How to Recognize an Algebraic Literacy Course

The next AMATYC journal will have an update on the New Life Project (over 100 colleges, over 800 sections, and something like 16000 students this fall semester).  In order to prepare that update, I spent a lot of time searching various web sites and following up leads.  That detective work led me to a number of colleges using “algebraic literacy” as a course title … when the course was just ‘intermediate algebra’; I also found some courses titled ‘intermediate algebra’ that were closer to algebraic literacy. #AlgebraicLiteracy #MathLit #NewLifeMath

This post is a brief “field guide” to help us recognize an algebraic literacy course.  Algebraic Literacy (the course) is one of the New Life math courses (AMATYC Developmental Mathematics Committee) developed in 2008 to 2010, based on the professional work from the last two decades.  The material below comes from our wiki (http://dm-live.wikispaces.com/Algebraic+Literacy ).

GOALS and FOCUS:

The Algebraic Literacy (AL) course prepares students for mathematics pathways which include college algebra, pre-calculus, and other courses requiring a background beyond the Mathematical Literacy (MLCS) course.

This is similar to an intermediate algebra course … on the surface.  There is a fundamental difference, however:  intermediate algebra is a derivative of an earlier “Algebra II” course from K-12, while Algebraic Literacy is engineered to meet the mathematical needs of college mathematics (backwards designed).

The focus of the AL course is on building understanding of mathematical systems with a dual emphasis on symbolism and application. The Algebraic Literacy course includes quantitative topics from areas besides algebra, which supports the needs of both STEM (Science, Technology, Engineering, and Math) bound students and other students.

A typical intermediate algebra course is heavily symbolic, with applications playing a minor role (and often using trivial applications with little payoff for preparing students).  The Algebraic Literacy seeks a balance between procedural fluency and higher level skills.  For some Algebraic Literacy courses, the applications form the context within which the mathematics is developed; for others, the mathematics begins first with applications integrated.  In considering applications, the Algebraic Literacy course includes problems with numeric solutions which would be solved symbolically in calculus.

PREREQUISITES

Basic proportional reasoning and algebraic reasoning skills, and some function skills, are required prior to the Algebraic Literacy course.

We do list 5 specific areas of prerequisite skills following this general statement.  However, the Algebraic Literacy course is designed to allow ‘co-requisite remediation’ at the appropriate level: Building on basic algebraic reasoning skills, for example, we aim for deeper understanding and solid symbolic skills.  By contrast, the typical intermediate algebra course presents a conflicted approach: students must show higher levels of symbolic mastery before enrolling but then intermediate algebra reviews many of those skills (without directly dealing with the development of reasoning directly).

More students are able to begin an Algebraic Literacy course than a typical intermediate algebra course.

CONTENT

1. Numbers and Polynomials
2. Functions
3. Geometry and Trigonometry
4. Modeling and Statistics
The content is intended to be integrated and connected.

In the Algebraic Literacy course, we would not see a chapter on “radicals and rational exponents”; we might see a section dealing with fractional exponents in an early sequence dealing with functions, including an application in half-life models … and a later section working on radical notation focusing on domain and range, followed by a section on translating between radical and exponential forms.  Either of these sequences of topics might also include geometry and/or trigonometry, and modeling concepts such as parameters.  Almost all topics will be presented as connected to one or more other topics, both conceptually and in terms of applications.

For most intermediate algebra courses, the content is usually 9 to 12 ‘chapters’ of material arbitrarily divided up … and separated.  A minimum of connections are made to other ‘chapters’.  Overall, the intermediate algebra course does not tell any story; the intermediate algebra course is a long series of vignettes only loosely connected by ‘category’.

By contrast, the Algebraic Literacy course tells a story of mathematical reasoning with both symbolic and application dialogues.  The design of the Algebraic Literacy course is based on being the first step along a path which includes calculus and/or other significant mathematics.  We seek to build covariational reasoning, a step up from Mathematical Literacy, on the path towards a good pre-calculus experience.

This field guide would not be accurate without  emphasizing a fundamental difference: Algebraic Literacy supports other STEM fields in addition to those needing the traditional Calculus Path.  This is primarily a distinction for the two-year college situation, where our programs often include mid-skill to high-skill fields (manufacturing technology, engineering technology, health careers, electronics, computer science, etc).  This inclusive approach is why Algebraic Literacy is not just algebra … geometry, basic trig, and statistics are included.  Most intermediate algebra includes some non-trivial geometry (right triangles, for example); however, you can recognize an Algebraic Literacy course by the presence of non-trivial geometric reasoning and symbolic representations, trig functions at a basic level, and enough statistics to interpret models developed from data.

Recognizing an Algebraic Literacy course involves multiple factors — goals, prerequisites, content, and the nature of the ‘story’.    A instinctive evaluation is based on this:

As a mathematician, can I get excited about teaching this course … is the focus on good mathematics, with the goal of developing abilities as opposed to “Algebra II all over again”?

We will see colleges move in this direction; I hope that you will consider joining the work!

 
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The Common Core State Standards and College Readiness

At the recent Forum on mathematics in the first two years (college), we had several very good presentations — some of these very short.  Among that group was one by Bill McCallum, a primary author of the mathematics portion of the Common Core State Standards.  Bill focused his comments on 9 expectations for the high school standards intended to represent college and career ready.

The expectations listed are:

• Modeling with mathematics
• Statistics and probability
• Seeing algebra as based on a few coherent principles, not a
  multitude of unrelated techniques
• Building and interpreting functions to represent relationships between quantities
• Fluency
• Understanding
• Making sense of problems and persevering in solving them
• Attending to precision
• Constructing and critiquing arguments

Of these, Dr. McCallum suggested that fluency is the only one commonly represented in mathematics courses in the first two years.  The reaction of the audience suggested some agreement with this point of view.

So, here is our problem:  We included all 9 expectations when the Common Core standards were developed.  We generally support these expectations individually.  Yet, students can … in practice … do quite well if they arrive with a much smaller set of these capabilities.  Clearly, the Common Core math standards expect more than is needed.

What subset of the Common Core math expectations are ‘necessary and sufficient’ for college readiness?

For example, even though it is critical in the world around us, modeling does not qualify for my short list; neither does statistics and probability.

We are basically talking about the kinds of capabilities that placement tests should address  Measuring 9 expectations (all fairly vague constructs for measurement) is not reasonable; measuring 4, perhaps 5, might be.

I think we should develop a professional consensus around this question.  The answer will clearly help the K-12 schools focus on a critical core, and can guide the work of companies who develop our placement tests.

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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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Pre-Algebra Just In Time

In my beginning algebra classes, I made a number of changes for this year.  Some are related to pre-algebra as a prerequisite for success in beginning algebra.  For quite a while, I have concluded that a pre-algebra (or basic math) class is an inappropriate prerequisite to beginning algebra based on analyzing data; my experience this semester might provide an alternative.  Students need not spend an entire semester ‘getting ready’ for algebra.

At the start of our course, we have a chapter which reviews operations on signed numbers with a minor emphasis on very basic algebraic expressions (like terms, distributing, etc).  This chapter is essentially a typical pre-algebra course contracted down to one chapter; historically, I concluded that both experiences (a course, a chapter) did very little to enhance the readiness of students for algebra.

Instead of spending any class time on signed number operations, we spent every minute of class time on algebraic language, syntax, and concepts.  We talked about adding changing coefficients but never exponents, and about multiplying changing both (depending) … and followed this up with a variety of problems for students to struggle with.  Much time was spent on translations, but not just in to algebra: we did a bit of work on translating algebra into words; even when students remember the ‘right thing to do’ (procedure) there is often a misunderstanding about what the expression meant.

Given the equation concepts we will be studying, we covered zeros in adding and subtracting.  Take some time to interview your students about a simple problem like this:

5y – 5y = ??

When this  would come up later in the course, something like 20% of my students would report that the answer is ‘y’ — the fives cancel out, leaving the y.  Curiously, textbooks do not have problems involving zero for combining like terms, even though this is critical for later work.

Class spent a lot of time (very frustrating for students) dealing with the different uses of the ‘-‘ symbol: opposite, negative, minus.    Some of this was imbedded in the translation work, and others in procedural work.  As instructors, we are incredibly careless about reading the ‘-‘ symbol, tending to say ‘negative’ when it is ‘opposite’ (like ‘-x’).  The central issue here is often “do we have any options about how to treat this particular ‘-‘?”

To assess this change, I used the same test from prior years with some added ‘difficulty’ — 3 added questions on expressions (including the zero in adding).  The initial assessment is that the new emphasis (pre-algebra just in time) helped students with algebraic proficiency without harming numeric skills.  The average score on the somewhat harder test was almost identical to the average on the prior (easier test).  Obviously, this is not enough to assess the merits of the new approach:  if the change does not help students later in the course, then the new process is not good enough yet.  I am especially eager to see if the ‘zero’ in adding has been improved.

My belief is that we could improve the outcomes of developmental mathematics by a fairly simple change:  do not require any ‘math’ or ‘pre-algebra’ before beginning algebra, just some basic numeracy is good enough.  Some students have a direct need to know arithmetic skills for an occupation, but this is a different need than ‘algebra’.  By placing almost all students directly into beginning algebra, we eliminate a math course for a large group of students — without producing harm.  [From general data I’ve seen, the chance of success in algebra AFTER a pre-algebra course is statistically equivalent to the chance of those below the placement cutoff … and the numbers are not good for either group.]

However, I am also sure that our current algebra courses need to do much more about basic literacy issues — translations, syntax, paraphrasing, etc — as well as a more complete treatment of procedures (zero in particular, but also adding vs multiplying).  We tend to move too quickly into applications of literacy (procedures for solving equations, for example) without building the conceptual foundation required for understanding.    We need “Pre-Algebra Just In Time”!

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