Flip? Reverse? Teaching?

For some reasons, math teachers seem to change their classroom behavior because of popular media … perhaps more than we do based on cognitive psychology, learning theory, and research.  It’ as if we think “a million ‘likes’ can’t be wrong” … and , as if we think that there is something new about the structures of ‘flip teaching’ or ‘reverse teaching’.

In a typical flip-teaching structure, students get their lectures outside of class time.  Class time becomes mostly individual work and small group time.  People who use this model report that it is popular with their students, and I have no doubt that this is true.  The anecdotes also tend to report that students are more engaged; I find this part a bit humorous — the structure pretty much demands more student activity.  I would bet that we could generally increase student ‘activity’ just by not talking. 

There are some questions about ‘flip teaching’ that are really questions about what we see as the goals of learning in our classroom.  Are we about basic skills?  Mastering procedures?  Reasoning?  Application?  Seeing a coherent whole?  I find it sad that we are drawn to structures that direct students towards the smallest aspects of mathematics instead of the largest.  We tend to worry a great deal about whether a student will be able to perform a known procedure.  (And, yes, I know that there are such things as high-stakes testing which tend to reinforce this bias.)

Flip teaching is a new name for an old idea.  My first teaching experience was in a program where developmental classes were run like that … students used materials before class time for instruction, and we focused on questions during class.  Originally, this program was individual classes in this format and eventually we blended all classes in to one large program.  I generally observed that students would mimic procedures, often without understanding — the same problem that we find in other classes.

Flip teaching is not a solution for learning mathematics.  If you need a new structure for motivational purposes, flip teaching can work for a while.  While you have this ‘break’, think seriously about real solutions that help the learning of mathematics.  You won’t find these solutions on YouTube or KhanAcademy … you will find them in AMATYC and NCTM and MAA and other professional groups.

I will point out that the emerging models — AMATYC New Life, Carnegie Pathways, and Dana Center Mathways — all have a focus on the learning environment in the classroom with the faculty “facing forward” (not reversed).  We use resources outside of class time, and we also emphasize directed activities during class to build understanding of mathematics.  Faculty are professional designers of learning experiences.

 
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Are Modules the Answer for Developmental Mathematics?

The number of institutions implementing modules in developmental mathematics continues to increase, which I expect to continue for another year or two.  Over the next 5 years, I expect most of these institutions to shift to other models and solutions for their developmental mathematics programs.  Perhaps you can think of some reasons why colleges would try modules now and then replace modules.

Our context for this problem is complex, with multiple expectations for developmental mathematics and multiple measures of current problems.  Modules are appealing because of the clear connection between a modular design and some measures of current problems — low pass rates and low completion rates in particular.  For a change to survive in a longer term, the methodology needs to address enough of the basic problems to be sustainable.

When we started the New Life project in the Developmental Mathematics Committee of AMATYC, we asked a set of national leaders in the field to identify the basic problems they saw.  In analyzing that input (done via email, primarily), the problems could be clustered in a few basic categories:

The content of developmental mathematics courses is not appropriate for the majority of students.

The typical sequence of courses has too many steps for students to complete in a reasonable amount of time.

The learning methods emphasized in most programs were not effective, and do not reflect the accumulated wisdom about learning and cognition.

Faculty, especially in developmental mathematics, were professionally inactive and they tended to be isolated.

Faculty were not using professional development opportunities, both due to lack of information and due to lack of institutional support.

Modules are often selected based on rationale of content and sequence.  However, when we look deeper at the content problem, the issue is a very basic one: the typical developmental mathematics sequence emphasizes symbolic procedures presented in isolation from both applications and other mathematics.  In other words, completing a developmental math course typically does not result in a significant increase in the mathematical capabilities of students … the learning was of the type that is quickly forgotten.

One reason, then, that modules will tend to be a short-term process is that the design does not generally address basic content problems.   A modular program makes it easier for students to complete; a consequence of this is that the content is deliberately compartmentalized and isolated.  Module 4 is independent of Module 3; the learning is not connected, nor is there (normally) a cumulative assessment at the end of a sequence (like a final exam). 

I am hoping that you are thinking … “Wait a minute, modules can do more — the learning can be connected, and we can have a cumulative assessment”.  Great, good job thinking critically.  However, every single modular implementation I am reading about focuses on the independence of the modules, and none have a ‘final exam’.  Some colleges will eventually try to address this problem.  The challenge is that doing so is fairly difficult, and will tend to increase cost.  [You might have noticed that cost was not a general problem as identified by leaders in the profession.]

The learning methods are also a problem in the typical modular design.  Modules have a high probability of using online homework systems; these systems tend to be limited to symbolic procedures.  More fundamentally, though, I see modular programs as missing the learning power of groups and language.  Modular programs tend to be individual-based; social settings, such as small group work, are either difficult to manage or just plain impossible.  Language (meaning speaking and writing) are often quite limited; as in traditional developmental programs, modules tend to emphasize the correctness of answers as a measure of learning … as opposed to quality of work, written explanations, or spoken explanations.  Therefore, I generally expect that modular programs will result in levels of learning that are statistically equal to the programs they replace; this (if true) is enough of reason for colleges to leave the module design in a few years.

Some modular designs have addressed some of the problems related to faculty … at the point of implementation, and in limited areas.  Not enough for long-term viability.  We, the faculty in developmental mathematics, have much to do.  The overwhelming majority of us are not engaged in any professional activity (beyond a few hours of work per year at our own campus); we generally do not attend conferences, we don’t join AMATYC and state affiliates; we don’t read professional journals, let alone publish in them.  We need to develop a deeper understanding of our profession; in particular, we need to be proficient in analyzing learning mathematics as a matter of mathematics and of cognition.  We need both deeper toolsets and the knowledge about best uses for those tools.  None of the modular designs I read about have a long-term strategy for supporting faculty.

The designs I often call “the emerging models” all deal with multiple problem areas, resulting in long-term viability.  The emerging models (AMATYC New Life, Carnegie Pathways, Dana Center Mathways) address content, sequence, learning, and faculty issues.  Over the next few years, you will begin hearing of institutions who had implemented modules switching to one of these emerging models.  We all are committed to helping our students, and these models provide a better solution.

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Diversion, Advancement, and Math

Developmental education continues to be a subject of much research, and even more discussion.  A few studies on remediation have become the drum beat for some foundations and projects that call remediation a failure.  As is normal in scientific research on a human activity, the landscape is much less clear … and much more attractive … than the ‘failure’ studies imply.

A recent report authored by Judith Scott-Clayton and Olga Rodriguez for the National Bureau of Economic Research describes efforts to study a broader range of possible outcomes.  The report is titled Development, Discouragement, Or Diversion? New Evidence On The Effects Of College Remediation and available at http://www.nber.org/papers/w18328.  Statistically, this report uses a discontinuity (regression-discontinuity, or RD) approach  as did several other studies; they expanded the research by using high school records and other college records. 

The major findings of the study are:

We find that remediation does little to develop students’ skills.

We also find relatively little evidence that it discourages either initial enrollment or persistence, except for a subgroup we identify as potentially mis-assigned to remediation.

The primary effect of remediation appears to be diversionary: students simply take remedial courses instead of college-level courses.

The first finding is consistent with other RD reports.  My own explanation for this discouraging finding is that developmental courses focus more on ‘facts’ than ‘skills’ or understanding; our students experience a long sequence of tasks related to what they need, but we do not generally provide a cohesive treatment that addresses the need directly.  The second finding shows some differences with other studies, which suggest that developmental courses might lower persistence.

The third finding is the one that raises concerns for me.  In their report, the authors suggest that diverting students from college-level academics might be a valid approach for a group of students with a lowered probability of success.  While I agree that there are some parts of the ‘developmental population’ with such extreme learning challenges that diversion is best, I do not agree in general that diversion is healthy or desirable. 

Is developmental education about diversion, or are we about advancement?  For us as professionals, this is one of those fundamental questions that determines our classroom behavior and expectations of students.  I see myself as an ‘advancement fanatic’; not only do I see advancement as the fundamental goal of developmental education (even mathematics), I believe in the advancement goal for every single student in spite of the predominant temporary evidence that this might not be reasonable.

One of my students said “You believed in me when nobody else would”.  I do not know if this student will reach her goal of becoming an elementary teacher; I hope she does … children deserve to have a teacher like her.  Even if she does not make it there, I believe that she can.  Quite a few of our students are in our courses because other people gave up on them.  I will not give up.

Diversion for developmental math students is the last resort for us.  Advancement is the thing.

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Algebra for Everybody, and Algebra for Nobody

Algebra is so basic that all students need to have a good grasp, and adults without this capability will be limited in their economic and social choices.  Algebra is so esoteric compared to daily life and work that only those in STEM careers need to bother learning it.  Both of these statements can be true; the apparent inconsistency is based on what is meant by the word ‘algebra’. 

As such things normally happen, there is another article suggesting that nobody needs algebra (see Andrew Hacker’s article at http://www.huffingtonpost.com/2012/07/30/in-new-york-times-op-ed-c_n_1719947.html) and a response by Borwein & Bailey (see http://www.huffingtonpost.com/david-h-bailey/algebra-is-essential-in-a_b_1724338.html).  Reflecting our society in general, we tend to view issues as a binary choice — if it is good, all people must; if it is not good, nobody should.  The Bailey & Borwein response is well written, and reflects a balanced point of view.

As a mathematician, I view algebra as a language system used to describe and manipulate features (whether known or not) of the physical world based on arithmetic operators .  Basic literacy in this language system is essential in both academia and ‘real life’; translations into and out of algebra are the most basic literacies, followed by different representations (symbolic, numeric, graphic).  

Unfortunately, the algebra of mass education tends to focus on procedures and complexity of limited value to anybody combined with a focus on solving algebraic puzzles, as if completion of a crossword puzzle is a basic skill for a language.    True to our current binary approach, people who agree with a literacy approach will invest great effort to avoid all procedures, complexities, and puzzles.  The truth is that we undertake these problems ourselves just for fun, and this is one element in our transition to being mathematicians; how are we to capture the attention of potential STEM students if we avoid the fun stuff? 

As a language system, basic literacy in algebra means that a person can read the meaning of statements; transformations to simpler forms is based on that meaning.  I failed to help my students in the algebra class this summer … I know that because students could distribute correctly in a product but failed routinely with a quotient.  [In case you are wondering, the problems involve a 3rd degree binomial and a 1st degree monomial; in the division case, many students ‘combined’ the unlike terms in the binomial instead of distributing.  This is a basic literacy error; very upsetting!]

Hey, I know … nobody needs to distribute algebraic expressions on their ‘job’ (except us!).  That type of  reason is enough for me to conclude that we do not need to cover additional of rational expressions (prior to college algebra/pre-calculus); that process is complex, and is based on a higher level of understanding of the language.  Distributing is a first-order application of algebraic literacy; avoiding that topic means that we present an incomplete picture of algebra as a language.  A pre-college mathematical experience needs to provide sound mathematical literacy — including algebra.

Everybody needs algebraic literacy, as part of basic mathematical literacy.  We can design courses that provide the needed mathematical literacy as a single experience — no need for a numeric literacy course (‘arithmetic’) and an algebra course and a geometry course; all of that, plus some statistical literacy, can be combined into one course.  This is the approach of the New Life model, and is imbedded within the Quantway & Statway (Carnegie), and in the New Pathways (Dana Center).  I encouage us all to include some transformations (‘simplifications’) in the algebraic language.

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