Category: Content of developmental math courses

45 Years of Dev Math

These are the materials from the November 11th presentation … history and future.

The presentation slides: Forty Five Years of Dev Math in 50 minutes web

The handout: 45 years of dev math in 50 minutes AMATYC 2017 S137

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Modern Dev Math

Let’s pretend that we don’t have external groups and policy makers directing or demanding that we make fundamental changes in developmental mathematics.  Instead, let us examine the level of ‘fit’ between the traditional developmental mathematics curriculum and the majority of students arriving at our colleges this fall.

I want to start with a little bit of data.  This chart shows the typical high school math taking patterns for two cohorts of students.  [See ]


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There has been a fundamental shift in the mathematics that our students have been exposed to, and we have reason to expect that the trends will continue.  We know that this increased level of math courses in high school does not translate directly into increased mathematical competence.  I am more interested in structural factors.

Intermediate algebra has been the capstone of developmental mathematics for fifty years.  At that time, the majority of students did not take algebra 2 in high school … so it was logical to have intermediate algebra be ‘sort of developmental’ and ‘sort of college level’.   By about 2000, this had shifted so that the majority of students had taken algebra 2 or beyond.

The first lesson is:

Intermediate algebra is remedial for the majority of our students, and should be considered developmental math in college.

This seems to be one lesson that policy makers and influencers have ignored.  We still have entire states that define intermediate algebra as ‘college math’, and a number that count intermediate algebra for general education requirements.

At the lower levels of developmental mathematics, the median of our curriculum includes a pre-algebra course … and may also include arithmetic.  Fifty years ago, some of this made sense.  When the students highest math was algebra 1 in most cases, providing remediation one level below that was appropriate.  By fifteen years ago, the majority of students had taken algebra 2 or beyond.  The second lesson is:

Providing and requiring remediation two or more levels below the highest math class taken is inappropriate given the median student experience.

At some point, this mismatch is going to be noticed by regulators and/or policy influencers.  We offer courses in arithmetic and pre-algebra without being able to demonstrate significant benefits to students, when the majority of students completed significantly higher math courses in high school.

In addition to the changes in course taking, there have also been fundamental shifts in the nature of the mathematics being learned in high school.  Our typical developmental math classes still resemble an average high school (or middle school) math class from 1970, in terms of content.  This period emphasized procedural skills and limited ‘applications’ (focusing on stylized problems requiring the use of the procedural skills).  Since then, we have had the NCTM standards and the Common Core State Standards.

Whatever we may think of those standards, the K-12 math experience has changed.  The emphasis on standardized tests creates a minor force that might shift the K-12 curriculum towards procedures … except that the standardized tests general place a higher premium on mathematical reasoning.  Our college math courses are making a similar shift towards reasoning.  Another historical lesson is:

Developmental mathematics is out-of-date with high schools, and also emphasizes the wrong things in preparing students for college mathematics.

We will never abandon procedures in our math courses.  It is clear, however, that procedural skill is insufficient.  Our traditional developmental mathematics curriculum focuses on correcting skill gaps in procedures aligned with grade levels from fifty years ago.  We appear to start with an unquestioned premise that remediation needs to walk through each grade’s math content from 5 decades ago … grade 8 before grade 9, etc.  This is a K-12 paradigm with no basis in current collegiate needs.

The 3- or 4-course sequence of remedial mathematics is, and always will be, dysfunctional as a model for college developmental education.

There is no need to spend a semester on grade 8 mathematics, nor a need to spend a semester on grade 9 mathematics.  When students lack the mathematical abilities needed for college mathematics, the needs are almost always a combination of reasoning and procedural skills.  If we can not envision a one-semester solution for this problem, connected to general education mathematics, we have not used the creativity and imagination that mathematicians are known for.  Take a look at the Mathematical Literacy course MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2 .  If students are preparing for pre-calculus or college algebra, take a look at the Algebraic Literacy course  Algebraic Literacy Goals and Outcomes Oct2013 cross referenced

Pretending that the policy influencers and external forces are absent is not possible.  However, it is possible for us to advocate for a better mathematical solution that addresses the needs of our students in an efficient model reflecting the mathematics required.



Assessment: Is this “what is wrong” with math education?

I have been thinking about a problem.  This problem is seen in too many of our students … after passing a test (or a course) the proficiency level is still low and understanding fragile.  Even accepting the fact that not all students achieve high levels of learning, the results are disappointing to us and sometimes tragic for students.

Few concepts are more basic to mathematics than ‘order of operations’, so we “cover” this topic in all developmental math classes … just like it’s covered in most K-12 math classes.  In spite of this, college students fail items such as:

  • Simplify    12 – 9(2 – 7) ÷(5)(3)
  • Write  3x²y without exponents

I could blame these difficulties on the inaccurate crutch called “PEMDAS”, and it’s true that somebody’s aunt sally is causing problems.  I might explore that angle (again).

However, I think the basic fault is fundamental to math education at all levels.  This fault deals with the purpose of assessment.  Our courses are driven by outcomes and measurable objectives.  What does it mean to “correctly use exponential notation”?  Does such an outcome have an implication of “know when this does not apply?”  Or, are we only interested in completion of tasks following explicit directions with no need for analysis?

Some of my colleagues consider the order of operations question above to be ‘tricky’, due to the parentheses showing a product.  Some of my colleagues also do not like multiple choice questions.  However, I think we often miss the greatest opportunities in our math classes.

Students completing a math course successfully should have fundamentally different capabilities than they had at the start.

In other words, if all we do is add a bunch of specific skills, we have failed.  Students completing mathematics are going to be asked to either apply that knowledge to novel situations OR show conceptual reasoning.  [This will happen in further college courses and/or on most jobs above minimum wage.]  The vast majority of mathematical needs are not just procedural, rather involve deeper understanding and reasoning.

Our assessments often do not reach for any discrimination among levels of knowledge.  We have a series of problems on ‘solving’ equations … all of which can be solved with the same basic three moves (often in the same order).  Do we ask students ‘which of these problems can be solved by the addition or multiplication properties of equality?’  Do we ask students to ‘write an equation that can not be solved just by adding, subtracting, multiplying or dividing?’

For order of operations, we miss opportunities by not asking:

Identify at least two DIFFERENT ways to do this problem that will all result in the same (correct) answer.

When I teach beginning algebra, the first important thing I say is this:

Algebra is all about meaning and choices.

If all students can do is follow directions, we should not be surprised when their learning is weak or when they struggle in the next course.  When our courses are primarily densely packed sequences of topics requiring a rush to finish, students gain little of value … those procedures they ‘learn’ [sic] during the course have little to no staying power, and are not generally important anyway.

The solution to these problems is a basic change in assessment practices.  Analysis and communication, at a level appropriate for the course outcomes, should be a significant part of assessment.  My own assessments are not good enough yet for the courses I am generally teaching; the ‘rush to complete’ is a challenge.

Which is better:  100 objectives learned at a rote level OR 60 objectives learned at some level of analysis?

This is a big challenge.  The Common Core math standards describe a K-12 experience that will always be a rush to complete; the best performing students will be fine (as always) … others, not so much.  Our college courses (especially developmental) are so focused on ‘procedural’ topics that we generally fail to assess properly.  We often avoid strong types of assessment items (such as well-crafted multiple-choice items, or matching) with the false belief that correct steps show understanding.

We need conversations about which capabilities are most important for course levels, followed by a re-focusing of the courses with deep assessment.  The New Life courses (Math Literacy, Algebraic Literacy) were developed with these ideas in mind … so they might form a good starting point at the developmental level.  The risk with these courses is that we might not emphasize procedures enough; we need both understanding and procedures as core goals of any math class.

Students should be fundamentally different at the end of the course.

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Equity and Stand-Alone Remedial Math Courses

One of the key errors that co-requisite (mainstreaming) advocates make is the treatment of ‘developmental mathematics courses’ as a single concept.  We would not expect college students who place into arithmetic to have comparable outcomes to those who place into intermediate algebra.  However, most ‘research’ cited with damning results uses that approach.  We need to have a more sophisticated understanding of our work, especially with respect to equity (ethnicity in particular).

A local study by Elizabeth Mary Flow-Delwiche (2012) looked at a variety of issues in a particular community college over a 10 year period; the article is “Community College Developmental Mathematics: Is More Better?“, which you can see at   I want to look at two issues in particular.

The first issue is the basic distribution of original placement by ethnicity.  In this study, ‘minority’ means ‘black or hispanic’; although these ethnicity identities are not equivalent, the grouping makes enough sense to look at the results.  The study covers a 10 year period, using cohorts from an 8 year period; partway through the 8 year period, the cutoffs were raised for mathematics.

Here is the ‘original’ distribution of placement by ethnicity using the data in the study:
Distribution by level Flow-Delwiche 2012 Original









After the cutoff change, here is the distribution of placement:
Distribution by level Flow-Delwiche 2012 New HigherCutoffs









Clearly, the higher cutoffs did exactly what one would expect … lower initial placements in mathematics.  However, within this data is a very disturbing fact:

The modal placement for minorities is ‘3 levels below college’ (usually pre-algebra)

This ‘initial placement’ data appears to be difficult to obtain; I can’t share the data from my own college, because we do not have ‘3 levels below’ in our math courses.  However, the fact that minorities … black students in particular … place most commonly in the lowest dev math course is consistent with the summaries I have seen.

We know that a longer sequence of math courses always carries a higher risk, due to exponential attrition; see my post on that    Overall, the pass rates for minorities is less than the ‘average’ … which means that the exponential attrition risk is likely higher for minorities.

The response to this research is not ‘get rid of developmental mathematics’; the research, in fact, shows a consistent pattern of benefits for stand-alone remedial math courses.  This current study shows equivalent pass rates in college math courses, regardless of how low the original placement was (1-, 2-, or 3-levels below); in fact, the huge Achieve the Dream (ATD) data set shows the same thing.  See page 46 of the current research study.

The advocates of co-requisite (mainstreaming) focus on the fact that 20% or more of the students ‘referred’ to developmental mathematics never take any math AND the fact that only 10% to 15% of those who do ever pass a college math course.  The advocates suggest that a developmental math placement is a dis-motivator for students, and claim that placing them into college math will be a motivator.  Of all the research I’ve read, nothing backs this up — there are plenty of attitudinal measures, but not about placement; I suspect that if such studies existed, the advocates would be including this in their propaganda.

However, there is plenty of research to suggest that initial college courses … in any subject … create a higher risk for students; it’s not just mathematics.  So, the issue is not “all dev math is evil”; the issue is “can we shorten the path while still providing sufficient benefits for the students”.    This goes back to the good reasons to have stand-alone remedial math courses (see ); although we often focus on just ‘getting ready for college math’, developmental mathematics plays a bigger role in preparing students.  The current reform efforts (such as the New Life Project with Math Literacy and Algebraic Literacy) provide guidance and models for a shorter dev math sequence.

Even if a course does not directly work on student skills and capabilities, modern developmental mathematics courses prepare students for a broad set of college courses (just like ‘reading’ and ‘writing’).  It’s not just math and science classes that need the preparation; the vast majority of academic disciplines are quantitatively focused in their modern work, though many introductory courses are still taught qualitatively … because the ‘students are not ready’.  Our colleagues in other disciplines should be up in arms over co-requisite remediation — because it is a direct threat to the success of their students.

Developmental mathematics is where dreams go to thrive; our job is to modernize our curriculum using a shorter sequence to give a powerful boost for all students … especially students of color.

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