ASPIRE in Math is an NSF-funded project (DUE 1916490) that is developing inquiry-oriented Introduction to Proof curricula and accompanying instructor support materials.
The ASPIRE in Math project has created modular course materials and a full suite of resources to facilitate its successful use in community college and university transition to proof courses. Here we briefly describe each of the four project deliverables.
Curriculum Modules (Modules).
1. real analysis module that engages students in reinventing fundamental concepts of real analysis through an exploration of root approximation methods and adapted for use in a transition to proof course.
Click "expand" for Outline of Real Analysis Module ---->
Roots of Polynomials
- $\rightarrow$ Intermediate Value Theorem (IVT) Conjecture
- $\rightarrow$ Bisection Approximation Method $\rightarrow$ Investigating Resulting Sequences
- $\rightarrow$ [Endpoint Option 1] Prove Increasing Unbounded Sequences "Tend to Infinity"
- $\rightarrow$ Monotone Convergence Conjecture $\rightarrow$ Define Convergence $\rightarrow$ Define Least Upper Bound
- $\rightarrow$ [Endpoint Option 2] Prove Monotone Convergence
- $\rightarrow$ Define Convergence / Sequential Characterization of Continuity $\rightarrow$ [Endpoint Option 3] Prove IVT
- $\rightarrow$ Intermediate Value Theorem (IVT) Conjecture
- Students begin by considering questions about whether polynomials of odd/even degree are guaranteed to have real roots.
- An Intermediate Value Theorem Conjecture emerges as part of an intuitive proof that a 5th degree polynomial must have a real root.
- Students develop a bisection method for approximating a root of a continuous function that has a sign change.
- Students investigate the various sequences generated by iterating the approximation method indefinitely
- Depending on the version of the Real Analysis Module being implemented convergence properties of different sequences are investigated (including output sequences generated when the function is not continuous).
- Endgame Version 1: In shorter versions of the module, (inspired by applying the Bisection Method to discontinuous functions) students define "unbounded above", "increasing", and "tends to infinity" on the way to proving that any sequence that is increasing and unbounded above will tend to infinity.
- Endgame Version 2: In medium versions of the module, students define "bounded above", "increasing", "convergence", and "least upper bound" on the way to proving that increasing sequences that are bounded above will converge.
- Endgame Version 3: In the longest version of the course, after establishing the monotone convergence theorem, students connect continuity to sequence convergence, and then prove additional theorems involving limits and continuity on the way to completing the IVT proof (by showing that the limit of the endpoints of the intervals must be a root of the function).
2. group theory module (developed for a previous project - link?) adapted for use in a transition to proof course that engages students in reinventing fundamental concepts of group theory through an exploration of symmetry groups.
Click "expand" for Outline of Group Theory Module ---->
3. mathematical language module focused on formal mathematical language with an emphasis on the use of the quantifiers “for all” and “there exist”.
Click "expand" for Outline of Mathematical Language Module ---->
Defining Unbounded Above & Eventually Constant $\rightarrow$ Defining Tends to Infinity
- Students engage in defining activities to think critically about how the structures “for all…there exists” and “there exists…for all” are used to encode two distinct processes. The lesson culminates with an activity in which students write an instruction manual for writing statements using both of these quantifiers.
- Students build on their understanding of the process encoded by the different quantifiers “for all” and “there exists” to define a concept with a triply quantified structure (for all…there exists…for all).
4. proof comprehension module focused on understanding the role of the structure of a proof and how it functions (or not) to prove a claim.
Click "expand" for Outline of Proof Comprehension Module ---->
Analyzing Existence Proofs $\rightarrow$ Identifying Hidden Lemmas in Proofs
- Students analyze drafts of two proofs of an existence claim. These proofs have opposite structure where one assumes the conclusion and the other uses a standard constructive existence argument. The value of “working backwards” as a part of the problem-solving aspect of proof production is emphasized.
- Students analyze drafts of two proofs with an emphasis on identifying gaps in arguments (and unwarranted conclusions) with a special focus on identifying “hidden lemmas” that could fill these gaps (and how to use these to either revise the proof or modify a conjecture).
Online Instructor Support Materials (OISM). We developed online instructor support materials that provide the instructor with all of the tasks as well as key information about i) student thinking (e.g., how they students will likely approach the tasks and difficulties they might experience), ii) rationale for each task/sequence (e.g., what is the goal of the task and how does it fit into the overall strategy), and iii) implementation suggestions (e.g., whether the task should be done in small groups and how to ensure opportunities to engage are equitably distributed among the students). All of this information is supported and exemplified by images and video from actual classroom implementations and links to relevant research produced by the project team. You can find the OISM at https://taafu.org/aspireinmath/ and email firstname.lastname@example.org to request login credentials.
Wiki-Style Textbook (Wiki-Text). We designed a Wiki-style online textbook resource that allows students and teachers to build an online custom text.
The instructor and students will be able to create and edit pages during and outside of class as they record and refine the definitions, conjectures, theorems, and proofs that the classroom community produces. This will promote the iterative refinement of the mathematical ideas and the development of precision in the use of mathematical language. The mathematical exposition generated in this way will have the advantage of reflecting the students’ own mathematical activity and their experiences in class. For the instructor, we have created templates that can be used to easily create your own courses. These include templates for different pages to include in your course wiki and templates of the task statements.
In this way, the Wiki-text serves as:
- a workbook containing all of the task sequences,
- a way to track and document the mathematical progress of the class, and
- a resource for students to use outside of class in much the same way they might use a textbook or lecture notes.
Check out a Wiki-Sample to see an excerpt of a Wiki-text that was used in an ASPIRE in Math course.
Professional Development Workshop (PD Workshop). We currently have plans to host a PD workshop Summer 2022 (Tentative dates 7/20, 7/27, and 8/1).
Click to learn more about the Summer 2022 workshop (and to apply to attend!) Summer 2022 PD Workshop
This workshop is designed to prepare instructors to effectively use project materials to teach a bridge course and includes 1) opportunities for participants to engage with the tasks themselves as the facilitators model recommended teaching strategies and 2) opportunities for the participants to engage in discussions and planning focused on implementing tasks successfully and adapting the materials for their own teaching context. Special attention is given to developing strategies to promote equitable outcomes by ensuring that the ownership of the mathematics and opportunities to engage and contribute are distributed to all of the students. We are also currently developing materials for an asynchronous self-guided online PD for future instructors interested in using our materials.
ASPIRE in Math Wiki (Resources for ASPIRE in Math Teachers)