Reflecting on Year Three Goals

So I am at that point in the summer where, with three weeks left to go until students are sitting in front of me, I am trying to juggle lots of math ed summer reading that I haven’t done yet, big-picture thinking for school leadership, making decisions about my classroom for the year, and concretely planning my first couple of weeks. I need to start to pull together everything I have thought/read/learned/talked about math education this summer and prioritize it into goals. 

But before I do that, I want to reflect on my goals from last year:

1. My students and I will build a culture of student ownership in my classroom (through SBG, group work, and student-generated questions).
2. I will give more distributed practice (through homework and intentional placement and revisiting of big concepts).
3. I will do a better job at building procedural fluency from conceptual understanding.
4. I will plan my lessons farther in advance in order to maintain some life-work balance.

Although I did not in any way achieve everything I planned, I did have some elements of success with each goal:

• Student ownership through group goal-setting and table captains
• Elements of solving equations and/or representing and comparing linear relationships (what I see as the main threads of eighth grade) appeared in every unit
• Linear relationships “challenges” as a form of spaced, varied, interleaved practice
• The piggy bank principle story and the soda machine analogy served as a really solid touchstone for conceptual understanding about equivalent equations and functions, respectively
• In general waiting longer to get to formalized procedures
• In the spring, staying at work until I finished for the next day led to actually having some time in my life where I wasn’t thinking about teaching

I also found myself focusing on a lot of things that I hadn’t anticipated in my goals:

• By the end of December, my students and I had created a classroom that had neither productivity nor joy. This led to my primary focus for the next couple months being on building honest, authentic relationships with students and leading class through partnerships with them (a coach from my grad program wrote up our work together here).
• I also was part of a group of teachers, supported by the Deans for Impact, that spent this spring thinking together about how to apply research about cognitive science and learning to the classroom. Besides continuing my work with spaced, varied, interleaved practice, I also got really into framing/motivating questions.
• And finally, in the last two months of school, I focused on open questions, thanks to some excellent PD led by the other math team leader in my school.  We used Marian Small’s differentiation book as the basis of a lot of our work.

The least “successful” part about my goals from last year was my elaborate plan for goal #1, where I worked out what the different components of each part was, how they built, and what I would introduce when. This didn’t leave me flexibility and just didn’t make sense once I actually got to know my classes. I was also super excited about student-generated questions last summer, but I did basically nothing with that. The idea itself or the routines I had to implement it were not broad enough for me to leverage students asking questions in order to be able to learn math.

 As I think about making goals for the next year, here are my guidelines for myself:

• I can’t change or do everything that I want to do. At TMC, Dylan Kane suggested that changing 10% of your practice each year was a reasonable expectation. This means I need to prioritize.
•  Choose goals that I think will have the most impact on my classroom. I think my criteria here are goals that I am excited/interested about, aren’t too hard to implement, and are can be repeatedly leveraged for kids to learn math.
• At least some of my goals should be concrete, but also have lots of more abstract implications. For example, at PCMI, we heard Sam Shah talk about his Sam goal of “have kids sit in groups.” That is delightfully concrete but there is so much underneath that.
• Have some idea of how to start trying to achieve that goal, but know that the how will probably change.
• Be open to new goals or switching goals that are necessitated as I get to know my new students and the new environments of my classes.

Reflecting on Practice 2016

In PCMI's Reflecting on Practice class this year, we focused on making connections. Here are some things that made me think and that I would like to try.

1. Pre-Assessment

One place that it is important for kids to make connections is when they are introduced to a new concept and they build their understanding from their prior understandings. We talked about alternatives to a written, individual pre-test that were primarily short ways to activate students’ prior knowledge and get a sense of how comfortable they are with the pre-requisite (or co-requisite) skills. For example, this is a WODB that could be done at the start of a unit about linear functions with eighth graders, to see what they remember about proportional relationships and to preview some of the new ideas.

In the past, I have generally not done this type of pre-assessment. I like to start my units with a task that students can use their intuition and prior knowledge to solve, but that introduces a new idea in the unit (for example, solving a contextualized system of equations where the difference between the two situations is very obvious). I like that because I get a sense of the different ways students are approaching the new concept before I have given them any sort of direction. I also like it because it is something that we can repeatedly return to as students start to formalize the new ideas. But I see the value in doing this type of pre-assessment as well, particularly if it is pretty quick and low stress. This prompts a wider range of vocabulary, procedures, and concepts than my one task might.

2. Contrasting Cases – from Star and Rittle-Johnson

The goal of this routine is to develop procedural fluency that is flexible. This is a format that is pretty similar to things I have done in my class before, but I like it because it formalizes the process a little more and provides a bank of question options to choose from. Students are first presented with two different methods for solving a problem, where the work is accurately done out. There’s then a sequence of questions, first for understanding, then comparing, then making connections. Finally, students choose which method to use and why for each problem in a handful of similar problems. Here’s an example that we did in class.

And here are the different question options:

This seems most valuable where one method is not always better/more efficient than the other. I think that this could either be done where the original problem is a toss up between the two methods, or the original problem is obvious but the follow-up problems are split.

3. Examples and Non-Examples

Another routine I could see using in my class is having a set of examples and non-examples. Having students individually choose one that they know is an example, one that is a non-example and explain why. Then have all of the options up on the board and have students place one color post-it on which one they chose as an example, another color on which one they chose as a non-example.  Then discuss as a class based on where there is the most conflict.

Here’s one we did where we had to determine whether there was enough information to tell if the triangle was isosceles.

4. Different types of comparison

This slide, from Day 2 in the third module, summarizes the different types of comparisons that we talked about.

For a lot of reasons, discussions in my class this past year were often about what steps to take to solve a particular problem. I don’t think that this is particularly valuable to students or a good use of discussion time (particularly when it is the only type of discussion happening). This is something that I want to work really hard to change this upcoming year and I think it will be helpful to think about for what type of learning goals might I use each of these types of comparisons.

5. Quiz Feedback through Highlighting Mistakes

We watched this teaching channel video where Leah Alcala explains how she gives feedback on tests by highlighting anywhere a student makes a mistake in their solving process. I like this in contrast to what I currently do (check or x next to each problem, with a handful of random comments) because I think that highlighting where the mistake(s) is gives an entry point into revision/correction that many students need. Students who are overwhelmed or stuck when they get back a quiz with just a check or x next to each problem have a place that they can start. I still think, though, that you would need to figure out how to support students who even when they know where the error is don’t know how to fix it because they don’t understand the concept/know the procedure. Having students work in groups to do the corrections is a step toward this, but I think as I am highlighting I would also need to keep track of what the misconceptions are, how many people have them, and whether or not they need to be addressed in the whole class, small group, etc.

6. Thinking Classrooms

Peter Liljedahl video-conferenced in to talk to us about thinking classrooms. In his research, he found student behavior depends much more on what’s happening in the rest of the school than what the teacher is doing. He defines what’s happening in the rest of the school as institutional norms: normative behavior we take for granted in our institution’s classrooms. And then he researched the best ways invert the institutional norms because the more opposite teachers were to institutional norms, the more engaged kids were. Here are what he found were the biggest disruptors, where the gear size represents the relative impact.

I am really intrigued by his framework and definitely want to read more of what he has to say here.  My overall PCMI commitment to change in my classroom next year is “Use the whiteboards [which I installed all around my room last year] so that students learn from multiple strategies.”

Resources/Further Reading:

GAISE (Guidelines for Assessment and Instruction inStatistics Education)

How People Learn

How Students Learn: Mathematics in the Classroom

Twitter Math Camp Take-Aways

A summary of what I want to remember from Twitter Math Camp this year. 

Important reminders:

- “We’re not empowering students, they already have power. We are cultivating their power and often need to get out of the way. “ –Jose Luis Vilson

- “Comments with a grade are a waste of time” – Julie Wright

- “Never skip the close” – Tracy Zager

- “Kids notice vertical patterns [in a table] intuitively, not so much horizontally” – Graham Fletcher

New things to consider:

- Mathematical structure definition from Jasper DeAntonio, Kaitlin Ruggiero, David Wees

- Intersection between “showing your work” and “doing the work” from Jose Vilson

- Subtracting a negative number is the most challenging part of integer operations. Here are some of the things that students need to understand from Christy Pettis and Aran Glancy

• “Students need to decide for themselves that adding a 0 [pair] is sensible and allowed”
• “Students need to decide that an equal # of [negatives] and [positives] really can be thought of as 0”
• “Students need to experience subtraction makes larger”

- The components of an instructional routine

Things I want to implement this upcoming year:

- Splitting up the table to reflect structure in a visual pattern

Resources to look into:

- Floats and Anchors game for integer addition and subtraction

- Snag It

- Sam Shah’s version of Explore Math

Lesley Take-Aways

Now that I have written up my take-aways from each session, I have gone through to figure out what are my overall take-aways.

Really important reminders:

- make sure to have some lessons that are based on the following formats: guided discovery, reasoning and proof, data collection and analysis, interesting application, interdisciplinary connection, problem solving, counterintuitive phenomenon

- algebra is generalized arithmetic

- all of our students deserve proper vocabulary and conceptual understanding, not cutesy tricks

Things I want to implement this upcoming year:

- Contemplate then Calculate Routine—choose tasks with multiple structural shortcuts

- conjecture board: record student conceptions before introducting a new topic and then have students see if they can disprove the conjectures through a counter-example

- have students predict the next step in a visual pattern after being given the first two

- open # line prompts as pre-assessment

- construct right triangles in order to exactly place irrational numbers on the number line

- 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)

- the expanded tracks problem for the Pythagorean Theorem

- visual approximation of square roots

Resources to look into:

- Appreciating Mathematical Structure for All by Mason, Stephens, and Watson

- Well Played 6-8 by Dacey, Gartland, and Lynch

- Make bar diagrams at Thinking Blocks

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…

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

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

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

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