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Adams, E. (2022). Teaching is a journey: Common vision leads to collaboration. Mathematics Teacher: Learning & Teaching, 115(7), 520-522. The National Council of Teachers of Mathematics. https://doi.org/10.5951/MTLT.2022.0062

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Oftentimes teachers from the same cohort leave their teacher education programs having seen the same visions for the future of education but work towards those visions (or not) inconsistently across their practices. 

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How to find support: (1) creating/attending book clubs, (2) reach out to master teachers/district coaches, (3) attend conferences with other similarly visioned people, (4) reach out to professors, (5) seek graduate education, or (6) find a new school/work environment.

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“I realized I could not become the teacher that I wanted to be without productive collaborative with other teachers who have a similar vision. I wanted to gain more knowledge about mathematics education and to be part of a community of teachers seeking to implement best practices” (521).

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Teachers need to collaborative with others who share a similar vision to reify that vision into practice.

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Addressing Levels of Cognition through Design

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Addressing Levels of Cognition through Design

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Kohler, B., & Alibegović, E. (2015). Assessing for learning: How to design formative and summative assessments that address different types of student cognition. Mathematics Teaching in the Middle School, 20(7), 424-433. The National Council of Teachers of Mathematics.

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How can we as teachers design test items to model the expectations of the Common Core State Standards to assess mathematical understanding while still offering opportunities to learn?

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The following lists each tier of learning and describes how to best assess it: (1) Construct a concept - prompts that require sorting/categorizing (examples/nonexamples), describe concept attributes, or create examples/nonexamples of your own; (2) Discover a relationship - look for evidence that students made discoveries by having them report narratives of the experience of discovery; (3) Simple knowledge - ask students to respond to questions based on what they remember; (4) Comprehension and communication - present novel prompts to students that ask them to formulate explanations involving literal and/or interpretative understanding of a technical mathematical expression/message; (5) Algorithmic skill - students are required to recall a sequence of steps and execute the mathematical procedure, emphasis the process over the outcome; (6) Application - students demonstrate proficiency in deciding how to solve problems, prompts must avoiding clueing students into how to solve the problem, students may need to decide how to handle extraneous information/data; and (7) Creative thinking - use synectics, the jusxtaposition of seemingly unrelated ideas, (e.g., How is pi like a snowflake? Write a paragraph with illustrations to explain.)

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Based on the cognitive learning levels defined by Cangelosi (2002), there are seven ‘tiers’ at which learning occurs, assessing which requires thoughtfully designed test questions. The following seven levels are ordered according to a learning progression that is meant to help teachers plan instruction, but the order is non-rigid: (1) Construct a concept - use inductive reasoning to distinguish examples of a particular concept from nonexamples, (2) Discover a relationship - Use inductive reasoning to discover that a particular relationship exists or why the relationship exists, (3) Simple knowledge - Remember a specified response (but not a multistep process) to a specified stimulus, (4) Comprehension and communication - A) Extract and interpret meaning from an expression, B) Use the language of mathematics, and C) Communicate with and about mathematics, (5) Algorithmic skill - Remember and execute a sequence of steps in a specified procedure or multistep process, (6) Application - Use inductive reasoning to decide how to use, if at all, particular mathematical content to solve problems, and (7) Creative thinking - Use divergent reasoning to view mathematical content in unusual, novel ways.

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Gutiérrez, R. (2015). Risky business: Mathematics teachers using creative insubordination. International Group for the Psychology of Mathematics Education, 679-686.

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Teachers most often engage in creative insubordination for one of the following reasons: (1) Changing the minds of others - teachers were optimistic that their actions would have some positive effect on the decision-making process of others, (2) Projecting an identity that one can be proud of - teachers found the risk taking to be worthwhile regardless of how they perceived the outcome of others because it reflected who they are, and (3) Modeling advocacy behavior for bystanders - teachers thought creative insubordination had the possibility of influencing others who were bystanders to stand up in similar ways to toxic narratives.

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“In an era of high stakes education and the persistence of racism, classism, and the politics of language in society, professional development for mathematics teachers has started to expand beyond developing forms of [PCT] that draws upon deep understanding of mathematics and students’ funds of knowledge to include an understanding of privilege, oppression, and political knowledge” (679). This environment has led many teachers to engage in ‘creative insubordination,’ but little is known about what drives teachers (not) to do so.

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“With respect to mathematics teaching, creative insubordination includes the following acts: creating a counter-narrative to the achievement gap; questing the forms of mathematics presented in the school; highlighting the humanity and uncertainty of mathematics; positioning students as authors of mathematics; challenging deficit narratives of students of color; renaming a course to reflect the fact that it only covers Western, Euclidian geometry, not all geometries that are practiced in the world; refusing to go along with procedures at a workshop that asked teachers to publicly endorse the Common Core State Standards in mathematics; and convincing a co-teacher that the mathematics being taught needed to reflect a more rigorous curriculum so that students understood why procedures worked” (680).

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“She felt the need to stand up to [a student who called her racist] in a similar way, less because she thought he was serious about thinking she was racist or because she was trying to change his mind, but because she wanted other students in the class to know what kind of teacher she was, someone who was comfortable talking about race” (683).

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This study examines the ‘why’ behind secondary mathematics teachers’ decisions to engage in creative insubordination, finding three main rationales for justifying taking risks: (1) changing the minds/practices of others, (2) projecting an identity one can be proud of, and (3) modeling advocacy behavior for bystanders.

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Nirode, W. (2019). Going over the test. The Mathematics Teacher, 112(6), 422-425. The National Council of Teachers of Mathematics. https://jstor.org/stable/10.5951/mathteacher.112.6.0422

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Going over tests is often a disengaging, unenriching process for students that often does not promote mathematical thinking.

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(1) Creating notes - grade tests question-by-question to get a better understanding of patterns in whole-class thinking, and track each the number of students who made each common misconception, (2) Generate discussion questions - based on these misconceptions, create discussion questions that situate the concepts to be reviewed by situating them within a slightly different, more relatable context, or by comparing/contrasting the concept with a different concept, (3) Pose discussion questions to students - ask students to work through the questions in small groups first, before coming together to discuss as a whole class (about 15-20 minutes for both), and ask students to relate each discussion question to a problem on the test, (4) Shift testing culture - instead of only using tests as summative assessments, use this structure with shorter quizzes, given more often, and give students less time.

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“I began to view almost all of my assessments as formative instead of summative” (425).

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Rather than going over tests question-by-question, by creating notes over common misconceptions on the test and generating three to four discussion questions for students, students can review the material on the test in a way that is mathematically engaging.

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Fukawa-Connelly, T., & Buck, S. (2010). Using portfolio assignments to assess students’ mathematical thinking. Mathematics Teacher, 103(9), 649-654. The National Council of Teachers of Mathematics.

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Mathematical writing engages students in rich, conceptual mathematics, but planning effective mathematical writing assignments is challenging and time-consuming.

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The portfolio format is always the same, which includes the following: (1) Cover and table of contents, (2) Cover letter - students summarize what they learned during the unit and identify what knowledge was new, what was review, and what questions remain, (3) Vocabulary - for each intentionally and directly taught unit-specific term selected at the beginning of the unit, students write the mathematical definition; a paraphrase of the definition in their own words; an example/drawing; a false example; and an explanation connecting the example and pseudo-example to the definition, (4) Skill descriptions - for each of the three to five teacher-selected skill objectives in the unit, students write a step-by-step description of how to “do” the skill and provide one or two challenging/relevant examples along with written explanations of the procedure and how the skill is relevant to the context; graphic organizations and sentence stems are very helpful here in conveying teacher expectations, (5) Unit questions - Short-answer-esque questions that vary in type but require students to explain conceptual understanding, and (6) Reflection - students reflect on their classroom participation, behavior, and quality of work during the unit and identify specific areas for improvement.

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“Our portfolio assessment design forces teachers to identify key unit skills, seek convincing evidence of understanding, and seriously engage students in their learning” (653).

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Rather than having students reflect on previously completed work, the Prospect Hill Academy model for mathematical portfolios has students create new work, the construction of which is woven into unit design and lesson planning over an extended period of time. Additionally, portfolio sections submissions are completed in phases similar to the composition process where (1) students submit a first draft, (2) the teacher provides feedback that is positive, respectful, detailed, and consistent, and (3) students create revisions based on the feedback they get.

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Early, K., Hammonds, K. E., Ratliff, B., Rosenhammer, M., & Martin, W. G. (2023). Rethinking wait time: What can 3 seconds do?. Mathematics Teacher: Learning & Teaching, 116(2), 108-114. The National Council of Teachers of Mathematics. https://doi.org/10.5951/MTLT.2022.0201

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Many teachers do not implement wait time properly, precluding opportunities for students to engage in rich discourse.

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Tips for successfully implementing wait time: (1) Count to three - build the habit of counting to three BEFORE you begin talking, (2) It takes time for students to get used to the new routines and norms - often students may change their answers because they think the absence of evaluation indicates incorrectness; be patient but also consistent with them, (3) Wait time will slow down the pace of class - the quality and length of discussions will increase at the cost of more class time; make wise decisions about which problems to tackle with discourse, (4) Call on students who may not volunteer ideas - some students may shy away from discussion/debate; make sure there are participation structures in place so their ideas can be heard too, and (5) The types of questions you use will start to change - more open-ended questions will naturally support the necessary wait time that will drive the conversation; take note of the kinds of questions that best promote productive discourse.

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“…Rowe (1972) found that wait time decreased students’ voice inflection during a conversation, suggesting an increase in the level of confidence present in students’ responses” (111).

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Intentionally implementing wait time into a mathematics classroom as a participation norm can open up the floor for students to engage in mathematical discourse.

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“Courage and Humility” of Disciplinary Mathematics

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“Courage and Humility” of Disciplinary Mathematics

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Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-63. https://jstor.org/stable/1163068

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While coming to know/doing mathematics in the discipline is often depicted as a zig-zag path moving back and forth between conjectures and they results they yield through mathematical investigation, reevaluating any assumptions that lead to contradictions, what it means to come to know mathematics in school is often associated with students’ sense-making of their teachers’ expectations to follow rules and procedures informing what it means to do mathematics as merely applying those rules and procedures on-demand.

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Redefining what it means ‘to know’ mathematics: (1) The problem is not the question and the solution is not the answer - posing problems designed intentionally so that students have the tools to answer but must work to come up with their own path to the answer without the teacher explaining how, (2) Students’ knowing as conjecture and proof - students embedding their hypotheses and assumptions into their answers and reevaluating those assumptions when they encounter something that doesn’t work, (3) Mathematical argument - framing students’ answers as conjecture until their argument is defended sufficiently that it gains favor among their peers; additionally, teaching them proper language to respectfully engage in argument and recognizing inconsistencies within that language between people as miscommunications, and (4) Modeling as a teacher - as the ‘expert’ in the classroom, the teacher should model the work of the discipline within the class structure, overall redefining common terminology (i.e., know, think, revise, explain, etc.) that students may have imparted with an inconsistent meaning so that it takes on the meaning of how it is used within the discipline.

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“The need for revisions does not become obvious, however, until one engages in the process of proof and discovers the shortcomings of one’s assumptions. The insufficiencies of the original assumptions come to be recognized as one tries to pursue the logical consequences rather than before the face: Refutations of the conclusions, often in the form of counterexamples, suggest revisions to the assumptions” (30).

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Intellectual authority being primarily consolidated within the teacher is not representative of what it means to ‘come to know’ and to ‘do’ mathematics in the discipline. Teaching students how to engage in mathematical argumentation and the process of testing conjecture and reevaluating assumptions with courage and humility rewrites this paradigm juxtaposition between the classroom and disciplinary spheres.

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Engaging in Mathematics Discourse with MLs

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Engaging in Mathematics Discourse with MLs

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Banse, H. W., Palacios, N. A., Merritt, E. G., & Rimm-Kaufman, S. E. (2016). 5 Strategies for scaffolding math discourse with ELLs: Eliminate obstacles to effective classroom communication with these research-tested suggestions. Teaching Children Mathematics, 23(2), 100-108. The National Council of Teachers of Mathematics.

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Access to discourse around mathematical ideas is critical for students to engage in to learn mathematics, yet some teachers find challenge in engaging students in discourse, particularly students who are simultaneously learning English in addition to mathematics.

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5 strategies for scaffolding math discourse for multilingual learners: (1) Ask open-ended (referential) questions, (2) Follow open-ended questions with close-ended (display) questions, as needed, (3) Scaffold students’ responses by revoicing (repeating, extending, and rephrasing), (4) Model mathematical vocabulary in context, and (5) Strive to include MLs in mathematical discourse each day.

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“Teachers may think that referential questions are too challenging for ELLs to answer and instead ask simpler, procedural questions. Although ELLs might require scaffolding to respond, asking questions that probe their understanding of mathematical concepts is important” (102).

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While discourse can be difficult to implement with students who are still mastering English, by making specific, intentional moves during instruction, discourse can become more accessible to students who need language scaffolds.

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Su, F. (2020). Mathematics for Human Flourishing. Yale University Press.

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Table of Contents: (1) Flourishing, (2) Exploration, (3) Meaning, (4) Play, (5) Beauty, (6) Permanence, (7) Truth, (8) Struggle, (9) Power, (10) Justice, (11) Freedom, (12) Community, (13) Love

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Cheng, E. (2004). Mathematics, morally. https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-morality.pdf

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“…[P]roofs, despite being revered as the backbone of mathematical truth, often contribute very little to a mathematician’s understanding” (1), i.e., errors are often found in published proofs that are socially accepted as true, one can read a correct proof and completely understand individual steps without understanding what they mean as a whole.

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While mathematical morality itself is more defined through example, the framework for three kinds of mathematical truth is as follows: (1) Belief | Believed truth, (2) Understanding | Moral truth, and (3) Knowledge | Proved truth.

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“Perhaps mathematical morality follows the same general pattern. Children learning mathematics at school have little sense of how mathematics ought to be. They are taught some rules, and if they are taught rules that are too crude, they get upset when it doesn’t work. eg adding up column by column without being told how to carry over” (24).

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Proof does not play the role of convincing mathematicians that something is true; rather, proofs are a social communication tool used to communicate ‘moral reasons’ which then convince mathematicians that something is true. There are three types of ‘truth’ within the mathematical context: (1) believed truth, (2) moral truth, and (3) proved truth. Further, while many people believe the goal of proof is to move a piece of maths into the proved truths section, Cheng believes that the real goal is to move it to the believed truths section, that is, believed by as many people as possible.

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Principles of Design for Mathematics Curriculum

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Principles of Design for Mathematics Curriculum

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Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the design of mathematics curricula: Promoting language and content development. Understand Language/Stanford Center for Assessment, Learning, and Equity.

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Inquiry Model for Addressing Inequity in Mathematics Curriculum

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Inquiry Model for Addressing Inequity in Mathematics Curriculum

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Waid, B. E. (2020). Supporting LGBTQ+ students in K-12 mathematics. Mathematics Teacher: Learning Teaching, 113(11), 874-886. The National Council of Teachers of Mathematics.

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Modifying Mathematics Tasks for Accessibility

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Modifying Mathematics Tasks for Accessibility

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Seceda, W. G., Medina, E., & Avalos, M. A. (2017). A framework for modifying mathematics tasks for accessibility. Teaching For Excellence And Equity In Mathematics, 8(1), 23-30. TODOS: Mathematics for ALL.

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Anticipating, Monitoring, Selecting, Sequencing, & Connecting Student Responses

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Anticipating, Monitoring, Selecting, Sequencing, & Connecting Student Responses

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Smith, M. S., & Stein, M. K. (2018). 5 Practices for Orchestrating Productive Mathematics Discussions (2nd ed.). The National Council of Teachers of Mathematics & Corwin Mathematics.

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Arranging Furniture & “Defronting” Classrooms

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Arranging Furniture & “Defronting” Classrooms

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Liljedahl, P. (2020). How we arrange the furniture in a thinking classroom. Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (pp. 70-81). Corwin Press.

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Liljedahl, P. (2020). Where students work in a thinking classroom. Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (pp. 56-69). Corwin Press.

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Liljedahl, P. (2020). How we form collaborative groups in a thinking classroom. Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning (pp. 38-55). Corwin Press.

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Horn, I. S. (2013). Designing for group work: The teacher’s role before, during, and after instruction. Strength in Numbers: Collaborative Learning in Secondary Mathematics (pp. 61-76). The National Council of Teachers of Mathematics.

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Horn, I. S. (2013). Fostering positive interdependence through activities and assessments. Strength in Numbers: Collaborative Learning in Secondary Mathematics (pp. 47-60). The National Council of Teachers of Mathematics.

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Horn, I. S. (2013). Providing access to meaningful mathematics: Groupworthy tasks. Strength in Numbers: Collaborative Learning in Secondary Mathematics (pp. 35-46). The National Council of Teachers of Mathematics.

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Gutiérrez, R. (2009). Framing equity: Helping students “play the game”and “change the game”. Teaching For Excellence And Equity In Mathematics, 1(1), 4-8. TODOS: Mathematics for ALL.

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“‘Equity’ is a hot topic in mathematics education these days. However, for many people, addressing equity issues rarely moves beyond the goal of closing the achievement gap (Gutiérrez 2008)” (5).

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(*) Dimensions: (1) Access - the resources that students have available to them to participate in mathematics, (2) Achievement - measured by tangible results for students at all levels of mathematics, (3) Identity - focuses on students’ pasts, emphasizes balance between self/others in a global society, and acknowledges ways students are racialized, gendered, and classed, (4) Power - comprised of issues of social transformation at many levels. (*) Dominant Axis (Access->Achievement) - prepares students to participate economically in society and privileging a status quo; measures how well students can play the game called mathematics. (*) Critical Axis (Identity->Power) - ensures students’ frames of reference and resources are acknowledged in ways that help build critical citizens so that they may change the game.

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“This equity diagram seeks not to simplify the complexity but rather to offer a useful ‘mapping space’ for ideas when trying to reflect on one’s practice… Take for example the issue of ‘power.’ While teachers in interviews may say they ‘want to empower students,’ they almost always mean it only as it relates to achievement, not with respect to helping students reach personal goals of excellence that may intersect with the doing of mathematics (e.g., helping their communities solve a local problem)” (6).

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“…[E]quity must be framed with both dominant and critical definitions. In working with teachers, I have found it useful to explicate four key dimensions (Access, Achievement, Identity, and Power) and to highlight the relationships and tensions between them” (5).

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Graham, E. J. (2018). Authority or democracy? Integrating two perspectives on equitable classroom management in urban schools. The Urban Review, 50(1), 493-515. https://doi.org/10.1007/s11256-017-0443-8

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Castrellón, L. E., Fernández, É., Reyna Rivarola, A. R., & López, G. R. (2021). Centering loss and grief: Positioning schools as sites of collective healing in the era of COVID-19. Frontiers in Education, 6(636993), 1-15. https://doi.org/10./3389/feduc.2021.636993

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Ladson-Billings, G. (1998). Just what is critical race theory and what's it doing in a nice field like education?. International Journal of Qualitative Studies in Education, 11(1), 7-24. https://doi.org/10.1080/095183998236863

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Decolonizing Education Through a “Humanizing Violence”

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Decolonizing Education Through a “Humanizing Violence”

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Leonardo, Z., & Porter, R. K. (2010). Pedagogy of fear: Toward a Fanonian theory of ‘safety’ in race dialogue. Race Ethnicity and Education, 13(2), 139-157. https://doi.org/10.1080/13613324.2010.482898

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Stengel, B.S., & Weems, L. (2010). Questioning safe space: An introduction. Stud Philos Educ 29(1), 505–507. https://doi.org/10.1007/s11217-010-9205-8

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Stengel, B.S. (2010). The complex case of fear and safe space. Stud Philos Educ 29(1), 523–540. https://doi.org/10.1007/s11217-010-9198-3

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Venet, A. S. (2024). Equity-Centered Trauma-Informed Education. Routledge.

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Table of Contents: (I) Bringing Equity to the Center - (1) Defining Trauma-Informed Education, (2) Defining Equity, (II) Shift 1: Adopt a Universal Approach - (3) Trauma Is More Than a Number, (4) Four Proactive Priorities for Decision Making, (III) Shift 2: Rethink Your Role as an Educator - (6) Build Relationships Rooted in Equity, (7) Cultivate Unconditional Positive Regard, (8) Make Connections, Respect Boundaries,  (IV) Shift 3: Move From Mindset to Systems Change - (9) Support Teacher Wellness, (10) Foster Professional Growth, (11) Work Toward Policy Change, (V) Shift 4: Change the World From Inside Your Classroom - (12) Examine the Curriculum, Disrupt Harmful Narratives, (13) Get to Work: Activism and Action as Healing

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“I care about you. You have value. You don’t have to do anything to prove it to me, and nothing’s going to change my mind” (98).

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Patterson Williams, A. D., Higgs, J. M., Athanases, S. Z. (2020). Noting for equity to sustain multilingual literacies. Journal of Adolescent & Adult Literacy, 63(4), 457-461. https://doi.org/10.1002/jaal.1025

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Lambert, R. (2021). The magic is in the margins: UDL math. Mathematics Teacher: Learning & Teaching, 114(9), 660-669. https://doi.org/10.5951/MTLT.2020.0282

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“Research in the field of special education in mathematics is influenced by behavioral and information processing theories of learning, whereas research in mathematics education is dominated by constructivist and sociocultural theories of learning. These differences in underlying theories of learning have created two quite different sets of recommendations for best practice; research in special education mathematics has focused almost exclusively on direct and explicit instruction, with little research on students solving complex problems without being given a procedure, as in mathematics education” (661-662).

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(*) Green // Affective Networks (the “why” of learning) - stimulate interest and motivation for learning via (1) supportive classroom environment and (2) meaningful, relevant mathematics; (*) Purple // Recognition Networks (the “what” of learning) - present information and content in different ways via (1) multimodal representation and (2) focus on core ideas; (*) Blue // Strategic Networks (the “how” of learning) - differentiate the ways that students can express what they know via (1) equitable feedback and (2) understanding self as a mathematics learner through strategies (mathematical, problem-solving, self-regulation, and executive functioning)

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“Empathy interviews in mathematics are one-on-one conversations with our students, in which we seek to learn about their experiences learning mathematics, build understanding of what barriers they have faced, and recognize what supports their learning” (663).

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UDL Math works to center students as problem solvers and teachers as designers. It is neither a passive process, nor an afterthought to curricular design - it is dynamic and must be rooted in an empathy for our students’ experiences.

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Mathematical Imagination & Embodied Cognition

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Mathematical Imagination & Embodied Cognition

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Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159-174. https://doi.org/10.1007/s10649-008-9150-4

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Themes emerged from analysis of mathematical imagination: (1) Juxtaposing displacements - Carlene’s reenactment of her imaginary activity with her body was not a direct replica, but rather a union of different spatial and temporal perspectives that enabled her to tell a story to make sense of her lived space-time. “…[T]his leads us to suggest that imagining resembles the creation of cubist paintings… Rather than animating scenes from a single point of view, such as an unedited regular photo shoot does, the imaginary activity appears to jump from one point of view to another, making the simultaneous sequential, and altering the temporal order of the events” (167-168), (2) Articulating necessary cases - the cases being disjoint and Carlene having live experience where certain noticings seemed obvious/given excluded other possibilities from the realm of imagination, enabling her to draw hard conclusions from her own imagination.

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“This paper is a preliminary analysis of some aspects of mathematical imagination in light of a 17-second episode in a 10th grade high school mathematics classroom” (159), specifically documenting and analyzing the utterances of a student, Carlene.

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(*) Imaging is proposed as “entertaining possibilities for action; entertaining… a state of readiness for the enactment of possible actions” (159), and is fully part of any perceptual and motor activity. Imagination is comprised of empirical possibilities (e.g., those based on empirical evidence) and pure possibilities (e.g., those immune to empirical evidence; those represented in fiction.) Pure possibilities entail symbol-use: perceptuo-motor-imaginary activity with symbols, which is partially why pure possibilities are immune to empirical evidence. (*) Embodied cognition posits cognition is not independent of the body, but rather conceptual understanding is interpreted by the sensory-motor system. For the purpose of interpreting imagination, all action is referred to as perceptuo-motor-imaginary activity.

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“In mathematics, many decisions are equally arbitrary, but some of them seem forced by their premises to be what they necessarily are. This contrast between literary fiction and mathematics does not imply that the former lacks coherence or ‘logic’ in the colloquial sense. The distinguishing trait we are trying to point out is that for an author writing literary fiction no outcome is strictly impossible; she excludes possibilities because they would make bad fiction, not because they could not be accommodated in principle” (160).

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Like imagination, mathematical imagination is also comprised of empirical possibilities and pure possibilities, but “possibilities held by mathematical imagination are regulated by, or embedded in, a sense of logical necessity in all its deductive and inductive modalities” (160). 

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Attending to Technology in Mathematics Classrooms

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Attending to Technology in Mathematics Classrooms

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Dick, L. K., McCulloch, A. W., & Lovett. J. N. (2021). When students use technology tools, what are you noticing?. Mathematics Teacher: Learning & Teaching, 114(4), 272-283. The National Council of Teachers of Mathematics. https://doi.org/10.5951/MTLT.2020.0285

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While digital objects are continually integrated and embedded into instructional practices, many frameworks for professional noticing in classrooms do not address student-technology interactions.

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Four components for professional noticing before deciding how to respond to students: (1) Attend to students’ spoken and written mathematical thinking, (2) Attend to students’ engagement with the technology, (3) Interpret students’ understanding from their engagement with the technology, (4) Interpret students’ understanding from their spoken and written mathematical thinking

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“In addition, when deciding how to respond to a student working in a technology-mediated learning environment, one must consider how to position the technology (or not) in one’s response to support the student in moving their mathematical thinking forward” (275).

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“When students are working with a technology tool, the ways they interact with the tool can offer insight into their mathematical thinking and learning. Noticing student thinking in such a context requires paying attention not only to what students say and write but also to the ways they engage with the technology” (274).

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Ma, J. Y. (2016). Designing disruptions for productive hybridity: The case of walking scale geometry. Journal of the Learning Sciences, 25(3), 335-371. https://doi.org/10.1080/10508406.2016.1180297

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Joseph, G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (3rd ed.). Princeton University Press. https://doi-org.proxy.library.vanderbilt.edu/10.1515/9781400836369

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Table of Contents: (1) The History of Mathematics: Alternative Perspectives, (2) Mathematics from Bones, Strings, and Standing Stones, (3) The Beginnings of Written Mathematics: Egypt, (4) The Beginnings of Written Mathematics: Mesopotamia, (5) Egyptian and Mesopotamian Mathematics: An Assessment, (6) Ancient Chinese Mathematics, (7) Special Topics in Chinese Mathematics, (8) Ancient Indian Mathematics, (9) Indian Mathematics: The Classical Period and After, (10) A Passage to Infinity: The Kerala Episode, (11) Prelude to Modern Mathematics: The Islamic Contribution

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Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment. International Journal of Educational Research, 51-52(1), 28-44. https://doi.org/10.1016/j.ijer.2011.12.009

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Math Curriculum Design (In)tensions & Curriculum Ergonomics

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Math Curriculum Design (In)tensions & Curriculum Ergonomics

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Dietiker, L., & Riling, M. (2018). Design (In)tensions in mathematics curriculum. International Journal of Educational Research, 92(1), 43-52. https://doi.org/10.1016/j.ijer.2018.09.001

# data[].attributes.custom_properties.BZy5eS0g
Example themes in tensions found in analyzed materials: (1) tensions raised by momentarily increasing student responsibility, (2) tensions raised by relying on students to have specific mathematical insights, (3) supporting teachers in maintaining student responsibility, (4) supporting student responsibility while increasing content specificity, and (5) protecting student responsibility by providing implementation options.

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Curriculum materials are designed with intention for how they will be implemented in a classroom and what thinking they will generate from students. In practice, teacher implementation of said materials and student perception of the materials are influenced by many factors which may not align with the curriculum designer’s original intentions.

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Dimensions of design tensions analyzed: (1) Who - Who is assumed to be responsible for the mathematical reasoning and the production of new mathematical knowledge in the class? {Student <-> Teacher}, (2) What - What is the nature of the mathematical content? {Unstructured and unspecified <-> Explicitly specific and detailed}, (3) How - How flexible or constrained is the implementation of the materials? {Rigidly prescribed <-> Teacher choice}

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“…[W]e generally believe these tensions are most problematic in terms of the who and the what dimensions. In contrast, we see the how dimension as an avenue in which curriculum designers can manipulate access for teachers to lessen anticipated tensions in the other two dimensions” (50).

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“…[I]nherent in design tension is a force felt by designers and users that motivates action by each to mitigate the difference. The less alignment there is between the visions of the users and designers, the greater the design tension and thus the greater need for relief” (44). This may cause either the designer to alter the materials or the teacher to adapt the materials themselves, their instruction in tandem with the materials, or their usage of the materials.

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Living Mathematx & Ecology of Knowledges

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Living Mathematx & Ecology of Knowledges

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Gutiérrez, R. (2017). Living Mathematx: Towards a vision for the future. Philosophy of Mathematics Education Journal, 32(1), 2-26.

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Historically speaking, mathematics has been used to do great harm to the planet and to other humans. While strides in mathematics education push for students to ‘solve real world problems,’ often times this manifests in the classrooms by reducing the complexity of real-world problems and multi-layered relationships between persons to mere abstractions (3).

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“Mathematx is a way of seeking, acknowledging, and creating patterns for the purpose of solving problems (e.g., survival) and experiencing joy” (12). The x is to be thought of as a variable that can represent anything - it shows movement. Mathematx is a verb, not a noun - and then, “Living mathematics means both that we live a version of mathematx as well as we are a living version of mathematx” (14).

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“Second, whereas mathematics tends to be thought of as a noun… mathematx is performance and, therefore, a verb. Just as identity is not something that you are, but rather something you do (Butler 1999), mathematx emphasizes the guiding principles and the process as opposed to the product” (13).

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“This, for me, has been one limitation of social justice mathematics, as it tends to assume we will keep intact as ‘classical’ what I refer to as ‘dominant’ mathematics rather than challenging whether that version or any single version should remain central” (3).

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Community-Based Model of Mathematical Creativity

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Community-Based Model of Mathematical Creativity

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Riling, M. (2020). Recognizing mathematics students as creative: Mathematical creativity as community-based and possibility-expanding. Journal of Humanistic Mathematics, 10(2),  6-39. https://doi.org/10.5642/jhummath.202002.04

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McTighe, J., & Wiggins, G. (2005). Understanding By Design (2nd Expanded ed.). Association for Supervision & Curriculum Development.

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McTighe, J., & Wiggins, G. (2013). What makes a question essential? Essential Questions: Opening Doors to Student Understanding (pp. 1-16). Association for Supervision & Curriculum Development.

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Burke, J. (2010). An intellectual rite of passage: Engaging students with essential questions. What’s the Big Idea?: Question-Driven Units to Motivate Reading, Writing, and Thinking. Heinemann.

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Berger, R., Rugen, L., & Woodfin, L. (2014). Models, critique, and descriptive feedback. Leaders of Their Own Learning: Transforming Schools Through Student-Engaged Assessment. John Wiley & Sons.

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Taylor, C., & Nolen, S. (2008). Assessment of valued performances. Classroom Assessment: Supporting Teaching and Learning in Real Classrooms (2nd ed., pp. 126-175). Pearson.

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Berger, R. (2019). A Culture of Quality: A Reflection on Practice (3rd ed.). EL Education Publications.

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Critical Response Process for Critique of Creative Processes

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Critical Response Process for Critique of Creative Processes

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Lerman, L., & Borstel, J. (2022). Critique is creative: The critical response process in theory and action. Wesleyan University Press.

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Tharp, T. (2003). The Creative Habit: Learn It And Use It For Life. Simon & Schuster.

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Humphrey, D. (1959). Design, part 4: The stage space. In B. Pollack (Ed.), The Art of Making Dances (pp. 72-90). Rinehart & Company.

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Humphrey, D. (1959). Design, part 3: The phrase. In B. Pollack (Ed.), The Art of Making Dances (pp. 66-71). Rinehart & Company.

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Humphrey, D. (1959). Design, part 1: Symmetry and asymmetry. In B. Pollack (Ed.), The Art of Making Dances (pp. 49-59). Rinehart & Company.

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Humphrey, D. (1959). The ingredients and the tools. In B. Pollack (Ed.), The Art of Making Dances (pp. 45-48). Rinehart & Company.

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Humphrey, D. (1959). The theme. In B. Pollack (Ed.), The Art of Making Dances (pp. 34-41). Rinehart & Company.

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“Land” vs. “land” & Pollution as Colonialism

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“Land” vs. “land” & Pollution as Colonialism

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Liboiron, M. (2021). Land, nature, resource, property. Pollution Is Colonialism (pp. 39-79). Duke University Press.

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Since the articulation of ‘assimilative capacity,’ or the theory that environments can sustain pollution below a specific threshold before harm occurs, bodies of water and other environments have been rethought of as vacuoles for disposing waste, giving people and companies the permission to pollute in the name of capitalism, and by extension, colonialism.

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Definitions: (1) Land (written with a capital L) - Land is a verb; “[Land] is about relations between the material aspects some people might think of as landscapes - water, soil, air, plants, stars - and histories, spirits, events, kinships, accountabilities, and other people that aren’t human” (43), (2) Nature (written with a capital N) - usually synonymous with land (lowercase L); “focus on only some aspects of relations, such as soil, water, animals, and plants, but not on human people, events, memories, spirits, or obligations” (48); Nature is separate from humans, (3) Resource (written with a capital R) - “unidirectional relations where aspects of land are useful to particular (here, settler and colonial) ends” (62).

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“When Nature becomes robust within limits and threshold theories of harm are dominant, land relations become managerial rather than reciprocal. In colonial understands of Nature, (certain) humans can protect, extend, augment, better, use, preserve, destroy, interrupt, and/or capitalize on robust-within-limits Nature. That is, Land becomes a Resource” (62).

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Land has been made pollutable by notion of property rights as property owners can deem pollution as the best purpose for the land. This whole chapter touches on the ties of colonialism to pollution.

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Edwards, C., Gandini, L., & Forman, G. (1998). The Hundred Languages of Children: The Reggio Emilia Approach - Advanced Reflections (2nd ed.). Ablex Publishing Corporation.

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Table of Contents: (I) Staring Points - (1) Introduction, (2) What Can We Learn from Reggio Emilia?, (II) Reggio Emilia’s Educators Describe Their Program: Interview With Lella Gandini - (3) History, Ideas, and Basic Philosophy, (4) The Community-Teacher Partnership in the Governance of the Schools, (5) Projected Curriculum Constructed Through Documentation - Progettazione, (6) The Role of the Pedagogista, (7) The Role of the Atelierista, (8) The Voice of Parents, (III) Reflections on the Interplay of Theory and Practice - (9) Educational and Caring Spaces, (10) Partner, Nurturer, and Guide: The Role of the Teacher, (11) Children With “Special Rights” in the Preprimary Schools and Infant-Toddler Centers of Reggio Emilia, (12) Curriculum Development in Reggio Emilia: A Long-Term Curriculum Project About Dinosaurs, (13) Negotiated Learning Through Design, Documentation, and Discourse, (14) Theory and Praxis in Reggio Emilia: They Know What They Are Doing, and Why, (15) Poppies and the Dance of World Making, (IV) The Extension of the Reggio Emilia Approach Into American Classrooms - (16) The Child in Community: Constraints From the Early Childhood Lore, (17) Existing Frameworks and New Ideas From Our Reggio Emilia Experience: Learning at a Lab School With 2- to 4-Year Old Children, (18) Bridge to Another Culture: The Journey of the Model Early Learning Center, (19) The City in the Snow: Applying the Multisymbolic Approach in Massachusetts, (20) Looking in the Mirror: A Reflection of Reggio Practice in Winnetka, (21) The Project Approach Framework for Teacher Education: A Case for Collaborative Learning and Reflective Practice, (22) Stories of Change from the St. Louis-Reggio Collaborative, (23) Reconsidering Early Childhood Education in the United States: Reflections From Our Encounters With Reggio Emilia, (V) Conclusion - (24) Conclusion

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Trawick-Smith, J. (2018). Early Childhood Development: A Multicultural Perspective (7th ed.). Pearson Education.

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Table of Contents: (1) Studying Early Childhood Development in a Diverse World, (2) Research and Assessment in Early Childhood Development, (3) Theories of Child Development, (4) Genetics, Prenatal Development, and Birth, (5) The Newborn, (6) Physical Growth of Infants and Toddlers, (7) Cognitive Development of Infants and Toddlers, (8) Infant and Toddler Language and Literacy, (9) Infant and Toddler Social and Emotional Development, (10) Preschool Physical and Motor Development, (11) Cognitive Development in the Preschool Years, (12) Symbolic Thought: Play, Language, and Literacy in the Preschool Years, (13) Social and Emotional Development of Preschoolers, (14) Physical Growth and Motor Development in the Primary Years, (15) Cognitive Development in the Primary Years, (16) Language and Literacy in the Primary Years, (17) Social and Emotional Development in the Primary Years, (18) Parents, Families, and Children: A Multicultural Perspective

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Lillard, A. S. (2017). Montessori: The Science Behind the Genius (3rd ed.). Oxford University Press.

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Table of Contents: (1) An Answer to the Crisis in Education, (2) The Impact of Movement on Learning and Cognition, (3) Choice and Perceived Control, (4) Executive Function, (5) Interest in Human Learning, (6) Extrinsic Rewards and Motivation, (7) Learning From Peers, (8) Meaningful Contexts for Learning, (9) Adult Interaction Styles and Child Outcomes, (10) Order in Environment and Mind, (11) Recent Research on Montessori Education, (12) Education for Children

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Convention on the Rights of the Child, November 20, 1989, https://www.ohchr.org/en/instruments-mechanisms/instruments/convention-rights-child

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Wright, W. E. (2019). Language. Foundations for Teaching English Language Learners: Research, Theory, Policy, and Practice (3rd ed., pp. 29-49). Brookes Publishing.

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Crosson, A. C., Tapu C., & McKeown M. G. (2022). Think like a linguist: Leveraging multilingual students’ expertise about language. Journal of Adolescent & Adult Literacy, 65(6), 481–493. https://doi.org/10.1002/jaal.1226

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Cioè-Peña, M. (2022). TrUDL, A path to full inclusion: The intersectional possibilities of translanguaging and universal design for learning. TESOL Quarterly, 56(2), 799-812. https://doi.org/10.1002/tesq.3074

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The number of students who are classified as both multilingual learners and as students with disabilities is increasing, and while conversations about the disproportionality of these classifications are important, less research has been devoted to the needs of these students. “…[N]early three-quarter million [EBLAD] students encountered segregation through placement in (1) specialized bilingual SpEd programs which deny them access to their enabled peers of diverse linguistic, racial, and ethnic backgrounds, or (2) through placement in monolingual integrated settings which grant them access to enabled peers but at the detriment of their multilingual development which can, and often does, result in social and communal exclusion beyond school” (801).

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Core Elements of TrUDL: (1) Multilingual and multimodal teacher-employed strategies focused on increasing student accessibility and comprehension, (2) Multilingual and multimodal student practices focused on flexibility and increasing student output, and (3) Culturally responsive practices that relate to student interest and identity in order to activate, and/or increase, engagement (805).

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“Additionally, unlike a placement in bilingual special education… TrUDL is a pedagogical practice that can be implemented in any classroom” (809-810).

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Translanguaging and UDL intersect in both pedagogy and theory and by capitalizing on this common ground, we can further support multilingual learners with disabilities.

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Keil, F. C. (2022). Wonder: Childhood and the Lifelong Love of Science. The MIT Press.

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Table of Contents: (I) The Cognitive Gifts of Childhood - (1) The Puzzle and the Promise, (2) Early Exploration and Discovery, (3) Working with Others, (4) The Mechanistic Mind, (II) The Big Sleep: Weakening Wonder - (5) - Developmental Disconnects, (6) Motivational Muddles, (III) Disengagement and Its Discontents - (7) Seductive Detours, (8) It’s a Wonderless Life, (9) Great Reawakenings

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Noddings, N. (2013). Why care about caring? Caring: A Relational Approach to Ethics and Moral Education (pp. 7-29). University of California Press.

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Larios, R., & Zetlin, A. (2022). Bilingual and monolingual parents’ counterstories of the Individualized Education Program (IEP) meeting. Urban Education, 57(7), 1207-1229. https://doi.org/10.1177/0042085918804003

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Espinoza, M. L., & Vossoughi, S. (2014). Perceiving learning anew: Social interaction, dignity, and educational rights. Harvard Educational Review, 84(3), 285-313.

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Alameda-Lawson, T., & Lawson, M. A. (2019). Ecologies of collective parent engagement in urban education. Urban Education, 54(8), 1085-1120. https://doi.org/10.1177/0042085916636654

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McDermott, R., & Varenne, H. (1995). Culture “as” disability. Anthropology & Education Quarterly, 26(3), 324-348.

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Ewing, E. L., Davis, B., & Guz, S. (2023). “I hope I make it”: Alternative school students’ attendance and the need for an expanded accountability. Urban Education, 58(6), 1383-1414. https://doi.org/10.1177/00420859211027618

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Centering Counterstories in Non-Dominant Family Knowledge

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Centering Counterstories in Non-Dominant Family Knowledge

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Barajas-López, F., & Ishimaru, A. M. (2020). “Darles el lugar”: A place for nondominant family knowing in educational equity. Urban Education, 55(1), 38-65. https://doi.org/10.1177/0042085916652179

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Generett, G. G., & Olson, A. M. (2020). The stories we tell: How merit narratives undermine success for urban youth. Urban Education, 55(3), 394-423. https://doi.org/10.1177/0042085918817342

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Giroux, H. A. (2001). Schooling & the politics of the hidden curriculum. Theory and Resistance in Education: Towards a Pedagogy for the Opposition (2nd ed., pp. 42-71). Praeger.

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Flores, N., & Rosa, J. (2015). Undoing appropriateness: Raciolinguistic ideologies and language diversity in education. Harvard Educational Review, 85(2), 149-171.

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Cherry-McDaniel, M. (2018). Skinfolk ain’t always kinfolk: The dangers of assuming and assigning inherent cultural responsiveness to teachers of color. Educational Studies, 55(2), 241–251. https://doi.org/10.1080/00131946.2018.1500912

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Lamboy, L., Taylor, A., & Thompson, W. (2020). Paternalistic aims and (mis)attributions of agency: What the over-punishment of Black girls in US classrooms teaches us about just school discipline. Theory and Research in Education, 18(1), 59-77. https://doi.org/10.1177/1477878520912510

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Chatzidakis, A., Hakim, J., Littler, J., Rottenberg, C., & Segal, L. (2020). Introduction: Carelessness reigns. The Care Manifesto (pp. 1-7). Verso.

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Scott, D. M. (1997). The resurgence of damage imagery: Representations of the Black psyche in an age of conservative reform, 1981-1996. Contempt & Pity (pp. 187-202). University of North Carolina Press.

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“Problem Child” vs. “Child with a Problem”

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“Problem Child” vs. “Child with a Problem”

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Freidus, A. (2020). “Problem children” and “children with problems”: Discipline and innocence in a gentrifying elementary school. Harvard Educational Review, 90(4), 550-572.

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Patel, L. (2019). Fugitive practices: Learning in a settler colony. Educational Studies, 55(3), 253–261. https://doi.org/10.1080/00131946.2019.1605368

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Moak, D. S. (2022). The politics of the federal education state: Faith in education and the turn toward punitiveness. From the New Deal to the War on Schools: Race, Inequality, and the Rise of the Punitive Education State (pp. 1-16). University of North Carolina Press.

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Lorde, A. (1984). Age, race, class, and sex: Women redefining difference. Sister Outsider (pp. 114-123). The Crossing Press.

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Dewey, J. (2004). Education as growth. Democracy and Education (pp. 63-78). Dover Publications.

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Cohen, D. K. (2005). Professions of human improvement: Predicaments of teaching. In M. Nisan & O. Schremer (Eds.), Educational Deliberations: Studies in Education Devoted to Shlomo (Seymour) Fox (pp. 278-294). Jerusalem: Keter Publishers.

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Biesta, G. (2009). Good education in an age of measurement: On the need to reconnect with the question of purpose in education. Educational Assessment, Evaluation, and Accountability, 21(1), 33-46. https://doi.org/10.1007/s11092-008-9064-9

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National Research Council. (2001). The strands of mathematical proficiency. Adding It Up: Helping Children Learn Mathematics (pp. 115-155). Washington, DC: The National Academies Press. https://doi.org/10.17226/9822

# data[].attributes.custom_properties.EAP2vV9Q
The 20th century was characterized by revolving themes in mathematics education, with foci shifting almost every decade. The themes have centered on computation, understanding of big ideas, tending to accuracy and precision with agility, and the communication of mathematical ideas; however, no single one of these emphases entirely encapsulates what we mean by mathematical competence.

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5 components (strands) of mathematical proficiency: (1) conceptual understanding - comprehension of mathematical concepts, operations, and relations, (2) procedural fluency - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately, (3) strategic competence - ability to formulate, represent, and solve mathematical problems, (4) adaptive reasoning - capacity for logical thought, reflection, explanation, and justification, and (5) productive disposition - habitual inclination to see mathematics as sensible, useful, and worthwhile, couple with a belief in diligence and one’s own efficacy. These strands are interwoven, but not all or nothing, and develop over time.

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“…[D]uring the past 25 years mathematics instruction in U.S. schools has not sufficiently developed mathematical proficiency in the sense we have defined it. It has developed some procedural fluency, but it clearly has not helped students develop the other strands very far, not has it helped them connect the strands. Consequently, all strands have suffered” (145).

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“Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we believe is necessary for anyone to learn mathematics successfully” (116).

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Teaching Mathematics for Social Justice (TMSF)

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Teaching Mathematics for Social Justice (TMSF)

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Berry III, R. Q., Conway IV, B. M., Lawler, B. R., & Staley, J. W. (2020). What is social justice, and why does it matter in teaching mathematics? High School Mathematics Lessons: To Explore, Understand, and Respond to Social Injustice (pp. 16-26). Corwin Mathematics & The National Council of Teachers of Mathematics.

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“When teachers focus only on teaching mathematics, disconnects can occur between the content and students’ passions and lived realities” (23). 

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Standards-Based Math Instruction (SBMI) > Complex Instruction (CI) > Culturally Relevant Pedagogy (CRP) > Critical Math Education (CME) > Teaching Math for Social Justice (TMSJ)

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“Thus, TMSJ has two goals: freedom from oppression through the development of mathematical literacy, and the freedom to act upon and impact the world through personal and social transformation. The first is a mathematics goal, and the second, a social justice goal” (22).

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“So why bring social injustices into the mathematics classroom? Besides offering critical issues for students to examine and serving as context for mathematics development and investigation… [t]eaching mathematics for social justice helps to build an informed society, connect mathematics with students’ cultural and community histories, empower students to confront and solve real-world challenges they face, and help students learn to value mathematics as a tool for social change” (23).

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Gutiérrez, R. (2018). The need to rehumanize mathematics. Annual Perspectives in Mathematics: Rehumanizing Mathematics for Black, Indigenous, and Latinx Students (pp. 1-10). The National Council of Teachers of Mathematics.

# data[].attributes.custom_properties.EAP2vV9Q
Mathematics education is riddled with dehumanizing practices including, but not limited to: 1) measuring categorizing bodies, 2) evaluation that does not honor complexity, context, or individuals’ own goals, 3) being asked to leave one’s identity at the door, 4) rule following as opposed to rule breaking or creation, 5) speed valued over reflection, and 5) separation of mathematical practice from politics/values/ethics.

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Dimensions of rehumanizing mathematics practices for students who are Latinx, Black, and Indigenous: (1) participation/positioning - recognizing hierarchies in classrooms and society and shifting the role of authority from teacher/text to other students, (2) culture/histories - acknowledges students’ funds of knowledge, algorithms from other countries, the history of mathematics, and ethnomathematics, (3) windows/mirrors - students can be taught to appreciate the view of others, (4) living practice - underscores mathematics as something in motion, (5) creation - students can invent new algorithms/forms of doing math that are consistent with their own values, (6) broadening math - doing math extends beyond classroom drilled algebra, calculus, number sense, symbolic representation, and favoring the general case, (7) body/emotions - depart from purely logical perspective of math and invite students to draw on parts of themselves (e.g., voice, vision, touch, intuition); attending to emotions, in particular, conjuring feelings of joy and, (8) ownership - students view mathematics as something one does for oneself, not just for others.

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“Many teachers have been trained to anticipate the misconceptions that students have so that they can address them in their lessons. Yet, students don’t have misconceptions. They have conceptions. And those conceptions make sense for them, until they encounter something that no longer works. They are only ‘misconceptions’ when we begin with the expectation that others need to come to our way of thinking or viewing the world” (2).

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“Beyond being seen as a legitimate participant (a “doer” of mathematics), a student should be able to feel whole as a person - to draw upon all of their cultural and linguistic resources - while participating in school mathematics. That is, every student should be provided with windows and mirrors (Style 1996) onto the world through mathematics (Gutiérrez 2007); they should see aspects of themselves reflected back (mirror) as well as obtain views of new worlds outside of their own (windows)” (1).

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Larnell, G. V., Bullock, E. C., & Jett, C. C. (2016). Rethinking teaching and learning mathematics for social justice from a critical race perspective. Journal of Education, 196(1), 19-29. https://doi.org/10.1177/002205741619600104

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1) TMSJ derives much of its epistemology and methodology from a social-justice-as-critical-consciousness perspective, i.e., teaching math for civic responsibility, and 2) TMSJ scholarship is also based on social-justice-as-instrumental-access perspective, i.e., teaching mathematics as a civil right.

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“[S]ome have questioned how equity-oriented work in mathematics education is communalized and the extent to which mathematics education research, practitioner, and policy literatures evince erasure (or “e(race)sure”) that effectively silences racialized discourses” (19).

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“Our intent in concluding the article with using CRT as a lens for analyzing current research is not to evaluate the contributions as each is significant in its own right. Rather, the analyses represent examples of CRT’s power to expose TLMSJ’s ‘blank and blind spots’ with respect to race, racism, and racialization” (27).

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“[E]mploy[ing] a race-critical perspective on the growing social-justice discourse as one way to shed light on the role(s) and operation(s) of race and racial injustice within that discourse” (19).

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Horn, I. S. (2013). Mathematical competence and status: What “being smart” means. Strength in Numbers: Collaborative Learning in Secondary Mathematics (pp. 19-34). The National Council of Teachers of Mathematics.

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“By the time they are in secondary school, students enter their mathematics classes with strong ideas about who they and their peers are as mathematics learners. They can tell you who is smart and who is not. They base these judgements on earlier school achievement, as well as on categories such as race, class, popularity, and gender. These assessments play out in the classroom. Some students’ contributions are sought out and heard, whereas others’ contributions are ignored. This imbalance obstructs productive mathematical conversations because an argument’s valuation comes from who is speaking and not what is being said” (19).

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(1) Noticing Status - looking for participation patterns, listening for ideas that get dropped during conversations, body language, distributions of materials/resources, inflated talk about self/others; (2) Status Interventions - establishing & maintaining norms, multiple-ability treatment; (3) Assigning Competence - help students see the strengths in their peers by praising students “being smart” where the praise is public, specific to the task, and intellectually meaningful.

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“If students arrive in the classroom with expectations about whose contributions are worth listening to, they will act accordingly” (24).

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“Disentangling achievement and ability may sound reasonable, but we need a new model for thinking about students we teach. Elizabeth Cohen’s (1994) work on complex instruction frames these issues around status, a concept that clarifies the conflation of achievement and ability. Status gives teachers rom to analyze this problem and respond through their instruction. In this context, we will use the following definition of status: Status is the perception of students’ academic capability and social desirability” (21). 

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National Council of Teachers of Mathematics. (2014). Essential Elements: Access and Equity. Principles to Actions: Ensuring Mathematical Success for All (pp. 59-69). The National Council of Teachers of Mathematics.

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“Equity in school mathematics outcomes is often conflated with equality of inputs. Providing all students the same curricular materials, the same methods of teaching, the same amount of instructional time, and the same school-based supports for learning is different from ensuring that all students, regardless of background characteristics, have the same likelihood of achieving meaningful outcomes” (60, as cited in Gutiérrez 2013).

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Includes examples of how to reframe unproductive beliefs about access and equity in mathematics to have more productive dispositions.

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“Achieving equity with respect to student learning outcomes will require that educators at all levels operate with the belief that all students can learn” (64).

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“Our vision of access and equity requires being responsive to students’ backgrounds, experiences, and knowledge when designing, implementing, and assessing the effectiveness of a mathematics program” (60).

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Sinclair, N. (2006). Developing a Mathematical Aesthetic Lens. Mathematics and Beauty: Aesthetic Approaches to Teaching Children (pp. 39-57). Teachers College Press.

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Butler, J. (2002). Subjects of sex/ gender/ desire. Gender Trouble: Tenth Anniversary Edition (pp. 3-44). Routledge.

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Foucault, M. (1978). The History of Sexuality (Hurley, R., Trans.). New York: Pantheon Books, Random House (Original work published 1976).

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Table of Contents: (1) We “Other Victorians,” (2) The Repressive Hypothesis - (2.1) The Incitement to Discourse, (2.2) The Perverse Implantation, (3) Scientia Sexualis, (4) The Deployment of Sexuality - (4.1) Objective, (4.2) Method, (4.3) Domain, (4.4) Periodization, (5) Right of Death and Power over Life

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Sedgwick, E. K. (1990). Epistemology of the Closet. University of California Press.

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Self-Actualization & Teacher Lens Development

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Self-Actualization & Teacher Lens Development

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Alspup, J. (2004) Am I a teacher? Exploring the development of professional identity. Language Arts Journal of Michigan, 20(1), 35-39. https://doi.org/10.9707/2168-149X.1245

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“We can make the induction phase of the new teacher easier by giving assignments or facilitating experiences that encourage the expression of various genres of integrative teacher identity discourse, including narratives and metaphors, resulting in experimentation and exploration of ideological, affective, and corporeal borderlands” (37).

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How can teacher educators better prepare pre-service teacher candidates for fieldwork?

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Examples of such assignments include: (1) candidates provide narrative or anecdotal evidence to support pedagogical decisions, (2) candidates compose reflective writings or teaching journal entries, (3) candidates role-play classroom scenarios or videotape themselves teaching and reflecting on the experience, and (4) encourage candidates to create visual/text-based metaphors that re-conceptualize abstract ideas and philosophies as concrete images.

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“In my study, if such self-actualization had not occurred, the pre-service teachers chose not to enter the profession immediately, and if they did take a teaching job, they were likely to experience future professional identity crises, just as I did so many years ago” (37).

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“I believe the teacher education and/or mentor teacher can intentionally use these genres of discourse to help new teachers develop workable and complex teacher identities” (36).

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Zepeda, C. D., Richey, J. E., Ronevich, P., & Nokes-Malach, T. J. (2015). Direct instruction of metacognition benefits adolescent science learning, transfer, and motivation: An in vivo study. Journal of Educational Psychology, 107(4), 954–970. https://doi.org/10.1037/edu0000022

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Results revealed that those who received the metacognitive instruction and training were less biased when making metacognitive judgments, endorsed higher levels of motivation after instruction, performed better on a conceptual physics test, and performed better on a novel self-guided learning activity. This study demonstrates that metacognitive instruction lead to better self-regulated learning outcomes during adolescence, a period in which students’ academic achievement and motivation often decline.

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Prior studies have not tested whether an instructional intervention aimed at improving metacognitive skills results in changes to student metacognition, motivation, learning, and future learning in the classroom.

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“Since metacognitive interventions teach students different skills to improve their understanding, students who learn about metacognition might be more likely to adopt and endorse mastery-approach goals to create consistency between their goals, knowledge, and behaviors” (955).

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“…metacognitive skills (e.g., planning, monitoring, and evaluation) and student motivation (e.g., beliefs, goals, and dispositions) interact to determine learning outcomes” (954).

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Spencer, M. B., Noll, E., Stoltzfus, J., & Harpalani, V. (2001). Identity and school adjustment: Revisiting the “acting white” assumption. Educational Psychologist, 36(1), 21-30.

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"Specifically, high-achieving African American adolescents are not only failing to identify with acting White values, but more than likely, have a better understanding of the irrelevance of the comparison for Blacks and in what seems to be the term’s lack of meaning, even for Whites" (29).

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There is an opportunity and achievement gap between White and African American students. Is the traditional solution (adopting a eurocentric approach) a deficit-oriented approach? How does this impact normative development (peer relationships, self-esteem/self-efficacy)?   

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"Accordingly, a lack of understanding of cultural contexts leads to a misinterpretation of minority youth behavior and development (23)."

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The central argument is that the "Acting White" assumption is not true and based on a deficit perspective that overlooks the cultural identity development of African American students. African American students can have high self-esteem, achievement goals, and a strong Afrocentric cultural identity while performing well academically, contrary to the traditional assumption.

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Sirin, S. R., & Fine, M. (2007). Hyphenated selves: Muslim American youth negotiating identities on the fault lines of global conflict. Applied Development Science, 11(3), 151-163.

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"Living with the haunting ghosts of ‘‘terrorist’’ looming around him, he, like so many young Muslim men, feels swallowed by a representation he can’t actively resist, lest he embody the hegemonic trope—young Muslim man filled with rage."

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Navigating identity as a hyphenated individual (Muslim-American) is hard enough. After 9-11, however, resolving this tension became signficiantly harder as Muslim-Americans faced marginalization and isolation. This study focuses on the specific effects. 

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"When one’s social identity is fiercely contested by the dominant discourse either through formal institutions, social relationships, and/or the media, one of the first places we can witness psychological, social and political fallout is in the lives of young people."

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Identity-formation ought to be studied in relation to history, government, society, etc. and is also very gendered.

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Mangual Figueroa, A. (2017). Speech or silence: Undocumented students’ decisions to disclose or disguise their citizenship status in school. American Educational Research Journal, 54(3), 485-523.

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* Students’ legal citizenship status affects what they feel they can disclose about themselves and their families. * Curricular material can support students in developing more robust reasoning about citizenship and its relation to broader historical and political contexts. * Classroom pedagogy may have the unintended consequence of prompting student talk about citizenship or causing them to feel silenced. * Closely examining student narratives may tell us more about the risks that students face and the resilience that they exhibit than we might notice at first glance.

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Even though undocumented students have the legal right to public education, the nature of the law makes students feel like they have to keep that secret for their famillies which affects their participation. Teachers also feel as though they can and should not speak to students about their immigration process/experience. 

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"There silences can also be eloquent, indicating when children perceive it is too risky to publicize their immigration status" (489). "These two kinds of citizenship—legal and cultural—are independent even though they can overlap" (490). 

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There is a difference in the type of silences that students employ when deciding how to converse about their immigration status: audible silences are when the class is not talking whereas inaudible silences are a strategic refusal to engage in a topic that the student perceives could put them at risk. Children are very deliberate when and how they speak about citizenship, which is influenced by the pedagogy of their classroom. 

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Queerly & Culturally Responsive Pedagogy

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Queerly & Culturally Responsive Pedagogy

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Brockenbrough, E. (2016). Becoming queerly responsive: Culturally responsive pedagogy for Black and Latino urban queer youth. Urban Education, 51(2), 170-196.

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1) Organizational mission has a component that specifically addresses the experiences of queer youth of color rather than solely focusing on the needs of all queer youth. 2) Comprehensive sex ed and workshops that address specific needs of the community. 3) The embracing of House Ball culture which is historically rooted in black, latino, urban, queer culture.  "These campaigns seek a shift in schools from abstinence-focused curricula to a sex-positive frame- work that empowers all students" (190).  

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Recent attention to "the plight of queer students... has also fallen short of addressing a range of dilemmas facing urban queer youth of color, whose needs extend beyond protection from homophobic victimization" (p 170). 

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"Queers of color, like any cultural group in the United States, are not homogeneous, but they do share the unique chal- lenge of finding space and voice in a society where their multifaceted other- ness requires ways of being and knowing that resist systemic assaults on their humanity" (174). 

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Queer youth of color need specific places for queer youth of color. The spaces need to be informed by their identities and actively doing something to support them. 

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Baines, A. D. (2012). Positioning, strategizing, and charming: How students with autism construct identities in relation to disability. Disability & Society, 27(4), 547-561.

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(1) Students with autism make a deliberate effort to be perceived positively. (2) The label "autistic" is seen as negative, and those who are labelled as "autisitc" try to distance themselves from others. (3) Experiences shape identities in complex and dynamic ways. (4) Teachers, researchers, and practitioners must find ways to disrupt patterns of interaction which position students with disabilities such as autism in ways that limit their opportunities to fully participate.

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There is an assumption placed upon students with learning disabilties (the article highights autism) that they are not capable of devloping their identity in similar ways to neurotypical students. The same is assumed for acaemic identities and interests. It is assumed that they are not capable of learning and developing academic interests. 

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"To explore how youth with disabilities develop academic identities, we considered how their access to rights and opportunities were shaped by their positioning in different contexts" (559).

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As "classroom teachers" we will not equitabily teach students with disabilties if we are also not adjusting our positioning as their teacher. Students with learning disabilities need to be met at a space that is equitable to them, and each domain teacher needs to meet them there. 