**Author:** Rachel Odoroff

**School/Organization:**

*Henry C. Lea School*

**Year:** 2020

**Seminar:** A Visual Approach to Learning Math

**Grade Level:** 7-12

**Keywords:** algebra, function, Math

**School Subject(s):** Math

This unit will explore problem solving in the middle school mathematics classroom by employing visual representations to support student thinking. Proportional reasoning, rate of change and rate problems that include initial values present some of the most critical concepts for students to master at this level. Students will view PowerPoint presentations designed to help students “see” a problem. Lessons for this unit will also capitalize on a key, middle school strength: the power of peers. Using cooperative learning strategies, students will have the opportunity to work together to solve rich tasks (math problems) involving rate of change problems that are both proportional and non-proportional. A written teaching objective for this type of problem might look like: *“Students will be able to interpret linear equations in two variables in order to solve a real-world word problem.”* The accompanying Pennsylvania State standard addressed in this unit is: Solve real-world and mathematical problems leading to two linear equations in two variables. (**M08.B-E.3.1.5)**

**Download Unit:** Odoroff-R-Visualizing-Math.pdf

This unit will explore problem solving in the middle school mathematics classroom by employing visual representations to support student thinking. Proportional reasoning, rate of change and rate problems that include initial values present some of the most critical concepts for students to master at this level. Students will view PowerPoint presentations designed to help students “see” a problem. Lessons for this unit will also capitalize on a key, middle school strength: the power of peers. Using cooperative learning strategies, students will have the opportunity to work together to solve rich tasks (math problems) involving rate of change problems that are both proportional and non-proportional. A written teaching objective for this type of problem might look like: *“Students will be able to interpret linear equations in two variables in order to solve a real-world word problem.”* The accompanying Pennsylvania State standard addressed in this unit is: Solve real-world and mathematical problems leading to two linear equations in two variables. (**M08.B-E.3.1.5)**

Here is an example of this type of problem:

Reprinted with permission from Pennsylvania Department of Education Released Item Bank at pde.gov.

Common Core (and PA Core) standards are built on the Standards for Mathematical Practice, which detail habits all mathematically proficient students should use. This paper and accompanying lesson plans will highlight the first standard: Make sense of problems and persevere in solving them. Corestandards.org contains this description of making sense and persevering: “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt…. This unit encourages students to try a visual “solution pathway.” Students may not need to use this strategy forever, but at this critical juncture, I hope that a visual solution will solidify their understanding and more importantly their ability to find a “point of entry.”

According to Pennsylvania Department of Education data, there is work to be done in the field of middle school mathematics education. With four reporting categories for student scores on the Pennsylvania State Standards of Assessment (PSSA): *Below Basic*, *Basic*, *Proficient*, and *Advanced*, students across 8^{th} grade score poorly overall. Statewide, close to 40% of 8^{th} graders scored *Below Basic* with another 28% scoring in the *Basic* category on the most recent PSSA (2019). Interpreting this data another way, more than two-thirds of 8^{th} grade students statewide are not considered to be proficient in math. Looking across grade levels from 3^{rd} through 8^{th} grade, 8^{th} graders also score at the bottom. The lowest percentage of students in the proficient and advanced categories belongs to 8^{th} grade. Data on PSSA results can be found here.

This does not come as a surprise to me, the 8^{th} grade standardized math test is hard. Common Core standards implemented in the last decade moved some standards previously taught in high school down to a middle school level with a stated goal of standardizing content across the United States and also trying to match what is being taught worldwide. According to the Core Standards (corestandards.org) website, the standards were designed to be, “*Inclusive of rigorous content and applications of knowledge through higher-order skills, so that all students are prepared for the 21 century,*” **and **“*internationally benchmarked, so that all students are prepared for succeeding in our global economy and society.” *

Henry C. Lea School is a Kindergarten through 8^{th} grade school in the West Philadelphia section of the city, serving approximately 600 students from the surrounding neighborhoods. Lea is a decidedly urban school in a large district with a wonderfully diverse student body. Children who come to Lea School arrive with a variety of strengths, backgrounds, and needs, and in some cases barriers to learning at grade level. The school has a significant portion of students who are learning English as a second language, with students coming from points throughout the globe. 100% of students are listed as “economically disadvantaged” according to the School District of Philadelphia’s website. Of all the students who participated in the PSSA at Henry C. Lea School last year, 88% scored in the ** Basic** and

Anecdotally, students in my class seem frustrated and overwhelmed by the rigorous math content presented in the curriculum and on grade level standards. It is not uncommon to work with a student and watch him or her struggle with basic math facts such as multiplication and addition. The School District issued curriculum (Envision published by Pearson) assumes that students have full mastery of the content from the previous grade. It incumbent upon the teacher to remediate and support where there are deficient skills, and by 8^{th} grade students have sometimes accumulated many skills that need remediation. Scaffolding lessons to help students achieve success is critical.

Enter the dreaded word problem. Students groan when they see them. One of my students said: “Oh, when I see those problems on the PSSA (Pennsylvania State System of Assessment), I just skip them.” Given the fact that these tests are a critical measure of student, school, and teacher performance, it is important for students to have strategies to have a reasonable chance of solving these types of problems. A quick look at released PSSA test items from the past several years shows the extent to which complex language understanding is required to unlock the answer, not just math proficiency. Here’s an example from 2019 PSSA test for 8^{th} grade:

Source: Pennsylvania State Department of Education Website

https://www.education.pa.gov/Documents/K-12/Assessment and Accountability/PSSA/Item and Scoring Samples/2019 PSSA ISS Math Grade 8.pdf

The question above is easily recognizable as a system of equations question with no solution but is also asks for an understanding of a constant rate of change when an initial value is different. Numerically it is not that difficult to solve, but faced with this question on a test, many students will simply become overwhelmed by the words. What is an initial fee? What is an hourly rate? What does it mean to charge an amount of money? A student has to read and comprehend the question mathematically and wade through 4 answer choices that have similar sentence stems. To a practiced mathematician/teacher, this problem is straightforward. To a student, it is not. From what I can see in my class, students *can* read the bicycle question at least on a decoding level. Understanding and breaking down the problem, pulling apart its deeper meaning and really figuring out what the problem is asking, *that* is a whole other matter. With Covid-19 causing school closures and disruption in education continuity, states are putting standardized tests on hold (possibly even for the next few years). These problems are still at the heart of algebra and represent some of the most important concepts from that strand. The questions are well written, piloted across the state on previous tests and address critical standards. They seem like a good bet, tested or not.

Finally, my students struggle to find real world applications for math concepts taught in 8^{th} grade. Of course they parrot the question every math teacher hates: “*When are we ever going to need this?”* Even though I do not have a definitive answer to that question, I do know that if I can make math more fun, more accessible and easier to *see, *my students will have a better chance at answering that question for themselves*. *

Of all the content standards required to be taught in middle school mathematics, without a doubt, concepts involving rate of change, proportional reasoning, slope and slope intercept are the most heavily emphasized. Examining the *Pennsylvania State Standards of Assessment* (PSSA) Reporting Categories, these types of problems can be found in the B-E (Expressions and Equations) and in the B-F category (Functions). By 8^{th} grade these categories comprise 60% of the eligible content on the annual test. Some of the critical objectives at the middle school level involving proportional reasoning include graphing equations of the form y = mx. (The Envision curriculum used at Lea School presents this in Seventh Grade as

y = kx, where k is the constant of proportionality. Finding the unit rate (or constant of proportionality, k) is also a critical real world application taught in Seventh grade. Moving on to Eighth Grade, students are expected to use their knowledge of proportional reasoning to develop an understanding of slope. The curriculum then introduces the idea of an initial value with a constant rate of change leading to problems in the form of y = mx + b. There are so many ways to represent these problems: with equations, through graphs and through word problems and with tables. A successful student can answer any of these types of problems and make connections between them. A successful student is also able to choose between a variety of strategies to solve these types of problems depending on his or her comfort level with the material and the type of problem. Here are three examples of word problems written for the PSSA.

By 8^{th} grade, students should also begin to develop an understanding of the rate of change accompanied by an initial value other than zero. (Equations and accompanying graphs with the form y =mx + b.)

Source: Pennsylvania Department of Education Released Item Bank 2019

M08.B-E.3 Analyze and solve linear equations and pairs of simultaneous linear equations.

In the past few years, moving into mid-career status and wishing to expand my teaching, I became interested in using different strategies for teaching math in the classroom, ones which would de-emphasize my role as guide, and strengthen student voice and engagement. My students are good at receiving direct instruction. They know how to raise their hands, to listen, to take notes, to view the board in the front of the classroom. They are developing a think-pair-share model when assigned a buddy and can articulate reasoning. However, I wanted to step off to the side and let my students work through problems in small groups rather than always modeling their solutions the way I told them to. There is so much that goes into getting this right! Would my students get along and be able to engage in respectful, ordered discourse? Would they have the supplies they need? Could they reach a successful conclusion as a group even if it is incorrect and could they show and articulate that conclusion as a group? Could they listen within their group and to other groups? How could I create activities and tasks that support this type of learning so that all students are engaged in standards based activities?

As part of the training I chose, I joined the *Responsive Math Program* at University of Pennsylvania’s Graduate School of Education (Cohort 3). Teachers from a variety of grades and schools in *School District of Philadelphia’s* *Learning Network 2* (West Philadelphia) meet monthly to work together on a specific learning protocol, called Responsive Math Teaching (RMT) instructional model. The model works as follows: The teacher presents a problem (called a task), which is rich and complicated and excellent for productive struggle. These problems have embedded content standards, and also require other skills to be able to solve. Students may need to guess and check or draw a picture or diagram. They may need to work backwards to solve or use a combination of several strategies. Students begin by reading the problem for themselves silently. The teacher then spends some time making sure that every student understands what the question is asking. That includes a teacher read-aloud of the question presumably designed to support all students in their understanding of what is being asked and to aid developing readers. Teachers ask a variety of questions aimed at getting students to reveal their understanding of what the question has asked and to clarify any misconceptions. Students are asked NOT to begin to solve yet!

Once the teacher has determined that the student understands what is being asked, students are instructed to begin to solve however they see fit with whatever strategy works for them. They should do so silently and individually for a given amount of time (maybe 3-5 minutes). They are then free to begin to collaborate with their peers, discussing how they are solving, sharing answers and pathways and debating. At this point, teachers move around the room observing, clarifying and sometimes giving a nudge if a student needs a little support. But the work is largely up to the students, the teacher does not give away the answer or show a way to solve. She might instead ask a question or point out something the student has already done.

Once students have been working for a bit, the teacher may ask some students to collaborate with other students in the room based on their problem-solving path. The final phase of this lesson is for students to work on a product that they can share with the rest of the class. This may take the form of a poster that students share or it may be as simple as sharing solutions by a document camera. Students are asked to make comparisons between different types of solutions or perhaps examine a misconception or an incorrect solution. The teacher asks open-ended questions such as: “can you tell me how you know this solution works?” or “how did you come to this solution or conclusion” or “tell me more about…”. Student thinking is made more explicit by the work they show and also the words they choose to explain their solutions. This cooperative learning model is grounded in the idea that as students struggle through a problem, they are called on to do the deep thinking they need which along the way requires them to use basic math skills and truly understand what is being asked.

Here lies an opportunity to model a visual representation of the problem they are solving, and then encourage them to create their own visual models on subsequent problems.

In an article titled ** Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning**, Professor Jo Boaler (Professor of Math Education at Stanford University and founder of Youcubed.org) looks at math visualization as a critical part of math development for students of

Boaler’s work also supports the cooperative learning model. With a firm commitment to equity in education, she sees small student learning groups as the key to keep all students moving forward. She argues that this model encourages every student to participate at the level he or she is able, supports rich dialogue about math, and provides real world context for problems. This is what employers really want: communication about a problem and its solution. (Boaler, 2016 p. 110)

Cooperative learning strategies are not new concepts in the field of education by any means. (While Responsive Math Teaching does not call itself cooperative learning, it does inclued some of cooperative learning practices.) In the 2007 book *Cooperative Learning in the Classroom: Putting it into Practice,* author Wendy Jolliffe proposes two main benefits to placing students in groups for schoolwork (as opposed to facing the front of the room and listening to the teacher talk). She argues that students are motivated by a team mentality to do “whatever is necessary to succeed and in a team this means that everyone needs to put in maximum effort….” This results in motivational and cognitive gains. She gives significant attention to setting up the classroom for this type of learning strategy, and I believe that many of these practices would be possible in online platforms if needed. (Jolliffe, 2007)

Merging this work together, students will be called upon to view problems on *PowerPoint*, possibly repeatedly watching the action. They will then have a chance for “private reasoning time” when they can process what they have seen and begin to develop their own strategies. By the heart of the lesson they will work with peers to confer on their answers and strategies for solving, creating a product to share with their class. Over the course of several iterations of this process, I hope students will gain the ability to visualize new problems and have a chance to use drawing and diagramming to solve and share with their peers.

Building off the work of the Responsive Math Cohort at UPenn’s GSE, students will learn a protocol for problem solving complex math questions through a series of steps designed to encourage the students to think independently and collaborate with peers to share different problem solving strategies. This includes sharing work for others in class to see. “Recognizing what students know and are able to do and leveraging that to move towards higher level reasoning and problem solving ensures equity and access to mathematics for all students. When the teacher responds simultaneously to student thinking and a mathematical goal, each and every student is recognized as a capable learner who can develop deep, meaningful, and flexible understandings.” From Responsive Math Website: https://www.gse.upenn.edu/academics/research/responsive-math-teaching

This unit will utilize PowerPoint to create a visual representation of a math problem in order to help students understand what is being asked.

Students work in groups of two to four in order to solve a problem. Students will be instructed in the rules and expectations for cooperative learning such as: one voice at a time, don’t give away the answer, no putting other people down. Groups should be heterogeneously mixed by gender, math ability, behavioral needs, and other diversities.

Students must show “product” or present to the class at the end of a lesson so that the teacher can get an informal read on what students have gleaned from the lesson. This increases accountability for all students.

Students share their work with the class. This could take the form of explaining to each other in smaller groups, explaining their solutions to other students, or explaining to the teacher.

Students may need to view and solve problems through online portals. This unit can be adapted for students learning remotely. Math problems can be posted online and students can be asked to follow the protocol of solving individually and then talking with peers through online chat rooms or meetings.

Significant time must be spent to teach student appropriate roles and responsibilities for each cooperative learning group. Students may need to practice roles with simpler tasks first to be able to master. Student groupings may take into account behavioral concerns and learning styles.

The Power Point presentations provided in this unit are designed to be a significant and important portion of the lesson. The emphasis of each presentation is on deconstructing the word problem itself, NOT on providing a method for solving or on teaching a specific path towards solving, allowing students to develop their thinking about how to solve. You could call this a set up for productive struggle. Each question chosen for these PowerPoint presentations was copied verbatim from the Pennsylvania State System of Assessment released item bank and is a multiple choice question. Note that I have not emphasized or included the answer choices here, though the final slides of each presentation are there for reference.

These presentations can be given in any order, dependent on teacher preference. The slides can also be given in any order depending on teacher choice. Perhaps a teacher may choose NOT to show the written problem first and just play the images, saving the question for last. Certainly the slides can be rearranged. The actual numbers involved in the problems are not particularly hard with minimal fraction use, so students can easily choose many pathways to solve: some common pathways include guess and check, building a table, creating a graph or equation.

Rachel’s note on the figures in the slides:* “I dove into this project with enthusiasm, and thoroughly loved creating the characters depicted in these slides. Each of them has a unique personality and all represent some qualities I love in my students: spunk, sense of humor, style, smarts, and hard work. I knew from the start that I wanted to represent the diversity that I see in my students and the wonderful diversity that exists all across America. There were some limitations based on the icons available and my own limitations with the program. I recognize that creating figures from varied racial backgrounds is complicated and requires thoughtfulness. I hope I have achieved that.” *

PowerPoint Version 16.37 was used to create these presentations on MacBook Air with Catalina 10.15.5 operating system. Some functionality may be lost on different programs, older programs or software.

There are three separate PowerPoint presentations: The swimming problem. The savings problem. The house care problem.

Objective: Students will be able to solve mathematical problems leading to equations in two variables.

Standards: Solve real-world and mathematical problems leading to two linear equations in two variables. (M08.B-E.3.1.5)

Time: 45 minutes should be sufficient to solve one problem. These are not hurry questions. Students may need additional time to talk about their solutions and to create presentations.

Materials: Powerpoint presentation, student copy of problem, pencils, calculators (optional, though PSSA would allow calculators for this type of problem), chart paper (optional), markers (optional). These lessons can be adopted for use in distance learning.

Set up for classroom**:** Students should be in groups of 2-4 to be able to easily work together. If this is not possible, students can work alone and still share answers. Those strategies are embedded in the lesson plan.

- Each presentation starts with a problem as presented on one of the PowerPoint presentations. Present the question on Slide One and let the students read the question quietly. Students should have a paper (or their own digital copy) of the problem as well.
- Ask students if they have any questions regarding what is written. Remember that the point of the slides is to deconstruct the question, so it might be an option just to let the presentation play and then ask questions. Obviously, the slide shows can be played multiple times.
- Clarify any words that may be confusing or need deconstructing by way of questions: What does it mean to swim a lap? What does it mean to mow the lawn? Can anyone describe what it’s like to be on a swim team?
- Slowly play through all the slides, allowing students to read them. Slides are designed to guide students through an understanding of each successive bit of information in the original problem and to develop a mathematical picture.
- Students should have a copy of the problem in front of them to take notes, jot down ideas. This could also be in digital format if digital or distance learning is the mode.
- Ask students NOT to solve quite yet. Make sure that the problem has been fully read by students and thoroughly discussed. The whole idea is to SLOW down and let students have plenty of time to understand what is being asked.
- Clarify what the question is for the specific problem. Ask questions to elicit answers. Ex. “What is this problem asking us to find?”

- Set a timer for 3-5 minutes, and ask for silence for that time to allow students to read and think. We call that Private Reasoning Time in my class.
- After 5 minutes (or so) students will often naturally begin to discuss the problem with each other. Establish a time frame that students will have to discuss and work together. Students can write answers on their paper in any way that makes sense to them. This is a rough draft. A neater version can be done later. Allow students about 15-20 minutes to work through the problem. Circulate the room answering questions, asking clarifying questions, deepening reasoning. Teacher may ask students to move to look at other types of solutions or to find similar reasoning strategies.

- Teacher can decide how best to present the work created by the students. Some options are:
- document camera
- collect student work and copy for others to see
- have students create a larger poster size of their solution (this usually tidies things up too and might deepen thinking).
- have students report orally how they solved
- set up a carousel situation where students can circulate and have one member of each team stay behind to explain the team solution.

- Discuss different strategies with students, look for commonalities. Ask students to clarify what they did. Ask students to see if they can clarify and explain the work of others. See if we can find an agreement on the answer!

While not a traditional exit ticket, have students take another look at the final slide of the presentation and really answer the question as they would see it on a standardized test. Students may also want to answer a separate question about their level of engagement or their ability to work together or what they learned by solving these problems.

One method for online learning is to launch the problem with students by “sharing your screen” to show the PowerPoint. Students can have a digital copy as well. Complete the launch section of the lesson plan above, steps #1-7. Allow students to work on the problem individually or meet in pairs virtually to discuss without teacher present. Give students a digital platform on which to show their work during that time. Google Draw has excellent tools. Students can also handwrite their answers and take a photo of those to share. Give students about 2 days to answer the problem and document their solutions, uploading them to an online shared site. Meet with students later in the week to share their solutions and discuss what strategies they used.

Pennsylvania Department of Education Website:

https://www.education.pa.gov/DataAndReporting/Assessments/Pages/PSSA-Results.aspx

Data on recent PSSA results statewide including breakdown by the four reporting categories.

Pennsylvania State Department of Education Website for Released Items

https://www.education.pa.gov/K-12/Assessment%20and%20Accountability/PSSA/Pages/Mathematics.aspx

Contains released test items from recent years for grade level 3^{rd} through 8^{th}.

School District of Philadelphia Website

Philasd.org

Contains information on enrollment and student demographics for each school in the district.

Core Standards Website:

Corestandards.org

Website dedicated to Common Core Standards information including how and why the standards were developed.

Jolliffe, W. (2007). Cooperative Learning in the Classroom: Putting it into practice. Chapter Title: “Cooperative Learning and How it Can Help.”

Sage Publications.

Well researched and thorough book on Cooperative Learning strategies in classrooms. Jolliffe argues that there are multiple benefits of this type of learning strategy: cognitive, behavioral, motivational. She also breaks down the process and explains how to teach students to work in small groups productively.

Works by Jo Boaler:

Website**: **Youcubed.org

Website devoted to Professor Boaler’s work at Stanford University. This site contains numerous articles and a myriad of interesting math activities and visualizations for students and adults.

Article:

Boaler J, Chen L, Williams C, Cordero M (2016) Seeing as Understanding: The Importance of Visual Mathematics for our Brain and Learning. Journal of Applied Computational Math 5: 325. doi: 10.4172/2168-9679.1000325

This article can also be found the website youcubed.com. Jo Boaler is a professor of Mathematics education at Stanford University. In this article she argues that the students should use visual strategies as much as possible in solving math problems and that there is brain research that backs up the connection between mathematical learning and the ability to visualize a math problem.

Book**:**

Boaler, Jo. (2016) Mathematical Mindsets: Unleashing students’ potential through creative math, inspiring messages, and innovative teaching. John Wiley and Sons (Jossey-Bass).

This book is a thorough exploration of Boaler’s philosophy and practical tips for working with cooperative learning in the classroom. She suggests a redefining of the way we should look at math instruction across the United States, hoping to make math more fun, more practical and more inclusive.

Letter of permission to use Released Items from Pennsylvania Department of Education website.

Good Afternoon Ms. Odoroff,

The released mathematics items on the Pennsylvania Department of Education website may be used for your work with the University of Pennsylvania. They were released for individuals to use.

I am reminding you that the released items are a sample of the content assessed by the PSSA in mathematics. If the content covered is important for your work, you may want to consider the standards or other descriptions of the content.

The items not released from the PSSA are to remain secure. Test Administrators or others are not to have reviewed the PSSA items found on the assessment. Also, the Test Administrators are to have knowledge of those items.

If you want to contact me, my email address is c-rblust@pa.gov.

Best Wishes,

Ross Blust