Math is Different

A student applies for a community college, having set a personal goal of earning an associate degree in a field with good employment prospects.  The college informs her that she needs to take a placement test to confirm that she has her basic skills in science and history; clearly, a student needs to have those basic skills before taking college courses that involve science or culture.

The student, Lindsay, takes the tests; she learns that she has to take two remedial science classes and one remedial history class before she can start her program; these courses correspond to 8th and 10th grade science and to 9th grade history.  With relief, she learns that she does not need to take a course like her 12th grade mathematics.

What’s that you say?  This is ridiculous and untrue?  Yes, clearly I’m making this up.  However, it is possible that our approach to mathematics in college is just as ridiculous.

We say, and do many others, that “Math is Different”.  Of course.  Does the set of differences justify the punitive approach we use for mathematics?  We place students in boxes, each with a label for the degree of deficiency.  These boxes have no known connection to college courses, justified by a belief that ‘high school’ must be mastered before ‘college’.  The most common math courses taken by college students were never designed to provide benefits in college; they are copies (sometimes poor copies) of outdated school mathematics imposed on students.

Do we have students who truly need remediation in mathematics?  Absolutely; the rate is probably large — over 20% of incoming students probably need some remedial math course before they have a reasonable chance of success in college courses in science or mathematics.  Some students come to a community college with extensive needs in mathematics, and need help with number sense, proportional reasoning, algebraic reasoning, basic ideas of geometry, and more.  Many come to us with weak skills in algebraic reasoning and basic geometry — combined with needs for other areas of mathematics.

Placing students into a sequence of courses covering years of school mathematics makes no sense in college.  Research suggesting that many students are equally successful placed directly into college courses reflects a design problem, as much as ‘remedial is not working’.    As an analogy — I had a flash drive stop working this week.  Now, a flash drive needs a port and an operating system; there is a sequence of things here.  Our approach to remediation is like installing a new cover on the flash drive so it looks more like the computer, instead of making sure that the system works together.

Redesign of developmental and introductory college math courses is not enough.  Instructional delivery systems will not solve our problems.

We need to look at root causes and basic relationships so we can identify student capabilities that will make a difference.  In developmental mathematics, the New Life Project has done this type of analysis; take a look at http://dm-live.wikispaces.com/ for information.  Not as much has been done for basic college mathematics (college algebra, pre-calculus, etc); the MAA CRAFTY materials provide a start — see http://www.maa.org/sites/default/files/pdf/CUPM/crafty/CRAFTY-Coll-Alg-Guidelines.pdf  for information.

Math is different; students are different.  Take a look at the differential pass rates among groups of students.  The types of students most of us really want to help — those lacking prior success (predominantly poor and minority students) have significantly lower course pass rates in our current courses.  Sometimes, the differential is so severe that completion of the sequence is a trivial number of such groups.  The conditional probability of “need 3 developmental math courses AND is black/African American” is somewhere around 5%, compared to about 18% for all students.  A cynic might say that the primary purpose of developmental mathematics is to make sure that the high paying jobs stay in the hands of the ‘haves’.  I do not believe that we want to block the upward mobility of students in our communities.

We need new math courses, courses designed to provide benefits; courses designed to provide equity to our students.

All other improvements in mathematics at colleges will be temporary relief at best.  The system is not designed to succeed, and that is the problem needing our attention.

 
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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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Serving all students

Lately, I have been reading a lot of college schedule books.  One interesting idea I found was a college that offered “Intermediate Algebra for STEM” as well as regular intermediate algebra.

However, I saw an overall pattern that is disturbing.

Math programs tend to respond to struggling students by either creating a new course in addition to the old OR by splitting existing courses into 2 or more parts.

Let me clarify the ‘splitting’ — although I did notice a few places with modules (same semester), what I saw more was partial courses to be taken over an entire semester.  Beginning algebra might be a deliberate 2 or 3 semester experience (if all parts are passed); intermediate algebra might be another 2 or 3 semester experience.  The only way this pattern can be justified is by meeting two criteria:

1. A pass rate approaching unity (>90% in every segment)     AND
2. A retention rate approaching unity (>90% continue to the next segment)

These conditions still only allow about 50% of those starting beginning algebra to complete intermediate algebra after a 4-semester sequence (.91^7 = 51.7%).  The more typical 60% pass rate, combined with a more typical 80% retention rate, suggests a whopping 7% of students completing both algebra courses.  Most students are not done with intermediate algebra; getting to pass a college math course would happen for about 3% of those who start a beginning algebra sequence.

The issue here is not whether something needs to be done.  A significant portion (a minority, but non-trivial group) of our students have excessive struggles in spite of good effort on their part in the presence of a good classroom.  My own gut-level estimate is that approximately 20% of students need something more to help them succeed in a traditional algebra course.

Clearly, this is one of the motivations for the New Life courses, Mathematical Literacy and Algebraic Literacy.  These designs provide a more diverse curriculum, with reasoning emphasized, which is meant to help more students succeed.

However, not all institutions are able to implement the Literacy math courses at this time; some institutions will take 5 years to reach that stage.  What should the rest of us do?

I suggest this principle as being a valid guideline for our work with all students (including those who struggle):

The math curriculum should provide a one semester experience at each ‘level’ for all students.

Our current levels are beginning algebra and intermediate algebra.  Courses before beginning algebra have their own issues, and we need to justify their  existence. A beginning algebra course should be one semester, whether a student needs minimal support or maximal support.  A intermediate algebra course should be one semester, whether a student needs minimal support or maximal support.

So, instead of a two-semester beginning algebra sequence, we could offer an expanded class time version of beginning algebra.  If we think students need twice as much ‘instruction’, then we could offer a 8-class-hour per week version of a 4-credit course; at many institutions, this translates into a 4 credit course with 8 billing hours.

Most of us introduce the sequence of partial courses in response to lower pass rates for a group of struggling students.  My college did this for about 10 years.  Our rationale was that only 25% to 30% of the struggling group passed the single course, compared to 60% for the remainder of the students.  However, our best efforts only achieved 60% pass rate in the two half-courses; this resulted in about 29% of the students completing both halves — about the same rate as completed the single course.

Helping struggling students is not about providing more courses in a sequence.  Helping struggling students is much more about what we do in one course, in one semester.  Whether ‘struggling’ comes from learning disabilities (the most common reason), or historical accidents of the student (no diploma, no GED), helping students should not hurt students chances of completion.  In the political parlance, we are talking about societies most vulnerable adults.  Our work should be about catching them up, not setting them further behind.

Let’s drop those sequences of partial courses.  The design can not succeed as a strategy.  Let’s create some truly helpful solutions that fit within one semester.

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Should a Math Class be an Approximation?

I was tempted to title this post “should high school math teachers be allowed to teach college?”, because that is what I was thinking about recently; of course, that is not really the issue.  The issue is: should a math class, such as developmental math, be an approximation of the mathematics or should it be precisely the mathematics?

Here is the situation that got me thinking about it … While I was at the AMATYC conference (and Dev Math Summit) last week in Anaheim, I had substitutes in all four of my classes.  Two of them were former high school teachers, one a retired state worker.  Both of the former high school teachers confused my students by their mathematical presentation.  In one case, the teacher said that students should shade all regions for each inequality in a system including the ‘double overlap’; this is an approximation to the mathematics — only the overlap area should be shaded.  In the other case, the topic was the imaginary unit and complex numbers; this teacher did not appear to say anything ‘wrong’ but focused entirely on the mechanics.

It’s probably obvious that no math class can achieve precision in all topics during the learning process.  Approximations come from various factors, some more malleable than others.  One factor is linguistic in nature … precision is based on language, and deep understanding of language comes with experience.  We can not expect an expert understanding of the language from novice users; however, I would like to think that we design courses and curriculum so that students will move steadily towards the expert level.  This is complex, perhaps impossible … but I think it is critical to invest energy in this process for students.

Another factor for precision is created in the modeling process we provide to students.  In the ‘imaginary’ number case last week, the instructor emphasized correct symbol manipulation as a proxy for understanding the topic.  However, the human brain does not store information in a purely symbolic form … the process involves a verbal statement (sometimes called ‘unpacking’) from the symbols.  A novice student has no knowledge connected to the symbols; my substitute confused students by not supporting a verbal (conceptual) framework.

To re-state the title …

Should a math class be a deliberate or accidental approximation?

At this point, we should be thinking something like “Well, what is the problem if a math class is an approximation (deliberate or otherwise)?”

Here is a key problem:

Correcting prior knowledge is more difficult than creating accurate knowledge.

You may have noticed that students’ understanding of fractions is resistant to our efforts of ‘correction’ (same with algebraic faux paus such as distributing a power over a sum).  We spend millions of dollars on instruction partially as a result of math classes being a approximations at some prior stage(s) of the student’s math history.  Every time a student is required to take a standardized test in math, we are seeing the direct results of approximations in math classes and the harm they cause students.

I have no delusions that excessive ‘approximations’ are limited to K-12 teachers; I’m sure that many of our college instructors and professors do the same kinds of things.  I am guessing, however, that school teachers are more prone to running approximate math classes (based on interviewing experienced teachers across levels).  Also, policy makers who focus on ‘skills’ often provide indirect motivation to make math classes more approximate, as does a focus on ‘teaching for the test’.

Here is the tension we face:

Approximations result in inaccurate or incorrect learning.

Perfect precision results in no learning at all.

This is the math teacher’s paradox.  Like most paradoxes, this one serves to sharpen our problem solving.  The solutions lie along a path where approximations are deliberately limited and then refined towards perfection over time.

Many of us seriously underestimate the amount of work needed to learn mathematics — both professionals and policy makers.  Resolving the math teacher’s paradox depends upon appropriate conditions; the most basic of these conditions is time.  “Covering” the Common Core or “covering” the algebra curriculum will tend to doom most students to suffering the consequences of repeated approximations to mathematics.

We’d be better off working on precision for a lot less curricular content.

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