Understanding the Data on Co-Requisite Remediation

We need to change how we handle remediation at the college level, because the traditional system is based on weak premises … and the most common implementations are designed to fail for most students.  Where we have had 3 and even four remedial courses, we need to look at one for most students.

Because of that baseline, the fanatical supporters of “co-requisite remediation” are having a very easy time selling their concepts to policy makers and institutional leaders.  The Complete College America (CCA) website has an interactive report on this (http://completecollege.org/spanningthedivide/#remediation-as-a-corequisite-not-a-prerequisite   ) where you can see “22%” as the national norm for the rate of students starting in remediation who complete a college math course.  With that is a list of 4 states who have done co-requisite models … all of whom show 61% to 64%.

One obvious problem with the communication at the CCA site is that the original data sources are well hidden.   Where does the ‘22%’ value come from?  Is this all remediation relative to all college math?  The co-requisite structures almost always focus on non-algebraic math courses (statistics, quantitative reasoning).  One could argue that this issue is relatively trivial in the discussion; more on this later.

What is non-trivial is the source of the “61% to 64%”.

One of the community colleges from a co-requisite remediation state came to our campus and shared their detailed data … which makes it possible to explore what the data actually means.  Here are their actual success rates in the co-requisite model they are using:

Math for Liberal Arts: 52%

Statistics: 41%

These are pass rates for students in both the college math course and the remediation course in the same semester.  Another point in this data is that ‘success’ is considered to be a D or better.

For comparison, here are similar results from a college using prerequisite remediation, showing the rate of completing the college math course for those placing at the beginning algebra level.

Quantitative Reasoning: 53%

Statistics:  52%

In other words, if 100 students placed at the beginning algebra level in the fall … there were about 52 who passed their college math course in the spring.  Furthermore, this college considers ‘success’ to be a 2.0 or better.  The prerequisite model here has higher standards and equal (or higher) results.

The problem with the data on co-requisite remediation is that only high-level summaries (aggregations) are shared. Maybe the state average for the visiting college really is “61%” when they have about 45% (they have more in statistics than Liberal Arts).  Or, perhaps the data is being summarized for all students in the college course without separating those in the co-requisite course.  One hopes that the supporters are being honest and ethical in their communication.

I suspect that the skewing of the data comes more from the “22%”.  The source for this number usually includes all levels of remediation followed to any college math course (including pre-calculus).  The co-requisite data is a different measurement because the college course is limited (statistics, quantitative reasoning).

Another interesting thing about the data that was shared from the co-requisite remediation college is this statement:

Only about 20 students out of 1500 in co-requisite remediation had an ACT Math score at 15 or below.

At my institution, about 20% of our students have ACT Math scores at 15 or below.  Nationally, the ACT Math score of 15 is at the 15th percentile.  Why does one institution have about 1% in this range?  Is co-requisite remediation being used to create selective admission community colleges?  [Not by policy, obviously … but due to coincidental impacts of the co-requisite system.]

Sometimes I hear the phrase “a more nuanced understanding” relative to current issues in mathematics education.  I suppose that would be nice.  First, though, we need to start with a shared basic understanding.  We can not have that basic understanding as long as the data being thrown at us is ill-defined aggregate results lacking basic statistical validity.

Perhaps the co-requisite remediation data has statistical validity.  I tend to doubt that, as we use a peer review process to judge statistical validity … we we know that has not been the case for the co-requisite remediation data we are generally seeing (especially from the CCA).  The quality of their data is so bad that there would be a failing grade in most introductory statistics courses for a student doing that quality of work.  It’s discouraging to see policy leaders and administrators become profoundly swayed by statistics of such low quality.

Reducing ‘remediation’ to one measure is an obviously bad idea.

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MichMATYC 2016 conference schedule (Saturday, October 15)

The MichMATYC conference planners at Delta College are doing the final tuning of the program for October 15 (2016).  Here is a almost-final schedule of sessions:



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The Right Answer is Not the Thing

This is not another post on assessment, though the content is related.  The central theme in this post is faculty being wise about the process of helping students navigate through mathematics in an efficient manner (something we might call “learning mathematics”) 🙂 .

As context, I want to share part of a lesson from our math literacy course.  Like many such courses, we both use accessible situations and recognizing patterns in the learning process.  This particular lesson uses interest (simple and compound) with these basic steps in the process.

  1. We deposit $500 in an account that pays 6% interest each year, on that $500.  Find the interest earned in the first 4 years by competing the table.  [The table shows a row for each year.]
    Find the total money by adding the interest and the original $500.
  2. We deposit $500 in an account that pays 6% interest each year, on the current balance (including prior interest).  Find the interest and current balance for the first 4 years by completing the table.  What is the total money for this account?
  3. Which account results in “more money” for us?
  4. We found the current balance by calculating “0.06 × 500 + 500”.  Is there a way to simplify this calculation so there is only one multiplication and no addition?

Of course, much time is used in the first two steps.  Students often have misunderstandings about percents, but these are motivational questions … as is #3.  However, the learning in the problem is all about the fourth step, which is looking for “1.06 × 500”.

Many teachers will present the 4th question in a manner that defeats the purpose of the question … “we added 6% to 100%; what do we get?”  This approach ‘works’ in that many students will see how we got the 1.06, and we feel good that they got the right answer.  Unfortunately, we just avoided all of the meaningful learning in this context.

First of all, students need to really know that percents do not have any meaning by themselves.  When we say “added 6% to 100%”, we have reinforced the basic misunderstanding that percents work like decimals in all situations.  It’s easy to determine if students have this misunderstanding by asking a variation of the classic question:

We had a 10% decrease in pay last year, and this year we got a 15% increase in pay.  Our current pay is what percent increase or decrease compared to the pay before the decrease?

This problem is tough for students because it does not explicitly state the core situation … that the base for each percent is the current pay … and we might think that this is the main reason we get the wrong answer “5% increase”. However, even when this fact about the base is pointed out, students continue to add the percents.

Secondly, the “we added 6% to 100%; what do we get?” question divorces the situation from the algebraic reasoning.  We’ve done adding of fractions, where a common base is required.  Somehow, with percents, we are comfortable leaving the base out of the problem when this produces more ‘right answers’.  Each of those percents has a base, which happens to be the same number in this ‘interest’ situation.  A more appropriate instructional move is to provide a little scaffolding:

Let’s write 0.06 × 500 + 500 this way:  (0.06 × 500) + (1.00 × 500)

Remember how we added 4x + 2x?  We got 6x.

Does that suggest how we might do the adding first?

Now, this instructional move will not make the problem easy.  The goal with this move is to connect the new problem to something fundamental in mathematics:  “like” things can sometimes be added.  Having the right answer without applying this concept is not learning any mathematics.

In our Math Lit course, this lesson introduces the concept of ‘growth factor’ which is then used as we identify sequences that are linear versus exponential.  That discrimination in sequences can get quite sophisticated, though we generally keep the level reasonable for the needs of the course and students.  The phrase ‘growth factor’ is used temporarily until we consider declining situations … however, this “adding to get one multiplication” is behind all exponential models.

Unrelated to the main point of this post, it’s interesting that many of us think of the number ‘e’ when exponential models are being discussed.  There are, of course, very good reasons why that is the most commonly used base in mathematics; unfortunately for the learning process, using base ‘e’ presents a disguise of the direct process involved in the situation … a multiplicative factor based on a percent increase or decrease.  I don’t see using ‘e’ prior to a pre-calculus course, in terms of helping students.

Back to the main point … whether you are teaching Math Literacy, Algebraic Literacy, or even the old-fashioned courses, “right answers” are a poor measure of the quality of learning.  The learning process itself needs to be richer and more valid than using a measure known to have limited validity.

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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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