Dev Math: Where Dreams go to Thrive

In response to data showing the exponential attrition of long sequences of developmental mathematics courses, some people are using the quote “developmental mathematics is where dreams go to die”.  This phrase has been one of the most influential statements in our field over the past 5 years — not because it is true but because people (especially policy makers) believe that it is true.

This is a normal political strategy: frame an argument in a way that there is only one answer (the one that ‘you’ want).  I’ve seen leaders at my own college use this method, often successfully. … and I imagine that you’ve encountered it as well.  As teachers at heart, this style of communication is not natural for us; we respond by reasoned arguments and academic research with a goal of getting everybody to understand the problem.

The difficulty is that leaders who use the “where dreams go to die” phrase have little interest in understanding the problem.  Their goal is to remove developmental mathematics as a barrier to student success.  The next phrase after “where dreams go to die” is often “co-requisite remediation”, with claims that this solution is a proven success because of all of the data.  Of course, our view of this data is a bit more restrained than the leaders and policy makers; this is not a problem for them, as they have the answer in mind — all we have to do is agree with it.

We must do two basic things so that we can really help our students succeed:

  1. Shorten and modernize our mathematics curriculum, both developmental and college level.
  2. Consistently use our narrative:  “Developmental mathematics is where dreams go to thrive!”

Much of the material on this blog, as well as the wiki (dm-live.wikispaces.com)is meant to help faculty with the first goal.  The new courses, Mathematical Literacy and Algebraic Literacy, allow us to provide great preparation for college level courses within an efficient structure which minimizes exponential attrition.

“Developmental mathematics is where dreams go to thrive”:  We need to articulate this accurate view of our work, which is valid even within the old-fashioned traditional curriculum with too many courses.  I’ve posted about some of the research with a ‘thrive’ conclusion:

Also, a great project at CUNY called “ASAP” gets a glowing external evaluation:  http://www.mdrc.org/project/evaluation-accelerated-study-associate-programs-asap-developmental-education-students#overview  The ASAP model is currently being validated at other institutions.  Please let me know of other research showing that dreams thrive in developmental mathematics.

We should add our own ‘thrive’ stories and data.  For example, at my institution, we had 6 students start in pre-algebra and the proceed up to Calculus I in a four year period … 5 of them passed Calculus I on their first attempt.  If we believe the ‘die’ narrative, you would expect zero or 1 of these to exist; I am sure that most institutions have similar results to mine where the data shows more of a ‘thrive’ result.

Our traditional courses must go; we must do the exciting work of renewing the curriculum based on modern thinking about mathematics combined with more sophisticated approaches to instruction and learning.

However, that work will generally be wasted unless we establish a ‘thrive’ attitude.    The two conditions existing together create a new system that serves students well.  Developmental mathematics is where all dreams go to thrive.

 

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Student Success & Retention: Key Ideas

I’m working on a project which involves a search for strong research articles and summaries, and that included some work on ‘retention in STEM’.  I have some references on that, later; however, I wanted to present some key ideas about how to keep students in class so they succeed and how to retain them across semesters.

Rather than look at certain teaching methods as ‘the answer’, let’s look at some key ideas with surface validity and examine their implications for teaching.

  • Students need to be working with the content over an extended period in order to be successful.

We know that learning is the result of effort, usually intentional.  Attendance is easily measured, but is not sufficient by itself.  The class needs to establish environments where students want to work with the material, and we know that grades are insufficient motivation for many students.

  • Non-trivial ‘success’ (positive feedback) based on effort is strong motivation for most people.

If success seems impossible regardless of effort, it is easy to see why students would stop working.  However, success regardless of effort is also likely to result in drastic reductions in effort.  As in most human endeavors, people need to see a connection between effort and reward.

  • A teacher’s attitudes are more important than specific methods.

A few years ago, I was trying some very different things in a class; in fact, I was not very proficient with some key parts of that plan.  However, my students responding to my attitude more than those methods.  As one student said, “Mr. Rotman would not give up on me!”  An honest belief that almost all students are able to succeed is strong motivation.

We need to see our classes as a human system, a community with a shared purpose.  Most people need relationships with a purpose … connections that help them deal with challenges.  I am not trying to be a friend to my students, but we do form a community which can support all members.

  • Every student contributes to the success of the class.

Not all students will pass a math class.  Some of those who do not pass are able to provide help to those who do pass.  This past semester, I had a student who did very poorly on written assessments who routinely helped the class understand concepts and procedures.  The contributions of a student are valued independently of their grade, and independently of any other measure or category (ethnicity, social standing, mastery of formal language, etc).

I have not mentioned any teaching methods; pedagogy does matter … but the pedagogy follows from other ideas.  I can not use the key ideas above if all I do is ‘lecture’ (though I do a fair amount of that).  My class must provide a variety of interactions in order for my attitudes to be clear … and for all students to have opportunities to contribute.  Establishing a community is social navigation, so students need times to talk with each other in smaller groups as well as the entire class.

Here are some good articles and summaries of retention in mathematics and other STEM fields; these studies focus on retention in programs as opposed to courses … though there are obvious connections between the two.

  1. Teaching For Retention In Science, Engineering, and Math Disciplines: A Guide For Faculty http://www.crlt.umich.edu/op25
  2. Increasing Persistence of College Students in STEM  http://www.fgcu.edu/STEM/files/Increasing_Persistence_of_College_Students_in_STEM.pdf
  3. Retaining Students in Science,Technology, Engineering, and Mathematics (STEM) Majors
    http://mazur.harvard.edu/sentFiles/Mazur_399966.pdf
  4. Should We Still be Talking About Leaving? A Comparative Examination of Social Inequality in Undergraduate Patterns of Switching Majors http://wcer-web.ad.education.wisc.edu/docs/working-papers/Working_Paper_No_2014_05.pdf
  5. Gender and Belonging in Undergraduate Computer Science: A Comparative Case Study of Student Experiences in Gateway Courses http://wcer-web.ad.education.wisc.edu/docs/working-papers/Working_Paper_No_2016_02.pdf

Success and retention starts with us, and depends upon both our attitudes and our professional knowledge.

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Brain-Based Math Learning

I have been amazed (and appalled) by the phrase ‘brain-based learning’.  The suggestion is that there is learning NOT based on how the human brain functions; like mathematics, the brain uses ‘existence proofs’ — if learning happened, the brain must have worked.

The point of this post is to talk about what we commonly report as facts about the human brain.  For example:

Learning occurs through modification of the brains’ neural connections.

This is just about the most basic statement we can make, and it is actually correct.  Of course, it does not lead to an easy-to-implement teaching method.

Take a look at the following statements with an eye towards truthfulness:

  • Individuals learn better when they receive information in their preferred learning style (e.g.,
    auditory,visual, kinesthetic).
  • We only use 10% of our brain.
  • Differences in hemispheric dominance (left brain, right brain) can help explain individual
    differences amongst learners.

Each of these statements is false; these statements are examples of ‘neuromyths’, a phrase used by the Organization for Economic Co-operation and Development (in “Understanding the Brain:
Towards a New Learning Science”, 2002).  In other words, experts in neuroscience have determined that these statements are false.

The first myth listed is dangerous, because it leads to easy-to-implement teaching methods which will not help learning (and can reduce learning).  Even if “learning style” was a valid construct with a solid research basis, matching a trait to a treatment has shown to be a very difficult design strategy for learning (based on decades of research on attempts).  However, the recent summaries I’ve seen on “learning styles”  are still showing concerns about the construct itself.  The phrase ‘learning styles’ is most often used by educators trying to influence others; learning theorists and cognitive psychologists will seldom use the phrase (and often react very negatively to the phrase).

So, what would “brain based math learning” look like?  This is equivalent to asking what math learning would look like.  To me, the key is to keep focused on the basic statements about the human brain — like the one above about modifying neural connections.  Each learning task in a college math classroom is an interaction between new information and existing connections in the brain.

  1. The default response by the brain is “what I have now is correct” and is reinforced by the new information
  2. The need for modifying existing neural connections is based on some level of conflict
  3. “Learning” occurs during the resolution of the conflict
  4. The strength of this learning is based on multiple factors, including the use of verbal conclusions and practice (amount and variety)
  5. The learning may create a new set of neural connections that store information in conflict with pre-existing information; which set is accessed in the future depends upon the processing of inputs
  6. Resolving conflicting neural information takes the most effort but results in the most stable set of knowledge

As an example, we used about 6 class days last semester in my intermediate algebra course on a better understanding of rational expressions.  Most students responded based on their existing (incorrect) ideas about fractions. The classes created enough conflict (mentally) that most students developed some new information about fractions.  Later (on the chapter test, or the final exam) some students retrieved the new (correct) information while other students retrieved the old (incorrect information).  In a perfect world, students would have further learning experiences based on these assessments.

We seldom have sufficient time for students to learn math in college when they have existing incorrect information.  At the developmental level, the New Life project courses (Mathematical Literacy; Algebraic Literacy) focus on reasoning and communication with a more defined content — allowing some additional learning time.

As a profession, we need to move beyond pseudo-science so that our pedagogy is based on a body of knowledge accepted by scientists specializing on the human brain.  For a single-source, you might try Applications and Misapplications of Cognitive Psychology to Mathematics Education  at http://act-r.psy.cmu.edu/papers/misapplied.html

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Using Mathematics: It’s Not Always About ME !!

In the traditional college mathematics curriculum, mathematics is used to solve problems which students do not care about.  Some reform curricula involve mathematics only for problems which most students care about.  Is one of these extremes naturally superior to the other?

Perhaps some researchers are already working on experiments to test that hypotheses.  My own conjecture on this might surprise a few people:

The net gain for students is higher in a curriculum which solves problems which students do not care about, compared to a curriculum focusing on problems students do care about.

The traditional curriculum normally focuses on individual students creating a symbolic statement (equation or function) for the problem, and then using this symbolic statement to determine all answers.  The reform curricula often engage students with informal group work around a context, looking for alternative strategies to find the answer; symbolic work comes later (often on a different class day).

Most reformers will assert that the group work in a context provides definite advantages in student learning.  The etymology of this assertion often has its roots in a constructivist point of view; the original researchers in this area were more interested in the social context and juvenile development.  We often conflate the issue by speaking of a ‘constructivist theory’ — there is no constructivist theory (since a theory provides predictions that can be tested with either positive or negative results); I’ve never seen research supporting constructivism in learning mathematics with adults.

However, there is a non-trivial advantage to the reform work with work on problems which students care about:

Students having the novel experience of working on problems they care about is exciting and motivating.

Seeing that process in class is exciting for instructors; sometimes, we become addicted to this experience to the point that we think students have to be dealt with in this manner all of the time.

Is a math class all about ME?? (a student)

Of course it isn’t.  Students are in college to either get an education or training (or both).  Getting an education is all about “not me” — understanding other points of view, analyzing problems, and solving … often with the person deliberately left out (objective point of view).  We might think that ‘training’ should deal with just problems which students care about … this view has two fatal flaws.  First, let’s assume that training exists to get a job (employment); how much of any job is something that the student personally cares about?  Sure, the student picks a program that they care about in general — but their job is going to involve a large portion of specifics which they don’t care about.

The second fatal flaw in the training point of view is ‘stability’ (or lack there of).  How many workers deal with the same types of problems for years at a time?  We are hearing from business and industry that they need a flexible work force — not one constrained by ‘it’s important to me’.

When I teach our traditional algebra courses (beginning & intermediate) I almost always make a statement such as the following:

Passing this math course means that you can apply mathematics to problems which you don’t care about, but you did so because somebody else said they were important.

The main downfall of the traditional curriculum is that it does not modify the pre-existing negative attitudes about mathematics [though I try 🙂 ],  Students have a negative attitude about mathematics and especially about ‘word problems.  Using problems which students care about can provide some scaffolding to get students out of their negative attitudes.

We can’t stop there.  For each problem students care about, we should have them deal with 2 or 3 which they don’t care about.  We need to make the connections between the processing done on the ‘care about’ problems and the symbolic tools of the trade (expressions; functions; known relationships [such as D=rt]).

At the developmental level, students will be proceeding to college courses.  College courses have a general expectation of dealing with symbolic statements.  Being able to determine solution to a specific problem is often a trivial exercise in itself.  Students need to see quantitative relationships and use appropriate symbolism to state that relationship.  We have no confidence that the majority of these situations will be innately important to the student; we do them a diservice to imply that the only mathematics they need is to find solutions to problems they care about.

We need to get rid of the traditional curriculum, recognizing that we achieved some good results within that.  We also need to moderate our use of ‘problems students care about’, and we need to make sure that we always keep the focus on the tools of the trade (relationships, symbolic statements, representations).

 
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