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

 Join Dev Math Revival on Facebook:

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).

 
Join Dev Math Revival on Facebook:

Intermediate Algebra is NOT College Math ! :(

I actually spend a fair amount of time looking at other colleges math courses, partly from my interest in seeing how many colleges are doing New Life Project courses (Mathematical Literacy, Algebraic Literacy).  From that work, it is clear that the landscape is changing in both beginning algebra and general education mathematics.  However, two patterns are still present:

1. We continue to offer one or more courses in arithmetic focusing on procedures.  The presence of these courses is a tragedy on our campuses, since they negatively impact exactly the student groups we want to help (minority, poor).  I’ve posted on these issues earlier this year.
2. We frequently classify intermediate algebra as a college course, and commonly use it as a general education requirement.  Using a course which mimics a high school course in this way is professional embarrassment.  That’s the topic of this post.

We all know that “intermediate algebra” varies considerably between colleges, states, and regions.  In some cases, the intermediate algebra course has content at the level of the Common Core Mathematics (see http://www.corestandards.org/Math/ ) within the algebra and functions categories.  In most cases, however, our intermediate algebra courses fall below those expectations.

Intermediate algebra is a remedial course!!

The primary distinction between K-12 algebra and intermediate algebra is assessment — the college intermediate algebra course most likely requires a higher level of performance by the student in order to earn a passing grade.   It’s like “So, you were supposed to have learned this stuff in high school but NOW you are going to have to REALLY know that stuff.”

However, in many ways, our intermediate algebra (IA) courses are inferior copies of the K-12 curriculum.  Our IA courses are still descendants of copies of Algebra II from the 1970’s; much emphasis on procedures and correct answers … not much dealing with reasoning.  Given that we don’t deal with most of the discipline issues that occupy a K-12 teacher’s time, we should to better.    The K-12 content has responded to a series of standards (NCTM, Common Core) while our intermediate algebra has been standing still.

The Algebraic Literacy (AL) course is a modern system to help students get ready for college mathematics.  However, AL is still “not college math”, even though AL raises the expectations for students.

Entire states use intermediate algebra (IA) as an associate degree requirement.  In Michigan, which lacks a central governing body for community colleges, most colleges use that as one option for degrees.

We can, and must, do better.  If students do not need a course like Pre-Calculus, then we should use quantitative reasoning (QR) or statistics for their degree requirement … or even a course like ‘finite mathematics’.

Personally, I think intermediate algebra must die (and soon).  The issue in this post is whether a K-12 level standard course should be used for associate degree requirements.  Beyond the criteria of ‘expediency’, there is no rationale for that use.  IA is remedial, not college level.

Let’s MOVE ON!!

 
Join Dev Math Revival on Facebook:

Mathematical Literacy WITHOUT a Prerequisite

Starting this Fall (August 2016) my department will begin offering a second version of our Mathematical Literacy course.  Our original Math Lit course has a prerequisite similar to beginning algebra (it’s just a little lower).  The new course will have NO math prerequisites.

So, here is the story: Last year, we were asked to classify each math course as “remedial, secondary level”  or “remedial, elementary level” or neither.  This request originates with the financial aid office, which is charged with implementing federal regulations which use those classifications.  Our answer was that our pre-algebra course was “remedial, elementary level” because the overwhelming majority of the content corresponded to the middle of the elementary range (K-8).  We used the Common Core and the state curriculum standards for this determination, though the result would be the same with any reference standard.

Since students can not count “remedial, elementary level” for their financial aid enrollment status, our decision had a sequence of consequences.  One of those results was that our pre-algebra course was eliminated; our last students to ever take pre-algebra at my college finished the course this week.

We could not, of course, leave the situation like that — we would have no option for students who could not qualify for our original Math Literacy course (hundreds of students per year).  Originally, we proposed a zero credit replacement course.  That course was not approved.

Our original Math Literacy course is Math105.  We (quickly!) developed a second version … Math106 “Mathematical Literacy with REVIEW”.  Math106 has no math prerequisite at all.  (It’s actually got a maximum, not a minimum … students who qualify for beginning algebra can not register for Math106.)  The only prerequisites for Math106 are language skills — college level reading (approximately) and minimal writing skills.

Currently, we are designing the curriculum to be delivered in Math106.  We are starting with some ‘extra’ class time (6 hours per week instead of 4) and hope to have tutors in the classroom.  Don’t ask how the class is going because it has not started yet.  I can tell you that we are essentially implementing the MLCS course with coverage of the prerequisite skills, based on the New Life Project course goals & outcomes.

We do hope to do a presentation at our state affiliate conference (MichMATYC, at Delta College on October 15).  I would have submitted a presentation proposal for AMATYC, but all of the work on Math106 occurred well after the deadline of Feb 1.

One of the reasons I am posting this is to say: I am very proud of my math colleagues here at LCC who are showing their commitment to students with courage and creativity.  We will deliver a course starting August 25 which did not exist anywhere on February 1 of this year.

 Join Dev Math Revival on Facebook: