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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Problem Solving … and Learning Mathematics

Our Math – Applications for Living course is sometimes used as a last option; students try passing the intermediate algebra class, and (after 2 or 3 tries) an adviser says that they have another option.  This is not true for all students in the course, though it is a common path to my door.  The result is a class with some very anxious students, and many who doubt their ability to solve ‘word problems’.

Math – Applications for living is all about problem solving; all topics are verbally stated.  We had an interesting experience last week when we did an example with a simple statement:

The distance from the Moon to the Earth is 3.8 x 10^5 km.  A light-year is 9.5 x 10^12 km; in one second, light travels 3 x 10^8 meters. How long does it take light to travel from the Moon to the Earth?

The problem presents to issues to resolve: the operation to perform, and making the units consistent (meters and km).  A few students knew to divide distance by speed to get time; if they did not already know this, it did not help much to solve the D=rt formula for t.  We explored the problem by working with rates (as we have been doing for most unit conversions); this helped a little more.

We got frustrated, however, with the km and meter conversion in the same problem.  After about 10 minutes of discussion, some progress was made.

In working through these struggles, more than one student said something like:

Can’t you just show us how to solve these in a way that we already understand?

Of course, it is exactly this gap between current understanding and present need that causes learning to happen.  As a problem solving issue, this is essentially a statement of what problem solving is … as opposed to exercises.  In the most encouraging manner, I told the class that this tension they are frustrated with — is the zone where we will learn something.  I stated, with emphasis, that if I did not create situations where there was a gap like this that they would leave the course with the same abilities as when they started.

I’ve been talking with faculty in some other programs at my college about the mathematical needs of their students.  The first thing they say is always ‘problem solving’, and they don’t mean solving a page of 20 ‘problems’ using the same steps.  The second thing they say depends on their program, and a surprisingly large number of them say ‘algebra’ is the next priority — in spite of the fact that algebra is often de-emphasized outside of the STEM-path.  In the Math – Applications for Living course, we use algebraic methods when useful, as it is when solving problems with percents.

In the larger context, all learning is problem solving.  A learner faces a situation where existing knowledge is not sufficient, and the gap is completed by some additional learning.  I believe that this statement is true regardless of the pedagogy a teacher uses, whether active or passive for the learner.   I do not agree with a constructivist viewpoint, especially the more radical forms; however, there is a basic element in the constructivist view that is true, I believe — knowledge is built as a result of gaps.  I believe that teachers can (and should) model the process of filling the gaps, and explaining the reasoning behind ideas that can help.  Learning math does not need to involve students stumbling through to discover centuries of mathematics; we can both guide and be a sage in the process.

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