Enhancing the Middle School Math Classroom with Visual Computation

Abstract

This article conducts an examination of the concept of visual mathematical computation in Polyup. This pilot research with 120 8th grade students and 34 7th grade students highlights both math engagement through visuals and its impacts on learning. This diverse district in Southern California is a not-so-unique setting with widespread socio-economic challenges that impact school engagement. In summary, the students saw an 80% increase in their completion rate with a 67% decrease in their completion time of math activities in this classroom. Further, measures of Academic Process (MAP) math aptitude test resulted in a 20% increase from pre-test to post-test comparisons in mathematical computation comprehension, as observed with Polyup activities performed twice a week for 6 weeks.

Introduction

With respect to math knowledge and skills, the 2015 PISA placed the U.S. at an unimpressive ranking of 38 out of 71 countries in math. On the 2015 National Assessment of Education Progress (NAEP) mathematics test, 75% of 12th graders, 67% of 8th graders and 60% of 4th graders scored below proficient. Already established globally as a prominent piece of the K-12 curriculum, mathematics provides an excellent domain for developing computational thinking skills. As discussed earlier though, traditional math teaching and learning approaches have failed to deliver the knowledge and skills necessary for success in the 21st Century for U.S. students. Growing out of these ideas, Polyup delivers on mobile devices a free, open, math-centered, computational thinking playground for visual learning.

During the 2018–19 and through the 2019–20 school years, Polyup was used in 7th and 8th-grade math class in Willard Intermediate School in the Santa Ana School District. According to Ed-Data.org, Willard has historically had approximately 90% of its student population receiving free and reduced lunch. During the 2018–19 school year, 20% of the school’s entire enrollment (708 students) were receiving Special Education Services. 23% were classified as homeless under the McKinney Vento Homelessness Act.

School districts in California that qualify for the so-called “State System of Support” would show low scores or little progress among defined student groups in specific categories that fall into a “red zone” on two or more educational indicators, from test scores to suspension rates and chronic absenteeism. In 2017, the state identified 228 such districts, but critics questioned the numbers, noting that test scores pointed to a far more widespread need for assistance. Since then, the dashboard has been adjusted. Santa Ana district (where Willard resides) has traditionally scored in the “red zone” according to the California Dashboard (CADashboard.org), marking it as one of the state’s lowest-achieving districts. In 2018, the entire student population at Willard was 153.1 points below standard.

After the first year of Polyup implementation (2018–19 SY), four out of the five defined student groups at Willard (All students, English Learners, Socioeconomically Disadvantaged, Students with Disabilities) moved out of the red zone for math for the first time in 4 years*. Other factors contributed to the shift included a focus on growth mindset (Boaler, 2018). “Much of the research on mindset has focused on changing students’ mindsets outside of any content teaching and learning; by contrast, this study examines an intervention that combines mindset with changed views of mathematics and mathematical engagement.” The work at Willard Intermediate served as an intervention to these particular defined student groups, “addressing the intersection of mindset and mathematics can improve students’ academic achievement, as well as students’ behavior and beliefs about mathematics.” (Boaler, 2018).

Efforts to align this pedagogical practice with district adherence to the CPM Mathematics Curriculum focused on two areas:

1. Initial learning of a concept is best supported by discussions within cooperative learning groups guided by a knowledgeable teacher.

2. Integration of knowledge is best supported by engagement of the learner with a wide array of problems around a core idea.

Combining the CPM curriculum with growth mindset pedagogical practice has been the impetus for the change in student outcomes at Willard and one math classroom leads the efforts for the improvement in student outcomes.

*Hispanic students at Willard Intermediate stayed in the “red zone” for the 2019 reporting year.

Visual Mathematical Computation

According to youcubed.org at Stanford University, “its labs have developed some really interesting new knowledge on the ways brains work mathematically. Prof Vinod Menon and his team have shown that our brains are made up of 5 different networks that are involved when we think about maths. Two of the networks are visual pathways — one is the ventral and one the dorsal visual pathway. Neuroimaging has shown that even when people work on a number calculation, such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is partly visual.”

Marc Petrie is a math teacher at Willard Intermediate and has been at the forefront of the changes the school is taking in its math teaching practices. Petrie’s use of Polyup’s graphical user interface has provided his students with more user-friendly access for generic math problems than a worksheet. Polyup’s top-down user interface format allows students to visualize the mathematical operation as it occurs. It allows teachers to focus on specific aspects of the problem-solving process that the teacher wants to emphasize, and provides a path for student validation within that student’s zone of proximal development. At the same time, coupled with the CPM curriculum, it has enabled Petrie to provide a much more meaningful application for students. For example, a typical order of operation problem on a worksheet will look like this:

4+3 × 8

For a student with a limited understanding of the order of operations, it is difficult to decipher the logic of the operators and the order required to solve the problem. A typical mistake would be for that student to work from left to right:

4+3=7, 7 × 8=56

This answer would be marked as incorrect, and it is common that the teacher would admonish the student about the order of operations. Based on research in student engagement in mathematics (Kong, 2003), the student would most likely end up leaving disengaged from the problem, the class, and math. Polyup attends to these concerns through its more intuitive workflow, and by having there be a pre-existing error in the problem, it allows the student to make the correction in the problem.

As a teacher, Petrie uses this process to engage his students’ abilities to meta-cognate. This process combines a visual representation of the problem with the more traditional symbolic notation, which provides two ways to view the problem. When the student self-corrects, it now generates a visual understanding of the process simultaneous to the symbolic logic, thereby enabling the student to perceive a greater level of math acuity than previously held.

This dual perception is one of the key drivers for Petrie’s use of Polyup to enhance growth mindset and a positive image of mistake-correction in the students he teaches.

In the example above, the student visualizes the process. The title of the problem, “multiply then add” provides another reminder for the necessary process. There are four numbers, 4, 3, 8, and 28, and three operators, +, *, and =. The students can see the flow of the problem in Reverse Polish Logic, and understand their goal is to take the three numbers and order them in such a way that they total to 28. Note that three of the numbers are bounded by a dark outline. They are locked and cannot be moved. This forces the student to work on moving 4 and 8, and the addition and multiplication operators, to reorder the problem to reach the goal of an answer of 28. If the student presses Poly, expression will execute as it is now written, and the result will be shown as 56. The student can then perform an error analysis, make corrections to reach the goal of an answer of 28 using the correct order of operations.

The student can now reorder the numbers and switch the operators. This will set up the following calculation:

First the multiplication, next the addition, and finally ending with a final validation.The process flow now shows the order of operation, multiplication followed by division. This is shown visually, without any direct instruction. The teacher, as a coach, can walk around and ask students questions that will help prompt the proper flow. This is also a good time to use pairs with one student working the computer and the other student acting as a scribe. The visual, interactive environment keeps the students engaged in the process, and this allows them to vocalize their thoughts before going in and trying to put in an expression.

This process allows the teacher to drive the student toward the desired outcome, rather than have the student make repeated mistakes until that person reaches the upper limits of frustration and shuts down believing that there is no way to get the problem correct.

This is done through the process of augmented intelligence, where the teacher is creating the problem in the desired format to lead the student to reach the desired outcome. It provides yet another valuable tool in the teacher’s automation arsenal.

In Petrie’s own words, “In my classroom, I have developed these programs called ‘chips’ in Poly jargon for two different large worksheets, equations in multiple steps with one variable and order of operations.”

Teachers like Petrie have found a way to create a mathematical excitement within their students and within their classrooms, embracing the standardized curriculum while allowing students to thrive in an interactive and visual environment. The mathematical thought of students is amplified and concretized, as they can not only make mistakes but also see the impact of their mistakes in the computation —as opposed to a worksheet setting, where they would just see a big X on their answer.

Research in Action

The teaching of mathematics is complex, and more so in places where diverse populations are direct indicators of student engagement and capacity. Using pedagogies that tap into adolescent minds requires teachers to have a deep understanding of both the mathematical knowledge that students are capable of and the engagement they need to feel connected to the classroom. Coupling edtech tools with a curriculum also requires teachers to be skilled at teaching in ways that are effective in developing mathematics learning for all students.

Action research is a way to examine issues within a school or district. While educators analyze their own teaching and learning environments on a daily basis, using action research to improve and to meet the diverse needs of their students provides great impact in learning and contributions to the field. McMillan (2004) describes action research as being focused on solving a specific classroom or school problem, improving practice, or helping make a decision at a single local site. Action research offers a process by which current practice can be changed toward better practice. Marc Petrie’s work with his students is the appropriate format for this study because of the emphasis that it would eventually have on his own teaching via the engagement and improvement in math through his students’ work.

Impact on 8th Grade

8th Grade CPM worksheets with 20 problems (120 students)

In August 2018, comparisons were made for the same eighth-grade student population in Petrie’s class. The classroom activities included solving problems on “traditional worksheets” and activities in the district adopted curriculum. After the start of the school year, the eighth-grade students then solved the same problems using the Polyup platform. With Petrie taking on the growth mindset work via Jo Boaler and her youcubed.org lab at Stanford University, Polyup’s role in his classroom allowed for ease of implementation. The results of their activities in the math concepts taught via Polyup was positive for Petrie and the students. The same students were able to complete the problems in Polyup in 30% of the time it took them to complete the problems in the “traditional” sense with worksheets. Accompanied by a 80% increase in comprehension (as scored in “in class” assessments), Polyup brought much engagement in math.

In the past, students would on average score 6 out of 20 on the worksheet set. Observationally, these deficiencies were competency and efficacy based. After Polyup was introduced and engagement as well as activities completed increased as previously stated, student assessment scores increased to 11 out of 20 on the same worksheet, an increase of approximately 80 percent. Students would take an average of 45 minutes to work through the worksheet (almost the entirety of the class time). Students were able to execute the same problem sets on Polyup, and complete the activities in 15 minutes.

Petrie found that students would engage about 85% of the time on the Polyup platform as compared to how he traditionally taught. While most of the eighth-grade students saw an increase in scores, there are always a few students who navigate elsewhere on their devices and impact engagement and scores. This particular student group would engage in the worksheets less than 50% of the time and would tend to give up on the problem set after attempting and failing at 6 to 7 problems.

In summary, Marc saw the following indicators while using Polyup

(August 2018 through May 2019):

• Comprehension of order of operations increased by 70% using Polyup as opposed to using a traditional worksheet.
• Completion rate increase — Completion rate rose to 85% versus worksheet because students remained actively engaged in Polyup machines. These students would tend to reach a frustration level with the worksheets then stop working on them.
• Time to Perform problems — Students intuitively understood the order of operations in Polyup with the Reverse Polish notation, so they were able to perform 45 minutes of worksheet problems in 15 minutes, or one-third of the original time.

Impact on 7th Grade

AVID students and the Impact of Polyup

AVID (Advancement Via Individual Determination) is a national nonprofit that works with districts in helping schools shift to a more equitable, student-centered approach. AVID trains 85,000 educators annually to close the opportunity gap, so they can prepare all students for college, careers, and life. AVID.org offers a variety of classroom activities, lesson plans, professional learning videos, and timely articles that are relevant to students. These tools help educators implement and refine instructional practices. They also help educators provide the key academic and social supports students need to thrive. Schools can utilize the professional learning modules and materials for in-service training and can access all of these resources year-round.

AVID and Polyup worked together to create math events online and provide AVID teachers resources for differentiating approaches to math instruction. Petrie was fortunate to have his seventh-grade section of math be a complete section of AVID students. There were 34 AVID students Petrie worked with, and 110 non-AVID students in the rest of the grade level. The AVID students used Polyup twice a week during the Fall semester of 2019. Much like Petrie’s eighth-grade students the previous year, the students saw an increase in engagement and increases in their assessments. The seventh-grade students scored an average of 20% greater on calculations than Petrie’s other math sections who did not receive this additional support from Polyup.

Polyup serves to create a visual representation of an arithmetic process. The structure of the problems lends itself to problem-solving and error correction (much of which his seventh-grade students needed). Along with AVID support, this subgroup benefitted from a unique approach to math learning. These factors provide students greater agency which leads to a higher level of growth mindset and mathematical understanding.

Conclusion

The action research model was conducted over a two year period with two distinct groups (eighth-graders in 2018 and seventh graders in 2017). During the initial six-month period starting in the 2018–2019 school year, Petrie used the students’ MAP test scores for a total of 120 eighth grade students. The data compares test results for the Fall MAP testing in September, 2018 to the Winter MAP testing in February 2019. Polyup was utilized as an intervention after the Fall MAP tests were administered. The Fall 2018 test scores serve as control data for the study.

Further research was conducted in September 2019 with Petrie’s seventh-grade students, and test results for 120 seventh grade students. This data compares the test results of AVID students who used Polyup on a regular basis with the students who did not.

While the conclusion of the seventh-grade student learning data was impacted by school closures due to COVID-19, it is clear through the data collected that Polyup meaningfully increased Petrie’s students’ engagement and led to improved results in standard assessments. Now, a response to the huge demand for the continued availability of growth mindset messages about mathematics from Petrie was needed remotely by his students. Petrie is working with Polyup to launch an initiative, not only for his seventh-grade students but also other students in the Santa Ana school district (in partnership with AVID) to harness the power of Polyup’s platform to transform mathematics education in his classroom, his district and beyond. One of his greatest efforts is doing everything he can: erasing damaging messages, learning deficiencies, widespread failure, and trauma at home.

References

Boaler, J., Dieckmann, J. A., Pérez-Núñez, G., Sun, K. L., & Williams, C. (2018). Changing Students Minds and Achievement in Mathematics: The Impact of a Free Online Student Course. Frontiers in Education, 3. doi: 10.3389/feduc.2018.00026

Kong, Q.-P., Wong, N.-Y., & Lam, C.-C. (2003). Student engagement in mathematics: Development of instrument and validation of construct. Mathematics Education Research Journal, 15(1), 4–21. doi: 10.1007/bf03217366

Lo, C. K., Hew, K. F., & Chen, G. (2017). Toward a set of design principles for mathematics flipped classrooms: A synthesis of research in mathematics education. Educational Research Review, 22, 50–73. doi: 10.1016/j.edurev.2017.08.002

Mills, G. E. (2011). Action Research: A guide for the teacher researcher (4th ed.). Boston Pearson.