Should ANY Adult take an Algebra Class?

In Michigan, we are working through a process to update the math courses that can be used to meet general education requirements.  We are using pathways concepts, and face the issue of intermediate algebra … and college algebra.  This has led me to ponder the question — is there a good reason for any adult to take an algebra class?

College courses, overall, are either for general education or for specialization.  Developmental math courses are a subset of the general education courses — they are pre-college level, and not specialized.  I know a few places have integrated developmental math into occupational courses; however, the majority of us do our developmental in a general context.

There is no need for an ‘algebra’ course in general education, whether developmental or not.  At the pre-college level, we focus on the mathematics that students need in college level courses.  Certainly, this preparation needs to include algebraic ideas, reasoning, and processes.  However, this basic algebra is a tool used in combination with other mathematics — whether geometry, statistics, networking, or other.  A developmental mathematics course might have more algebra than other domains, but will never serve students well if the only content is basic algebra.  Mathematical reasoning is not isolated bits of knowledge.

At the college level, a general education course is meant to provide breadth to a student’s understanding of the world.  An intense focus on algebra in a course for this purpose is misleading at best; more commonly, such an intense focus on algebra for general education creates barriers to completion with a course widely viewed as being disconnected from the real world.  A general education math course needs to be diverse, and show relevance.

The other broad category is ‘specialization’, usually related to a particular program or major.  The ‘algebra’ we are using in this discussion is a subset of polynomial algebra, which is nobody’s specialization; none of us teach such algebra courses because we were inspired to earn an advanced degree in the content.  This specialization, practically speaking, is justified by the study of calculus.  Even in a traditional calculus course, algebraic understanding is just one of the basic factors in success.  Visualization, flexibility, and breadth of knowledge are important as well.  We often provide separate courses in ‘college algebra’ and ‘trigonometry’ (with little geometry in either one), and then wonder at why students can not integrate their knowledge and apply it to new situations.

With all of the intense focus on developmental mathematics, we tend to not think about the curriculum at the next level … and whether it serves students well.  These courses in college algebra, trigonometry, and pre-calculus have completion rates that ‘compete’ (in a negative sense) with developmental courses; only the small ‘n values’ involved keeps this problem out of the attention of policy makers and grant-making foundations (is there a difference between those two?).  We have much work to do.

I do not believe any student should be faced with an algebra course.  Mathematics is much more interesting than that, and more diverse.  Let’s put a variety of good stuff (good mathematics) in every course a student takes.  We might even inspire significant numbers of students to take more mathematics.

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More Math Lit Materials Available

The second textbook for the Mathematical Literacy course is available.  Great news!!

The new book is by Brian Mercer and David Sobecki, published by McGraw Hill; the title is  Pathways to Math Literacy ( © 2014).  I had an opportunity to see some of the material prior to publication; you will find this text to be a good contrast with the other book currently available (some info below on that one).

This new book has a web site for instructors at http://successinhighered.com/pathways/   Take a look … they have information about the book that is actually helpful, and they have a resource page for faculty considering ‘pathways’ (whether you use their book or not).

Also available is the initial Math Lit book by Kathy Almy and Heather Foes, published by Pearson ( © 2013).  You can get information on that book at http://www.pearsonhighered.com/product?ISBN=0321818458  and at Kathy’s blog (http://almydoesmath.blogspot.com/ .  I noticed that a recent post on that blog is a recording of a webinar on implementing pathways; very helfpul.

As of this semester, we are up to about 170 sections of Mathematical Literacy across the country (this is not counting the Quantway™ sections).  Now that we have a second commercial textbook, even greater growth is possible!

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Is Mathematics a Science?

My college has recently completed a ‘reorganization’ of programs and departments.  As a result of this change, mathematics is now in the same administrative unit as science. Is this a good fit?

Although we share much, I have seen some interesting differences.  One striking difference is this:

Mathematics faculty are expected to be flexible generalists.

Science faculty are expected to be specialists.

We are likely to be posting one full-time position in mathematics, and at least 2 full-time positions in science.  As the programs talked about requirements for the positions, mathematics consistently kept flexibility as a top priority — to be able to teach a variety of courses.  Science faculty, on the other hand, consistently listed specific backgrounds — micro-biology versus biology, physics versus geology, etc.  I have asked about why this is the case, irritating a few of my friends along the way; the rationale basically boils down to ‘we need a specialist to teach x’.

In mathematics, we sometimes seek a specialist — like a math for elementary teachers course, or statistics.  The vast majority of math faculty (full-time) are qualified (in our view) to teach any of a dozen courses.  Science faculty seem to keep themselves in a box, where they may have 3 to 5 courses that they can teach.  I am not sure which approach is superior, but I do know that the situation is related to the other observation about math & science.

Science, in general, does not do developmental.

Students in K-12 have had a variety of science.  When students arrive at college, the college-level science courses they take are determined by their program — not by ‘deficiencies’.  Certainly, students who have struggled in science select programs that will provide them with lower-level science courses.  Every student begins chemistry with a college-level chemistry class; every student begins biology with a college-level biology class.  [My college had, at one time, a developmental science course — never a large population.]

Part of this is the acceptance of ‘science’ as a set of (almost) independent disciplines (sometimes competing disciplines).  Students will generally take courses in 2 science disciplines.

Mathematics is seen by policy makers as a single, continuous strand.  At the bottom is arithmetic; at the top, calculus … in between, lots of algebra, a little geometry, and some trigonometry.  There is “one mathematics”; there are “multiple sciences”.

Of course, this ‘one mathematics’ is an incorrect view.  First of all, that image confuses a sequence of prerequisites for a content structure; only parts of algebra are needed for calculus, as is the case for geometry and trigonometry.  Students in occupational programs are the ones who might get to experience the other parts of these mathematical disciplines.  We, the faculty, reinforce this incorrect view by testing and placing all students along this single continuum (including the requirement for remediation of arithmetic and algebra).

Secondly, there are mathematical disciplines that are relatively unrelated to calculus preparation … disciplines that are used extensively in the modern world.  Students are more likely to interact with network problems than they are common denominators.

As we talk with career experts and other programs about what their students need, what topics do we ask them about?  I suspect that 99% of the discussion focuses on the ‘calculus continuum’ (arithmetic to calculus, via algebra).  Do we ask about topics that are not in developmental math courses?  Topics that are not in introductory college courses?  I’ve not seen that done.

Could we envision a world where there really was no need for developmental mathematics (in the sense of repeating school mathematics)?  Unless students need calculus for their program, would it be possible to start with “basic quantitative reasoning” or “introductory statistics” or “math for electronics” for students less prepared?  Better prepared students, perhaps, could take “applied calculus” or “diverse mathematics for college” or “statistics and probability”.   Students needing calculus could take “general calculus” as a preparation for a calculus sequence. These questions, perhaps, are related to the nudge that some state legislators are giving us when they limit developmental education.

Although mathematics is the “Queen of the Sciences” (historically), our practice of mathematics is not so much a science.  A science is based on a collection  of methods applied to related sets of objects (like chemistry does); mathematics does consist of several disciplines.  However, we do not function like a science, nor do we provide students with preparation for scientific thinking within our math classes.

Mathematics in college is not a science.  Would we serve our students better if it were?  What would that look like?

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Our Security Blanket is a Problem for Our Students

Acceleration is both a buzz-word and a set of solutions in developmental mathematics.  In a basic way, the New Life model is based on acceleration to college mathematics for most of our students.  The courses in the New Life model — Mathematical Literacy and Algebraic Literacy — are being well received; dozens of colleges have implemented one or both courses.

However, we are resisting a simple change that promises significant improvement at little risk — eliminating any college course prior to the level of beginning algebra or mathematical literacy.  I’m talking about courses called pre-algebra, basic math, and or arithmetic.  I believe that these courses have insignificant benefits while presenting risks to students.

The vast majority of these courses focus on procedural skills in a few content domains (decimals, fractions, percents, very basic geometry, and perhaps extremely limited algebraic skills).  Historically, these courses are a relatively recent development from a remedial point of view:

The myth that we must fill all student deficiencies before they can take a college-level math course.

We all have deficiencies; human beings have a capacity to function in spite of them.  We tend to accept without question the surface logic that says a student needs to master arithmetic before they can master algebra.  [The New Life courses do not de-emphasize algebra; our focus is on diverse mathematics and understanding, including algebra.]  A course like beginning algebra or Math Lit continues to be one of the key gatekeepers to college success.

At the global level, I have never seen any study reporting a large correlation between pre-algebra (or arithmetic) skills and success in beginning algebra; sure, there are a few studies (including my own) that show a significant correlation … due primarily to large sample sizes.  Significance does not show a meaningful relationship in all cases.  A correlation of 0.2 to 0.3 is only connected with 5% to 10% of the variation in outcomes; other student factors (like high school GPA) have larger correlations.

At the micro level, we often justify a pre-algebra course by justifying the components.  Fractions are needed before algebra, because the algebra course covers rational expressions.  Other content areas have similar rationales.  This justification has two major problems:

1. The need in the target course is artificially imposed in many cases (‘needed for calculus, so we do this in beginning algebra’).  [This is a pre-calculus course has the responsibility for this need.]
2. The pre-algebra content is almost always a procedurally bound, right answer obsessed quick tour with no known transfer to an algebraic setting.

When the New Life model was developed, we did not assume any particular content connections.  We looked at the content of Mathematical Literacy, and determined that nature of the knowledge needed before students would have a reasonable chance of success.  The list of prerequisites to Math Lit is quite short:

• Understand various meanings for basic operations, including relating each to diverse contextual situations
• Use arithmetic operations to solve stated problems (with and without the aid of technology)
• Order real numbers across types (decimal, fractional, and percent), including correct placement on a number line
• Use number sense and estimation to determine the reasonableness of an answer
• Apply understandings of signed-numbers (integers in particular)

For the vast majority of students, any gaps in these areas can be handled by just-in-time remediation.  This list certainly does not justify a prerequisite course.  A similar analysis from a beginning algebra reference would yield a similar list, I believe.

In spite of what we know, we continue to offer courses before beginning algebra or Math Lit, and continue to require students to pass them before progressing in the sequence.

This has been a long-debated topic in AMATYC — why does an arithmetic-based course need to be a prerequisite to algebra?  Essentially, I think this is our problem — these courses are security blankets for us.  We feel like we are doing the safe thing and helping our students by giving them this ‘chance to be successful’; we believe that these courses offer real benefits for students, even though the data is pretty clear that they do not (in general).

It is uncomfortable, perhaps even scary, for us to consider the possibility that all students be placed into beginning algebra or Mathematical Literacy.  We worry about the risk.  We seem unconvinced that another math course in a sequence is creating known risks and problems for our students.

We can easily see the problem by a simulation.  Let’s assume that 70% of the students pass pre-algebra, that 80% of those continue to beginning algebra (or Math Lit), and 60% of these pass.

Enter pre-algebra, pass beginning algebra  … about 34%

Compare this to these same students starting out in beginning algebra.  There is no sequence; the percent who pass beginning algebra is simply the pass rate for a group with somewhat higher risk.

Skip pre-algebra, pass beginning algebra … about 40% to 50%

The real world is not as rosy as the first scenario.  At my college, less than 50% of our pre-algebra passers complete beginning algebra (and a fourth of these barely pass, having little chance at the next level).

Related to this issue is the body of research on the connection between placement into developmental mathematics and completion of college.  One such study is by Peter Bahr (http://www.airweb.org/GrantsAndScholarships/Documents/Bahr%202012%20Aftermath%20of%20Remedial%20Math.pdf)  A consistent finding in these studies is that completion is inversely proportional to the ‘levels below college’ that students are placed at — even if they pass the math courses.

We should be very upset by the situation.  Few researchers talk about this, but we know.  Pre-algebra (and arithmetic) courses tend to have a higher (sometimes much higher) proportion of minority students, as well as people with employment and economic problems.  Community colleges are supposed to be about upward mobility; instead, we’ve created a system which has been shown to keep certain groups from advancing.

Let go of that security blanket called pre-algebra (or arithmetic).  Take the very small risk of helping a lot more students get though their mathematics and their program.  Completion leads to economic opportunity.  Let’s get out of the way, as much as possible!

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