Author: Vicki Baker
School/Organization:
Philadelphia High School for Girls
Year: 2020
Seminar: A Visual Approach to Learning Math
Grade Level: 912
Keywords: Algebra 2, calculus, powerpoint, visual math
School Subject(s): Math, Algebra, Calculus
This curriculum unit uses animation and transition techniques learned in the TIP seminar to enhance PowerPoint presentations presented for Calculus and Algebra 2 lessons. The animations are engaging for students and the lesson format supportive of visual learners “whole picture” approach to learning. The student work includes animation of transformations and two introductory integration lessons using the principles of Riemann sums.
Download Unit: VickiBaker.pdf
In my seminar, “A Visual Approach to Learning Math”, we explored techniques to create animations in all levels of mathematics which “develop fun and engaging approaches to our classroom math lessons”. Specifically, we discussed how to animate mathematics while being mindful of aesthetics – making certain the graphics are pleasing to the eye, exciting to watch and appropriate for the medium of choice. We learned how to draw and animate mathematics using PowerPoint morph transitions and animation at a variety of levels joining art, numbers, and math. The seminar showed us how to break complex problems into simpler ones and identify questions students might ask and then use that knowledge to get to the core of the idea using animation. I will utilize seminar material about animation to create high school math lessons emanating from our discussions which bring mathematical concepts to life using movement. This curriculum unit focuses on math content and by using the techniques learned in the seminar as a template, transforming lesson delivery to include animation and movement of screen elements. These lessons are designed so that students in Algebra 2 and Calculus classes can have a visual experience with mathematical content rather than only experiencing it in a static 2dimensional space. The objective is to create visually appealing lessons which better engage students, especially visualspatial learners.
I teach high school math at The Philadelphia High School for Girls, a school with a rich history of academic excellence. Founded in 1848 to “prepare teachers for the common schools of Philadelphia,” Girls’ High, as it is affectionately known, was the first municipally supported secondary school for girls in the United States and was called the Girls’ Normal School. In 1893, the Philadelphia High School for Girls separated from the Girls’ Normal School, and the foundation for today’s college preparatory curriculum was laid. The school continues its legacy as a school for academically talented girls, providing young women with outstanding opportunities for scholarship, leadership, and service. Its motto, “Vincit qui se vincit” (He conquers who conquers himself), is a key centering point for our students maturing into young woman of purpose and honor (Cutler, 2012).
Research is clear that there is no one clear learning style for BIPOC student or any other group of leaners. There are studies which guide our teaching and inform educators on key elements which are helpful for reaching more students.
By the time my students enroll in high school they have taken high school preparatory courses and have experienced a modicum of success. They are a group of highly able learners who bring energy and a thirst for learning to education. When they interact with the academic content and each other they question and engage is high level conversations for both the content and the work they are given. Even with their strong educational background, my students struggle to apply and extend the mathematical concepts to real life word problems.
While strong math students in previous settings, across all grades, my students’ math skills are often rudimentary and reflect rote learning. Their math knowledge consists of algorithms and formulas of which they have little practical knowledge, and they do not see how those skills apply to real world applications. My students read the problem and focus narrowly on the question. They focus on remembering “the right way,” the “formula” or “algorithm,” or “the prior lesson” required to solve it. While my students have solid math skills and a strong grasp of math fundamentals, they lack the ability to apply these skills to new math content and to extend that knowledge to think critically about the math. My students fail to learn deeply and lack the ability to apply the recent skill later when it is applied to a more complex problem. As a teacher I experience these student gaps despite published stories trumpeting significant progress in student achievement by students in Philadelphia School District and the United States.
In February 2020, the School District of Philadelphia announced the fourth consecutive year of improvement citywide on the School Progress Report (SPR). The School District analysis said, “After years of investments, we are seeing increases, particularly in climate and (academic) progress”. “The SPR is the District’s primary tool to measure progress towards the Action Plan 3.0 anchor goals on gradelevel literacy and college and career readiness. The SPR evaluates schools in multiple areas, including student achievement, student progress, school climate, and for high schools, college, and career readiness. It is a transparent, actionable way to gauge the progress and achievement of a school. Schools across the city also made improvements in the individual domains used to calculate SPR scores, which also matter greatly to student success. Since 20142015, the percentage of schools have increased in major domains: 58 percent, in Achievement; 67 percent, in Progress; 82 percent, in Climate; and 60 percent of schools serving grades nine through 12, in College and Career Readiness.” Girls’ High’s SPR also improved though not as dramatically as those experienced citywide. For the progress category which measures growth on standardized assessments and progress toward graduation, Girls’ High scored 84% which corresponds to the “model” category, the highest of the three overall rating categories. Of the fourth categories measured this was the only category to earn the “model” rank.
This is great news and would suggest that students entering high school are better prepared and ready to show significant gains in all subjects including mathematical understanding. The reporting is also positive for students nationally. Impressive increases have also been seen in SAT and ACT Math scores and Advanced Placement examination participation in terminal high school courses such as calculus and statistics.
However, while we celebrate these record high NAEP scores and increases in SAT and ACT achievement – despite a significantly larger and more diverse range of testtakers – other recent data at Girl’s High demonstrate that we are not moving forward. In the PSAT and SAT, Problem Solving and Data Analysis category, Girls’ High students have not been scoring well . As shown in the tables below, even after three years, scores are flat year to year.
PSAT Data Sophomores 2018
PSAT Data Juniors 2019
The NCTM Principles to Action report also indicates there are still large disparities among racial and ethnic minority groups. The report says: “…too few students – especially those from traditionally underrepresented groups – are attaining high levels of mathematics learning. In a math classroom, a focus on critical thinking and problemsolving skills greatly increase math proficiency.
My students are as diverse as the City of Philadelphia having applied and gained admission to Girls’ High from diverse neighborhoods and middle schools of every type (public, private, parochial, magnet and neighborhood). Our diverse student population while representative of Philadelphia neighborhoods across the city has become increasingly dominated by Black, Indigenous and People of Color (BIPOC). These BIPOC students represent multiple cultures and ethnicities. Dr. Christopher Emdin, in his book, “For White Folks Who Teach in the Hood…and the Rest of Y’all Too, Reality Pedagogy and Urban Education”, proposes that educators embrace students’ cultural differences by adopting pedagogy which connects positively with their differences and “become an ally who is working to reclaim their reality”.(Emdin, 2016) This unit will not unpack all of the changes needed to offer sensitive, culturally relevant environment and lessons for BIPOC students but recognized that their learning styles compared to the more Eurocentric educational structures which dominate American education and doesn’t always support BIPOC student learning styles.
The acceptance the importance of learning styles and brain modalities has become common place among educators, administrators and even parents. In f act, in his article Ethnocentric Origins of the Learning Style Idea, Thomas Fallace states:
In recent decades, the idea that teacher should align their instruction with students’ particular learning style, cognitive style and/or learner preference has become commonplace in the literature on effective teaching. (Fallace,(2019).
Its popularity withstanding, many recent studies written in educational literature have questioned whether aligning teaching with student learning styles is beneficial and appropriate. (Hushmann & O’Loughlin (2019); Knoll, Orani, Skeel, & Van Horn,(2017), Pashler etal., (2009); Knoll (2017) . The notion that students were different along characteristics which benefitted from individualized instruction began after World War II and intensified in the 1960’s during what is called the progressive movement (Cremin, 1961; Zilversmit, 1993). While the research focus acknowledged both different learning styles and intelligence, later the focus changed eliminating learning difference considerations and related intelligence and achievement fueling conclusions that learning differences among ethnic populations were an indication of intellectual inferiority based on race. (Fallace, 2019) The arguments regarding biological justifications of inferiority driven by racist ideas and the cultural relativistic stance which held that social inequalities, unjust social policies, cultural differences, and biased assessments explained the achievement differences began to gain acceptance in the 1940’s and continued throughout the 1960’s (Fallace, 2019). Eventually while some research eluded to a Black learning style most research avoided the distinction regarding race possibly to avoid the controversy related to learning styles and racial superiority compared to the dominant culture. The research of Dunn, etal (1975)and Kolb (1976) established what the educational community considers the standard learning styles philosophy but makes no mention of learning styles and ethnic or racial groups. (Fallace, 2019).
Some African American scholars continue to pursue the idea that Black students may have a learning style that is significantly different enough to effect learning particularly when pedagogy is designed primarily for White students Hale, (198)2, Shade (1982)
There are seven learning styles which educational literature defines relate to student learning.
The Seven Learning Styles
Visual (spatial):You prefer using pictures, images, and spatial understanding.
Aural (auditorymusical): You prefer using sound and music.
Verbal (linguistic): You prefer using words, both in speech and writing.
Physical (kinesthetic): You prefer using your body, hands, and sense of touch.
Logical (mathematical): You prefer using logic, reasoning, and systems.
Social (interpersonal): You prefer to learn in groups or with other people.
Solitary (intrapersonal): You prefer to work alone and use selfstudy.
Source: https://www.learningstylesonline.com/overview/
Learning style preferences are not uniform across any group of students regardless of racial group however researchers have consistently found trends useful to consider for racial groups. Pat Guild reports in the article, Learning Styles of African American Children: Instructional Implications that researchers HaleBenson (1986), Shade(1989) and Hilliard (1989) have reported three kind of information about culture and learning styles among Black and Brown students learning math:
The lessons created in this unit will aid visual spatial learners while also reaching auditory sequential learners. How are students with these learning styles different? Auditory sequential learners prefer sequential reaching methods where they listen (auditory). They easily recall math facts, memorize steps to complete math problems answer the practice problems with ease and may not learn the concepts deeply or understand the underlying mathematical concepts without specific work to learn it. (Hass 2003). Visual spatial learners by contrast need to translate auditory inputs into visual images to learn and apply the information. They learn concept holistically rather than in parts. (Haas, 2003; Silverman, (2002) The visual spatial learner creates a picture, video, photograph, icon, or another image to. aid in the translation process. (Freed, Kloth, & Billett, 2006; Haas, 2003; Silverman, (2002). They “see” real world applications more easily than other learners and often understand complex problems more readily than simpler ones because they can start with the whole and then tackle the problem.
Auditory Sequential Learners
Left Hemisphere/Left Brain Learners

Visual Spatial Learners
Right Hemispheric/Right Brian Learners

· Think primarily in words
· Hand a good sense of time · Are step by step learners · Follow oral directions we · Are well organized · Memorize linear instructions and arrive at one correct answer · Progress readily from easy to difficult material (Silverman, 2002) 
· Think primarily in pictures
· Relate well to space but no time · Are whole concept Learners · Read maps well · Have unique methods of organization · Learn best by seeing relationships or patterns · Learn complex concepts easier than simple ones (Silverman, 2002) 
Each of the lessons created for this curriculum unit have elements beneficial for both auditory sequential learners (steps) and visual spatial learners (whole problem).
Backward design is also beneficial strategy both for the teacher and the learners of different learning styles. Wiggins and McTighe Backward Design, planning with the end in mind, helps the teacher establish where she is going and then lay out the lessons to achieve the goal. In these lessons the essential question represents with the student will know when they complete the lesson. Particularly for the visual spatial learner knowing the big picture first, where the learning is going is helpful in helping them to frame the work ahead as pictures and chunks of knowledge. There are also directions stepby step for the auditory sequential learners to follow methodically as they complete the lesson.
Students will learn
My curriculum unit will provide opportunities for students to experience the math content with visual animations and to use manipulatives designed specifically for the lesson so that they have a personal experience with the content. Curriculum unit lessons will allow students to see mathematical transformations physically move across the smartboard based on the changes in mathematical functions in xy space. These animations will bring the mathematical concepts to life visually much like the instantaneous movement they encounter in numerous video representations.
My students struggle with concepts and seeing past the numbers to higher understanding allowing them to think critically about the content. The first lesson, for Algebra 2 students, provides a visual representation of translations. Specifically, seeing the animations and then crating animations of vertical translations. I have chosen vertical translations because the “k” values can be derived directly from the quadratic equation. By comparison, the horizontal translation and the “h” variable changing in the opposite sign compared to its direction is a major stumbling block best addressed once the idea of movement and a numerical change causing that movement are firmly established. Students will make a flip book to demonstrate the “k” value changing and the graph moving. They will be able to see the graph move and ultimately relate it to the formal definition of quadratic equations.
The second and third lessons will focus on calculus with an introduction to Riemann and an accumulated area/Riemann sums word problem. My calculus students struggle to understand the approximation of area using the rectangles filling the space under the graph of the function and the xaxis. The idea that the approximate area gets better as the number of rectangles increases is understood a rote level as a process but not a transferable skill to carry them into calculus and the fundamental theorem of calculus. Students begin calculus with an indepth study of limits. Historically as we enter the study of integrals and draw on limits and Riemann sums, students do not make the connection . It is therefore extremely difficult to successfully complete the integration word problems and reallife application problems.
Lesson Plan 1 – Transformations of Quadratic Equations: Making a Math Transformation Flip Book
Essential Question(s):
Lesson Objective(s): Students will demonstrate knowledge of function families
Previous Learning: Students have prior knowledge of basic transformations from middle school mathematics. They were introduced to transformations in the previous chapter.
New Vocabulary: quadratic function, parabola, vertex of a parabola, vertex form
Previous Vocabulary: domain, range, transformations
Teaching Materials: PowerPoint –quadratic translations, student warm up sheet, student flipbook sheet
Materials for Students: graphing calculator, graph paper, supplies for making math transformation flip book.
Warmup:
Students will use a graphing calculator and graph each of the quadratic functions given. They will match each quadratic function with its graph. They will describe their reasoning by list what they noticed the differences and similarities are for each function. (teacher supplied from textbook). Ask students to discuss their findings in groups of 2 to 4 students. When the groups report out to the larger group, encourage comments comparing the graphed problems to the parent function.
Example of problems you might select from your textbook:
Introduce New Vocabulary
The following vocabulary will be new to most students: quadratic function, parabola, vertex of a parabola, vertex form. Define and provide visual representations of each word. Most algebra 2 textbooks and the internet have descriptions and images for each word
Core Concept: Horizontal and Vertical Translations
Referring to the warmup graphs discuss the form of the parent function equation f(x) = x^{2} with one of the problems. Example .
Introduce the general form of the quadratic equation.
Tell your students that the parent function of a quadratic equation f(x) = x^{2} has that form because it has its vertex on the origin (0,0).
The general equation of a quadratic function is:
Comparing equations define a, h, and k spending time on the k values for the parent function and those of the problems from the warmup.
Students will make flipbook for vertical translations.
What is a flipbook? A flipbook is a small stack of index cards where each card contains a pictures or image. When you flip through cards an animation appears of the pictures or drawings.
Students will use a flip book to animate translations of quadratic equation graphs. Working in pairs students will use their cellphones to film each other’s animations.
Supplies:
Instructions: (make a copy for each student)
Congratulations, you have created a vertical translation of the quadratic parent function y = x^{2!}
Lesson Plan 2 – Part 1 Accumulation Function and Riemann Sums
Essential Question(s):
Lesson Objectives:
Previous Learning:
Student will have experience graphing functions, finding area of geometric figures and function notation.
Previous Vocabulary: General derivative, graph functions, area, triangle, rectangle, limits.
New Vocabulary: Accumulation function, Riemann sum,
Teaching Materials:
PowerPoint presentation “Introduction to Riemann Sums,
Student materials (see list below and appendix),
Student materials:
Student Handout 1 – Triangle (4 copies)
Coin sheets (1 each half dollars, quarter, nickels, pennies)
Student Worksheet
Glue Stick
Scissors
Warm up: Textbook (select problems requiring graphing and interpreting graphs
Core Lesson:
Introduce the accumulation function
Students should think of the accumulation function as an “area over an interval” of a function. For some input x, the value of F(x) is the area under the function from a point “a” to x. Use the PowerPoint presentation to demonstrate the accumulatio of area in a triangle using coins.
Lesson Activity Instructions
Using the instructions below, guide student work finding the area of the triangle and then the area of each triangle using the four coins.
Teacher Guidelines Using Student wroksheet  
Using the triangle given, measure the length and width and calculate the area of the triangle using area formula.
½ length x height = Area ½ * _____ * _____ = _________ 
Example: Diagram from student worksheet

For each of the four coins, half dollars, quarters, nickels, and pennies, cut out the coins from the coin sheet. Arrange the coins one at a time in separate triangles in rows and columns as shown to the right below. Calculate the area of the triangle represented by the coins using the left column  
Coin: Half Dollar
Number of Coins _______________ Diameter of Coin _______________ Radius of Coin _________________ Area of Coin ___________________ Triangle Area Using Coin: (Coin Area) x (Number of Coins) = ________ Approximate coin area outside triangle = __________ Final Triangle area: _______________ 

(see student handout for further instructions using additional coins)
Writing in Math/Discussion
Ask student to use their own words to compare and contrast the triangle areas found for each coin. What might they conclude about the effect of changing the size of the coin on the area of the triangle? What effect does each coin have on the area around the coin in each triangle? What general conclusion can be reached about smaller coins and the accuracy of the area estimation? What prior calculus concept is demonstrated here? (guide to discussion to apply limits to this problem)
Lesson Plan 2 – Part 2 Word Problems Using Riemann Sums
PowerPoint to introduce the word problem
Essential Question(s):
Lesson Objectives:
Previous Learning:
Student will have experience graphing functions, finding area of geometric figures and function notation.
Previous Vocabulary: General derivative, graph functions, area, triangle, rectangle, limits, accumulation function, Riemann Sums.
New Vocabulary: Partitions
Teaching Materials:
PowerPoint presentation “Velocity Word Problem and Riemann Sums”
Word Problem
Graph the function described in the word problem. Have students explain the significance of the directions.
Students should find the area from t= 0 to t=12 for 12 partitions. The area for each partition is given in the graph. The area for increased numbers of partitions investigate how the area approximation changes with increasing numbers of partitions.
Students will complete repeat the problem using 24 and 36 partitions and compare the areas of each calculation.
Unit Bibliography
Anderson, J.A. (1988). “Cognitive styles and multicultural populations”. Journal of Teacher Education, 39, 29.
Cutler, W. (2012). Public Education: The School District of Philadelphia Encyclopedia of Greater Philadelphia. Retrieved from https://philadelphiaencyclopedia.org/archive/publiceducationtheschooldistrictofphiladelphia/
Durodoyle, Beth, and Bertina Hildreth. “Learning styles and the African American student.” Education, vol. 116, no. 2, 1995, p. 241+. Accessed 2 Mar. 2020.
Freed, J., Kloth, A., & Billett, J. (2006). Teaching the gifted visual spatial learner. Understanding Our Gifted, 18(4), p. 36
Unit Bibliography
Anderson, J.A. (1988). “Cognitive styles and multicultural populations”. Journal of Teacher Education, 39, 29.
Cremin, L. (1961). The transformation of the school: Progression in American education 18761957. New York, NY Alfred Knopf
Cutler, W. (2012). Public Education: The School District of Philadelphia Encyclopedia of Greater Philadelphia. Retrieved from https://philadelphiaencyclopedia.org/archive/publiceducationtheschooldistrictofphiladelphia/
Durodoyle, Beth, and Bertina Hildreth. “Learning styles and the African American student.” Education, vol. 116, no. 2, 1995, p. 241+. Accessed 2 Mar. 2020.
Freed, J., Kloth, A., & Billett, J. (2006). Teaching the gifted visual spatial learner. Understanding Our Gifted, 18(4), p. 36
Visual Math Improves Math Performance
The Math Leaning Center
https://www.mathlearningcenter.org/resources/lessons/visualmathematics
Mathematics courses for grades 510. These pdf format math courses aid in core math subject and visual math skills
Khan Academy Left and Right Riemann Sums