MATH 310: Patterns & Structures in Mathematics
Course Description
This course is required for undergraduates seeking the Liberal Studies degree and for those seeking a major in Foundational Mathematics Education. The course focuses on proportional reasoning, data analysis, and algebraic thinking, what they are and how we can develop them in our students. Algebra and algebraic ideas are being introduced into lower and lower grade levels, so it is imperative that elementary and middle school teachers understand how to develop these habits of mind in young students. As an introduction to this material, we will investigate the basic ideas of proportional reasoning, which can form a foundation for moving into algebra, probability, and data analysis.
Course Design & Modality
This course is offered online asynchronously and is is organized in weekly modules.
Weeks 1-2: Intro to course and tools, review of prerequisite content knowledge
Weeks 3-5: Unit 1 Rainbows
Weeks 6, 8-9: Unit 2 Rainbows
Weeks 10, 13-14: Unit 3 Rainbows
Roughly half of the class are transfer students, and their prerequisite coursework lacks the rigor we expect here at Chico State. Review the first two weeks enables students to learn the technological tools of the course while also refreshing/learning key concepts from the prerequisite courses that will be used throughout the course. Students also then have a place to reference when the concepts come up again later.
Unit 1 on Proportional Reasoning is the core mathematics of the middle grades, and the tools and reasoning here provide the basis for Units 2&3.
Elements of Course Design: Philosophy
I believe all work I assign is worth doing and benefits my students, and so with complete/incomplete grading, students can complete and submit work up through the Friday of the last week of classes without penalty. I still have deadlines and a late work policy, though, which establishes expectations with that flexibility.
I also firmly believe that content mastery should not be expected right after a unit, so there is no penalty for earning a Content Goal in May vs January. I only care that they have a solid understanding of the content by the end of my class.
Student Engagement
WEEKLY RAINBOWS
🔴Red is a self-check of the previous Rainbow’s Individual Practice
🍐Pear Deck interweaves textbook-like slides with videos and student response prompts.
📚Readings from practitioner journals; Practitioner videos; Podcasts on pedagogy
🔵Blue is Individual Practice where students do work by hand and upload pictures or build solutions digitally
Math Together is an exploration of a tool or resource related to the content
OVERVIEW & TO DO LIST
Every Rainbow starts with the learning objectives and a shortened to-do list that students can use to keep track of which assignments they have submitted.
Every Rainbow ends with a summary of the ideas as they relate to teaching and a brief overview of the content of the coming weeks.
INTRO ACTIVITIES
Intro activities are designed to connect students with the instructor and each other and to prepare them for the expectations of the course.
I use "This is Us" Padlet: I post first!
Required messaging of instructor on Pronto
Intro Survey & Syllabus Quiz
Podcast on Study Strategies & reflection assignment–which of these have you used? Which do you plan to use in this course?
MATH TOGETHER
Students are able to communicate with one another and see each other’s work on Pronto but also in a weekly Math Together. Students are typically instructed to produce some work and then to comment on slides of their classmates.
MIDTERM CHECK-IN
This survey helps remind students of expectations and also provides an opportunity for them to give me feedback on what is working and what is not for them.
Assessment of Student Learning
SELF-CHECK
Each week, 🔴Red is a self-check of the previous Rainbow’s Individual Practice. Students are given a PDF of “Featured Responses” but are then required to choose one problem from their original work to improve based on the FRs. They have to upload a screenshot of the original work, a screenshot of updated work, and to write a summary of the changes they made and why.
CONTENT GOALS
After each Rainbow is due, students have access to a self-check and then have Content Goal assessments (in a form similar to Individual Work) which are untimed and open resource–the main restriction is that they can’t talk to anyone about the exact problems on the assessment or problems with the same numbers.
There are 1-3 CGs each Rainbow, 7 per unit, 21 total over the semester.
Most students will not earn the CG on the first draft–mastery shouldn’t be expected right after learning something! Students receive feedback on an individualized Google sheet, and later there is a 2nd attempt at the same goals with slightly different problems. Students need not attempt any CGs they have already achieved.
The “Final” is all 21 CGs open for three days - students only need attempt CGs they have not achieved. Students happy with their grade don’t need to do any! Completing CGs can only bring a grade up.
New this semester is grading on a 0-3 scale. This is because students were frustrated with Content Goals being all or nothing. I think it still needs adjustments, but it did resolve a lot of student anxiety.
0 - nothing of mathematical value
1 - there was some mathematics of relevance
2 - Correct solution with work but without thorough explanation; thorough explanation but missing part of the problem
3 - Correct (unrelated arithmetic errors aside) with thorough explanation.
Technology & Tools
Pronto: the main tool for communication between students and between instructor and students. I also tripled the number of students who regularly checked in to the Pronto feed by embedding it on the course homepage. Each week, a handful of students (different by week) take to the class chat to post their work and ask questions. Everyone’s experience is enriched, and I have a better sense of where students are struggling.
For digital tools and platforms, we use Pear Deck, Desmos, Polypad by Mathigon, Google Slides, and Google Docs all regularly. We also use and investigate a variety of tools and resources for specific mathematics content, such as SolveMe Mobiles. Everything we use is free or has free levels and is also cross-platform so that my students not only use the tech to learn but they then can also use the tools in their own classrooms.
By far, the biggest change this term is that instead of paying for a textbook, students purchased a mathematics manipulative set. I shifted all content demonstration to be from articles available in JSTOR, practitioner articles and videos, and my own writing, diagrams, and videos. This enabled students to spend less money and also have better tools.
I remade nearly all of my videos using a document camera and the manipulatives from the kit so that students could see their use. I had previously used digital manipulatives (and still make them available and use them sometimes), but since the content is so challenging, representations and abstractions were too far out of reach for many students. Working with the same tools they will use with their students both gave them a sense of how to use them with children and also helped them better understand the mathematical concepts.
Accessibility & UDL
The manipulative kit has also expanded the tools students use in their independent practice and content goals. Students can upload photos of handwritten work and/or of manipulatives, enhance those with digital annotation or not, or do the work entirely digitally. When written explanations are required, they can type or handwrite those.
Articles posted for reading are always “Read & Annotate” assignments. Students are given a few guiding questions and are then asked to:
use the Canvas tool to annotate or
download the file and annotate with a tool of their choice or
print it out and annotate by hand and scan and upload their work.
Elements of Experiential Learning & Connections to the Professional Field
This is a class designed for preservice teachers whose certification will allow them to teach middle school mathematics. Every facet of the course is focused on preparing them for that–videos from teachers, videos of students doing mathematics, articles for practitioners, etc.
Many students do not anticipate teaching beyond upper elementary, and so I have incorporated connections between K-5 mathematics and the course topics so my students 1) see how this material is relevant to their intended work, 2) are better prepared to teach K-5 in a way that will support the vertical trajectory of children’s mathematical learning, and 3) gain a better understanding of the content by having an entry point that is more comfortable for them.
Equity, Diversity & Inclusion
Practitioner journal articles include photos of children of color doing mathematics
Classroom videos feature diverse classrooms and teachers
Latinx (Fall) or Black (Spring) History Month assignment to learn and share about a Latinx/Black mathematician and to reflect on how more BIPOC and women mathematicians can be featured in school mathematics.
Practitioner journal articles that address cultural mathematical funds of knowledge, such as Probability Games from Multiple Cultures
The focus of our LBST math courses in general is to shift the perspective on what mathematics is and who is capable of doing it. There are a number of mindset activities and readings designed to both improve our students’ sense of self-efficacy in mathematics and to communicate to them that a teacher’s fixed mindset about a student can be very limiting for that student.
Training in Course Design
I have completed QOLT as well as the UDL FLC*.
While my favorite part of any professional development is the discussion and collaboration with colleagues across disciplines, I found the central tool for each of these to be useful as well. It is helpful to consult the QLT checklist during course creation to have a way to remember the many different parts that need to come together, not just content-wise but in relation to the student experience. The UDL Guidelines break down learning by part of the brain and by stage of learning, and that’s extremely helpful for me to be able to take where I’m seeing students struggle and review areas where I can strengthen what I provide for their needs to be met.
*I am cheating, though. I do have a doctorate in Curriculum and Instruction: Mathematics Education
