Category: cognition

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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Rational Expressions … Easy Reducing, or “They’re Just Symbols”

For our intermediate algebra course, I am grading the test on rational expressions; in general, this is my least favorite test to grade.  One of my students provided a bit of unintentional humor, however; at least, it was funny for a minute.

Here is the problem and the student’s work:

Reducing Rational Expression the Eazy Way

 

 

 

 

 

So, what you are seeing here is that the student combined the numerators (fine) and added a half space in one term in the denominator, which caused “4k – 5” to be seen as “4 k-5”.  Reducing fractions the easy way!!

Fortunately, most students are actually doing okay with this test.  This problem has been on my tests before, and this is the first time I’ve seen that ‘method’.  I think this illustrates a generality about our students and fractions of any kind:

Students will deal with fractions at the symbolic level only, whenever possible.  Meanings are not attached, in general.

If you are curious, I start our work with rational expressions with an activity where we explore the meaning of the ‘fraction bar’ and the role of factors in simplifying.  That activity does shift students thinking a bit towards the meaning, though clearly not always :).

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Discovery Learning versus Good Learning

As people look at improving mathematics courses in college, we tend to look at some methodologies as naturally superior to others; we often fall in to the trap of criticizing faculty who use “ineffective” methods (traditional ones).  Some of my discomfort with the current reform efforts in developmental mathematics is the focus on one category of teaching methods … discovery learning.  #CollegeMath

At the heart of the attraction for discovery learning (and it’s cousins) is a very good thing — an active classroom with students engaged with the material.  It’s no surprise to find that research on learning generally concludes that this type of active involvement is one of the necessary conditions for students learning the material (in any discipline).  We can find numerous studies that show that a passive learning environment results in low learning results for the majority of students.  One such study is “The Effects of Discovery Learning on Students’ Success and Inquiry Learning Skills” by Balim (http://wiki.astrowish.net/images/e/e1/QCY520_Desmond_J1.pdf). In this study, the control group was (perhaps intentionally) very passive; of course, discovery learning produces better results.

It feels good to have our students engaged with mathematics.  By itself, however, that engagement does not produce good learning.  Take a look at a nice article “Correcting a Metacognitive Error: Feedback Increases Retention of Low-Confidence Correct Responses” by Butler et al (http://psych.wustl.edu/memory/Roddy%20article%20PDF’s/Butler%20et%20al%20%282008%29_JEPLMC.pdf) The role of feedback is critical to learning, but most implementations of discovery learning suggest that the teacher not intervene (or even correct errors).

Good learning does not happen from constantly applying one teaching method; teaching needs to be intentional, and modern teaching tends to be diverse to the extent that our work is research based.  I can see the benefits of incorporating some discovery learning activities within a class, along with other teaching modes.  See a study of this for college biology “The Effects Of Discovery Learning In A Lower-Division Biology Course” by Wilke & Straits (http://advan.physiology.org/content/25/2/62)

I use some discovery learning activities in my classes, and have found that I need to be very careful with them.  Here is my observation:

When students are asked to figure something out, they tend to apply similar information they have (correct or erroneous) and the process tends to reinforce that prior learning.

For example, I use an activity in my intermediate algebra class to help students understand rational expressions at a basic level — focusing on the fraction bar as a grouping symbol and on “what reduces”.  The activity provides a structured sequence of questions for a small group to answer.  Each group tends to use incorrect prior learning, even when the group is diverse in terms of course performance.  Even the better students have enough doubts about their math that they will listen to the bad ideas shared by their team; the only way for me to avoid that damage is to be with each group at the right time.

So, I have taken the discovery out of this activity; I now do the activity as a class, with students engaged as much as possible.  Even when done in small groups, students tend to not really be engaged with the activity.

I notice that same self-reinforcing bad knowledge in our quantitative reasoning course.  I use an activity there focused on the basics of percent relationships — percents need a base, and percent change is relative to 100%.  Many students do not understand percents, and the groups tend to reinforce incorrect ideas.  I continue to use that particular activity, as the class tends to be a little smaller; I am able to work with each group, during the activity.

Some of the curriculum used in the reformed courses are intensely discovery learning (often with high-context).  We need to avoid the use of one methodology as our primary pedagogy.  Do not confuse the basic message of replacing the traditional math courses with the pedagogical focus used in some materials.  To achieve “scale” and stability, our teaching methods need to be more diverse.

 
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Meaningful Mathematics … Learning Mathematics

Among the pushes from policy makers is ‘meaningful math’ … make it applicable to student interests.  Of course, this is not a new idea for us in mathematics; we’ve been using ‘real world applications’ to guide some reform efforts for decades. #realmath #collegemath #learningmath

Some current work in reforming mathematics education in college is based on a heavy use of context, where every new idea is introduced with a situation that students can understand.  We know that appropriate contexts with meaning to students helps their motivation; does it help their learning?

Before I share what I know of the theory and research on these approaches, I would like you to envision two types of textbooks or instructional materials (print, online, or whatever).

  1. Opening up the start of a lesson or section, repeatedly and randomly, generates a short verbal introduction followed by formal mathematical symbolism related to the new idea(s) in almost all cases.
  2. Opening up the start of a lesson or section, repeatedly and randomly, generates a variety of contextual situations and mention of a mathematical idea or tool that might be used.

Many traditional textbooks are type (1), while a lot of current reform textbooks are type (2).  Much of the change to (2) is based on instructor preferences, and I am guessing that much of the resistance to change is based on instructor preferences for (1).

Let’s take as a given that contexts that are accessible to students improve their motivation, and that we have a goal to improve student motivation; further, let us assume that we share a goal of having students learn mathematics (though that phrase means different things to different people).  There seem to be a number of questions to answer dealing with how a context-intensive course impacts student learning of mathematics.

  • How uniform is the impact of context on different learners?  (Is there an “ADA-type” issue?)
  • Do students learn both the context and the mathematics?
  • Is learning from a context more or less likely to be used and transferred to new situations?

I know the answer to the first question, based on research and experience: The impact is not uniform.  You probably understand that there are language issues for quite a few students, perhaps based on a class taught in English when the primary language is something else.  However, most of the contexts have a strong cultural factor.  For example, a common context for mathematics work is “the car”; there are local cultures where cars are not a personal possession, as well as cultures outside the USA where cars are either generally absent or relatively new (and, therefore, people know little).  The cultural problems can be overcome with sufficient scaffolding; is that how we want to spend time … does it limit the mathematical learning?  There are also ADA concerns with context: a sizable group of students have difficulties processing elements of ‘a story’ … leading to problems unpacking the context into the quantitative components we think are ‘obvious’.

The second question deals with how the human brain processes different types of information.  A context is a type of narrative, a story; stories activate isolated memories and create isolated memories.  To understand that, think about this context:

You are standing on a corner, and notice a car approaching the intersection.  When the light turns red, the car applies the brakes so that it stops in about 2 seconds.  You estimate that the distance during the stopping process is about 100 feet, and the speed limit is 30 miles per hour.  Let’s look at the rate of change in speed, assuming that this rate is constant through the 2 second interval.

The technical name for a story in memory is “episodic memory”.  This particular story might not activate any episodic memories for a given student; that depends on the episodes they have stored and the sensory activators that trigger recall.  Some students will respond strongly and negatively to a particular story, and this does not have to depend upon a prior trauma.  More of a concern are students who have some level of survival struggle (food, shelter, etc); many contexts will activate a survival mode, thereby severely limiting the learning.  Take a look at a report I wrote on ‘stories’ http://jackrotman.devmathrevival.net/sabbatical2006/2%20Here%27s%20a%20story,%20Ignore%20the%20Story.pdf

What happens to the mathematics accompanying the story?  If we never go past the episodic memory stage, the mathematics learned is not connected to other mathematics; it’s still a story.  In the ‘car stopping at an intersection’ story, the human brain might store the rate of change concepts with the rest of this specific story, instead of disassociating the knowledge so it can be used either in general or in new ‘stories’ (context).  Disassociating knowledge is another learning step; many context-based materials ignore this process, and that results in my biggest concern about ‘problem based learning’.

Using knowledge (Transfer … question 3) depends upon the brain receiving sensory input that activates the knowledge.  This is the fundamental problem with learning mathematics … our students do not see the same signals we see, ones that activate the appropriate information.  For this purpose, traditional symbolic forms and contextual forms have the same magnitude of difficulty: the building up of appropriate triggers to use information, as well as creating chunks of information that work together.  We need to be willing to “teach less mathematics” so that we can focus more on “becoming more like an expert with what we know”.  For more information on learning mathematics based on theoretical (and research-based) points of view, see http://jackrotman.devmathrevival.net/sabbatical2006/9%20Situated%20Learning.pdf and http://jackrotman.devmathrevival.net/sabbatical2006/6%20Learning%20Theories%20Overview.pdf 

I can’t leave this post without mentioning a companion issue:  Contextual learning is often done by ‘discovery’.  Some reform materials have an extreme aversion to ‘telling’, while traditional materials have an extreme aversion to ‘playing around’.  From what I know of learning (theory and research), I think it is safer to take the traditional approach … telling does not provide the best learning, but relying on discovery often results in even more incomplete and/or erroneous learning.  Just for fun, take  a look at http://jackrotman.devmathrevival.net/sabbatical2006/8%20Telling,%20Explaining,%20and%20Learning.pdf

Looking for a brief summary of all of this stuff?

Contextualized learning comes with significant risks; use it with caution and a plan to overcome those risks.
Mathematics is, by its nature, a practical field; all math courses need to have significant context used in the process of learning mathematics.

These ideas about context are related to the efforts of foundations and policy influencing agents (like CCA).  We need to keep the responsibility for appropriate instruction … including context.

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