Here are some suggestions I have for educational/mathematical reform. The
list is in no way complete.
1. For the illiterate population, work with the entire family together.
Children of illiterate parents tend to be illiterate.
Parents/caretakers are not as embarrassed to learn the basics
of reading/writing and arithmetic when helping their
children learn. Children learn better when their
parents/caretakers work with them. Along this line, form
alliances between Colleges and K-12 institutions.
2. Concrete, practical "life skills" are not at odds to abstract
concepts. In fact, abstract ideas are learned from concrete
approaches. For example, young children who learn to group
objects together in groups of tens with leftovers, learn the
base 10 system well. Very young children can understand the
essence of addition and subtraction of 1 and 2-digit integers
by grouping cherios. We unfortunately, still too many times
teach the abstract processes and assume they will bring the
abstract ideas to the "real" world.
There is no reason to give students the Area = Length x Width
formula, then have them use it for a lot of problems,
thereby making finding areas an abstract concept. (This
does introduce the formula and some algebra substitutions.)
Instead, students can form rectangles from squares, record
the length, width and number of squares. The students soon
"discover" the formula for area. Simultaneously, the student
gets a concrete understanding of linear vs. square units,
the concept of substitution into a formula, and the development
of a particular formula.
4. Along the same line, have students integrate topics. For example,
have students discuss/write a paper about the connections
between quadratic functions, their graphs and quadratic
equations. This takes manipulation skills to a higher level
and greatly increases the critical thinking component of math.
Some questions for discussion--
How can you tell from the graph of y = ax^2 + bx +c
whether ax^2 + bx +c = 0 has real or complex roots?
If a>0 and the vertex is at (-750,295), are you likely
to "find" the graph of y = ax^2 + bx +c in a standard
window? Explain.
Given a picture of the graph of y = ax^2 + bx +c that
does not cross the x-axis---Does ax^2 + bx +c factor?
Explain.
How, and under what conditions can you use the graph of
y = ax^2 + bx +c to estimate
the solution to ax^2 + bx +c = 0?
Follow questions with problems to work. Throw in some
chalenges like: graph M = 27.9r^2 + 1.8r + 2896. What
window was used that made the graph look like a parabola?
3. Assigning lengthy projects for students to do can have several
benefits. The projects can be practical where students see
the relevance of math and life. The projects can require
stringing together several skills and techniques of analysis
so that higher level problem solving and critical thinking
skills are needed than in the "drill and kill" manipulation
exercises.
4. Work across disciplines. When working with sampling or percentages,
use real data that is used for reports/case studies in other
classes such as Psychology, Sociology, Economics.
5. Revise the curriculum. Most of the algebra we teach is not the
best prerequisite for calculus. Continue the discussion of
the balance between manipulation skills and concepts and
the part that technology plays.
Martha Haehl
Maple Woods Community College
2601 N. E. Barry Rd.
Kansas City, Missouri 64156
(816)437-3147