AMATYC Presentations about New Life Math

To see the presentations and handouts for my sessions on New Life at the 2013 AMATYC Conference and Developmental Mathematics Summit, see the page AMATYC 2013 and Developmental Math Summit 2013    [also known as https://www.devmathrevival.net/?page_id=1807]

Probem Solving in a Digital Age

Whether we ‘flip’ a classroom, use an online homework system, or refer students to Khan’s Academy … our students are using task-oriented videos.  In addition, students have a tendency to see ‘look it up online’ as a substitute for learning something.  As we become immersed in (and dependent upon) the digital age, can we still work on problem solving or critical thinking?

One of the sessions at this year’s AMATYC conference dealt with the topic ‘stop the assault on critical thinking’.  In the session, they played the roll of a short video on subject (related rates in calculus, maximizing a function in pre-calculus, or unit conversions in a liberal arts math class).  The audience experienced something like a typical 3 minute video on that topic, and then we talked about how this supported critical thinking (or not).

The next session was one by Jim Stigler on ‘using teaching as a lever for change’, though he talked more about the futility of identifying specific teaching activities as being ‘effective’.  Dr. Stigler did include 3 aspects of teaching that are connected to improved learning — productive struggle, connections, and deliberate practice.  Learning is a complex process, and the presence of these 3 factors in the learning environment are connected to improved learning.

So, there is a connection between this research-based observation and the concern about critical thinking, I think:

Discrete learning experiences like short videos focused on successful completion of a task, based on clarity and being easy to follow, are guaranteed to limit both overall learning and critical thinking.

Mathematicians hold critical thinking as a goal to be valued; we want students to be able to flexibly apply knowledge to novel situations and interpret results.  This seems to be a basic problem.  Our students expect math to not make sense, that they could not figure something out; task-oriented videos support this self-defeating belief.

We can not hide from the digital age, even if we wanted to.  However, we can improve our understanding of the factors that contribute to the learning opportunities for our students.  A balanced approach appropriate to each course can help students through the learning process — including the struggle, the connections, and the deliberate practice.  We might even see these digital resources as just-in-time remediation to be used occasionally, rather than seeing the digital material as the basic course.

We need a more subtle understanding of how our teaching can contribute to student learning.  A single belief or methodology will not succeed for our students, no matter how good of an idea we might have.

To see a presentation by Dr. Stigler similar to what he did at the conference, see http://www.salesmanshipclub.org/downloadables/scyfc-Stigler.pdf

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Liberal Arts Math … College Algebra … ??

Once upon a time, colleges and universities wanted a math course for students in non-science fields.  The initial math for liberal arts course was designed for this purpose … a little bit of this, a little bit of that, and light on formality.

Once upon a time, colleges and universities wanted a math course for students who might or might not need calculus.  Since the content focused on ‘algebra’ and it was not remedial, the resulting course was called college algebra.

We might be better off if both titles were banned from the collegiate landscape.

In a way, both courses have a ‘this is not the other math’ type of implication.  We should be able to articulate a positive statement (and title) for what the courses are about.  This is not to say that all courses with these titles are ‘bad’ in some way (though some are bad in some ways).  I know of a few liberal arts math courses which are contemporary in design, with a focus on reasoning and some formality.  Some college algebra courses are actually high quality pre-calculus courses.

Back in the day, there actually were many programs that were not scientific.  Even fields like biology were considered non-quantitative, as were social sciences.  This landscape has changed in fundamental ways over the past 30 years; the fields that require no quantitative background are small in number.  Instead of ‘liberal arts’ math, we should use variations on the more modern ‘quantitative reasoning’ title.

College algebra is a mess.  It’s defined by what it is not (not remedial), and the title is used for 3 basic types of courses (general education, pre-calculus, and prep for pre-calculus).  If we need a general education course, we have better alternatives available now than we did 30 years ago (think quantitative reasoning and intro statistics).  Using college algebra for general education ensures that the course will primarily be a barrier to students completing a degree, and likely makes the course very challenging for faculty to teach.  It’s not unusual, of course; a major university close to me uses college algebra as their primary gen ed requirement.

If a college algebra course is meant to be pre-calculus, then we use the better title — pre-calculus.  Calling it ‘college algebra’ when it is meant to get students ready for calculus implies that the primary factor in calculus success is algebra beyond the remedial level.  I hope that this is not the case!

And, if college algebra is meant to be preparation for pre-calculus, there are larger questions.  Is the course non-remedial?  Are we adding a course to the sequence to have more classes to teach and fewer students completing?  If there is a valid reason for having both college algebra and pre-calculus, I have never seen it … and would appreciate seeing such a reason elaborated.

No, I don’t think we need either title anymore.  They once served a purpose, but Liberal Arts Math and College Algebra are obsolete.  The sooner we stop using them, the better we serve our students.

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Is Calculus Instruction Changing (or Curriculum)?

In our applications for living course, we are finishing our work with statistics.  One situation involves students deciding which situation is likely to involve statistical significance:.

Rolling a die 40 times and getting 5 threes             OR

Rolling a die 400 times and getting 25 threes

This is a tricky thing, as students often focus on the sample size only.  Although this problem presents situations where the larger number is connected with significance, there is no general pattern that says ‘larger sample sizes means significance.’

Within our curriculum, developmental mathematics has dominated the news and much of our work for a long time.  There definitely is a larger sample size; is there a difference in statistical significance between developmental math and calculus courses?  In a basic way, yes, there is — developmental math serves a large group of students with multiple academic problems, while calculus serves a group of students with general academic success.  A 60% pass rate in calculus is not good, and is statistically significant given that most students in the calculus courses are expecting a high grade (while developmental math students often expect low grades).

You can try this as I did — search for ‘reform in calculus college’ or similar terms.  Most of the results of this search will be historical artifacts from the 1980’s and early 1990’s.  What’s with that??

I think we have fallen into the large number fallacy — a larger sample size indicates significance (dev math), instead of analyzing each situation separately.  We should be able to expect an 80% pass rate in calculus 1, given the academic skills of students who typically enroll in that course.  My own college gets about 60% pass, and this seems pretty normal.

For programs which require a 4-semester sequence (Calc I – II – III plus diff eq), a 60% pass rate means that a maximum of 13% will ever complete the sequence.  The likely values are far less — some portion of students are lost between courses even after passing.  I suspect that observed values will be between 5 and 10% — which, coincidentally, is the same range as a developmental math sequence.  [These low values are the result of ‘exponential attrition.]

Recently, I did learn of some work of our friends in the MAA on calculus.  It’s not on reform; rather, their focus is to analyze data to identify what we are doing and what is more successful — the Characteristics of Successful Programs in College Calculus (CSPCC) project.  The web page for their work is http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/characteristics-of-successful-programs-in-college-calculus

David Bressoud is a lead ‘PI’ on this work; he’s written a few articles about their work for the MAA Newsletter, so you may have seen those if you belong to the MAA.  The web site (above) has links to those articles, as well as papers written about their work.

One of the co-PI for the work is Vilma Mesa, who has done quite a bit of work on community college mathematics.  Dr. Mesa did a session at our recent MichMATYC conference on some of the data from community colleges for the CSPCC work, including the contrast between homework and exams in terms of assessment level … and an analysis of prompt types and expected response types (verbal, symbolic, graphic, or multiple).

I encourage you to read the material on the CSPCC web site.

The larger question is this:  Are we doing anything of substance to make basic improvements in calculus?  Or, is our ‘best shot’ using Mathematica and/or MatLab with our students?  I hope that is not the case.

If you are doing some reform in calculus, I hope that you will share your work — the good and not so good.  “Developmental Math” should not be having all the fun!!

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