3.3.1. Reflection
Reflection as a structure to develop Teacher Identity. Reflection, helps teachers address, redress, and assess their own teacher identity as it relates to equity [
13]. Russell [
29] shares several principles of reflection for teachers including: begin with teaching and trust, engage in listening to whoever is teaching, share what each person learned in the experience, think about their own thinking—metacognition. Reflection should be based on experiences and should permeate everything that occurs in teacher education. One way to explore and analyze the identities and strengths of middle school students is using writing and reflections, discourse, and other forms of communication in mathematics [
30]. Teachers can start by focusing on their own mathematical identities, their problem-solving tactics, and their explanations of mathematical phenomena. When teachers look back at their own experiences, they begin to interrogate how they perceived math, the struggles they had, and/or the joys of learning mathematics.
Mathematical Autobiography. The purpose of writing a mathematics autobiography is to unpack one’s experiences with mathematics both in and outside of the school setting. This is a particularly important exercise as we frequently bring preconceived ideas and beliefs to our mathematics learning and teaching. These ideas are a composite of our life experiences, our cultural background, and our educational experiences. Some examples of how these experiences may impact one’s general feelings towards mathematics could include the student who has parents who did not like mathematics and therefore any mathematics homework brought home is met with stress from the adults. This stress then influences the student’s attitude towards engaging with math problems. Or there is the student who had an incredibly passionate 8th grade math teacher who inspired them daily to see mathematical beauty in everyday experiences. There is of course a wide range of potential experiences across this spectrum, and it is important for teachers to know how and why they hold their current beliefs to then be conscientious about how they will influence adolescents in their classrooms. To accomplish this reflection exercise, we concur that a mathematical autobiography should include two important components: reflection of experiences with learning and teaching mathematics, and how those experiences have influenced the students’ ideas about how mathematics should be taught in school.
While this list is not comprehensive, it is a starting point to reflect on the following:
Self-perceptions about your mathematical abilities and understandings;
Feelings and attitudes toward mathematics;
Characterize your learning needs;
Beliefs about what it means to teach and learn mathematics;
Important events in your mathematical life (in and out of school);
Describe your best and/or worst mathematics teachers and how they influenced you mathematically;
Where you are now in regard to both mathematics learning and mathematics teaching;
How your background and life experiences influence your teaching and learning.
Digging back into our experiences as learners of mathematics can be quite powerful. Having the opportunity to reflect on educational experience at a detailed level and identifying moments in that journey that either caused a spark or potentially influenced a negative mindset towards mathematics help us know our teacher identity.
Bonner [
31] shares research “that real and sustaining relationships and trust cannot be built in the mathematics classroom without first focusing on knowledge (about students, mathematics, culturally connected mathematics, and content) or communication (communicating in culturally connected ways, making mathematics accessible to students, communicating care to students, and allowing students to communicate in comfortable ways)” [
31] p. 388. By inviting students to reflect on their own growth, this practice can begin early and develop throughout a teacher’s journey and potentially help novice teachers identify their future students’ Funds of Knowledge, and build the capacities for culturally sustaining mathematics teaching that begins with building relationships and communication.
To help spark this reflection and provide an avenue for describing one’s experiences with mathematics, the following prompt could be used to begin the autobiography, “Math is like…” Begin with three options to which teachers can relate and ask them to select one that best describes their experiences—Math is like (1) an automobile, (2) a fast-food restaurant, or (3) a drink. The simile expands into a more in-depth reflection of their autobiographies as teachers provide explanations and examples. Then, each can begin to construct their own analogies revealing more about their experiences and perspectives. For example, “Math is like a puzzle with many individual parts that once together can form a meaningful picture.” “For me, math has been like a roller coaster as some topics I can coast through while others cause me to struggle in the climb.”
Journaling. A tool that compliments the mathematical autobiography is journaling. Journaling for both teachers and students allows for a more intentional view into teacher identity as well as students’ mathematical conceptual understanding, pedagogical considerations, and emotional experiences. Journaling allows for deeper thinking to happen during class or outside of the school or classroom environment. In other words, it allows for the opportunity to engage in individual discourse. The topics students reflect on can and should relate to skills related to communicating students own power and equity. It can help them consider their own ideas and develop ways to become self-aware thus building more confidence. Using journaling as a springboard for discourse makes conversations and concerns a safe place to share.
For math teachers, journaling can provide a gateway to becoming culturally self-aware and responsive in our classrooms [
32]. Creating a culturally sustaining mathematics classroom must include relationship building with communication [
16,
31]. Getting to know students allows teachers to examine the knowledge base of their students, and enhances relationships. The mathematical autobiography and journaling allow teachers to gather knowledge of their students and can be used to begin mathematical conversations based on background knowledge, ideas and insights as well as allowing students to reflect on their own mathematical journey.
3.3.2. Literacy Integration
In the mathematics classroom, we should consider the cultural and community resources that reflect students. Finding ways to celebrate diversity in mathematics classrooms can be challenging for some. Decisions teachers make can enhance or stifle this. Teachers can examine curriculum resources through an equity or critical lens for whose voice is being shared. Integrating literature in mathematics offers windows, mirrors, and sliding glass doors for students [
33]. Students can experience similarities (mirrors) to reflections of their own lives and differences (windows) to learn about others through storylines, characters, settings, or situations.
During planning, teachers can consider the connection between equity and literacy [
34]. Delpit [
35] defines equity as power and culture in the classroom. Gorski defines equity as “a commitment to deepening individual and institutional understandings of how equity and inequity operate in organizations and societies, and the individual and institutional knowledge, skills, and will to vigilantly identify inequities, eliminate inequities, and actively cultivating equity” [
13] p. 1. Standards and resources for middle grade teachers exist that address identity, diversity, justice, and action. Learning for Justice, formerly Teaching Tolerance, provides a document titled Critical practices for anti-biased education that addresses critical engagement. Critical engagement is defined as “requiring questioning, forming and challenging opinions, and feeling outrage or inspiration” [
36] p. 10.
In order to engage and empower students, teachers need tools for scaffolding that invite multiple cultures, current events, and questions to ponder into their mathematical classrooms. Specifically, examining how to use social issues in problem solving tasks is a tool for addressing equity. Very often when presenting mathematical concepts, teachers need to find and manipulate materials, ponder and ask real-world and relevant questions, speculate mathematical connections related to personal and cultural events, and share their modeling of problems and solutions with their students. While there are many resources available through the National Council of Teachers of Mathematics (NCTM), state agencies, and companies, teachers can create their own questions and explore multiple supports to empower their students. Using an equity lens challenge teachers to ensure that examples and illustrations represent multiple perspectives; and challenge teachers to provide opportunities to deepen students’ conceptual understanding, fluency, and problem solving. Teachers can use images, examples, and text that show a variety of cultures to illustrate mathematical thinking, mathematical problems, mathematical concepts. Guided by Ellerbrock and Vomvoridi-Ivanovic [
6], consider power and participation, academic support, and using cultural and community assets as the basis for examples and tasks.
Text. Texts associated with literacy integration come in all forms. Stories, images, data, and books can provide scenarios for meaningful problem solving and cultural awakenings. Teachers can use current data and images as planned supports to create questions that illuminate their future students’ insights into how mathematics is used in everyday media to create algebraic expressions that challenge how data is communicated. The following link provides insights into how businesses track young adolescent device use in order to market products. Teachers can write an algebraic expression to reflect the article. Teachers can then consider, “What would middle school students say?”
https://screentimelabs.com/blog/how-advertisers-target-your-teens-smartphone-and-how-to-cope/ (accessed on 5 December 2022).
Cultural/Community Based Funds of Knowledge. As the population of students of color continues to rise in our schools, teachers need to present curriculum from multiple perspectives. Providing insights into other cultures provides a context that the world is a truly fascinating place full of beautiful stories and environments, artists, and mathematical phenomena. Teachers need to examine mathematics first, through their own culture and community assets, and then consider their own thoughts on how to present material. To promote equity in classrooms, they must be intentional about providing resources and opportunities to examine ways to promote a growth mindset for themselves and their students. A growth mindset is “the belief that intelligence is not fixed and can be developed” [
37] n.p. Developing a growth mindset has been shown to be a predictor of achievement across all levels of socioeconomic status [
37]. Dweck [
38] provides encouragement for teachers to recognize that learning to teach takes time and that mistakes will happen; but over time, she states, we all get better at what we do. Therefore, teachers need to embrace a growth mindset as they learn to plan lessons with an equity focus.
Adolescent Literature as a support for engaging students in interdisciplinary thinking. Centering on literature is a great tool for creating interdisciplinary or integrated lessons. Literacy is a tool that provides students with “windows” into other worlds [
38]. Using literature helps them develop their own skills for tapping into their backgrounds and experiences. Literacy addresses equity by “developing analyzation skills and content knowledge to intervene when biases arise; examining the connections between inequity and oppressive policies and practices that are occurring now; and rejecting the deficit view of marginalized populations from outcome inequities (e.g., suspension rates or test scores)” [
39] n.p. Teachers can extend ideas and find books where students can see themselves and make connections to the mathematics curriculum. Using texts, teachers can empower students and promote participation with young adolescents. The following are examples of young adolescent and children’s literature that may be incorporated in math lessons to promote equity in young adolescent learner’s mathematical classroom learning experiences.
Hidden Figures [
40], based on the popular movie by that name is a book that shares the contributions of African American women mathematicians, “Dorothy Vaughan, Mary Jackson, Katherine Johnson, and Christine Darden, who participated in some of NASA’s greatest successes, like providing the calculations for America’s first journeys into space. And they did so during a time when being black and a woman limited what they could do. But they worked hard. They persisted. And they used their genius minds to change the world. This book explores the story of four female African American mathematicians at NASA, known as “colored computers”, and how they overcame gender and racial barriers to succeed in a highly challenging STEM-based career” [
41]. Not only is this story inspiring to young mathematicians, teachers and students can relate real-life applications of mathematics.
Patterns in Peru [
42] takes us on a South American adventure with hints of Incan history. Using the book in mathematics class, the teaching and learning of patterns and functions can come alive. The characters in Patterns in Peru are on an adventure using their own problem-solving skills to find their way using a series of patterns. The final pattern is a growing pattern which leads to a discovery of a vital detail of the adventure. Mathematically speaking, the discovery of this pattern opens a window to the importance of functions and how they work. The story-line naturally leads to opportunities for algebraic thinking with patterns and equations. To explore the growing pattern on the ancient wall from the story, students can use color tiles (physical or virtual manipulatives) to construct and model the puzzle posed in the adventure. In building the pattern, students discover how two variables consistently grow together, thus revealing a linear function. In analyzing the function, graphing adds another layer. Utilizing graphing calculators and/or DESMOS [
43], students investigate the relationship between the variables and see the linear formation. Extensions of the constructions can include building tiles in various growing configurations such as an L-shape discovering the linear function 2x − 1 and an H-shape for 5x + 2. Empowering students even further, the teacher can facilitate the activity in such a way for students to share how their individual approaches to the problem may differ, but the resulting linear function remains the same. Graphic representations of patterns that are seen in the story can be constructed and support students’ mathematical discourse.
One Grain of Rice [
44] is a mathematical folktale, set in India, that offers opportunities for cultural explorations while investigating an exponential function. While connecting with the storyline, students experience how quickly Rani, the main character, cleverly earns rice from the raja. Rani suggests as her reward, “Today, you will give me a single grain of rice. Then, each day for thirty days you will give me double the rice you gave me the day before” (p.13). Students can discover the exponential function and write it algebraically. Physically building parts of the pattern with rice enhances the experience and captures elements of student engagement. Working collaboratively also supports students’ social-emotional characteristics.
Math Curse [
45] can inspire students to investigate the world through the beauty and wonder of mathematics with the bonus of the humorous perspective of a young adolescent. The authors spark students’ interests by highlighting experiences through unusual questions. In the book, “…Mrs. Fibonacci says, ‘You know, you can think of almost everything as a math problem’. On Tuesday, I start having problems” p. 2. We invite teachers to examine Math Curse and create mathematical expressions or equations to reflect ideas within the text. Algebraically speaking, an equation can be derived based similarly to the time management scenario presented in the book. For example, write an equation based on waking up at 6 and arriving at school by 8. Students can write an equation like n + 6 = 8 where n is the unknown amount of time. The equation is set up to illustrate the scenario mathematically. Statistically speaking, students can create a graph, similar to ones in the book, of how much time they spend each week on social media. Convert that number over a ten-year period. Using Math Curse as a base, teachers and their students can be inspired to create weird and wonderful math questions based on their own experiences and settings. For example, “The Mississippi River is about 4000 km long. An M&M is about 1 cm long. There are 100 cm in 1 m, and 1000 m in 1 km. Estimate how many M&Ms it would take to measure the length of the Mississippi River” [
45] p. 12. Teachers and students can generate similar questions based on their personal surroundings or locations of prominence. For example, if the total shoreline of Myrtle Beach measures about 60 miles or 97 km, how many M&Ms would it take to measure the shore?
Literature integration can provide scenarios for simulations and explorations with mathematical manipulatives. These are great tools for engaging students in learning mathematics. Providing opportunities for active learning with manipulatives—algebra tiles, color tiles, geoboards, and more—provides a deeper understanding of concepts. Using manipulative tools physically or virtually can provide students with visual models to build conceptual understanding. Explorations with manipulatives provide opportunities for student discovery and inquiry. In a culturally sustaining mathematics classroom we recognize the need to “play” and the importance of the “joy” of examining challenges [
14]. When teachers incorporate student use of manipulatives, games and songs, these planned supports help students build conceptual understanding, as well as procedural fluency.
3.3.3. Place-Based, Problem-Based Learning
The third tool relates to using culture and surroundings as a framework for planning. A cultural lens must permeate our planning. Using a cultural lens can guide teachers in creating engaging lessons that highlight the cultural and community assets in their communities, their states, their regions, and the world. According to VanderArk, Liebtag, and McClennen [
46] in
The power of place: Authentic learning through place-based education, placed-based education embeds learning everywhere. It centers on individual learners, incorporates local and global thinking and requires design thinking to find solutions. For middle school teachers it can be interdisciplinary in nature, and should be inquiry-based to help students develop an understanding of their place in the world. This culturally sustaining practice addresses all the characteristics and the assets young adolescents bring to the mathematics classroom.
A pedagogical strategy that compliments placed-based learning is problem-based learning (PBL). PBL is a student-centered strategy that challenges learners to examine real-world problems [
46,
47,
48]. Problem based learning encompasses five steps. Students must consider a problem, gather information and organize data, consider how students can be grouped, facilitate student work, help students communicate their findings, and assess the processes used [
4]. It is our recommendation that teachers engage in research on PBL and participate in an experience before engaging with students.
Problem-based learning uses problems that young adolescents are interested in or are relevant to their communities, and are culturally centered. Place-based focuses specifically on the challenges or benefits that exist in a community. Very often problems are interdisciplinary. Discursive narratives in STEM, described as images, examples, and views not representing multiple cultures, discourages students of color from seeing themselves as scientifically minded [
49,
50]. Problem-based learning dismantles stereotypes and presents images and insights from multiple perspectives. Place-based learning encompasses the characteristics and assets of all students. Teachers can create opportunities for students to engage in collaborative problem solving, critical thinking, and dismantling of discursive narratives. The following examples illustrate the impact place and problems can have on students’ problem-solving abilities.
Water Quality as a Place Based Activity. Water samples are collected from various rivers and streams and are labeled as separate villages. Students work in groups to perform an analysis on sample water from their village and compare it to results of the tap water from the school. They use Probeware to analyze pH, turbidity, dissolved oxygen, and conductivity. Each set of Probeware is set up at stations that students go to with their water samples. Their results are recorded on charts and on individual worksheets. Students make observations about their safe “village” water samples. If asked why they would not drink the water even though it tested safe, they could remark upon the debris floating in the samples. The lab continues with a filtration exercise. Students use a 2-L bottle (already cut), pea gravel, sand, and coffee filters to build a filtration device and test their village water. They compare the filtered water with a jar of the original, unfiltered sample. A discussion follows on how each material works as different levels of filtration (i.e.,: Why we would not use just pea gravel as a filter). The discussion continues with how “mother nature” naturally filters our drinking water via aquifers and how much of the drinking water we use every day comes from the ground rather than surface water.
Mapping the Coordinate Grid as a Problem Based Activity. This activity could be used in conjunction with students reading Hidden Figures and also reflects rockets that are being launched and returned to earth by NASA. The activity is called Coding with Texas Instrument Rover Robots and Coordinate Planes. Coding Mission: In order for you to prepare for a safe lift-off to return to earth and receive the next colonists, you must get your rover to the launch pad site. The terrain on Mars is treacherous; consisting of mountainous regions, craters, and extreme sand storms. The trip to the launch pad will take 50 Earth days. The shortest distance to the launching pad will deplete half of the Rover’s battery life since it is in the direct path of Martian sandstorms and the solar cells will not work. So, you must find a different path to take. The path to the launch pad must begin at the Communications Hub and include two stops to a re-cell station to charge the rover battery. The path must avoid the burial pit and must include a stop at the commissary to secure the food sources in case of a bad storm. Code a model path using Rover or Smartquriz that demonstrates the path the rover should take to get to the launch pad in time.
Epidemics. The impact of an epidemic is studied across many fields. Learning about the science of the virus helps us prevent and treat infections that cause epidemics and pandemics. We use math to see trends and track incident of infection during outbreaks. Social studies show us the history of epidemics and how historical factors affect the spread of contagious diseases. Literacy provides us a way to report information and what we have learned about a disease and its effects on a population. Students create a 3D model that uses each core subject (language arts, math, science, and social studies). On a cube, they must include the following information. First, they must choose a virus, bacterium, fungus, or parasite to study; name and draw or print a picture of the virus, bacterium, fungus, or parasite; share who discovered the disease and a brief history of that person; a map showing the areas of the outbreak; a write-up of the history of the disease; the difference between a pandemic, epidemic, and an outbreak; a description of the disease (what it is, how it is spread, what the symptoms are, and any treatment or prevention measures); a graph showing the trends of outbreaks over a period of time; and provide a citation paper of all sources used.
Using place/problem-based learning allows students to address concerns and values that exist within a community. The literature describes how lessons in STEM benefit the academic achievement for a range of students including English Language Learners and gifted students by intentionally broadening participation of all students [
51,
52,
53]. Andreescu, Cordeiro, and Andreescu [
53] talk about problem-based learning as “an approach with a deep respect for the value, abilities, and strengths of each student by raising expectations beyond the standard and providing guidance in a supportive environment” p. xviii. They describe problem-based learning as student-centric, highly collaborative, multi-levels of complexity, relies on range, rigor and resilience, and is fun. These elements relate directly to students’ social, emotional, and psychological needs.
Students work collaboratively, on issues they deem critical, and have the potential to illustrate their own power in making a difference in their schools and communities. Place/problem-based lessons dismantle discursive narratives which often serve as barriers to broadening diverse students’ participation in STEM. Both allow students to see themselves as scientists and mathematicians.