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’.

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.

Pre-Calculus, Rigor and Identities

Our department is working on some curricular projects involving both developmental algebra and pre-calculus.  This work has involved some discussion of what “rigor” means, and has increased the level of conversation about algebra in general.  I’ve posted before about pre-calculus College Algebra is Not Pre-Calculus, and Neither is Pre-calc and College Algebra is Still Not Pre-Calculus 🙁 for example, so this post will not be a repeat of that content. This post will deal with algebraic identities.

So, our faculty offices are in an “open style”; you might call them cubicles.  The walls include white board space, and we have spaces for collaboration and other work.  Next to my office is a separate table, which one of my colleagues uses routinely for grading exams and projects.  Recently, he was grading pre-calculus exams … since he is heavily invested in calculus, he was especially concerned about errors students were making in their algebra.  Whether out of frustration or creative analysis, he wrote on the white board next to the table.  Here is the ‘blog post’ he made:

This picture is not very readable, but you can probably see the title “Teach algebraic identities”, followed by “Example:  Which of the following are true for all a, b ∈ ℜ.  In our conversation, my colleague suggested that some (perhaps all) of these identities should be part of a developmental algebra course.  The mathematician part of my brain said “of course!”, and we had a great conversation about the reasons some of the non-identities on the list are so resistant to correction and learning.

Here are images of each column in the post:

When we use the word “identities” in early college mathematics, most of us expect the qualifier to be “trig” … not “algebraic”.  I think we focus way too much on trig identities in preparation for calculus and not enough on algebraic identities.  The two are, of course, connected to the extent that algebraic identities are sometimes used to prove or derive a trig identity.  We can not develop rigor in our students, including sound mathematical reasoning, without some attention to algebraic identities.

I think this work with algebraic identities begins in developmental algebra.  Within my own classes, I will frequently tell my students:

It is better for you to not do something you could … than to make the mistake of doing something ‘bad’ (erroneous reasoning).

Although I’ve not used the word identities when I say this, I could easily phrase it that way: “Avoid violating algebraic identities.”  Obviously, few students know specifically what I mean at the time I make these statements (though I try to push the conversation in class to uncover ‘bad’, and use that to help them understand what is meant).  The issue I need to deal with is “How formal should I make our work with algebraic identities?” in my class.

I hope you take a few minutes to look at the 10 ‘identities’ in those pictures.  You’ve seen them before — both the ones that are true, and the ones that students tend to use in spite of being false.  They are all forms of distributing one operation over another.  When my colleague and I were discussing this, my analysis was that these identities were related to the precedence of operations, and that students get in to trouble because they depend on “PEMDAS” instead of understanding precedence (see PEMDAS and other lies 🙂 and More on the Evils of PEMDAS!   ).  In cognitive science research on mathematics, the these non-identities are labeled “universal linearity” where the basic distributive identity (linear) is generalized to the universe of situations with two operations of different precedence.

How do we balance the theory (such as identities) with the procedural (computation)?  We certainly don’t want any mathematics course to be exclusively one or the other.  I’m envisioning a two-dimensional space, where the horizontal axis if procedural and the vertical axis is theory.  All math courses should be in quadrant one (both values positive); my worry is that some course are in quadrant IV (negative on theory).  I don’t know how we would quantify the concepts on these axes, so imagine that the ordered pairs are in the form (p, t) where p has domain [-10, 10] and t has range [-10, 10].  Recognizing that we have limited resources in classes, we might even impose a constraint on the sum … say 15.

With that in mind, here are sample ordered pairs for this curricular space:

• Developmental algebra = (8, 3)           Some rigor, but more emphasis on procedure and computation
• Pre-calculus = (6, 8)                         More rigor, with almost equal balance … slightly higher on theory
• Calculus I to III = (5, 10)                   Stronger on rigor and theory, with less emphasis on computation

Here is my assessment of traditional mathematics courses:

• Developmental algebra … (9, -2)         Exclusively procedure and computation, negative impact on theory and rigor
• Pre-calculus … (10, 1)                           Procedure and computation, ‘theory’ seen as a way to weed out ‘unprepared’ students
• Calculus I to III … (10, 3)                      A bit more rigor, often implemented to weed out students who are not yet prepared to be engineers

Don’t misunderstand me … I don’t think we need to “halve” our procedural work in calculus; perhaps this scale is logarithmic … perhaps some other non-linear scale.  I don’t intend to suggest that the measures are “ratio” (in the terminology of statistics; see  https://www.questionpro.com/blog/ratio-scale/ ).  Consider the measurement scales to be ordinal in nature.

I think it is our use of the ‘theory dimension’ that hurts students; we tend to either not help students with theory or to use theory as a way to prevent students from passing mathematics.  The tragedy is that a higher emphasis on theory could enable a larger and more diverse set of students to succeed in mathematics, as ‘rigor’ allows other cognitive strengths to help a student succeed.  The procedural emphasis favors novice students who can remember sequences of steps and appropriate clues for when to use them … a theory emphasis favors students who can think conceptually and have verbal skills; this shift towards higher levels of rigor also serves our own interests in retaining more students in the STEM pipeline.