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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Benny, Research, and The Lesson

The most recent MathAMATYC Educator (Vol 6, Number 3; May 2015) has a fascinating article “Benny Goes to College: Is the “Math Emporium” Reinventing Individually Prescribed Instruction? ” by Webel et al.  This article describes research in a emporium model using a popular text via a popular online system.  A group of students who passed the course and the final exam were interviewed; some standard word problems were presented, along with some less standard problems.

At the heart of the emporium’s approach to teaching and  learning we see the same philosophy that undergirded Benny’s IPI curriculum: the common sense idea that mathematics learning is best accomplished by practicing a skill until it is mastered.

I would phrase the last part differently, though you probably know what the authors mean … this is more of ‘the common mythology that mathematics …’ (common sense implies a reasonableness that seems lacking, given students attitudes about mathematics).

The phrase “Benny’s IPI” is a reference to a prior study by Erlwanger (1973) wherein the author looked at an individualized prescribed instruction (IPI) system; Benny was a similarly successful student who left the course with some very bothersome ideas about the types of topics that were ‘covered’ in the course.  In both studies, the primary method involved 3rd party interviews of students.

The current study had this as a primary conclusion:

We see students who successfully navigate an individualized program of instruction but who also exhibit critical misconceptions about the structure and nature of the content they supposedly had learned.

Although I am not a fan of emporium-related models, I am worried about the impact of this study.  These worries center on what the lesson is … what do we take away?  What does it mean?  The research does not compare methodologies, so there is no basis for saying that group-based or instructor-directed learning is better.  The authors make some good points about considering the goals of a course beyond skills or abilities.  However, I suspect that the typical response to this article will be one of two types:

• Emporium models, and perhaps online homework systems, are clearly inferior; the research says so.
• Emporium models, and online homework systems, just need some adjustment.

Neither of these are reasonable conclusions.

I spend quite a bit of time in my classes in short interviews with students.  Most of my teaching is done within the framework of a face-to-face class combining direct instruction with group work, with homework (online or not) done outside of class time.  Typically, I talk with each student between 5 and 15 times per semester; I get to know their thinking fairly well.  Based on my years of doing this, with a variety of homework systems (including print textbooks), I would offer the following observations:

1. Misconceptions and partial understandings are quite common, even in the presence of good ‘performance’.
2. Student understanding tends to be underestimated in an interview with an ‘expert’, at least for some students.

I have seen proposed mathematics that is equally wrong as that cited in the current study (or even worse); granted, these usually do not appear when talking to a student earning an A (as happened in the study) … though I am reluctant to generalize this to either my teaching or the homework system used.  Point 1 is basically saying that the easy assessments often miss the important ideas; a correct answer means little … even correct ‘work’ may not mean much.

Point 2 is a much more subjective conclusion.  However, I routinely see students show better understanding working alone than I hear when I talk with them; part of this would be the novice level understanding of mathematics, making it difficult to articulate what one knows … another part is a complex of expectations — social status — and instructor expectations by students.

Many of us are experiencing pressure to use “best practices”, to “follow the research”.  The problem is that good research supports a better understanding, but almost all research is used to advocate for particular ‘solutions’.  This is an old problem … it was here with “IPI”, is here now with “emporium”, and is likely to be with us for the next ‘solution’.

The “Lesson” is not “use emporium”, nor is it “do not use emporium”.  The lesson is more important than that, and involves each of us getting a more sophisticated (and more complicated) understanding of what it means to learn mathematics.  Most teachers seek this goal; the problems arise when policy makers and authorities see “research” and conclude that they’ve found the solution.  We need to be the voice for our profession, to state clearly why it is important to learn mathematics … to articulate what that means … to develop courses which help students achieve that goal … and use assessments that measure the entire spectrum of mathematical practice.

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What does ‘sin(2x)’ mean? Or, “PEMDAS kills intelligence, course 1”

My department is starting a conversation about pre-calculus and intermediate algebra.  I’m very pleased we are taking this step, and it is great that I do not know where our work will take us.

In our discussion yesterday, one of the concerns expressed was that students tend to not understand functions … including function notation.  People referred to the classic error:

sin(2x) = 2*sin(x)

The problem runs deeper than functions and function notation.

√(a²+b²) = a + b              and it’s corollary

(a + b)² = a² + b²

We cause these problems ourselves by allowing and even encouraging students to learn procedures by categorizing symbols, without needing to apply the meanings of those symbols.  In arithmetic or pre-algebra, this occurs in both order of operations and basic variables.

“Use PEMDAS to determine the correct order of operations”

“To add like terms, combine the numbers in front”

In a basic algebra course, the comparable statements for exponents and radicals might be:

“Negative exponents mean you write the reciprocal”

“Take out the perfect squares”

Much of our work in classes deals with getting students to correctly process the symbols we place before them.  When they generate mostly correct answers, we conclude ‘they understand’.  Seldom is that the case … because we seldom take the time to focus on the meanings of these objects along with the various correct choices we have in working with them.

When I say “PEMDAS kills intelligence”, I am using the pneumonic as a place-holder for the prescriptive procedures that we focus on.  As mathematicians, we are all about choices.  When we see:

3x(x – 2) + x(x – 2)

We think of two choices (combine ‘like terms’ first, or distribute first).  The PEMDAS-mentality boxes students in to the ‘one correct [sic] way’ approach to mathematics; the PEMDAS approach also encourages students to perform procedures on symbols without much regard for what that particular expression or statement meant.  We use these approaches in all kinds of math classes, from elementary classrooms to university classrooms, and it has got to stop.

In recent years, some of the reform efforts have de-emphasized symbolic work … partially as a response to this problem.  I applaud that work, and have contributed to the efforts.  However, sometimes we over-react and provide too little symbolic work.  We have course which emphasize ‘functions’ but never use basic function notation [f(x)], let alone variations such as ‘sin(x)’.  An irony is that most technologies that students use for our math courses (calculators, apps, web sites) generally use function notation.

Maintaining a strong focus on procedures and correct answers encourages a PEMDAS-mentality, causes problems for us later, and (I would suggest) limits student motivation to learn mathematics.  Think how much better it might be to have a balanced approach, where the key principles are:

1. Meaning
2. Properties
3. Choices
4. Application
5. Extension and feedback on prior steps

Some of my colleagues have said that students should “do mathematics” in math classrooms, though they are mostly talking about step 4 (application).  I also believe that students should “do mathematics” in every math class by using all levels (1 to 5 in my list) with all topics.  If we are not willing, believe that student’s can’t, or think that we do not have time … well, then we should question whether we are really committed to teaching that mathematics.

Most of our collegiate math courses are overly ‘full’, not too full of topics but too full of wasted effort.  We focus so much on “simplify” and “solve” in the basic courses that students use the PEMDAS-mentality; of course they won’t remember most of it, and of course they can’t apply ideas to other contexts — we are training them to just process the symbols.

So, if you have been wondering what I would have us do to replace “PEMDAS” for basic expressions, we should focus on four items:

• Meaning of each expression
• Inherent priority of each operation (a generally predictable list, based on level of abstraction)
• Properties for the type (meaning) of expression
• Choices for this expression

It is almost useless to know that a student can correctly calculate “8 – 3(5)”.  Value comes from knowing that there are multiple procedures to correctly calculate “6(2x) + 8(4x)”.  It is also almost useless to know that a student can correctly solve “12 – 5x = 7”.  Value comes from knowing that we have choices for that equation and for “8(x – 2) + 4x = 48”.  [I also suggest that PEMDAS itself is both incorrect and incomplete.]

Mathematics did not become so valuable because we know how to correctly arrive at an ‘answer’.  Our work is indispensable because we can present alternatives, and in some cases one of those alternatives provides great benefit to people, companies or societies.  That is ‘doing mathematics’, and is the type of experience I want for our students … whether in pre-algebra, pre-calculus, or anything else.

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