Towards Effective Remediation: Quantity and Pacing

I’ve been having conversations about arithmetic and similar topics (at the Achieving the Dream conference, and online at MATHEDCC).  In some cases, the conversation was the result of telling people about the New Life model (see https://www.devmathrevival.net/?p=1401).  In other cases, we had been discussing other issues.

So, I have been thinking about related issues.  As a result, I have some ideas of how to frame a conversation about developmental mathematics that might help us make progress.  To start with, we often describe developmental mathematics by using names of courses or by listing topics with implied outcomes.  In our New Life work, we actually started by examining the types of mathematics that students need to succeed, and dealt later with course names and lists of topics.

Our curricular designs are based on our assumptions and goals, which are often unstated.  One of the most problematic assumptions is:

It is effective to deal with 8 general topics, with 10 to 15 outcomes within each, in one course.

There are two fundamental problems with this.  First, the courses we design cover so much ‘material’ that we prevent the learners’ brains from dealing with the associations and connections that are part of learning; this results in most students focusing on just remembering what to do, rather than making sense of it.  Second, the design is based on the absence of prior learning (good and bad) in the learner; this is obviously not true in almost all cases.  Time is needed for us and students to identify where there are conflicts between prior learning and current need, and time is needed to deal with these conflicts.  The result is that we add another layer of ‘learning’, one that is weaker than prior learning; students after our courses are notorious for returning to wrong methods and ideas after our course is done … because we do not provide a method of correction.

We need to slow down; learning is much more complex than having a list of 80 to 120 outcomes.  Since we need to go slower, we must be strategic about what areas to focus on .. trying to do it all (or even most of it) means that we are willing to accomplish little of significance.

This strategic work should be based on our judgments as mathematicians about which mathematical ideas are most important in particular cases.  Do we want work with fractions, or do we want work with proportional reasoning?  If we want both, what are we willing to give up … percents or linear models (as examples)?

We need to do some critical thinking about our goals and purposes, and apply our problem solving skills so that our courses are effective learning experiences for our students.

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Math Lit at Achieving the Dream Conference

I am currently on my way … to Anaheim for this year’s Achieving the Dream conference.  On Wednesday, I will have a poster at the “Emerging Ideas” event (11:00am) about the Mathematical Literacy course (and the AMATYC New Life work); Thursday, I am part of  a workshop (1:45pm) on developmental math … my part is the Math Lit course, and we will have extended time for discussions and questions.  This is my first “AtD conference”, and I am really looking forward to the opportunities and dialogue.

So, I have been thinking about how progress is made in academia — about how a basic change is accepted by large numbers of faculty and implemented at their college.  The AtD “mantra” uses phrases like “move the needle”, “acceleration”, and “progression with completion”; within the official communications of AtD and related foundations (Lumina, Jobs for the Future, etc) these phrases are repeated, and much conversation centers around engaging faculty in this work.  Parallel to this, the groups provide some outstanding professional development on theory and practice related to developmental education.

My hope is that the work of the New Life project touches and excites the values and beliefs of mathematicians and math educators.  Certainly, part of this is developing a better set of vocabulary phrases to communicate about our values and beliefs; the name ‘mathematical literacy’ is one effort to develop such a phrase.  However, vocabulary alone does not produce any change of significance; many prior efforts have failed because a new phrase was layered on to an existing curriculum (like ‘basic skills’, ‘application focused’, ‘mastery learning’).

I am convinced that our survival depends upon basic changes in our curriculum — and in our ideas behind the design of the curriculum; I believe that these basic changes will only happen as we all engage in conversations and even arguments about what things mean and what is really important.  Sure, we will need some resources, which means that we need to convince foundations and grant sources that our work is important; this will mean the strategic uses of phrases like “algebraic reasoning” and others like we use in the New Life work.  However, this is much more about our profession and our work together than it is about better words.

Progress occurs after dialogue; progress will happen when we actively seek to engage all members of our profession in a deep conversation about purposes and values, goals and beliefs.  Indeed, nothing can stop progress from happening if we can do so.

If you are coming to AtD 2013, I hope we have an opportunity to have some of that conversation!

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Mathematical Literacy: Student Capabilities

In our Mathematical Literacy course, we are working through concepts from a numeric point of view with less emphasis on algebraic (symbolic) statements.  This weeks’ content dealt with ratios, scaling rates up or down, linear rate of change and exponential rate of change.  Our work might indicate what our students are capable of, in a general way.

This course is ‘at the same level’ as beginning algebra, which means that we share prerequisite settings for math, reading, and writing; the students are similar, in many ways, to a typical beginning algebra class.  The Math Lit class also has a few students who did not meet all three prerequisites (due to some system problems at the college).

It’s true that students struggled at times in class.  One of those struggles dealt with language processing; we are using nutrition labels as a context for working on rates and scaling.  When students needed to read specific questions and then extract information from the label, most students did not see what they should do.  This is not a matter of mathematical ability or skills; in fact, students who have passed our beginning algebra class often exhibit the same pattern when I see them in the applications course (Math – Applications for Living).  A few students are having trouble with the scaling ideas, which is a non-standard approach; however, since they usually know an alternate method this is not a big issue.

Although I have not done an individual assessment yet, students did not seem to have any trouble with the concepts of linear and of exponential change.  We did numeric examples in two settings, and I observed groups and individuals — no issues spotted.  Most students are having difficulty connecting a situation to a symbolic model — both linear and exponential.  In the case of linear, we did “the salary is increased by 5%” … all of them could calculate the result for a given salary, but few of them could make the transition to the symbolic model (new = 1.05S).  The same kind of thing happened with exponential models.  Since we are not emphasizing symbolic work (yet!), this gap is not a big problem (yet!).

I’ve dealt with exactly the same issue in the Applications course (symbolic models for linear and exponential change), and observed the same proportion of students having difficulty.  The traditional beginning algebra course has an insignificant impact on students’ abilities to write symbolic models for situations — except when the correct key words are used in the problem.  If the problem is stated in a way that “normal” people talk every day, students can not make the connection to symbolic forms (in general).

In some ways, this was a discouraging week.  The difficulty with language is very frustrating; my judgment is that students (and people in general) are far less skilled with the written word than in prior decades.  Basic verbal skills like parsing and paraphrasing are not normally seen.  The transition to symbolic forms seems like such a small step, so that difficulty is troubling to a mathematician.  Our course is designed to build these skills over the course by visiting similar ideas from different points of view; I can hope it gets better!  However, I find it encouraging that these students — even the ones who lack all the prerequisites — are having no more difficulty than students who passed our beginning algebra course.

This Math Lit course is a good class for a mathematician to teach; we deal with basic ideas in detail and work on transfer of knowledge, with an emphasis on problem solving (as opposed to exercises and repetition).   In that work in-depth, we can see where students really do not get the idea and work on creating better mathematical knowledge.

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Best Wrong Answer Ever!! How to not graph a function

I never laugh at a student, though I often try to laugh with a student.

Today, we had our first test in our intermediate algebra class.  In this class, I like to extend the very simplistic work the textbook does with graphing functions; we cover this in class, and students have a small set of practice problems.

Well, on one student’s test, I see this:

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