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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How well is “Mathematical Literacy” Doing?

The AMATYC “New Life Project” has a curricular vision that includes two new courses that can replace the traditional developmental mathematics courses.  The first of the new courses is Mathematical Literacy for College Students (often shortened to just Mathematical Literacy, or even MLCS); this course shares content with the Carnegie Foundation Pathways (Quantway™ and Statway™) and with the Dana Center (Foundations of Mathematical Reasoning).

With the New Life Project, our work is based on faculty making choices and working with publishers to develop materials.  Currently, two commercial texts are available for MLCS (either published or soon-to-be published); some faculty have also custom published their own materials, or adapted existing materials.  The initial MLCS pilots started about two years ago.

As of fall 2013, here is a summary of the known MLCS course implementations (number of sections):

The actual total is definitely higher than this (153 sections), as we know of other colleges using one of the new Math Lit textbooks.  A few more colleges are implementing MLCS in Spring 2014, and several more colleges are implementing Fall 2014.

I think it is worth noting that all of this progress is being made without special grants; no mandates are involved, and we have no ‘staff’ in the New Life project.  What we do have — dozens of dedicated faculty, willing colleges, and publishers willing to work with us.

The New Life Project is a voluntary effort (AMATYC Developmental Mathematics Committee) with considerable collaboration with the Carnegie Foundation and the Dana Center — especially in providing curricular expertise to those organizations.  We can be proud of this progress and our work together.

The use of Mathematical Literacy will continue to grow.  Our work will increase the emphasis on the second course — Algebraic Literacy; for information on Algebraic Literacy, you can see a presentation from this year’s Summit on Developmental Mathematics at https://www.devmathrevival.net/?page_id=1807

More information on both New Life courses is available over at the New Life community home — dm-live (https://dm-live.wikispaces.com/)

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Symbols as Window

Like humans in general, our students develop expectations based on experience.  Habits form, often without awareness or conscious effort.  Behaviors exhibit, which are used to measure knowledge.  In assessments, we often confuse correct behavior with correct knowledge.

Symbolic work can be difficult for novices.  We (experts) see large amounts of information in short symbolic statements.  For a novice, symbols are like a map to a city never visited — yes, we can remember how to get from point A to point B on the map … but without any understanding of what these points mean in the city.

On a recent test in my beginning algebra classes, two mistakes were made by at least 20% of the students (one or both):

• -3² + 5² = 2^4 = 16
• 8^6 divided by 8^2 = 1^6

The first error is a coincidental ‘right answer’ for a very wrong method.  The second one, not at all.  Both involve over-generalizations of ‘same number’ rules.  Obviously, there is a very high probability that the students making one or both of these errors have low study skills or habits (like not doing any practice outside of class).

My concern is not these particular students, nor these particular errors.  My concern is our overall approach to mathematics.  We tend to take one of these approaches to symbolism in mathematics:

1. Emphasize symbolic procedures, and measure understanding by correctly completing more complicated problems.
2. Emphasize context and reasoning, and measure understanding by correctly completing related problems with differences in details.

Some reform models take approach #2 to the extreme — very few symbolic procedures are introduced, and most of what is done is arithmetic; algebraic models are used but carried out with technology more than symbolic procedures.  We need to learn how to balance the ‘symbols’ and ‘reasoning’ aspects of mathematics — and be willing to embrace both as critical in all mathematics courses.

Clearly, there is much (perhaps a majority) of our traditional algebra curriculum that involves symbolic work without a purpose now or in the student’s future.  I seriously doubt that solving a radical equation by squaring each side twice will ever be a survival skill in a student’s future.

Just as clearly (to me, at least), many of our students will need good understanding of various symbolic structures in mathematics, in future science courses (hard science and soft science).  Terms, exponents, coefficients, subscripts, groupings, equations, inequalities … are involved in stating properties in sciences and in using predictive models.

When we assess the mastery of symbolism, we need to deal with much more than ‘correct answers’.  In the ideal situation, assessments would be done in a one-on-one verbal interview so the expert can probe into the novice’s understanding based on the individual learner.  Lacking that luxury, we will need to use diverse assessment tools that deal with process and connections, as well as answers.

Sadly, I had integrated some of this assessment into the beginning algebra class about two weeks  ago — dealing with the adding terms error (first error above).  On a worksheet, students were faced with adding like terms (10x^4 + 6x^4) before we had dealt with them formally in class.  Something like half the students added the exponents as well as adding the ‘terms’ (coefficients).   About 40% of these students apparently maintained this erroneous method up until the test.

Correct answers are only correlated with correct knowledge; students are always seeking the simplest rules for achieving correct answers — which can lead to totally wrong rules.  Mathematical symbolism can be a window into the houses where students keep their math knowledge.  Too often, however, symbolism is confused with the knowledge and correct answers stop the assessment process.  

We need to slow down our courses.  Learning mathematics is not a fast or spontaneous activity.  Learning mathematics is hard work for both us and our students.

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