Maths

From arithmetic to algebra

Slightly edited version of a presentation at the University of Oregon, Eugene, OR February 20, 2009 H. Wu

Why can’t our students achieve introductory algebra? This presentation specifically addresses only introductory algebra, which refers roughly to what is called Algebra I in the usual curriculum. Its main focus is on all students’ access to the truly basic part of algebra that an average citizen needs in the hightech age. The content of the traditional Algebra II course is on the whole more technical and is designed for future STEM students. In place of Algebra II, future non-STEM would benefit more from a mathematics-culture course devoted, for example, to an understanding of probability and data, recently solved famous problems in mathematics, and history of mathematics.

At least three reasons for students’ failure: (A) Arithmetic is about computation of specific numbers. Algebra is about what is true in general for all numbers, all whole numbers, all integers, etc. Going from the specific to the general is a giant conceptual leap. Students are not prepared by our curriculum for this leap. (B) They don’t get the foundational skills needed for algebra. (C) They are taught incorrect mathematics in algebra classes. Garbage in, garbage out. These are not independent statements. They are inter-related.

Consider (A) and (B): The K–3 school math curriculum is mainly exploratory, and will be ignored in this presentation for simplicity. Grades 5–7 directly prepare students for algebra. Will focus on these grades. Here, abstract mathematics appears in the form of fractions, geometry, and especially negative fractions. (If you have any doubts about why geometry is abstract, try defining a polygon correctly.)

Graphically, we can present the situation this way: Algebra

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To go from grade 5 to grade 8, one can gradually...