WHAT IS THE APPROPRIATE MATHEMATICS THAT COLLEGES STUDENTS SHOULD KNOW AMATYC Conference November 20, 2015 Phil Mahler & Rob Farinelli

Activity #1 Consider the topics in Beginning Algebra, Intermediate Algebra, and College Algebra On the PINK index card, write your most favorite topic to teach from these classes. On the GREEN index card, write your least favorite topic to teach from these classes.

WARNING! You may not agree with parts or all of the things that we have to say. In fact, it may really tick you off! However, please stay for the rest of the conference. The purpose of this talk is to begin a dialogue. Even though we like to think so, we don’t have all the answers …

What does “Algebra” mean to us?

What does “Algebra” mean to our students Solve, factor, graph, compute A series of disjoint topics connected together by exams Something that I have to take, but will never use in my major (or my life) Rules, processes, procedures to be memorized long enough to take some assessment Symbol Manipulation

What Are the Topics You LIKE to Teach? Why do you like them? Do your students like them? Do they emphasize the “why” as opposed to the “how”?

What Are the Topics that You Do NOT Like to Teach? What is it about these topics? If you do not like them, what do your students think? Are these topics just presenting the “how” and not the “why”?

Why do students take these courses? Some (small portion) are getting ready for “serious” mathematics – pre-calculus, calculus, statistics, etc. Quite a few are taking these courses so that they can register for their required mathematics course (often just one). Many should be getting mathematical preparation for science and technology courses. We believe that all students should be ‘mathematically literate.’

Are we teaching the full spectrum? 1. Where does the concept come from? 2. How is it used abstractly? 3. How is it used in real life? Teaching 2 or 3 by themselves is essentially useless because students would not see the point of it or would not be able to see how it can applied outside a specific scenario.

Developmental Mathematics The purpose of remediation was viewed as bringing a student “up to the level” (in content and skills) of a typical high school mathematics curriculum Ignoring geometry, even the applied geometry  This emphasized algebraic manipulation for everybody. In High School that makes sense. We shouldn’t close career paths if we can help it, at that level.

Developmental Mathematics (con’t) But when a person comes to a community college They have some idea of a meta-major – it can often be determined upon entry whether or not they will need calculus If they don’t need calculus, they don’t need half of the algebra we have taught to these students They do need to develop the connection between language, problems, and arithmetic and simple algebra sooner than a STEM major, due to a hopefully shortened curriculum.

Developmental Mathematics (con’t) So let’s look at where the student is going instead of where the student has been! What is the math needed for their goal? If a student didn’t master factoring in high school, must they master it now? Most students won’t be taking a college level mathematics course that needs this, or similar, skills. Liberal Arts Mathematics Mathematics Modeling Statistics

Developmental Mathematics (con’t) If a student is terrible at pencil and paper arithmetic algorithms – why? They probably are not too current and perhaps never were at their “tables” They may have never mastered multiplication algorithms and long division. They have been using a calculator for the last half-dozen or more years. If we push that student to work on these pencil and paper algorithms, and notably those of fractions, what happens? If they were terrible at the tables they probably won’t get them now. Only 10% of these students ever take a gatekeeper (college level, transferable) mathematics course. When they are finally allowed to use a calculator, they do. Exclusively.

Developmental Mathematics (con’t) Don’t teach pencil and paper algorithms! Do teach how to get a correct answer with a calculator and know it’s probably correct. Do stress fractions, and ratios and their descriptive values. Do teach ratio and proportion. In other words, teach the “why”!

Non-STEM Majors What is important for these students? Concept of a variable Concept of a formula Solving simple linear numerical and literal equations Modeling with linear functions What are appropriate courses? Mathematics for Liberal Arts Quantitative Reasoning/Quantitative Literacy Statistics How do we get students there?

STEM & Business Majors These students need to get to calculus. Is the current path the best way to get students there? Let’s go back to the list of topics that were in the “least favorite” categories …

Where is the disconnect? Behavioral Add ½ + ¼ Solve 3x – 2 = 19 In a function, the x is the independent variable and the y is the dependent variable. Conceptual Without performing the division, explain why 5 ÷ 1/7 is a greater number than 5÷ 2/3. How do you “undo” the operations on x to solve the equation? How fast I wake up in the morning depends on (is a function of) how much sleep I get

Important Questions for the Mathematics Community What topics are essential? What topics need “refreshed”, in other words, taught in a more meaningful, interesting, and relevant manner? How do we promote critical thinking and problem solving? How do we overcome a generation of students who have become good test takers? How do we get students to own their mathematical knowledge instead of renting it?

Thanks for listening! Contact Information: Phil Mahler Rob Farinelli