Where is the Position Paper on Co-Reqs? Math in the First Year?

Two major movements are “sweeping” across the college landscape — co-requisites in mathematics (and English), and “college math course in the first year”.  Those who have “drunk the cool aid” see both changes as progress, while an academic response continues to be more of minor interest and waiting for real data on the impact of the changes on real students.  In this context, a lack of clear communication is equivalent to a yielding of the discussion.  In my view, AMATYC has done exactly that.

I want to make sure that you know of my long and generally positive relationship with AMATYC.  I first attended a conference of the “American Mathematical Association of Two-Year Colleges” in 1987. I had been teaching for about 14 years at that point, and was being impacted by a loss of enthusiasm for the work.  That conference was a major turning point for my professional life, as well as the start of many relationships that continued for decades.

AMATYC produces position and policy statements on a variety of topics, and these generally lag behind the need — understandable, given the processes that allow for broad input from members  This lag time is normally a few years … when ‘handheld calculators’ were first an issue, the AMATYC statement on them was developed about 3 or 4 years later.  When issues came up about credentialing, the statement on qualifications came out about 4 years later.

We are now in the 5th year of the co-requisite infusion.  (Infusion suggests an external agent seeking to modify the internal functioning of a body.).  I don’t believe anybody in AMATYC is even considering a position statement on co-requisite mathematics.  Instead, the conferences are increasingly populated by sessions sharing experiences with implementations.  In the absence of official statements, the presence of multiple sessions on a practice amounts to an implicit approval of that practice.

Do we, as professionals or members of AMATYC, support co-requisite remediation without qualification?

Like good political strategy, the news cycle is being dominated by one ‘side’.  Our silence … individually and as AMATYC … relinquish the power to those seeking to disrupt our work.  [And, yes, the groups have stated that they are working to disrupt our work.  I won’t name them, as I do not want to provide any more PR for them.  It’s time for our side of the story.]

I’ve posted before on the practice of co-requisite remediation; Why Does Co-Requisite Remediation “Work”? and TBR and the Co-Requisite Fraud as well as The Selfishness of the Corequisite Model.  Of course, we should consider new models.  Of course, our use of fundamentally new structures impacting students must be based on scientific evidence and research — not ‘data’ grown specifically to support a particular practice.

The other ‘movement’ (Math in the First Year) is based on some of the worst uses of statistics I have ever seen in academia.  Of course, I’ve posted on that: Policy based on Correlation: Institutionalizing Inequity.  The flaws in this movement are so obvious that I would expect any basic statistics student to spot them within a few minutes.

Do you see the flaw of ‘first year’ in this image?

Does AMATYC have a position statement on ‘math in the first year’?  Nope.  There is a policy on placement, but not on a practice which places students in situations where they can suffer academic harm.  “Math in the first year” basically says that if you get a random student to pass a math course early like the best students do, that all students accrue the benefits of the best students (academic grit, financial stability, etc).

I always strive to be honest, and that involves me divulging that my relationship with AMATYC has faded away.  Perhaps the current leadership is actively working with the committees to develop policy statements.  However, I do know that the latest ‘standards’ (IMPACT) have very little to guide our decisions on such basic issues; the web site for IMPACT has a link, but no content.  What does all of this silence say about us?

Will we continue to be silent?

Was That a Good Class?

I have a class this semester (called ‘summer’, even here in Michigan!), and I have my normal worry about a class.  Are students learning?  In other words, was that a good class?

This cartoon is a bit oriented towards elementary school settings, but the ‘teacher’ comment applies to college settings … our teaching does not mean students are learning.  This observation has resulted in huge increase of so called ‘active learning’ strategies.  I use the phrase ‘so called’ because I see this is a redundant statement of the obvious … learning, by its nature, is active.  I also say ‘so called’ because activity does not mean there is learning.

So, back to my class.  I am using teams every day, with structured activities to support student learning.  Most days, I leave class “feeling good” about how we are doing.  Students are talking and doing, and everybody is engaged with the material.   When a student needs help, their teammates contribute … as do students on other teams; I never know who will do the helping … there is a good level of support in the class. [I use the tag line in class “no student left behind”.]   It’s clear that we have established a social structure in which students are comfortable.

However, this only confirms the ‘active’ part of learning, not the learning itself.  I can take the easy way on this and look at test scores to see if learning is taking place.  However, the starting point (what students already knew) is only known at the general level — not at the granularity of a test.  It would be easy for me to conclude that students are doing well because the test scores are relatively good.  (And they are.)

In the social sciences, there is an awareness that symbols are often confused with what they represent.  Everybody wants the latest smart phone because it’s got a cool image, whether a given person has any need for the features (or not).  The appearance of health is confused with the presence of health.  In a classroom, I think we frequently confuse activity with learning.

I don’t have any magic to reconcile this quandary.  I suspect that being continuously aware of the risk is the best strategy to avoid the pitfall.  Perhaps we need to be less worried about visible activity and pay more attention to the cognitive processes within the learners.    The best measure may be the direct assessment of a conversation with a specific student about a specific mathematical idea.

It’s nice (and fun) to have a very active class with students engaged with their team and the entire class.  This ‘niceness’ likely has only a limited connection to the learning taking place.  When I read articles … attend sessions … study presentations & reports … concerning active learning methodologies, I am left with the impression that these opportunities are very popular along with a perception that most practitioners will not do a good job implementing the ideas.  In fact, I am not sure that I am doing a good job implementing them.  Copying a pedagogy is dangerous practice; I seek to understand the process at a deep level, and look to my students for feedback on whether a pedagogy was successful.  My class atmosphere certainly contributes to very good attendance, though I need to maintain my critical thinking about the processes and outcomes.

We need to have a complex understanding of the interaction between a pedagogy … such as team-based learning (or “PBL”, or “flipped”, etc) … and the needs of students related to the mathematics to be learned.  In many cases, the only gain for implementing a pedagogy is that it reduces boredom for our students (and us); the lack of boredom is certainly not an indicating of any learning taking place.  In fact, I think that many pedagogical ‘methods’ copied from others serves the purpose of fundamentally limiting the depth and rigor of learning:  we focus on a sequence of steps in a process at the expense of understanding mathematics.

Effective teaching is not accomplished by feel-good methods, and learning is not measured by the level of activity visible in a classroom.  Dealing with this complexity is the core issue of our profession as mathematics educators.

Our Biases versus What Students Need

So we are thinking about our fall classes.  Shall we structure them like we’ve done for the past n years?  Shall we do something different?  Perhaps we recently saw a presentation that inspired us.  Such questions deal with the fundamental problem of teaching:  Are the methods I use reflecting my biases about what ‘should’ be’, or are the methods designed to meet the deeply-understood learning needs of my students?

Perhaps it is your belief that collaborative learning is the key.  Why is that?  You might have an analysis which looks like this:

In turn, this type of image is based on research conducted by people who have definite ideas about how learning ‘should’ take place.  Much of the basis provided for collaborative learning is based on the faux theory of ‘constructivism’ in which each student ‘creates’ their own learning.  See http://archive.wceruw.org/cl1/CL/moreinfo/MI2A.htm ]  In the radical form of this philosophy (it is definitely not a theory), there is no external standard for the learning being correct or complete — it is an individual process with internal criteria.  Many advocates of collaborative learning — whether in K-12 or higher education — have a strong constructivist bias (unstated, in some cases).

What do our students need, specifically?  Often, we rely on easy images like this:

In fact, I have colleagues in my department for whom this image is critical to their classroom practice which has remained unchanged for ten years or more.  Just for fun, do a search for “learning pyramid research” or “learning pyramid myth”.  The fallacy of the pyramid is obvious, yet it holds influence over our practice.  [Ironically, most people remember learning about the pyramid in a ‘lecture’.]

What do my students need?  What do your students need?  Start with the obvious answer — they need to learn mathematics.  Specifically, we have a defined package of mathematics illustrated by a set of learning outcomes.  Do they know some of it before our class?  Very likely.  Do they have misconceptions about it? Almost always.  Can we identify those misconceptions?  Oh, yeah.  What does it take to reduce the misconceptions and build a better understanding?  Well, that is the fun part … because the needed treatment varies with the material being learned and the set of students in front of us.

We all approach the teaching problem with some biases — mine involve a socratic process, which is (like constructivism) a philosophy not a theory.  Perhaps you think the key is to make sure every student is responsible for completing their portion of a group process, and to shift this portion over time.  Perhaps you think a crystal clear presentation is the most important element.  These are all ‘wrong’ to some extent.

My point is that the situation is simple to describe:

• Learning is always an active process
• Talking and explaining (by students) is linked to the quality of their learning
• Our expertise is used to structure a successful learning process

In other words, don’t let your own biases guide or limit your instructional practices.  Instead, focus on key principles like those above and use your expertise to design a learning process.

I’ll give an example from our Math Literacy class, to illustrate what I mean.  We were learning “dimensional analysis” (DA).  As you know, becoming skilled at DA involves basic fraction concepts along with some analysis.  Here is the instructional plan:

1. Students work in teams with the directive “no student left behind” (everybody learns)
2. A series of questions is provided; each one is answered with the direction that everybody agrees; of course, I’m checking in with each team
3. Early questions deal with reducing one fraction; few students understood ‘common factors’ in reducing.
4. One of those questions is then presented in factored form (they are all constants) to get more students thinking about common factors.
5. Another question puts a unit (like “ft”) in the position of the common factor to see if students recognize that a unit can cancel.
6. A dimensional analysis problem is presented with a structure (steps) included; students fill in the blanks to see how to ‘get the answer’.
7. A brief example shows the idea of analyzing units.
8. A dimensional analysis problem (simpler variety) is presented without any structure.
9. A ‘lecture’ component involves the articulation of the principles involved with more examples.  [By the way, the analysis part of DA is easier for my students than the fraction part.]
10. An ending team activity checks to see if every student “got it” so they can complete the homework.

You can see elements of ‘scaffolding’ in this plan.  What may not be as visible is that there are opportunities for identifying misconceptions; for example quite a few students did not think the factored form of a fraction product was equivalent to the ‘original’ form.  Either team members, or the teacher (me), will re-direct in this situation.  Of course, it is important to be realistic — a 5 minute conversation will not overcome years of misconceptions about fractions (or any other mathematical topic).  What I want you to notice is the level of instructional analysis involved; in the case of this DA lesson, I spent a solid 3 hours working on the design and the documents … for a topic I have ‘taught’ many times before.

In the time I have been using this “no student left behind” team approach, I have not encountered a student who did not participate in the process.  Nobody has the additional stress of more leadership on a team than they are ready for; many students develop those skills and become more comfortable.  My team assignments are stratified random samples — each team has low, mid, and high skill students; these teams are shuffled twice later in the semester.  Each class tends to form a learning community, and I end up not being sure how much they would need me.  [One student commented on the course evaluations that “we could not help but learn”.]

Remember, my goal today is not to boast of my great teaching prowess (I try, but I know my weaknesses).  My goal is to suggest that you build a process designed for students to learn each day where you apply your expertise to uncover misconceptions as every student is engaged with the mathematics.    Use teams.  Use lectures. Most of all, make sure your math class is inclusive for every student regardless of their mathematical ‘worth’.

What to do: Intermediate Algebra Dies

What do we do when we terminate our intermediate algebra course?  A new course is necessary, with a focus on reasoning and communication — a more rigorous course (see The Rigor Unicorn).  What to do?

I’ve written before about the necessary demise of intermediate algebra as a college course (see Intermediate Algebra Must Die!! and Intermediate Algebra … the Barrier Preventing Progress).

The traditional narrative is that algebra is a barrier to college success.  Actually, the barrier is obsolete algebra courses (developmental and pre-calculus) which focus on drill more than understanding, and focus on artificial applications rather than fundamental relationships and concepts.  Mathematical Literacy forms a great starting point for a modern curriculum.  When we ‘kill’ intermediate algebra, the solution is to offer an algebraic literacy course (see Algebraic Literacy Presentation (AMATYC 2016).

My colleagues continue to show a dedication to a modern curriculum.  Within the past 5 years, we have dropped both pre-algebra and beginning algebra courses, and replaced them with a Math Lit class in two formats — regular and ‘with review’ (for students with especially weak numeracy skills).  Last month, we made the decision to eliminate our intermediate algebra course.

Temporarily, we will use a revised “fast track algebra” course.  That fast-track course has existed for 3 years, side-by-side with intermediate algebra.  However, the fast track algebra course still uses out-dated content and lower expectations.  Why?  Because there are not available algebraic literacy materials.  Actually, there aren’t any materials dealing with algebra focusing on communication and reasoning.  It’s like the books (and HW systems) are stuck in 1995 in terms of content.  [It’s only 1995 instead of 1975 because of a little bit of technology that some books incorporate.]

Our obituary sadly reads as follows:

After a long life, perhaps too long, the intermediate algebra course at ____ will be removed from life support on December 31, 2019, surrounded by many family and friends.  Intermediate algebra was preceded on the path to the math after life by basic math, pre-algebra, and beginning algebra.  Surviving are a temporary Fast Algebra course, a Math Lit course, and several college level math courses which are also on life support (although unaware of that fact).  A memorial service will  be held at some time in the future when a modern (current) algebra course can take it’s place to serve our students.

We face this change without a sense of closure.  There is some grief at the old course going away (though that was deserved), but there is a dissatisfaction with continuing the same type of algebraic work.  We are generally pleased with the learning occurring in Math Lit, and want a similar course to follow it … which might be called algebraic literacy or algebraic reasoning.

This is “our” problem, where “our” refers to faculty involved with mathematics in the first two years.  We have not written book materials to support a modern (algebraic literacy or reasoning) course.  Publishers have not pursued this, partially because of huge transitions in their “business model”.  That is, however, no excuse for our lack of movement.  There are smaller publishing companies that could undertake this work (XYZ for example) and we also have options with “OER”.

Is the lonely death of our intermediate algebra due to our disinterest?  A lack of understanding?  Are the enthusiasts for a modern course all too ‘seasoned’ (ie, old) to have the energy to write stuff?  Are the younger professionals only thinking about what they have to teach next week and next semester?  There are issues of professional involvement and responsibility behind this lack of newer materials; that is “on us”.

If you have seen value in an algebraic literacy type course, consider developing materials.  Network to find like-minded colleagues.  Collaboration and technology make the work of developing materials much easier.  Where are the people who will create the next level of new materials in developmental and pre-calculus?  Are you one of them?