Category: Professional Motivation

The Power of Understanding Math

One of the most rewarding experiences we can have is when a student exceeds our expectations.

This is a story of a student who initially struggled with everything and is now being successful within an individualized course structure.  In this class, I never ‘lecture’ to a group of students.  Class time is used for studying, help, consultation, and testing; we call it the “Math Lab” though it’s not what most people mean by that phase.  The common meaning of “Math Lab” is a drop-in help center open to a variety of students in a set of math classes.  Our Math Lab is a way to take a few math courses … our math help for other classes is separate from the Math Lab.

This particular student (I’ll call him Philip) was clearly having trouble on the first day.  He did not want to use the online homework system, and that was not a problem for me.  However, he opened the book to the first page of the first section and had lots of questions about the names for types of numbers, about order of numbers on a number line, and (shortly after) about adding signed numbers.  The second day brought questions about the meaning of words in statements of properties, and about the meaning of variables.

It’s not that most students “get” these things, nor that they do not need to work on them.  What was unusual was the level of the struggle (basic) along with the sheer quantity of questions.  I never tell students what my prognosis is for them (I’m sometimes wrong) but I thought this student was going to spend weeks on every chapter.

Philip did, indeed, spend weeks on chapter 1 … a chapter about real numbers in a beginning algebra course.  Following those weeks, Philip then missed several classes due to medical problems related to his PTSD and physical injuries.  With over 6 weeks gone, Philip had only tried that first chapter test.  He was about to encounter the chapter on linear equations and applications, a classic “speed bump” for students struggling to learn algebra.

Somewhere in the month after that, however, Philip began making consistent progress.  In fact, he was getting through the third chapter faster than many students.  That progress has continued, and Philip is very likely to pass the course.

The main point is that something in the way Philip dealt with the struggle made a difference in how he succeeded in the entire course.  Philip works towards understanding everything, including ideas the are relatively minor.  He writes down lists of both vocabulary to learn and problems that he needs help with.  My guess is that his turn-around from struggle to success was caused by his hard work at understanding (and not just knowing what to do).

We all have students in this level of course who interact with the material at a low level; for them, it’s more about remembering what to do than it is about understanding.  I think Philip’s intense effort at understanding provided him with a cumulative positive improvement in the ability to learn new material.

Like most of us, I strive to have all students look for that understanding in learning mathematics regardless of the specific math course.  With other students, I end up trying to pull them someplace they have no intention of going (understanding) while Philip approached the material that way without any influence from me.

As a minor point in this post, I will point out that a struggling student such as Philip will be lost prior to getting any success.  Taking several weeks on one chapter is not an option within a fixed-pace class; instead of accumulating benefits, struggling students accumulate bad grades on assessments.  Our Math Lab, with its focus on individual learning, allows a struggling student to truly become a successful student.

A fixed-pace class has a limited capacity for helping struggling students; they need to be within a relatively small range of struggle in order to succeed.  Our Math Lab expands that range considerably (though there are still limits).

Understanding … a focus on understanding … enables students to obtain power in mathematics by raising their level of functioning to a higher point.

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Dev Math: Where Dreams go to Thrive … Part II (Evidence)

Developmental mathematics is where dreams go to thrive; we have evidence that even the traditional courses help students succeed in college.  The narrative suggested by external political forces is often based on a simplistic view of students which is out of touch with reality.  Let’s help by spreading the word on a more complete understanding.

Students who need to take developmental math courses have a wide range of remediation needs.  Peter Bahr’s study on pathways with single or multiple domains of deficiency ( concluded that the basic college outcomes (such as earning a degree) show equivalent outcomes for groups of students (needed remediation versus not).

A totally different analysis by Attewell et al 2006 (see also reaches a conclusion of equal results between groups in many ways.  Many studies of remediation are simple summaries of enrollment and grades over a short period of time.  The Attewell research was based on a longitudinal study begun on 8th graders in 1988 (thus, the acronym “NELS: 88”) done by the National Center for Educational Statistics.  Over an 12 year period, the study collected high school and college information as well as additional tests and surveys on this sample.

A key methodology in this research is ‘propensity matching’ — using other variables to predict the probability of an event and then using this probability to analyze key data.  For example, high school courses and grades, along with tests, were used to calculate the probability of needing remediation in college … where a sample of students with given probabilities did not take any remediation while another sample did.  An interesting curiosity in the results is the finding that low SES and high SES students have equal enrollment rates in remedial math when ‘propensity matched’.

Thrive: Key Result #1
Students taking remedial courses have a higher rate of earning a 2-year degree than students who do not take remedial courses with similar propensity scores for needing remedial courses.  Instead of comparing students who take remediation with the entire population, this study compared students taking remediation with similar students who did not take remediation.  The results favor remediation (34% versus 31%)

In the bachelor degree setting, the results are the other direction — which the authors analyze in a variety of ways.  One factor is the very different approach to remediation in the two sectors (4-year colleges over-avoid remediation, 2-year colleges slightly over-take remediation).   However, the time-to-degree between the two groups is very similar (4.97 years with remediation, 4.76 years without).

Thrive: Key Result #2
Students taking three or more remedial courses have just slightly reduced results.  This study shows a small decline for students needing multiple remedial courses: 23.5% earn 2-year degree, versus 27.5% of similar students without multiple courses.  The Bahr study, using a local sample, produced equivalent results in this same type of analysis.

It’s worth noting that the results for multiple remedial courses are pretty good even before we use propensity matching:  25.9% complete 2-year degree with multiple remedial courses versus 33.1% without.  This clearly shows that dreams thrive in developmental mathematics, even among students with the largest need.

Thrive: Key Result #3
Students taking 2 or more remedial math courses have results almost equivalent to other students.  The predicted probabilities for students with multiple remedial math courses is 23.8%, compared to similar students without multiple remedial math (26.7%).

Note that this study was based on data from prior to the reform movements in developmental mathematics.  Even then, the results were reasonably good and indicate that the remediation was effective at leveling the playing field.

Thrive: Key Result #4
This is the best of all:  Students who complete all of their math remediation have statistically equivalent degree completion (2-year) compared to similar students (34.0% vs 34.7%)

This result negates the common myth that taking multiple remedial math courses spells doom for students.  The data shows that this is not true, that completing math remediation does what it is meant to do — help students complete their degree.


I encourage you to take a look at this research; it’s likely that you will spot something important to you.  More than that, we should all begin to present a thrive narrative about developmental mathematics — because that is what the data is showing.

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The Case for Remediation

Today, I am at a state-wide conference on developmental education (“MDEC”), where two presenters have addressed the question “is remediation a failure?”.  As you likely know, much of the recent conversation about developmental mathematics is based on a conclusion that the existing system is a failure.  The ‘failure’ or ‘success’ conclusion depends primarily on who is asking — not on the actual data itself.

The “failure” conclusion is presented by a set of change agents (CCA, CCRC, JFF); if you don’t know those acronyms, it’s worth your time to learn them (Complete College America; Community College Research Center; Jobs For the Future).  These conclusions are almost always based on a specific standard:

Of the students placed into developmental mathematics, how many of them take and pass a college-level math course.

In other words, the ‘failure’ conclusion is based on reducing the process of developmental mathematics down to a narrow and binary variable.  One of today’s presenters pointed out that the ‘failure’ conclusion for developmental math is actually a initial-college-course issue — most initial college courses have high failure rates and reduced retention to the next level.

The ‘success’ conclusion is reached by some researchers who employ a more sophisticated analysis.  A particular example of this is Peter Bahr, who has published several studies.  One of these is “Revisiting the Efficacy of Postsecondary Remediation”, which you can see at

My findings indicate that, with just two systematic exceptions, skill-deficient students who attain college-level English and math skill experience the various academic outcomes at rates that are very similar to those of college-prepared students who attain college-level competency in English and math. Thus, the results of this study demonstrate that postsecondary remediation is highly efficacious with respect to ameliorating both moderate and severe skill deficiencies, and both single and dual skill deficiencies, for those skill-deficient students who proceed successfully through the remedial sequence.  [discussion section of article]

In other words, students who arrive at college needing developmental mathematics achieve similar academic outcomes in completion, compared to those who arrived college-ready.  There is, of course, the problem of getting students through a sequence of developmental courses … and the problems of antiquated content.  Fixing those problems would further improve the results of remediation.

One of the issues we discuss in statistics is “know the author” … who wrote the study, and what was their motivation?  The authors who conclude ‘failure’ (CCA, CCRC, JFF) are either direct change agents or designed to support change; in addition, these authors have seldom included any depth in their analysis of developmental mathematics.  Compare this to the Bahr article cited; Bahr is an academic (sociologist) looking for patterns in data relative to larger issues of theory (equity, access, etc); Bahr did extensive analysis of the curriculum in ‘developmental math’ within the study, prior to producing any conclusions.

Who are you going to believe?

Some of us live in places where our answer does not matter … for now, because other people in power roles have decided who they are going to believe.  We have to trust that the current storms of change will eventually subside and a more reasoned approach can be applied.

In mathematics, we have our own reasons for modernizing the curriculum; sometimes, we can make progress on this goal at the same time as the ‘directed reforms’.  Some of us may have to delay that work, until the current storm fades.

Our work is important; remediation has value.  Look for opportunities to make changes based on professional standards and decisions.

I’ll look for other research with sound designs to share.  If you are aware of any, let me know!


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Our Students, Respect and Appreciation

So, I received a phone call today that really upset me.  Like most teachers at any level, “my student” is not just a reference … it describes the connection we feel to the people in our classes.  This phone call made me think, and changed how I think about my students.

This student (call her “Tami”) is in my beginning algebra course.  She’s not doing especially well, and has missed a class or two.  When she was not in class today, I did not think that much about it.

Tami left a message on my phone while I was in class.  I did not catch all of what she said, so I called her back and this is what she said:

I’m sorry that I was not in class today.  I wanted to make sure that you would not drop me.  I was in the emergency room this weekend because I got stabbed in the neck.

I thought about that a little bit … here is a person who had a real threat to her safety and continued survival, and she’s calling me about her math class.  How do my flimsy excuses for not taking care of responsibilities stack up against that?

Some people might be thinking “Jack, you’re so naive … did you think that the student might be either lying or ‘enhancing’ the truth?”  Actually, I did think of those possibilities; I’ll know more when I see Tami in class.  In the meantime, I chose to trust my students by default; that is not always warranted, but it sure helps in the efforts to build a positive classroom environment.

Sometimes, we are very quick to presume that students do not come to class because they don’t care.  Certainly, that is the case for some students … though I have more students who attend class in spite of the fact that they don’t care.

I realize that this is not a unique experience; you might have had a similar experience where a student had a ‘survival’ level experience and still showed some commitment to their math class.  However, the experience reminded me that many of our students deserve our respect and appreciation for dealing with the huge challenges in their lives … and still try to work on their math class.  For some, math class becomes their one safe space in a world of threats and chaos.

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