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!

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.

Bias in Mathematics Education: Did You See an Elephant?

People in the profession of education — including mathematics education — are prone to exhibit some common modes of reasoning.  We tend to value linearity within learning, compliant students, and evidence which supports our current outlook.  Until we overcome this bias in evidence, there is no hope to make real progress for our students.

A concept used in social science research (which is what education is) is ‘confirmation bias’.  Although the image above refers to ‘facts’, for our purposes the word ‘evidence’ might be a better fit (and I also include the phrase “established scientific research”).  We are so cursed by this bias that we seldom are aware that we are extremely biased.

Some examples:

• At a conference, we select sessions dealing with what we are currently working on … ‘what we want to hear’ becomes a guarantee of what we hear.
• In our department, we discuss issues almost exclusively with colleagues who are known to agree with us on problems and solutions.
• When we read professional material, we seek out mathematics or pedagogy that we are already using.

My concern today is not the ‘other’; the concern is us.  Although it is certainly true that Complete College America (CCA) and the organizations bringing us the “Core Principles” of remediation are suffering from severe confirmation bias, their problem would not be able to impact us … unless we are already in a weakened rhetorical state.

Our theories are often as immature as the mythical blind mind finding out what an elephant is like — we experience 1/100th of the entire domain, and conclude that we have a theory for the entirety.  Something like “students have short attention spans, so never try to have a prolonged exploration of a complex topic” or “yes corequisite remediation works after all” or “showing students I care will result in them learning”.

Not only do we have confirmation bias about the learning process, but we have the same type of bias about mathematics itself.  If you don’t rebel at the phrase “mathematics hasn’t really changed”, you have not been paying attention.  If you expect that mathematics remains stagnant, that is exactly what you will see — in spite of overwhelming evidence which conflicts that point of view.

The phrase “growth mindset” is all the rage. Apparently, this only applies to students.

Every Student Learns … Every Day!

There was a period in my teaching when the core principle was “deep assessment”.  This “Deep Assessment” idea was that every key outcome within a test would be assessed three times BEFORE the test for each student, in class … at the intro level when starting the topic, at an intermediate level after the first usage, and at a higher level as part of the review for the test.  I would tell my colleagues that I assessed the important ideas 3 times, and they seemed to think this was good … and so did I, until I thought about my observations.

Sure, it helps students to have multiple opportunities (assessments) on key ideas and get instructor feedback.  I would spend considerable time grading these assessments, and writing feedback.  This very logical structure did, in fact, work for a portion of my students.  As I thought about this, however, most of the students who benefited were doing fairly well before my class.  You know, they were mostly reviewing stuff they once knew well.

However, the ‘deep assessment’ strategy missed some of the students in the middle (of need), and missed almost all of the students with the greatest need.  Do our classes exist to serve just some students, or all?  Hopefully, you think about that question on a regular basis.  There are direct connections between that question and the posts made recently about equity Policy based on Correlation: Institutionalizing Inequity.

My current guiding principle is “everybody learns every day”.  I seek to provide some benefit to all students.

Many readers are going to be thinking … “What’s so different about that? Don’t we provide the opportunity to benefit every student in each class?”

Nope, we don’t.  Think about this … “learning” depends upon readiness and engagement, combined with communication.  We fail to address the readiness almost all of the time.  By this I am not referring to course prerequisites or placement tests; those are gross measures of overall abilities, and have very little to do with learning.

I’m referring to a thorough analysis of specific knowledge and understanding needed to learn a certain topic.  Let’s look at “basic function ideas” as you might cover in an intermediate algebra or college algebra course.  Learning basic function ideas (notation, interpretation, points) at an introductory level.  The readiness includes:

• input versus output
• simplifying expressions
• substituting values
• horizontal and vertical number lines
• ordered pair notation and meaning
• point plotting as opposed to slope

The image above shows ‘puzzle pieces’ between the person and the learning.  Vygotsky used a phrase “zone of proximal development”, which is related to what I am talking about.  [Vygotsky was primarily a developmental psychologist, so his results are indirectly related to current learning sciences.]

The ‘ready to learn’ criteria is always there.  If we ignore it, we only serve part of the students.  On the other hand, if we tell students that they need to ‘review’ something before the new stuff, we expect the weakest students to do the more complicated process without our direct support and advice.

I’m teaching developmental math, not ‘college level’, so my dive into this is really intense.  Every class day, we start with a team activity which both checks on the readiness and begins the process of learning today’s stuff.  We might spend 20 minutes doing the activity, followed by 10 minutes of reviewing it as a class; my goal is to get everybody ready, and have everybody learn every day.  Small teams (3 to 5) does a pretty good job of keeping everybody involved, and making sure that everybody is learning.

In a college level course, we could still use a team activity on readiness.  Depending on the topic, we might only need 10 minutes doing it, and 5 minutes reviewing it.  In other cases, the ‘readiness to learn’ activity might occupy the majority of the class time.

I can’t tell you that my ‘plan’ is perfect; that’s a unreasonably high standard (even for me 🙂 ).  However, I can tell you that this “everybody learns everyday” approach does wonders for attendance and participation.  My students with the greatest need still have gaps, but they are smaller.  The ‘middle’ students tend to look more like the high-quality (reviewing) students.

We know that ‘attendance’ is highly correlated with success in mathematics.  Students with greater learning needs get easily discouraged when our classes do not provide them with much learning — either due to lack of readiness (at the detailed level) OR due to our class structure not engaging every learner.  “Everybody learns everyday” minimizes this systemic risk, without harming the higher achieving students.