The Modules

• Like our supplemental textbook, these modules were designed to be used in any real analysis course for pre-service and in-service mathematics teachers, no matter what textbook the course uses. However, the modules refer to content (e.g. definitions, theorems) in Stephen Abbott's textbook Understanding Analysis (2nd edition).

• These modules were designed to be run in a workshop environment, in which students get a chance to discuss their answers to module questions in small groups.

• Each module takes approximately 80 minutes of workshop time.

• For each module we have created three documents:**

  • • A document for the instructor of the course with general notes on running the module, additional commentary worth mentioning in class, expected responses from PISTs, and assessment criteria for assigned homework problems. To gain access to these instructor copies, please contact us.

    • A document to distribute to students after the workshops, with the preamble, the real analysis content relevant for the module, and statements of homework problems. These student copies are available to download below.

    • A document with only the pages that students need during the workshop. These workshop copies are available to download below.

• In this project, we also developed supplementary GeoGebra files for use in a real analysis course. You can access all of these files here. In addition, there are links to specific GeoGebra files in module materials.

• Below you see a timeline with some (but not all) of the content of a traditional introduction to real analysis course. This timeline highlights the definitions and theorems covered in each ULTRA module.

ULTRA 1

Pedagogical goal: The teacher will be able to justify why fractions are sometimes preferable to decimals by highlighting the complexities of dealing with decimals, including that two infinite decimal expansions can represent the same number.

Connection to the real analysis content: By seeing the theorem that a=b if and only if |a – b| < ε for all ε > 0, the student can appreciate that two distinct infinite decimals can represent the same value. In particular, all real numbers with terminating decimals have an equivalence class of size two (e.g., 0.24999… = 0.25).

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ULTRA 2

Pedagogical goal: The teacher will be able to justify how some successive iterative approximation methods commonly used in high school mathematics can provide an arbitrarily close estimate to the value that we want to construct.

Connection to the real analysis content: Successive approximation techniques can be understood as producing a sequence of values. If the maximum distance between each term in the sequence and the value being approximated converges to zero, the sequence will converge to the value being approximated.

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ULTRA 3

Pedagogical goal: The teacher will be able to justify and illustrate with appropriate examples the potential error than can accumulate if one rounds values in the middle of a calculation rather than at the end.

Connection to the real analysis content: The proofs of the algebraic limit theorems deal with identifying the way that errors from various approximations accumulate (as the terms of a sequence converging to a limit can be understood as successive approximations to that limit). The same line of reasoning can be used to see what happens to the error bounds of approximations as you add two approximations, multiply two approximations, and divide two approximations.

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ULTRA 4

Pedagogical goal: Teachers will be able to recognize the logic implicit in students’ mathematical assertions. By recognizing the implicit logic in students’ mathematical assertions, the teacher will be able to evaluate the validity of these assertions as well as clarify the assertions so that students will recognize their logical structure.

Connection to the real analysis content: By exploring the monotone convergence theorem (and a theorem about subsequences), the teachers will be asked to unpack the logical structure of these theorems to understand exactly what is being asserted, including statements in any logically equivalent form.

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ULTRA 5

Pedagogical goal: Teachers should be able to recognize that some concepts like trapezoids and continuity are defined in different ways in high school mathematics. Further, these different definitions may result in different sets of objects being associated with them, and may alter the statements one can make about them.

Connection to the real analysis content: The class will compare the deductive consequences of standard ε−δ definition of continuity in real analysis with other plausible definitions of continuity. The aim will show how the truth of claims about continuity depends on the definition that is chosen.

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ULTRA 6

Pedagogical goal: When students present arguments in mathematics courses, teachers should be able to recognize the implicit assumptions that are contained in students’ arguments. Teachers should use this recognition to evaluate the validity of the argument and to provide feedback and clarification to his or her class on the students’ argument.

Connection to the real analysis content: Applying the Intermediate Value Theorem requires checking that certain conditions are in place. By looking careful at the Intermediate Value Theorem and its proof, students will recognize how continuity is a necessary assumption when applying the Intermediate Value Theorem, even though this assumption is often not explicitly stated when the theorem is applied.

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ULTRA 7

Pedagogical goal: Teachers should be able to introduce the arcsine function to students as the inverse of the sine function when the sine function is restricted to the domain [-π/2, π/2]. Further, teachers should be able to evaluate students’ work and provide feedback if they fail to account for all solutions when applying the inverse trigonometric function to solve an equation.

Connection to the real analysis content: We discuss a theorem that states that a continuous function restricted to an interval has an inverse only if the function is strictly monotonic on that interval. This justifies the choice of restricting sine to [-π/2, π/2] to find an inverse as this interval is the largest possible interval containing 0.

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ULTRA 8

Pedagogical goal: Teachers should be able to justify why functions are not differentiable at points of discontinuity. Most proofs of this statement are not necessarily explanatory.

Connection to the real analysis content: The formal definition of derivative and its accompanying theorems and proofs can be understood in terms of approximations to the derivative by slopes of secant lines. The notion of successive approximation and secant lines can be used to give accessible explanations for why some functions are not differentiable.

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ULTRA 9

Pedagogical goal: Teachers should recognize that many of the explanations or arguments that they and their students provide are limited in scope (e.g., “exponents are repeated multiplication” is only applicable if the exponent is a positive integer). Teachers should be precise in their language and make students aware of the limitations of scope in the explanations or arguments they provide.

Connection to the real analysis content: Common proofs for the power rule of differentiation are limited in scope and, frequently, only apply to the positive integers. Typical differentiation rule proofs will be used as lemmas to extend the scope of the power rule and its proof, identifying the set of numbers to which these proofs apply.

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ULTRA 10

Pedagogical goal: Teachers should be able to model complex objects with simple objects and make use of this approach in their teaching. For instance, we sometimes compute the area of a circle by modeling it as a many sided regular polygon.

Connection to the real analysis course content: When working with Taylor series, we model complicated and unwieldy functions with polynomials. This highlights how we can often work with complex objects by modeling them with simpler ones.

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ULTRA 11

Pedagogical goal: Teachers should be able to introduce Cavalieri’s principle in their geometry class and leverage it as an area-preserving transformation (in addition to the more-common “cut-reassemble” transformation) to justify area formulas.

Connection to the real analysis content: Reimann integration and Cavalieri’s principle are similar in that they both can be understood as successively approximating the area of shapes with increasingly thin rectangles.

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ULTRA 12

Pedagogical goal: Teachers should be able to explain the Fundamental Theorem of Calculus in a manner that provides some justification for why it would be true.

Connection to the real analysis content: The proof of the Fundamental Theorem of Calculus is provided using graphical arguments that can be made accessible for high school students.

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