# UTPA STEM/CBI Courses/Mathematics Teaching and Learning/Mathematical Knowledge for Equitable Teaching

Course Title: Mathematics Teaching and Learning
Lecture Topic: Mathematical Knowledge for Equitable Teaching
Instructor: Dr. Aaron T Wilson
Institution: University of Texas Rio Grande Valley

## Backwards Design

Course Objectives
* Primary Objectives- By end of this course project students will be able to:
** Operationalize instances of knowledge for equitable mathematics teaching
** State elements of teachers’ knowledge that are central to equitable mathematics teaching and situate the elements within contemporary research-based frameworks of math teachers’ knowledge
* Sub Objectives- The objectives will require that students be able to:
** Design math teaching vignettes that are:
*** self-contained
*** rich in authentic teaching context and in mathematics content
*** an intertwining of content and teaching theory
* Difficulties- Students may have difficulty:
** Identifying the intersection of mathematical knowledge for teaching (MKT) and theories that prescribe instructional moves appropriate for bilingual students and of teaching for social justice
** Connecting teaching theory with specific mathematics topics
** Writing valid choices for teaching moves given specific context and content elements
* Real-World Contexts- Students in this course are all practicing or pre-service mathematics teachers and, as such, they must:
** Apprehend the complexity of teaching mathematics in linguistically and culturally diverse classrooms
** Be prepared to generate multiple instructional and assessment strategies suitable for such classrooms.
** Be able to state specific difficulties, misunderstandings, background knowledge and issues that will arise in such classrooms.
Model of Knowledge
* Concept Map
** Teaching mathematics for social justice
* Content Priorities
** Enduring Understanding
*** Mathematics students in US schools who are English Language Learners (ELLs), on the one hand, face linguistic challenges in apprehending instruction and communicating mathematics and, on the other hand, they carry funds of background mathematics knowledge as well as linguistic knowledge of their first language, both of which serve as significant resources for them to grasp instruction and communicate about mathematics.
*** Mathematics classrooms are an appropriate place and mathematics is an appropriate topic through which teachers can lead students to question institutional structures that perpetuate inequities in their students’ societies.
** Important to Do and Know
*** Operationalizing instances of equitable mathematics teaching in this way has connections with the field of psychometrics and psychometric methods are useful for this work.
*** Minimizing latent variables by over-specifying the context is key to effective production of valuable vignettes  ***
** Worth Being Familiar with
*** Teacher licensure exams are conceptually related to this challenge.
*** The profession of teaching may benefit (elevation of status) by implementation of more rigorous practice-based examination measures.
Assessment of Learning
* Formative Assessment
** The “Generate Ideas” component of the Legacy Cycle provides one source of data for Formative Assessment
*** A) The Instructor must attend to student productions at this stage
** The “Research and Revise” component of the Legacy Cycle provides the second Formative Assessment data source. Student groups should submit measurement items to the instructor for review.
* Summative Assessment
** An Exam is given in the “Test Your Mettle” component of the Legacy Cycle (see below).
** Student groups share products in the “Go Public” component of the Legacy Cycle (see below).

## Legacy Cycle

OBJECTIVE
By the end of this course project, students will be able to:
* Operationalize instances of knowledge for equitable mathematics teaching
* State elements of teachers’ knowledge that are central to equitable mathematics teaching and situate the elements within contemporary research-based frameworks of math teachers’ knowledge
The objectives will require that students be able to:
* Design math teaching vignettes that are:
** self-contained
** rich in authentic teaching context and in mathematics content
** an intertwining of content and teaching theory
THE CHALLENGE You and your colleagues in the Psychometrics Department of the Texas Education Agency (TEA) have been tasked with drafting the next generation of teacher licensure exams. Policy-makers are calling for exams that are more reflective of actual content and actual tasks of teaching mathematics. In particular, your group will work closely with the Equity Taskforce to ensure that the measurement items respond to two particular needs: teaching math to students who are English Language Learners (ELLs) and teaching mathematics for social justice. Working with a partner, this project will require you, to create three multiple-choice test items designed to measure whether real practicing teachers do or do not know how to A) teach specific mathematical concepts from high school mathematics high school students who are English Learners and also B) teach mathematics for social justice.
GENERATE IDEAS
In class, partners will brainstorm ideas with the goals of:
* Arriving at a shared understanding of the important definitions related to this task (ELLs, social justice, math teacher knowledge, test items, etc.)
* Identifying specific mathematics content elements needed by high school teachers
* Envisioning teaching scenarios (actual or hypothetical) in which to situate the tasks
MULTIPLE PERSPECTIVES
This stage last the duration of at least one week, but usually two or three. Partners must read the research literature concerning A) math teachers’ knowledge, B) ELLs in the math classroom, C) teaching for social justice. Their own brainstormed ideas from the previous stage is an invaluable starting point, but in no case has their mere prior knowledge been sufficient to complete challenge task. Their understanding must be enriched by learning from research.
RESEARCH & REVISE
After brainstorming and reading research, students are ready to craft measurement items. This is the first time in the Legacy Cycle that students will directly address the challenge. In this stage partner groups must: Develop three multiple choice test items that have the following properties:
* Originality and creativity: the items were written by you and your partner and involve a genuine educational context that you have experienced, witnessed or imagined yourself, OR one that you have heard/read about and have translated into a test item
* Mathematicality: the items clearly involve a specific mathematics topic learned in high school
* ELL/Social Justice Issue: the items require respondents to know something about teaching ELLs or teaching math for social justice. That is, correctly responding to the item requires knowledge either of difficulties faced by ELLs, competencies possessed by ELLs, of strategies for teaching ELLs, or knowledge used by teachers to engage math students in social justice challenges.
* Format: the items are multiple-choice. Each item begins with a question stem followed by three good distractors and one correct answer.
TEST YOUR METTLE
After student groups have generated their psychometric measurement items, they are given a test composed of validated MKT items and also of high quality items produced by past students of this project. MKT items may be selected from released LMT items found here: http://hub.mspnet.org//index.cfm/17925?
This test is summative. The instructor should generate an answer form in which respondents must state the correct answer to the test item and also correctly state the domain of knowledge that the test items intends to measure.
GO PUBLIC
After all groups have submitted their test items, the instructor will generate a combined test including all student-written items. On a particular day, the class will orally “take the test” that was written by student groups. Partner groups present their item and defend it as a measurement tool for a specific domain of knowledge, after all students have a chance to attempt to provide the correct answer. This stage is a lot of Fun! Students enjoy Going Public with their test items.

## Test Your Mettle Quiz

The instructor selects MKT test items and ELL/Social Justice items for a brief test. For example:
* First MKT Test Item Ms. Hurlburt was teaching a lesson on solving problems with an inequality in them. She assigned the following problem. x < 9 Marcie solved this problem by reversing the inequality sign when dividing by – 1, so that x > – 9. Another student asked why one reverses the inequality when dividing by a negative number; Ms. Hurlburt asked the other students to explain. Which student gave the best explanation of why this method works? (Mark ONE answer.) State the correct answer: _________________ State the knowledge domain for which this item serves as an indicator: ________________.
** a) Because the opposite of x is less than 9.
** b) Because to solve this, you add a positive x to both sides of the inequality.
** c) Because –x < 9 cannot be graphed on a number line, we divide by the negative sign and reverse the inequality.
** d) Because this method is a shortcut for moving both the x and 9 across the inequality. This gives the same answer as Marcie’s, but in different form: –9 < x.
* Second MKT/ELL Item Mr. Canchola has asked his students to write the solution to the following problem: 12 billion minus 7 billion Which INCORRECT solution would most likely have been made by Fernando, an English Language Learner from Mexico City? (Mark ONE answer.) State the correct answer: _________________ State the knowledge domain for which this item serves as an indicator: ________________
** a) 5,000
** b) 5,000,000
** c) 5,000,000,000
** d) 5,000,000,000,000