Prologue … What was this blog?

This blog is officially ‘inactive’ as far as new posts and comments are concerned.  This ‘prologue’ post is my final commentary.

Goals and intentions:
When the three major professional efforts to change developmental mathematics began in 2009 (AMATYC’s New Life Project, Carnegie Foundation’s “Quantway & Statway”, and the Dana Center’s “Math Pathways”) people in the profession of developmental mathematics were facing a unique combination of opportunities and external forces.  This blog was begun with a goal of supporting fellow faculty members in their adaptations and challenges.  The words “hope” and “engagement” were central to my thinking at that time.

• Hope:  Dev math educators were being criticized and marginalized within the higher education community.  I was struck by the general comparisons to the gay community as represented by the film “Milk”, and particularly by the statement attributed to Harvey Milk that “you have got to give them hope”.  Without hope, people do not work toward shared goals and our profession is therefore diminished.
• Engagement:  Among administrators, faculty are sometimes classified as “green light” (will go along with doing something new) or “red light” (resists efforts to make changes).  While I recognize that such distinctions have some validity, I believe that the community can not leave any groups ‘behind’.  Nelson Mandela, in the movie “Invictus”, makes a point of including former ‘enemies’ when building collaborations that resulted in a new government.  I believe that progress is only made to the extent that all groups of faculty are deeply engaged in a process of critical thinking related to the problems in our shared space.

At the time I began this blog, the focus was on developmental mathematics (as seen in the name ‘dev math revival’).  If I was starting again, I would instead focus on the broader set of problems of ‘math in the first two years’.

Results (which types of Posts were ‘read’?)
In the 8.6666… years of maintaining this blog, I have published a bit over 500 posts.  Being a mathematician with a statistician’s interests, I have obviously watched the data on which posts got read.  I would not have anticipated the patterns observed:

1. The most popular posts dealt with ‘college algebra’ or pre-calculus.   Two of the top five posts (as measured by ‘hits’) are in this category.  Although not anticipated, I find this interest reassuring given how obsolete that part of our curriculum is.
2. One post (about ‘plus four’ and statistics in math ed) is the all-time leader for hits.  My guess is that there were hundreds of desperate stat students looking for help on adjustments in statistics and their search turned up my post.
3. The second most popular type of posts dealt with co-requisite remediation.  When the blog was begun, this terminology did not yet exist; however, the apparent popularity of this type of post is consistent with the goals which were in mind for the blog.
4. “Math Lit” was a big draw for the blog in the beginning; not so much anymore.  I worry that the reason is that people are either stuck in the traditional curriculum or they’ve been mandated to to co-requisites.  Given the logical limitations to co-requisites, and the generally awful nature of the traditional curriculum, this decline of interest is discouraging.

From time to time, I have posted about mathematics directly –such as “PEMDAS” (the terrible thing that it is) or basic algebra topics.  Some posts were about the teaching of mathematics.  Neither of these groups were terribly popular, and I am not concerned about that.  [The community has lots of resources for those types of ‘posts’, many of whom are more skilled and more articulate then I.]

Farewell
My retirement means that I will no longer be making posts.  The blog will remain visible, and might even help some people in the future.  As found for writing in general, my work in writing for this blog has helped me clarify my reasoning and sharpen my conclusions.  I am grateful for my colleagues who have taken the time to read and possibly engage with some commentary (even when you & I disagree).

Thank you!

Curriculum and Pedogogical Reform In College Mathematics: Regression

What SHOULD we teach?  How SHOULD we teach?

Those questions underlie discussions of professional standards.  In fact, AMATYC had a new project in the early 1990s with the acronym “CPR” — Curriculum and Pedagogical Reform for the first two years (sometimes listed as “CPR-MATYC” to emphasize the AMATYC connection).  The leaders of this work were knowledgeable professionals at the forefront of college mathematics who wanted to provide a set of standards that would help lead the practice of teaching college mathematics across the country.  The leaders knew that some elements of such standards would conflict with common practice, and that curricular inertia would cause others to have a negative initial reaction.  However, the leaders also sought to create the best possible reference for the profession.  Although they could not know this, such standards also provide a roadmap to counter external threats developing a decade or so later.

The result of CPR was the original AMATYC Standards, given the title “Crossroads”.  That document does, in fact, contain a fair amount of direction on “should” in terms of content and pedagogy.

Updates to the standards, however, have avoided direct statements on should.  The 2nd standards (Beyond Crossroads) focused on a process cycle instead of updating the content and teaching standards in the original.   The latest standards (#3, “IMPACT”) goes further away from ‘should’ statements to become primarily a collection of trends and practices.

For those curious about such matters, I am thinking about this now because I am sorting through old (and really old) files as part of retirement.  Reading it again, I remember excitement of the original CPR work (I did some reviewing and position paper writing) compared to the discouragement of working on the most recent standards.  I’ve been fortunate to have had a role in all 3 AMATYC “standards”, though I find myself discouraged by the trends in the actual products.  Apparently, we collectively think it is fine to teach awful mathematics in despicable ways as long as you incorporate an occasional ‘cool’ trick in class.

Is this the best we can do?

Mathematical Reasoning … Can We Recognize It? Do We Allow It?

My department has been discussing the concept of ‘rigor’, which usually invokes some variant of ‘mathematical reasoning’.  Definitions of either concept often involve communication and flexibility, though our practices may not encourage any of this as much as we would like.

In general, if a learner is simply showing the same behaviors (and mathematical analysis) that have been described … justified … and demonstrated during the class then I do not see much rigor.  Building mathematical reasoning involves exploring something new, and sometimes shows in failed attempts to solve a problem

Of course, the label ‘failed’ is an artificial description based on some arbitrary standard (like a correct ‘answer’).

Recently,  I graded my final batches of final exams.  I want to share two examples of mathematical reasoning which pleased me, in spite of the fact that the solutions submitted by the students were at odds with the expectations on our grading rubric.

Here is the first, in our Math Lit course:

Our rubric called for students to use the area formula once — since that information is shown in the problem.  We actually did very little work in this class with compound shapes.  I was pleased with this student’s analysis, which exceeded anything shown in class.

The second is from our Intermediate Algebra course (which is extinct as of next month):

In this case, our rubric was based on the expectation that students would ‘clear fractions’ (which ain’t the “good thing” it once was).  A handful of students did a reasoning process like that shown above AND recognized that the equation was a contradiction.  [The specific work above is from a student who only got a 2.5 grade in the class.  She has a lot more potential, which I did tell her.]  Although I don’t have any evidence, I can be hopeful that the emphasis on reasoning during my classes contributed to these students seeing a ‘different way’ to solve these problems.

In the algebraic situation, which is more important — to ‘always check solutions’ (awful advice!) or to ‘recognize a contradiction’?  Mathematically, there is no contest; connecting type of statement (contradiction in this case) with the solution set (none) is a fundamental concept in basic algebra … a concept actively discouraged by much of our teaching.

Using the phrase ‘mathematical reasoning’ does not mean that we build any mathematical reasoning in our students.  Our courses cover too much mechanics; “a thousand answers and a cloud of dust’.  Let go of trivial procedures (like extreme factoring of polynomials and simplifying radicals of varying indices with complicated radicands, or memorizing a hundred trig identities).

The easy choice is to emphasize procedures and answers.  The fun choice is to emphasize reasoning and analysis — even in basic courses.

School Mathematics can NOT be Aligned with College Mathematics

How do we help students become ready for college mathematics?  How do avoid students earning credit for learning that should have occurred before college?  Perhaps our conceptualization of these problems is flawed in fundamental ways.

As I write one of my final posts for this blog, I am pondering history and future … and the intersection called the present.  Some of this pondering has been pleasant reflection, while much of the pondering has been either professional regret or stimulating conjecture.  I hope to put some of each ‘pile’ in this post.

[Here, “common core” is a place-holder for school mathematics.]

As usual, a problem and its solutions are based on definitions.

• School mathematics is defined operationally by the curricular materials and accepted pedagogical practices.
• School mathematics is usually characterized by a closed system focused on experiencing a constrained subset of mathematics at constrained levels of learning.
• College mathematics is ill-defined with conflicting goals of historical course content and service to the discipline.
• College mathematics is characterized by a closed system serving history competing with components seeking to build mastery of modern mathematics.

The fact that one system is reasonably well-defined while the other is ill-defined suggests that any goal of alignment is unreasonable.  In other words, the reasonable-sounding effort to create a smooth transition from one level to the next is foolish.

Just as groups sought to deliberately disrupt the work of developmental education, groups using ‘alignment’ are also seeking to disrupt the world of college mathematics.  In their view, college mathematics should be more like school mathematics where the system is well-defined operationally by a limited collection of curricular objects (‘courses’).  The presumption is that the core of the college mathematics system is valid and that we can apply the school mathematics process to standardize the alignment.

All of this ignores two related and critical flaws:

1. School mathematics was (nominally) designed to prepare students for college mathematics.
2. College mathematics (as known today) is a collection of obsolete tools along with a bit of valuable mathematics.

At a CBMS meeting a few years ago, I raised the question “When are we going to question the college mathematics courses consisting of excursions into issues that we don’t care about?”  Some in attendance thanked me for saying that we should change the applications in our courses; sadly, that is not at all what I was saying.  I was suggesting that much of college mathematics presented mathematics that we no longer care about as mathematicians.

Advocating for alignment does not mean such alignment is possible; it’s not.  Advocating for alignment does not mean that people support our curricular goals; they have their own agenda (not ours).

Before we worry overmuch about ‘alignment’, we had better make basic corrections to our own system.  College mathematics could be an exciting world for our students to explore with colorful vistas combining symbolic and computational methods supported by conceptual knowledge.  Do not look to MAA and AMATYC to ‘tell us’ how and when … our organizations are too fearful of offending part of ‘us’.

Build local alliances to support experimentation in modernizing mathematics in college.  Do not let ‘alignment’ lock you in to an obsolete and harmful set of mathematics courses.