US: Deficiencies in High School Mathematics

Previously referred to, this document from Caltech, not only talks about deficiencies in US high school mathematics instruction, but also the remedial measures Caltech is taking.

To remind you:

The transition from high school to college presents problems for all students, but for some students it is particularly challenging. At Caltech, many newly admitted students lack the background in mathematics that is necessary to succeed in Ma 1a. Unfortunately, few of them are even aware that their background in mathematics is deficient. This is not their fault. The mathematics curriculum in high schools is less rigorous than it was even a few decades ago. In conversations with Caltech students who have struggled with freshman mathematics, most report that they were star math students in high school, which of course is a major reason why they were offered admission to Caltech in the first place. Many of them, however, have never seen mathematics as it is taught at Caltech.

The following is how Caltech is making up for the deficiency, so that there is no need to speculate on what the deficiencies are.

Dean Kiewiet contacted Professor Roberto Pelayo, a Caltech Ph.D. in mathematics who is currently on the faculty of the University of Hawaii and who, for the past several years, has taught in the Caltech Freshman Summer Research Institute. The outline for an online course, Transition to Mathematical Proofs (TMP) that incoming freshmen could take at home this summer before their arrival at Caltech.

The TMP course outline is as follows:

Structure of TMP

TMP provides an online resource for incoming Caltech freshmen that engages them in mathematical writing and in obtaining experience in constructing simple, but well‐written and logically sound proofs.

This online resource consists of approximately 10 modules, each consisting of a set of lecture notes, exercises, and a mechanism for submitting work. This submitted work is evaluated by teaching assistants and then electronically returned to the students.

Course Content

TMP focuses on strategizing and composing mathematical proofs using topics regularly found in Math 1a (e.g., continuity, differentiability, sequences) to demonstrate the various proof‐writing techniques. In particular, TMP assists students in making the transition to a more analytic paradigm of calculus that focuses on proving theorems rather than performing computations.
Module Topics
* Formal Logic
* Set Theory
* Functions between Sets
* Cardinalities of Sets
* Subsets of the Real Line
* Continuity of Functions
*  Differentiability of Functions
* Applying Continuity and Differentiability Theorems
* Sequences

Learning Outcomes

Incoming freshmen who successfully complete TMP will be able to:

* Write simple but logically correct proofs that utilize appropriate terminology and notation.
* Understand various proof methodologies, including direct proof, proof by contradiction, proof
by contrapositive, and induction.
* Manipulate sets using the various set‐theoretic operations and theorems.
* Compute the cardinality of various classes of sets.
* Prove when a function between sets is injective, surjective, and bijective.
* Prove or disprove that a real‐valued function is continuous.
* Use the definition of the derivative to prove various differentiation properties.
* Use various differentiability and continuity theorems in proofs beyond calculus.
* Prove when a sequence converges or diverges.