Ban Intermediate Algebra!?

Sometimes, there is a fine line between ‘reasonable interpretation of reality’ and ‘bad idea’.  Should we ban intermediate algebra in colleges?  Would it hurt anybody … help anybody … would anybody notice?

My current ‘reality’ includes teaching an intermediate algebra course that is quite traditional, except for us using an ebook (to save students money, and provide equal access).  This course has the usual combination of topics — functions, absolute value statements, polynomials, factoring (lots), rational expressions, rational equations, rational exponents, radicals, radical equations, quadratic equation methods, and quadratic functions (along with a variety of word problems, which are mostly puzzles).

In case you did not know, I have been teaching for quite a while (something like 39 years).  Originally, intermediate algebra was taken primarily by those who needed pre-calculus … and most of them needed calculus.  For a variety of reasons, the vast majority of my current students are not in this category; for them, intermediate algebra is part of their general education process.  [At my college, intermediate algebra is the MOST commonly used course to meet a gen ed requirement.]

Outside of the small minority of my students who actually need calculus (a group which should be larger), most students are not well served by an intermediate algebra course.  The traditional course does little to enhance their mathematical literacy or reasoning, with its focus on symbolic procedures; the traditional course does not contribute to the GENERAL education of students, since it is fairly specialized (polynomial arithmetic and related symbolic procedures).

For many of my students, intermediate algebra is where their dreams and aspirations wither and die under the negative influence of a curriculum which does not serve their needs.  Even for those who need pre-calculus, the traditional intermediate algebra course does not signficantly increase their mathematical proficiencies.  [The procedures learned are soon forgotten, and not much else was learned in the first place.]

Let’s ban intermediate algebra.  In its place, we should offer a version of the New Life “Transitions” course.  The Transitions course learning outcomes focus on providing mathematical preparation as part of a general education, especially if the student will take science courses (biology, chemistry, etc).

If you do not know about the Transitions course, take a look at the learning outcomes listed at https://dm-live.wikispaces.com/TransitionsCourse.   This course focuses on concepts and connections between concepts, so that students gain more than procedures.  The particular outcomes were chosen to be part of the general education of students needing science courses; some ‘STEM enabling’ outcomes are listed as an option for a course preparing them for pre-calculus.  The “Instant Presentations” page here has a presentation on the Transitions course; see https://www.devmathrevival.net/?page_id=116 

Of the two New Life courses, the first course “Mathematical Literacy for College Students” (MLCS) has generated more interest as an alternative to a traditional beginning algebra course.  I find this interesting, since we could argue that intermediate algebra is a worse match to student needs.  Curiously, the Transitions course is somewhat similar to some materials that are already on the market … which means that implementing Transitions avoids some of the challenges faced by those working on MLCS.

Some of you have been thinking “hey, we are required to use intermediate algebra as the prerequisite for all college-credit math courses”.  Well, I know … our profession needs to work on that problem.  Presently, the Developmental Mathematics Committee (DMC) in AMATYC is working on a position statement related to this problem; see http://groups.google.com/group/amatyc-dmc 

Obviously, I do not really expect us to ‘ban intermediate algebra’ (though I can dream!!).  Perhaps some of us can help our students by using the Transitions course as an alternative for those students who to not need pre-calculus.

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The Calculator Issue

I was talking with an editor from one of the larger publishing companies earlier this week, and one of the issues the editor saw as critical was “the calculator issue”, which this editor saw as both basic and divisive in the profession.

I would like to start by asking “When do YOU reach for your calculator?”

For my own work, and perhaps yours, I use calculating devices for several categories of work.  First, if the quantities are ‘complex’ and a precise answer is needed.  Second, if a procedure will need to be repeated more than twice (like finding a table of values for a function).  Third, if the calculation deals with a high-stakes question (like grades for my students).  Fourth, if the situation involves the exploration of ideas which are still in the ‘learning process’ (like a new mathematical concept or a review of long-lost treasures).  There might be a few other situations.

You might wonder why I start with our use of calculators.  So often, the comments we make are about our students’ “over-use” or “dependency” on calculators; we see calculator use as creating a risk for learning mathematics.  Many of us do allow calculators, and even embed their use in the learning process.  Some of us forbid their use, and some of us have a blended approach.  Most of us, however, believe that there is an issue with calculator dependency.

My conclusion is that the problem is not with calculators being used.  The problems occur when student attitudes about mathematics and their own efficacy create motivation to use calculators when the human brain is a better device.  If a student is doing a problem in the homework, or an example in class, and reaches for their calculator to add two one-digit numbers, this is part of the problem — the human brain is a better device, and the use of the calculator in this situation provides a clear ‘bad at math’ message about the student (and the student is the one sending the message).

Of course, we can not ignore the impact of culture on the use of tools, even calculators.  Some students temporarily feel ‘smarter’ when they use a technological tool frequently; I suppose this is not so much ‘smarter’ as ‘good’.  In the case of mathematics, the cultural bias towards technology combines with a norm that “it is okay to be bad at math” to encourage over-use of calculators.  These comments about culture are not universal for our students, some of whom come from cultures where very little technology is available … some even come from cultures with a positive attitude towards mathematics.

If this analysis is correct, then the issue is not whether we allow calculators or not.  The first basic issue is our outlook on learning — maximizing understanding, connections, and abilities to work with quantities … these are traits of the emerging models of developmental mathematics.  Students should develop their strategies for good uses of calculating technology, and they can do this as long as we do not focus so obsessively on ‘correct answers’; if we assess representation and communication, the use of calculators will not create problems.  As you probably know, technology does not really “solve” problems either; problems are solved by people and what we do.

I encourage you to ponder your approach to calculators in your classes.  Are you creating a calculator policy to make a personal statement, or are you creating a calculator policy which reflects your understanding of how technology affects learning of important mathematics?  Does your calculator policy encourage, or does it discourage, the development of mathematical proficiency in your students?

These issues are complex, and will not be solved by a simple yes/no policy on calculators.

 
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Math – Applications for Living VII

Our class, Math119 (Math – Applications for Living) is in the middle of our work on statistics.  The last class included finding a margin of error and a confidence interval for a poll … like those pesky political polls we are constantly hearing about. 

So, here is the situation.  This month’s poll showed 63% of respondents supported one candidate, based on results from 384 people; last month, the same poll reported 58% supported that candidate.  The article stated that the candidate is enjoying the increased support … is that a valid conclusion?

As you know, this relates to two issues.  First, the standard error for a proportion like this is found with the statistical formula:  

Tests of significance are then based on z values for a normal distribution; the most common reference is z = 1.96 … creating a margin of error representing a 95% confidence interval.

In our class (Math119), we use a quick rule of thumb to combine these two ideas into one statement which just uses the sample size — and this rule of thumb works pretty good for the types of proportions normally seen in polls (p values between 10% and 90%).  The rule of thumb for the margin of error is just the reciprocal of the square root of the sample size     

For the poll data, the sample sizes are both about 400.  The rule of thumb gives an estimate of 5%, which is very close to the actual value (approximately 4.7%.  In our class, we make a reference to the presence of the more accurate formula, but we use only this rule of thumb.

In this poll example, we create the confidence interval … and conclude that there is no significant difference between the polls.  The confidence intervals overlap; even though the new poll has a larger number, it is not enough of an increase to be significant (with this sample size).

We also have talked about selection bias and other potential problems with polls, and have begun the process of thinking about the impact of sample sizes on things being ‘significant’ (whether they are meaningful or not).

 
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Education as Transformation

Much is made these days of ‘value-added’, including the use of student ‘gains’ on standardized tests in the evaluation of teachers.  In colleges, we have defined courses in terms of student learning outcomes … which might reflect a comparable view of higher education (similar to K-12 and emphasis on ‘skills’).

“It must be remembered that the purpose of education is not to fill the minds of students with facts…it is to teach them to think.”  [Robert M. Hutchins]

What is the primary mission of colleges?  We all want our students to get better jobs, and would also like them to have a better quality of life.  Can these goals be achieved by the accumulation of discrete skills and learning outcomes?

Education is what remains after one has forgotten what one has learned in school. [Albert Einstein]  

Community colleges tend to serve the less-empowered segments of society.  People often cite mathematics as a key enabler of upward mobility, with some demographic studies to support this position.  These correlational studies produce a false impression of the processes involved.   The motto is not ‘algebra for all’ … the motto is ‘building capacity to learn and function’.

Education… has produced a vast population able to read but unable to distinguish what is worth reading. [G.M. Trevelyan]

Education should be a transformative experience.  Independent thinking, reasoning with a variety of methodologies (including quantitative), and clear communication should be evidence of this transformation.  In a community college, we can not strive for the same level of transformation as a university or liberal arts college education; however, we stand in the critical first steps for students along this path.

Education is the ability to listen to almost anything without losing your temper or your self-confidence. [Robert Frost]

In developmental mathematics, we have too often been content to provide little snippets of essentially useless knowledge — procedures to deal with a variety of calculations.  Even though it is not easy, and there is always a discomfort involved, our students are capable of much more.  Without reasoning and clear communication, these procedures will not benefit students (beyond a data bit that says they ‘passed math’).

Education is not filling a pail but the lighting of a fire. [William Butler Yeats]

As we work together to build a better model for developmental mathematics, we need to appreciate our place in the education of our students.  A good mathematics course produces a qualitative change in students. We can measure some aspects of this process by examining the reasoning and communication processes that students use.  However, there is no sure-fire and objective measure that says  a student has made progress.  We will develop better tools for this — including some focused on quantitative literacy and reasoning.  The challenges of measurement should prevent us from keeping our proper focus; we need to work to make the important measurable.

Education is the key to unlock the golden door of freedom. [George Washington Carver]

The pre-algebra/introductory algebra/intermediate algebra model of developmental mathematics needs to be re-made into a valid curriculum.  We can include mathematics that is practical, and that is an improvement — however, it  is not sufficient.  A central goal of developmental mathematics needs to be the improvement of quantitative reasoning and communication … a contribution which will enable our students to be educated, free people in a world facing diverse challenges.

 
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