Publications:
A Framework
for Adult Numeracy Standards:
The Mathematical Skills and Abilities Adults Need
To Be Equipped for the Future
ALGEBRA: PATTERNS AND FUNCTIONS
My high school algebra class was really hard. I didn't know what I was doing and
I felt like I was the only person without a clue.
I've used every math skill I ve learned with the exception of algebra.
Algebra is the gatekeeper.
Overview
Should algebra be on the "honest list?" The Conference on Adult Mathematical Literacy
voted on "informal algebra" as one of four basic topics to include in adult numeracy
education. NCTM's Curriculum and Evaluation Standards, The Massachusetts ABE Math
Standards and other reform movements include it as a critical skill. But say the word
"algebra" to any group of adults (ANPN focus groups, for example) and the reaction
is negative, with personal stories of frustration and sheer agony spilling out.
Is the general perception that "there's really no use for it" a signal that we drop it
from the list of instructional topics or is the answer simply to improve instructional
practice? This area of mathematics presents a challenging dilemma, because what
appears to be a case of the experts want it but the people say no, may be more a case that
the two groups are talking about apples and oranges.
When adults reflect upon what it means to do algebra, they tend to recall formal methods
of equation solving, age problems, and a lot of x's and y's. But mathematics educators
at all levels have begun a dialogue with a very different emphasis, one that "moves away from a tight focus on manipulative facility to include a greater emphasis
on conceptual understanding, on algebra as a means of representation, and on algebraic
methods as a problem-solving tool." (NCTM, 1989, p. 150) They are not talking about
the mechanical high school algebra but algebraic reasoning that allows us to think
about and express patterns, relations and functions and which ultimately gives many
more people access to technology (e.g., spreadsheets and relational data bases).
Key Findings
Many frustrations are connected to past experiences with algebra. "I remember my
father standing over me at the dining room table attempting to drill into my head
the algebra x, y, and x + y. I couldn't understand how anyone could understand it
and why anyone would want to." (Stakeholder) So many adults cite algebra as a major stumbling
block in their earlier mathematics education, the place where they got stuck. "Math
is pretty easy (division); but algebra, forget it." "Algebra in ninth grade was
hard for me. The teacher would do a problem and I was lost. I really felt out of place."
"Algebra is hard." Whatever the reason, many learners report an incredible disconnect
at the
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point when algebra is traditionally introduced. "I never had any trouble
with math in school until I got to algebra." " Math was pretty decent, and then when
you got to algebra it was like they totally switched it all the way around."
There is a widely held notion that algebra is not practical, relevant or useful. "What
is it used for? You don't use it unless you're teaching it or you're going into
some kind of manufacturing type deal where you actually make diagrams, but otherwise
it's no use. I use math everyday, fractions and so on and so forth, but I just don't
use algebra or geometry."
Algebra is a bridge between arithmetic and more broadly generalized mathematical situations.
Mathematics is the study of patterns. "Learning to recognize and analyze patterns
and number relationships connects math to the world." (Massachusetts ABE Math Standards, p. 42) These generalizations can be expressed in the notation of formulas
and graphs.
Many life and work experiences can be expressed in algebraic terms. While most adult
learners do not see the relevancy of algebra, and many teachers see the academic
relevancy, employers and other workers do see application to today's workplace.
"Our union has a formula to calculate union dues. Dues are based on a weighted average, because
not everyone in the union made journeyman at the same time. Some journeymen make
$14 an hour, and some make $22 an hour. So you have to take so many $14 an hour,
so many $16, so many $18, so many $20 and take a weighted average. And there's another
equation: twice your weighted average plus your life insurance plus your union payments..."
(learner) One person mentioned an unusual algebra application. "I read an article about how they determine how long someone will actually stay in prison or jail.
There is a formula that they use that factors in things like good behavior and work
release. With this formula, the sentence that someone would get from the court can
be reduced a certain number of days, so the inmate can get out earlier than originally
expected." (learner) The Massachusetts ABE Math Standards hold that "life experience
has afforded adult basic education teachers with a broad base of real-world ties
which can be readily linked to the concepts of equation, function, variable, and graph."
(p. 46)
Algebraic thinking skills are crucial if adults are to compete in the global economy;
therefore, all adult learners should have the opportunity to improve in that area.
While the SCANS Report doesn't get very specific in math content areas, there is
support within the document for going beyond the very basic skills and including thinking
skills. And beyond that, the "five competencies" that build upon the foundation
skills imply the need for some algebraic competence, especially in the areas of "information", "systems" and "technology." SCANS stresses the need for organizing, interpreting
and communicating information and employing computers as a tool for those tasks as
well as the ability to "discover a rule or principle underlying the relationship
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between two or more objects and apply it in solving a problem." Identifying and expressing
pattern, relation and function are the algebraic skills imbedded within these competencies.
When the teachers who wrote The Massachusetts ABE Math Standards say "the opportunity to study algebra should be available to any adult basic education learner
who may have missed it due to past educational experiences," (p.46) they are calling
for a chance for adult learners to compete in the global economy.
Algebra impacts the competency of workers, parents and citizens. Workers who are
involved with technology cited several examples of algebra use. "I did marketing
analysis using Lotus 123 to forecast exponential marketing sales vs. quotas to determine
sales regions. In the manufacturing area I was 20 years old and three top managers
couldn't figure out an algebraic formula and one of them very jokingly said, Here,
see if you can solve this, as they all laughed at the thought. However, in one minute,
I solved the problem to determine their daily production which they couldn't do. " (learner)
" We're teaching students at the Great Lakes (Naval Base) all different facets of
shipboard life..., and one of the big things we teach up there is electronics...
Ohm's law, it is simple basic algebra."
We heard from many parents enrolled in adult basic education classes who don't like
the fact they can't help their children. " They're (her children) always asking me
why I don't know about algebra. So I told them about what happened in the past. (My
daughter) can see how automation is coming in with computers and all - and she knows if
you don't know it, you'll fall through the cracks, be on the street." " Back when
I was growing up, we didn't have to (take) things like algebra and geometry, and
now you have to know these things. Like my son comes home with papers that I can't even
do." "My kid's doing algebra in 6th grade now. I'm trying to help him out."
When asked to talk about math skills needed to be a successful community member, one
learner replied, While basic math skills, e.g. adding and subtracting, are obviously
important, it is also useful to have a working knowledge of algebraic language.
I am surprised by the opportunities I ve had to use algebra mind, which I was unaware
of before I acquired it. For instance, I know now that God didn t create complex
mathematical problems; the Math Wizard did. Because I have an understanding of
how these formulas happen, I am able to apply that knowledge to creating formulas that work
for me. Algebra mind has also made a contribution to my daily thought process.
I find myself thinking more critically and analytically, which is a nice side effect
of all the problem solving ... In my opinion, any person who thinks more systematically
and analytically is bound to be a better citizen ... crime rate, parenting, population,
welfare and many other social concerns. Better thinkers think ahead. Algebra mind should be taught as part of social reform.
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Implications for Teaching and Learning
Improve algebra instruction by providing effective staff development . Teachers need
to relearn algebra through the lens of patterns, relationships and functions, how
it's applied now to the real world, and need to know a diversity of approaches. The
learners' curriculum should reflect these same elements.
Introduce all learners to algebraic concepts by making links to the learner's experiences.
"I learn better if I start off with something I already know. When I worked in
marketing...If you go back to the basic formula and link it to an easier way. Because the more I learn the easier it gets. Link it to something you already know and
you'll get it, you'll remember it."
Pay attention to instructional pace, vary teaching strategies and strengthen the development
of concepts to improve algebra instruction. Learners suggested that the source of
the trouble might have to do with the pace of the instruction. "To me, an improvement might be to slow down...and maybe dwell a little more on it." "There are so
many concepts to grasp at one time, you need more time." One stakeholder suggested
that a better understanding of the large concepts might have helped. "I remember
finding it (9th grade algebra) the first time I was frustrated with math. And I would get
the right answers but not the way the instructor got them. I have since learned
that I am highly intuitive. I probably skipped some steps along the way but it was
frustrating and difficult for me to go back and fill in to tell him how I got there. I
have since (worked with) some algebra materials which were sort of an aha, that
s what I was supposed to be learning. I think it would be more fun and exciting
to study those math concepts now because I would have a much clearer idea of why I was learning
and what I was doing and there'd be things like missed calculations that computers
and calculators could help with so I could really focus more on the concepts."
Connecting to the Four Purposes
Algebra supports the key purposes for literacy. How can algebra be a door-opener
rather that the gatekeeper to higher education and well-paying jobs? Skills and
knowledge in the area of algebra help adults access information that is presented
in written and oral mathematical symbols. Conversely, the ability to represent information and
relationships with algebraic symbols, graphs, or everyday language strengthens voice.
The ability to reason algebraically (to think logically), to recognize patterns and
generalizations provides a scaffold for problem solving and decision-making.
Effectively taught, algebra can be a source of empowerment for adults as parents,
workers, and citizens. It should not be the "sieve" and the hurdle that keeps people
out.
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