Problem Solving for Middle School Teachers

Initial Publication Date: July 5, 2023
Author Profile

Diana Cheng, Towson University

Location: Maryland

Abstract

The primary goal of this class is for the students to creatively engage in mathematical investigations which provide the opportunity to integrate and apply previously learned mathematical knowledge to non-routine problems. We examine a variety of strategies and skills that can be used to explore the fascinating world of mathematical problem solving. The course aims to cultivate students' habits of mind with respect to dissecting challenging mathematical problems for the purpose of solving them, as well as extending them by posing new related problems. Students are expected to justify their solution methods through both written and oral presentations, demonstrating knowledge of reasoning and proof, mathematical communication, and mathematical connections. Each student is expected to take an active role in all parts of the class. The assignments and activities are designed to enhance student understanding, appreciation, and practical experience of exploring and solving problems.

Student Goals

Student Goal 1: Develop a key question that can be solved using middle school level mathematics, and identify and select relevant quantities to consider in solving this question.

Student Goal 2: Lead peers to formulate a mathematical model to represent the solution process while investigating the key question.

Student Goal 3: Lead their peers to interpret the results of the mathematical investigation and analyze classmates' work.

Research Goals

Research Goal 1: Use social justice as a context for mathematical modeling.

Research Goal 2: Conduct educational action research on a micro-teaching lesson.

Context

The three-credit, cross-listed undergraduate (300 level, undergraduate students preparing to become teachers, primarily in middle schools) / graduate course (500 level, in-service teachers) has a wide range of enrollment. In pre-pandemic years, I had up to 20 students in this course with majority undergraduate students. More recently, the class is about half graduate students and half undergraduate students (around 5 of each). The pre-requisite for the undergraduate course was successful completion (grade of C or better) in Calculus 1, which is called MATH 273 at our institution. Calculus 1 is also a required course for all pre-service middle school teachers who have selected mathematics as a concentration.

Target Audience:Major, Upper Division
CURE Duration:Half a term

CURE Design

The students work in small groups to design and implement a mathematical modeling lesson (as of Spring 2022, I have been asking students to use a modeling lesson that relates to the context of social justice). They are allowed to adapt a lesson from an existing source (using a reference written by another teacher ensures that the lesson context will be one that applies to middle school students), and make the lesson relevant for their students by using their school data and more updated information than might have been available to the textbook authors. For example, to adapt a lesson on the Pythagorean theorem as it relates to wheelchair ramps (Gutstein & Peterson, 2013, pp. 136-137), the students took photographs of the various ramps at their schools and local businesses so that their middle and high school students could evaluate whether these ramps meet accessibility recommendations.

The course does not have a requirement to try out the final project (an originally designed mathematical modeling lesson) with middle school students. However, I am offering to interested undergraduate students the opportunity try out their lessons in after-school sessions (by partnering with in-service teachers in the school district in which our university is situated) if they wish to do so, outside the scope of their course responsibilities.

Core Competencies:
Nature of Research:

Tasks that Align Student and Research Goals

Research Goals →
Student Goals ↓

Research Goal 1: Use real-life scenarios as contexts for mathematical modeling.

Research Goal 2: Conduct educational action research on a micro-teaching lesson.


Student Goal 1: Develop a key question that can be solved using middle school level mathematics, and identify and select relevant quantities to consider in solving this question.

- Complete existing model-eliciting activities (MEAs) published in teacher practitioner journals or books to gain a better sense of the types of problems that can be solved with middle school mathematics in everyday contexts (e.g., Hirsch & Roth McDuffie, 2016; Gutstein & Peterson, 2013).

- Consider the guiding questions for developing mathematics lessons that matter to students written by Berry et. al. (2020, p. 28).

- Research real-life problems through web-based resources

- Define  parameters / constraints based on Common Core State Standards (2010).

- In consultation with in-service teachers and a graduate assistant, refine  educational activities, including considerations about how the lesson contributes to students' learning goals and connect to issues or contexts that are relevant to students.

- Meet with the instructor before lesson implementation  to discuss ways to improve the lessons.


Student Goal 2: Lead their peers to formulate a mathematical model to represent the solution process while investigating the key question. 

- Create sample solutions to the modeling problem, which further refine the activities (frequently in solving the problem themselves, the students would realize that some constraints needed to be better defined or wording of the prompts would need to be improved).

- Find web-based resources (such as Desmos.com or Geogebra.org) or use physical manipulatives as additional tools to help solve the problems (for example, in a gift wrapping paper task related to surface area of a rectangular prism, classmates were asked to bring wrapping paper and a rectangular box to wrap).

- Lead classmates to solve the problems during the micro-teaching lesson.

- Consider the lesson using Aguirre & Del Rosario Zavala's (2013) Culturally Relevant Mathematics Teaching lesson analysis tool.  This tool includes lesson analysis dimensions of Mathematical Thinking (intellectual support for the students, discourse during the lesson, student engagement), language (academic language support such as defining all key words, especially for English Language Learners), relation to culture and community, and use of social justice topics within the lesson.

- Rate own lessons based on some subset of dimensions and suggest ways to improve their lesson for a future implementation to better address all the dimensions.


Student Goal 3: Lead peers to interpret the results of the mathematical investigation and analyze classmates' work.

- Critically reflect on the solutions developed during the microteaching lesson.

 - Upload screen shots or photographs of  work for the solution process

- Conduct a gallery walk of the different solutions

- Discuss what might be done differently if the lesson were conducted in-person compared to online.

- Grade classmates' work.

- Complete written reflection on classmates' work.

- Use Anhalt & Cortez's (2015) rubric for work shown on MEAs, rate classmate's justifications and explanations that demonstrated understanding of concepts, connections between mathematical ideas and the real-life scenarios, the completeness of the work shown, evidence of thoughtful reasoning, the use of multiple representations and appropriate concepts for the problem, and correctness of numeric calculations.


Instructional Materials

More instructors' notes are available here: 
https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1581&context=tme

Syllabus from Spring 2023 (Acrobat (PDF) 701kB Apr24 23)
Final Project Description (Microsoft Word 2007 (.docx) 133kB Apr24 23)
Sample Social Justice Mathematics Lesson (Acrobat (PDF) 5.3MB Apr24 23)

Assessment

 

Student Sample Reflection (Microsoft Word 2007 (.docx) 325kB Apr24 23)

Instructional Staffing

The graduate students who are enrolled in the cross-listed version of this course are mentoring undergraduate students. In Spring 2023 semester, I also had a graduate assistant (an in-service teacher) who assisted a group of two undergraduate students completing their final project together. The graduate student is an in-service high school mathematics teacher who has worked with me during both her undergraduate and graduate research, and is familiar with mathematical modeling for social justice contexts through taking courses in the graduate program.

Author Experience

Diana Cheng, Towson University

Dr. Diana Cheng is a professor in the Department of Mathematics at Towson University, Maryland, and serves as the Graduate Program Director for the Masters of Science degree program in mathematics education.  She earned her Ed.D. in Curriculum and Instruction from Boston University, M.Ed. in Learning and Teaching from Harvard Graduate School of Education, and S.B. in mathematics from the Massachusetts Institute of Technology. In the past year, her work on justice-centered pedagogy was supported by the Mathematical Association of America for community outreach activities, Howard Hughes Medical Institute and the US Dept of Homeland Security for pre-service teachers' undergraduate courses, and the National Science Foundation for in-service teachers' professional learning experiences. She hopes to help teachers shape the world to be a more inclusive environment for all and to empower students with a sense of agency to enact change.


Advice for Implementation

Student feedback has shown that having a hierarchy of topics that range from easy to discuss to more challenging issues would be helpful. For example, pre-service teachers might find introducing cultural mathematics would be an easier topic to start with rather than a more contentious topic such as racial profiling.

Iteration

Classmates are expected to provide feedback to each other, to help revise lessons. Students are also expected to reflect on the student actions with respect to social justice that they expect their participants to engage in.

Using CURE Data

Students have an opportunity to present at several venues both internal and external to Towson University. Internally, there is an annual Student Research Forum with a poster session conducted on campus. Externally, students have presented at the Association of Maryland Mathematics Teacher Educators conference with their work in previous semesters of this course. Other possible venues for dissemination include Mathematical Association of America's MathFest conference and writing teacher practitioner articles.

Resources

The mathematical content standards in the middle and secondary levels that we used are from the following reference:

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Available at http://www.corestandards.org /wp-content/uploads/Math_Standards.pdf

I have had students read through these articles:

Anhalt, C. O., & Cortez, R. (2015). Mathematical modeling: A structured process. The Mathematics Teacher, 108(6), 446–452. https://doi.org/10.5951/mathteacher.108.6.0446

Aguirre, J. & Del Rosario Zavala, M. (2013). Making culturally responsive mathematics 
teaching explicit: A lesson analysis tool. Pedagogies: An International Journal,
8(2), 163-190. http://dx.doi.org/10.1080/1554480X.2013.768518.

This article was written by another graduate course instructor about teaching mathematics teachers about social justice mathematics lessons:

Bartell, T. G. (2013). Learning to teach mathematics for Social Justice: Negotiating Social Justice and mathematical goals. Journal for Research in Mathematics Education, 44(1), 129–163. https://doi.org/10.5951/jresematheduc.44.1.0129

This contains a Likert-style assessment that can be used as pre- and post- tests for the students taking the CURE course:

Enterline, S., Cochran-Smith, M., Ludlow, L. H., & Mitescu, E. (2008). Learning to teach for Social Justice: Measuring change in the beliefs of teacher candidates. The
New Educator, 4(4), 267–290. https://doi.org/10.1080/15476880802430361

These books can be used as textbooks or references: 
Berry III, R., Conway IV, B., Lawler, B. & Staley, J. (2020). High school mathematics
lessons to explore, understand, and respond to social injustice. Corwin Press.

Gutstein, E. & Peterson, B. (2013). Rethinking Mathematics: Teaching Social Justice by
the Numbers (2nd ed). Rethinking Schools Publication. https://rethinkingschools.org/books/rethinking-mathematics-second-edition/

Hirsch, C. & Roth McDuffie, A. (Eds.). (2016). Annual Perspectives in Mathematics Education: Mathematical modeling and modeling mathematics. National Council of Teachers of Mathematics.

This link contains the materials for a presentation of a final project developed by three of the in-service teachers who took my MATH 525 Spring 2022 course: 
Heath, M., Ciscell, J., Corum, K., Lynch, W., & Wohlfort, J. (2022, November 30).
Engaging with social justice mathematics modeling lessons [Conference
session]. National Council of Teachers of Mathematics Regional Conference,
Baltimore, MD, United States.
https://www.nctm.org/uploadedFiles/Conferences_and_Professional_Development/Regional_Conferences_and_Expositions/Baltimore2022/Schedule/Baltimore2022_Program.pdf

 




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