Math Walks

On March 24, 2020, everything stopped. Well, not everything, people were still walking (that is pretty much all they could do.) Covid 19 hijacked our normal routines, school was closed, gatherings canceled, and going to the store was a delicate act of trying to get toilet paper, while remaining 6ft away from everyone else. 

As I walked down our neighborhood street (trying to escape the Zoombie feeling from too many online meetings), I saw a mom and her 2 kids doing “PE” on a walk. “Go to the corner and do 5 jumping jacks. Now 3 windmills.” This ignited a spark of bringing math to daily walks!

Public problem solving had intrigued me since I read about Sara VanDerWerf’s “Can you Solve This?” a few years earlier. I posted Sara’s generously shared problems outside my classroom the past few years and watched with delight as students (not just mine), parents on campus, and teachers problem solved. I also vaguely remembered seeing Math Graffiti on a Twitter post a year or so ago. 

If I could leave a little bit of math on daily walks, I could not only give parents a way to incorporate some math, but maybe I could try and change math into curiosity, wonder and problem solving, even just a little.

I set out with my leftover chalk from an activity I did with my past math classes on my first math walk based on Which one Doesn’t Belong problem. I posted this on Twitter.

The next day, I saw a couple taking a picture and heard them talking across the street. What really delighted me was a student created “Which one Doesn’t Belong” one just a block past mine.

I decided to share this with other teachers in my district, and a few began to put these in their neighborhood too!

I was invigorated by how people responded. I began to take daily #mathwalks, highlighting some of my favorite ways of accessing math problems I learned from different math educators. I bought more chalk, lots of chalk. During this time, I found others who had also posted chalk problems for people through Twitter connections. In 2016 (I think) Leslie University incorporated math and movement with chalk for the earliest learners with a focus on counting. Twitter even had a hashtag for #publicmath and #sidewalkmath which I started to use in my tweets as well to connect with more people.

I was thrilled to see other people duplicating drawings or creating their own. It was suddenly not just my neighborhood! Although watching my neighbors take pictures and problem solve as family was equally interesting, since almost every time I walked down my street I could see problem solving in action. I got several texts and a few emails asking “Is that you leaving math everywhere?” I guess I have a reputation.

When the translations started coming in French and German I was very tickled. Someone even asked if they could duplicate a similar site in French! 

I love seeing the responses, especially when they involve excitement or creating. What I didn’t realize was the connections I would make when people stopped to share. I have been touched by people sharing that this is what motivated them to talk a walk, diverted their mind from a loved one passing, and was the topic of their dinner time conversation.

Young mathematician an coder was inspired by a math walk to create a video on how to explore the chalk picture in Python.

The excitement from this little girl made my day of someone replicating a maze from

I absolutely love when someone expresses that they are inspired by math in a chalk drawing.

This is an email thread I received shortly after I started.

On Thu, Apr 16, 2020 at 3:24 PM Jim Sullivan wrote:

Elena,

Wonderful. I did pretty good on the series but am struggling with the maze.  I can see how it has to end but I guess I will just have to try reverse engineering from there.

I am glad she is not on my walking route.  I might never get home.

Jim

On Thu, Apr 16, 2020 at 8:35 AM Elena Sullivan wrote:

My friend Traci Jackson is a math teacher here in the Poway Unified schools. During the quarantine, she is leaving math problems on the sidewalk for people to solve as they go out their (frequent) daily walks. I must confess that I have not actually stopped to solve any of these math problems, but whenever I happen upon one of them I think to myself, I bet Jim would enjoy this! 

She has a website where she posts them all. You can look at it here:

https://sites.google.com/powayusd.com/math-walks/home

Other teachers bringing math walks

https://twitter.com/mmelynnwallace/status/1251826694886961154?s=20

https://twitter.com/5BMT5B/status/1251570175289237506?s=20 

https://twitter.com/5BMT5B/status/1250135181442383872?s=20 

https://twitter.com/shammanteach/status/1250161928510398465?s=20 

Math Walk Event

Pictures of my math walk in action:

Mentions

http://ontariomath.blogspot.com/2020/06/math-links-for-week-ending-june-5th-2020.html

https://mailchi.mp/7a59f9956f05/humanizing-math-class-means-teaching-math-like-the-humanities-2687201?e=f4980a9dbb

https://mathforlove.com/2020/04/sidewalk-math-challenge/

https://www.kqed.org/mindshift/55961/how-sidewalk-math-cultivates-a-playful-curious-attitude-towards-math

Do You Want to Dance with Me?!

I have trouble keeping a beat, but that doesn't keep me from trying to do the Floss or Orange Justice in front of students. In fact, that is exactly why I dance with them before a big test.

It demonstrates what I ask of them every day:

1) Take risks, in front of others.
2) It is okay if others are better, you can still dance (problem solve.)
3) Release the pressure valve, get oxygen to your brain, and laugh, so you can think clearly.
4) Being uncomfortable is okay.
5) We are in this together.

I can't imagine how intimidating it is for a student to listen to me explain a big long proof, matter of factly, step by step, and without hesitation. What students don't realize is that I have done this same proof (or very similar) 3 times already today, and maybe 5 times last year and the year before and the year before.... What is second nature to me is a full on Pirouettes for them.

Remember this and share that information with them! Tell them how many times you have taught this and that it wasn't always this easy. Tell them about a time when you struggled with a problem, or better yet, one you are struggling with now.

I want them thinking, "Well if Ms. J is willing to do the Orange Justice, I can try proofs (and I won't look half as silly.)"

I let students know at the beginning of the year, that spelling is tough for me. I want to be sure anything I write is spelled correctly, but I need their help as this is an area of growth for me (maybe this blog writing will help!)

Being vulnerable with your areas of growth opens a door. Students know they don't have to start dancing Nutcracker proofs when they walk in your room, they just have to be willing to half raise a hand to Floss.

Widening the Threshold and Keeping Them in the Room

The title "Low Floor and High Ceiling" seems to be the gold standard with regards to the type of problems we should all be presenting to our math students. This label is given to a task that is easily accessible to everyone and can be taken to high levels.

My students will often get caught up in the problem wording or are overwhelmed multiple steps of problem and don't really know where to begin. These students just need wider (not necessarily lower) access to the problem. Additionally, many students see multiple step problems as boxes to check off as they complete each step rather than exploring the mathematics of the problem. When students finish the assigned problem parts, they leave the "mathematical" room (no matter how much was left to explore.)

I am not necessarily looking for low floor, high ceiling problems, but rather how to widen the threshold and keep them in the room.

CPM is a problem based curriculum, so the rich problems I needed are already written. I set out to explore how to open these problems further to make them more accessible, increase the exploration, and still meet the lesson goals.

Two ways I tried opening up problems:

• I removed the subparts (parts a-d) and asked students to explore the mathematics in the problem. I used the a-d parts as pocket questions to ask as needed as I circulated. This help to be sure all teams were hitting the mathematical goal.  Removing the subparts seemed to be most successful if I gave students some time as a whole class to notice and wonder about the problem before sending them off to their teams. I found that I also needed to spend some time really exploring the problem before giving it to my classes. I was so used to the curriculum questions, I didn't always explore the many directions students would go. 

• I explained a situation (usually orally and/or with pictures) and had students make up their own questions. Students were very engaged in both creating their own and solving other teams' questions. The only tricky part for me was how to quickly assess their learning. Since the teams all had different questions, they all had different answers. I did find I was really listening to their thinking, though it was tough to hear from all the groups in the allotted time for the lesson. I found this strategy most successful if students recorded their responses through Flipgrid, so there was time for peer assessment as well as my own in video form. I usually had to wait until the next day to close a lesson, but it was very powerful to reference student responses that explained the lesson goal.

Observations of opening up problems:

• Open problems worked best on a common team non-permanent surface (whiteboards) to encourage the exploration and allow for easy revision and addition to their thinking. The student work visual was also really helpful for the class discussion to show the main ideas for the day.

• Students stayed on core problems for the same amount of time (the time needed for my last team to hit the learning goal) and no one "finished" first, because there were not set number of problems to finish. This kept the class together and at the same time allowed me to differentiate my teaching with different questions for each team. 

• This benefit has lead to another equally important result, students' misconception of speed being connected with mathematical thinking. With a set number of problems or subproblems, students know who finishes first and measure themselves accordingly.  In this case of open problems, if a team says they are finished, the perception is the opposite, they didn't investigate the problem deeply.

• Students engaged in and explored mathematics more. By not giving a set of questions, this gave value to exploring a concept rather than completing the work.

• Communication and collaboration increased within and between groups. A student couldn't just read a problem, write the answer and move on.

• Ownership increased in both learning and mathematics. When students wrote questions or found mathematics in a situation, they made a personal connection, rather than reading and answering a pre-written question.

• My personal lesson planning changed to include more differentiation in my lessons. This is especially true with for my students with great deal of math background. I think and plan what questions I can give to them to push their thinking further.

In exploring how to open problems up for students, I feel like I have widened my own threshold for teaching mathematics, and will definitely be staying in the room to explore more ways this will benefit my students!

CMP Digging Deeper Day 3

June 22, 2016


Algebra, a word that strikes fear in many and is the topic of many jokes about how irrelevant to real life high school mathematics is.  I was good at Algebra, or what I though Algebra was.  I could follow a set of "like" problems so well, that I spent the majority of class writing and folding elaborate origami notes after finishing early (which were occasionally intercepted and read aloud to the class for entertainment.)

Algebra: Function based verses Equation based?

In answering this question we began our work today with Thinking with Mathematical Models Problem 1.3.
Students are given a situation of building trusses with rods (see above picture.) This is a bridge with a length of 7 trusses using 27 rods.  Student think about different length trusses in terms of a pictorial, table, graph, and final an equation.  There are many ways to push algebraic thinking by just beginning with different ways of seeing the picture.


 

Truss	Rods
7	27
1	3
2	7
3	11

Looking at the picture, some students see that for each truss added, four rods are needed. Three to create the triangle and one more connecting rod (except the first one which is missing one (-1). Students may use this picture to find the equation R = 4T-1 without even using a table or graph (although they could certainly use them as well.) This change for each truss can be represented in the  picture, table, graph and equation.

The equation R= 4T -1 shows that for every truss added, there are 4 more rods (except the first)

That is not the only way to see the problem. Students can also count the triangles and multiply by 3 to get the rods and then add the connectors (which will always be one less than the trusses because the top is one less).  R = 3T + T -1.  This is equivalent and gives a powerful visual example as to why.  

They could also count the total number of triangles and subtract out the double triangle pieces (since they have 2 in common for each triangle. R= 3(2n-1) -2(n-1)

We looked at the connections between different representations.  We then examined an entire unit of 7th Grade Moving Straight Ahead without having them sequenced for us. We explore linear functions, but the one piece we had the most trouble with was the equations section using pouches and coins. Where does this fit in?  Then we read a article from Texas Mathematics Teacher explaining the 2 different methods often used to teach algebra, equations and functions.

Texas Mathematics Teacher

CMP is a functions based curriculum that connects solving equations with what students know about functions.  In the case of the truss/rod problem questions are asked if there are 27 rods, how many trusses can be build.  Students can use the table (construct values using rate of change and initial value), graph (find 27 rods on the y axis and find where the line is at 27), or equation (substitute 27 in and solve) to answer the equation that is build from this specific instance. This grounds solving equations in the relationships defined by the function in multiple ways with more meaning to students.

To end our evening, we were given several articles to think about including one on questioning and assessment (with an in-depth example exploring the circle area and circumference standard in 7th grade.)

Questioning our Patterns of Questioning

Assessing for Learning

Day 2 CMP Digging Deeper

June 21, 2016

Orchestrating Mathematical Discussions

I have control issues. I like to be in control of making sure students learn. I would love to use my imagined superpower to open their brains and force them think and reason. I find this personality trait is common to many teachers. Several people have asked me how I am dealing with that using inquiry based learning in math. Honestly, orchestrating productive mathematical discussions gives me more control over learning than any traditional way I ever taught. You can't really force students learn (that doesn't stop us from trying though- or wishing for superpowers!) Students need to invest and connect in order to truly learn. During the summary of the problem students are primed and ready to make those connections and finally I can facilitate a semi-controlled learning opportunity by previously anticipating responses, monitoring discussion, selecting student work, sequencing the order, and finally connecting the ideas! Superpower!

Today we revisited our work from the orange juice problem and connected it directly to the excerpt we read on Orchestrating Mathematical Discussions We were given student work from the Connected Math site (from these we were given 7 different fairly randomly chosen pieces of student work (pieces 2,3,7,8, and 18). Then we were asked to look at the work from each of the 7 groups and choose which pieces of student work would be a part of our summary (selecting).  After selecting, we then we worked together to figure out how to best sequence them in a class discussion as well as questions we would use to connect. This was a very valuable experience, especially with the collaboration of our table groups. It was definitely not easy, but low risk (we weren't doing this on the spot) and high benefit (we saw our problem from new perspectives and got to wear our teacher hat without messing up our first period!)

We debriefed our choices with other groups; some chose a different order and also different groups' work to present. It was interesting to see how many different orders would still produce a math "story" and allow for students to recognize connections.

Linear relationships both proportional and non prorportional

As promised we continued to connect proportional reasoning beyond the Comparing and Scaling Unit from 7th Grade.  We explored problem 2.1 in Moving Straight Ahead where two boys race (one with a head start.) This problem is great opportunity for students to explore the difference between a proportional and a linear relationship that is not proportional.  

Inversely proportional relationships

In 8th Grade Thinking with Mathematical Models, students look at inverse variation (y=k/x) using several circumstances (holding the area constant while the side lengths both change- first explored in 6th grade, and holding the distance constant while the rate and time change problem 3.2.)  I have always understood inverse variation based on graphs and equations, but when I looked at the relationship between each variable in a table I really developed a deeper understanding. 

In a proportional relationship, when you multiply x by 2, you also multiply y by 2. However in an inversely proportional relationship you multiply x by 2 and divide y by 2! Of course this makes sense when you look at the equation  xy=24  (length x width = 24) because in order to remain equivalent, if you multiply x by 2, you must divide y by 1/2 to keep the equation balanced.  You are essentially multiplying by 2/2, just multiplying x by the numerator and dividing y by the denominator. 

We didn't teach this investigation this year due to time constraints, but I can't wait to see the connections my students will make. 

Proportional Relationship connection in 8th grade

Then we broke out into groups once again and explored other places that connected to revisiting proportional relationships.  Each group took on an upcoming lesson (jigsaw) to see how our work on proportional reasoning could be connected: Thinking with Mathematical Models 3.3, Looking for Pythagoras Problem 5.2, Growing Growing Growing Problem 1.3, Frogs, Fleas, and Painted Cubes Problem 4.3 and Say it with Symbols 2.1. Below are the results.

By the time I left today, I was even more convinced at how connected this math curriculum really is. This also seems to give me a little more of my desired "control" over student learning!

CMP conference day 1

June 20, 2016 CMP Summer Workshop Digging Deeper

Dot Walk

Our class began today with dot posters! After being given a sheet of sticky dots, we placed dots on chart paper diagrams and graphs to indicate our place. For example, a Venn diagram indicated what grades we taught, a coordinate plane poster showed years taught on the y axis and years teaching CMP on the x axis.  We made a dot plot according to our educational role (teacher, coach, administrator) and we plotted where we were from on a almost recognizable "map" of the United States. 

After all the dots were placed, our instructors led us in a Notice and Wonder where we explored patterns and addressed the yellow dot outside our Venn diagram (administrator) and the blue dot living in the ocean (he was from China.)

I see this as a great activity to use returning from summer break. I will need to think more on the categories I will use (maybe a coordinate plane looking at books read and movies watched?)

Survey of Beliefs and Fishing

We then examined our math education beliefs using a survey developed by NCTM in Principles to Action. While we all knew what our answers should be, we had a lively discussion on specific words in questions 7 (master) and question 2 (exactly.)  Knowing the right answers (principles) is at least a step towards implementing (action) mathematics in classroom that will lead to effective learning.   It's tough to see kids struggle, but if we always catch the fish (math) for them, without us they will starve (or at be unable or willing to even start a unique problem.)

How do we teach them to "fish" in math? By not answering their questions....directly. Answer  their question with another question to see just how much help they need baiting their hook! Maybe even wait until the class summarizes the problem if it is appropriate. CMP has a definite lesson flow. Launch the problem, let the students explore, and summarize as a class at the end. Both explore and summarize are excellent places to ask students questions to move them along and check for correct mathematical reasoning. Make sure their questions get answered, but help them get their on their own or you will be fishing all year for them.

Proportional Reasoning, it all Connects

With class sets of 6th, 7th, and 8th grade books we began to look at the critical concept of proportional reasoning.  This is an important step in mathematical development, moving from an additive relationship (more means you are adding) to multiplicative (comparing ratios for more per item). After a quick write on our understanding of proportional reasoning, we were given a problem to work from Comparing and Scaling in 7th grade where students are mixing juice.

Which one is the the strongest? We were given chart paper to explain our reasoning. Our facilitators  walked around, writing down some of our comments and answering questions with questions to move our thinking along. 

Several of us had done the problem already, and took a bit of a back seat. One important comment came up about students having no idea what concentrate was (especially liquid, because they were used to powder.)  

After all posters were up, we started a class discussion. First we looked at the most common solution (creating a part to whole ratio and then looking at percentages.) Then we examined and questioned several others (them most controversial looking at a digram representing 1 cup of water (unit ratio). We also examined students who just divided the first number by the second to get a decimal (2/3= .6 repeating) and then turned it into a percent and why it is not the same at the percent of part to total (40%). We discussed scaling up and down (unit ratio) as well as comparing to benchmark fractions.

We then brainstormed at all the information students could use to access this problem. Then we examined the CMP3 books to see where we could find those concepts.  Below is a summary.

Summary of concepts and where they are found

Orchestrating Productive Discussions

Our last activity of the day involved one of my favorite, and most challenging books, 5 Practices for Orchestrating Productive Mathematical Discussions.  The basic premise of this book involves pre -planning discussions including planning all the ways students will answer the problem, how the teacher will respond to any misconceptions with questions or counter examples, as well as how to sequence the discussion to make a mathematical story that connects smoothly. Quite a task when you have student variables! It is easy to see how amazing and overwhelming this concept is. In several of my lessons I have attempted to chart anticipated student responses. I will continue my work on this next year, as I find charting it makes my discusssions more relevant, efficient, and, when it goes well, beautiful!  The following is my example from the pizza problem in Comparing and Scaling investigation 2.

Chart of student responses and sequencing

Our day ended with a Ted Talk by Diane Laufenburg on How to Learn from Mistakes. 

Academy of Best Practices: Day 2 Assessment

Today's focus was on assessment and team building (although I don't think that team building was the stated focus, it is just when I picked up on today.) We started the day by finding our teams using multiple representations of a linear function.  We each took a card and had to find matching representations.

Once we were in our teams we were given time to create certain three dimensional shapes with only a string, a picture and us as team members. I found this particularly challenging, since it takes me a little longer than think spatially (I also have impressively weak fine motor skills.)

We were then given a problem that was intended to challenge us to think and to stretch our own abilities to work in teams (dealing with an invented base (-4) and translating the first 25 base 10 numbers.) There were 4 parts with increasing complexity. To be honest, I struggled with not moving onto the next part of the problem. I really wanted to see where the problem lead, but I needed to take time to make sure everyone in my group could do the same.

Both activities gave me such compassion for the difficulties participating in group work. My group had to wait for me and pull me along during the string construction activity and I needed to not sacrifice staying with my group for the sake of the problem (the problem will still be there tonight.) I did experience satisfaction and shared excitement as we solve the problem together. Worth it!

We then talked about assessment and rubric building. We were given a problem as a team to work. Then we were asked to build a rubric together for this problem. This was quite a lively discussion. I was thankful I could contribute due to the many rubric discussions my wonderful 7th grade math team had last year! After this, we each took a stack of papers and used our rubric to grade them (I kind of felt bad for all the hard work students put in only to be graded by us and recycled later.)

Once we decided our score, we put it on the underside of the paper in the right corner and placed it in the center. After we each graded a stack, we took a stack from the center we had not already graded, but another person at our table had. We couldn't see their score and we graded it and put our new score on the top left underside corner, then flipped it to compare. If we disagreed, we put it in the middle of the table to discuss later. Our rubric was pretty precise, and yet we did differ on several papers.

Each paper led to a conversation and final score. Of course doing this for every problem would take forever, but one is doable. I really enjoyed seeing all different perspectives and totally missed one student's drawing and gave her a 2/4 when really, she completely understood and represented the problem in a way that was not immediately apparent to me.

Motivational: used to encourage, no guidance. Example "Good job" " :)"
(I use this quite a bit)

Evaluative: measure, score, numbers." -1", "21/25", "88%"
(Yep, I do that too.)

Descriptive: what needs to be done, "Read directions", "Calculate answer", "Check your work"
(Yes...)

Descriptive Effective: move them to the next level, should make the learner think. "How could you use the table/graph and chart to find the change in x and the change in y."
(Only rarely do I use this.)

Descriptive Effective helps students take the next step in learning. Again, doing this for every question would take forever, but we can definitely do it for a critical skill problem.

Here is our attempt at descriptive feedback.

We spent some time modeling strategies to use with groups to build understanding. Below is our on going list with arrows checking of what we have covered in two days.  I hope to revisit this list to add diversity to my team strategies.

We also used Algebra tiles to work with negative expressions and equations. We discussed how confusing the minus sign seems to be with students and all the definitions we use for this simple dash. I had never used a 4 sided mat to model inequalities involving negative and positive expressions, and found it very interesting to see how the opposite of a negative is positive using this visualization.

We ended class with a walk and talk with our "critical partner" (a partner chosen at random with the intention of collaboration about our action plan goal for the year.)The walk and talk is a strategy where you are given a topic and time limit and leave the room to walk and talk about it. It is lovely to get out of the seat you have been in for several hours and talk about the same thing you would be talking about at your seat. I must say, it does give me some anxiety to use this in the classroom.

I can't wait to see what tomorrow brings!