – KEY

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Blogging Initiative Kick-off

My initiative at the beginning of the school year to blog more has not gone well (only two posts in nearly 90 days of teaching!). With the help of my mentor Teri Owens and Explore MTBoS I am making this pledge:

I, Laura Lee, resolve to blog in 2016 in order to open my classroom up and share my thoughts with other teachers. I hope to accomplish this goal by participating in the January Blogging Initiation hosted by Explore MTBoS.
You, too, could join in on this exciting adventure. All you have to do is dust off your blog and get ready for the first prompt to arrive January 10th! #MtBoS #ExploreMTBoS

 

 

Classroom Management – How to not let them “get me”

I need to develop my top three tenets of classroom management–three simple things I can come back to on days like today when I let them beat me, when I lost the respect of my students and myself. I don’t want to be the teacher that rules with intimidation and threats. These, for some reason, become my defaults on days that I’m tired or not on my A-game. These can’t be excuses. I teach humans, humans who want to do well. The best advice I’ve received so far (best meaning, I’ve tried it and it works) are the following:

• Replace 3 minutes of something they don’t like with 3 minutes of something they like (Dan Meyer)
• Routines – establish routines and stick to them-slow is smooth and smooth is fast (Terry Wyberg)
• Follow through – set expectations for students, communicate these to students and then follow through when these expectations aren’t met (Kim Campbell)
• Engaging lessons – yeah this is ideal, but unfortunately not an everyday reality

The follow through is what I struggle with the most. I let one kid get away with not meeting an expectation and all of the sudden I’ve been overtaken by 20 of them talking at once, wiggling around the room, a low hum buzzing throughout the room. Where’s it even coming from? How many chromosomes do 7th graders share with hummingbirds? The incessant flapping of wings at speeds that create a low, incessant hum makes me think that someone should start mapping these two species.

In all seriousness, though, if I followed through more, they wouldn’t have a chance to get to me and then I wouldn’t turn into this evil dictator that rules from an insecure place of power and authority over compassion and fairness. Recognize the good, don’t focus on the bad, proximity speaks louder than words (even if those words are on the microphone).

New Goal for 2015-16–Embrace the Math Talk

Ok, so after actually starting a week of both middle school and my final grad school class, I have a new goal for 2015-16–embrace math talks. As part of my final class to get my masters, my adviser suggested I start utilizing pattern talks as a way to build understanding of slope and y-intercept among my 8th graders. I’ve dabbled in math talks before, attended sessions at conferences, read blogs and articles online. I love the premise and I whole-heartedly believe in their potential, but completely lack the confidence in myself to facilitate the kinds of talks I’d read about.

It wasn’t until I started to read Number Talks: Helping Children Build Mental Math and Computation Strategies by Sherry Parrish that my efficacy started to increase. She suggested starting with 5 small steps that suddenly made math talks seem much more attainable:

1. Start with smaller problems to elicit multiple strategies
2. Anticipate solutions and strategies and be prepared to share a strategy from a “previous student”
3. Ok to put a student solution on the “back burner” – this was really important for me to read as I thought about my lack of confidence in leading a discussion.
4. Limit talk to 5-15 min. This is really important for maximizing engagement. I don’t know if it’s a teacher thing, but I have this gut instinct for more, make it longer, talk about it more, when really I should just move on, end it, do a follow-up activity, but end the talk. Do the talk in the last 10 min of class if you’re worried about sticking to this.
5. Be patent with myself and my students. Be willing to make mistakes, learn from them, and try again.

I use four terms to differentiate between four different types of talks I’ve been having:

• Pattern talk: putting a pattern on the board and asking students to make predictions about the next step
• Figure talk: scaffold up to a pattern talk by putting just a figure on the board and asking students “What do you see? How do you see it?”
• Number Talk: mental math computation like -75 + 12
• Math Talk: overall term for all of these different talks

Reflections so far:

8th grade

Overall Goal for the year: build understanding of slope and y-intercept using pattern talks

Started with a figure talk as a way of building up to a pattern talk.

Waited until all thumbs were up (it was pretty fast; this wasn’t meant to be hard).

All of the different strategies I recorded:

The first kid I called on thought I was insane. She just counted each tile one-by-one and didn’t understand why I was so interested in how she saw the figure. I think she assumed everyone did the same thing. Other kids immediately raised their hands, though, eager to share that they had seen it much differently.

What I did:

• Asked students to hold a thumb up in front of their chest when they had an answer; hold up a second finger if they had two different ways of getting the same answer
• Called on one student to share # of tiles and then asked if there were any other solutions before asking for strategies
• Anticipated different ways students would see the tiles
• Pre-planned how I would record student thinking

What I didn’t do:

• Ask “What do you see? How do you see it?” This was originally suggested to me by some research, but a colleague phrased it like that in her first class and got crickets. By changing the question to the one posted, we got much richer strategies and solutions.
• Have students talk to each other about different strategies
• Follow-up – ask students to do something after the talk to hold them accountable

What I want to do next:

• Pre-plan some student-to-student communication either by asking students to re-state in their own words what they heard a classmate say or by planning some Turn/Pair/Shares where students have to talk to a partner about a particular strategy. Not sure how to make this a rich, meaningful experience yet.
• Try a pattern talk now that they have seen that there are different ways of looking at a figure.

7th grade:

Overall goal: adding and subtracting integers and rational numbers (I know, integers are rational numbers, but I always feel like I need to delineate)

Number talks that have been productive:

2 + 2 + 2 + 2 + 2 + -2 + -2 + -2 =

These encouraged the idea of using opposites to add integers and connection between adding a negative and subtracting a positive. This also served as a great segway to talking about subtracting with the balloons and weights model.

As I think about planning math number talks, there are 5 things I’ve decided I need to think about beforehand (at this point anyway):

1. What strategy/concept am I trying to highlight?
2. Anticipate solutions – wrong and right
3. How will I record solutions?
4. How will I get students to talk to each other about different strategies presented during the talk?
   1. will I ask a student to repeat what they heard someone else say in their own words?
   2. Will I plan a turn/pair/share question to get students to discuss some aspect or make connections between different strategies?
5. Follow-up: What will I ask my students to do to hold them accountable to Math Talk?
   1. hold them accountable to a particular strategy or accountable to whole talk?