Why are categories so useful?  When we think about things, especially when we try to understand why things are the way they are, we often try to put things into categories.  We like to decide that certain elements fit categories A and B; we match certain processes to categories H and W; and then we conclude that the outcome turned out be a result in category Z.

This reliance on categories, on categorizing things or inventing new categories with which to label things, is something we do all the time.  I don’t have an answer as to why categories might actually be useful.  I don’t even have an answer as to why we believe categories are so useful.  But I do have some thoughts about why categories matter for scientific discovery.

 

Open Access Isn’t Universal Access

 

It all started when I was looking around for open access articles to read about scientific discovery.  I recently lost the paid subscription access I had to most journal articles.  So I had to switch over entirely to only open access articles (i.e., those that don’t live behind a paywall).  This led me to spend a week wandering around the internet looking for good quality free resources.

Finding free journal articles didn’t turn out to be the problem.  Finding relevant free journal articles did.

There is still no consistency to what peer-reviewed articles and pre-prints are freely available.  Sometimes an entire journal is open access.  Sometimes only articles the author paid to make open access are freely available.  And sometimes only articles the journal considers prestigious or very high impact are made freely available (as a form of advertising).  How much free access is the “right amount” of free access is an issue that both publishers and scientists continue to wrestle with.

So on one particular day, instead of looking for articles I needed and then checking to see if they were freely available, I spent a little time searching for free articles and then looking to see if they might be relevant.  I just needed to get a sense of what was out there.

It turned out to be time well spent because I came across some fascinating research in an area completely unknown to me: how to help kids with learning disabilities solve math word problems.

 

Learning Disabilities and Schemas

 

I’ve come across articles about how to improve student problem solving performance before.  I have worked in academia and, especially in physics, how to help students do better in class is a popular topic.

What was interesting about this research though is that I found it because I was actually looking up a definition of the word “schema.”

A schema, in psychology, is a way of mentally organizing and sorting information to help you make sense of the world based on previous experience.  We can have internal schemas about all sorts of things, like how to determine if you aced an interview, what’s appropriate behavior at a wedding, or why certain activities help you relax on vacation.

In early childhood mathematics education schemas have a more specialized meaning.  In that context, schemas refer to specific kinds of templates or recipes taught to children to allow them to solve word problems.

One of the earliest  and best known general math problem solving schemas was given by mathematician and educator George Pólya in his book How to Solve It (1945).  His math problem solving schema involves four steps: (1) clarify the problem, (2) create a plan to solve it, (3) execute the plan, and (4) check your solution.

In a 2011 article reviewing the literature on using schemas with children at risk of or with learning disabilities (both math and reading disabilities) author Sarah Powell talks about how specific (explicit and teacher-led) and lengthy (weeks to months) instruction on how to apply a schema to solve word problems can improve student performance.

I know you’re dying for me to get to the point and link this back to scientific discovery.  Here’s how that might work…

 

The Discovery is in the Transfer

 

Powell draws out of the research literature two key themes that were a lightbulb moment for my perspective on how to train yourself (or others) in scientific discovery skills.

The first key theme is that the schema training worked best when students were first asked to categorize the type of problem that needed to be solved.  For example, students were given word problems where they needed to add things together (“totaling” type problems), or subtract things (“comparison” type problems), or multiply things (such as “shopping list” type problems.  (We’re talking about 3^rd and 4^th graders in the U.S. education system, so just 8 to 10 year olds here.)

When students were just taught how to solve each type of problem using a schema, but not to identify problem types, they did well.  But when students were taught how to identify what type of problem they were dealing with and the corresponding schema, they did even better.  This was called the “schema-based instruction” approach.

The second theme Powell found in the literature is that this performance could be boosted even further if students were given explicit instruction on how to apply the schemas they already knew to novel problems.  By explicitly I mean that they were given specific guidance on how novel problems might differ from familiar problems and that students were also taught how to link novel problems to familiar problems and then apply the already known schemas for the familiar problems to these new problems.  This was called “schema-broadening instruction”, as in the students increased their breadth or broadened their ability to apply what they were already taught.

I think this is fascinating.  Do you see the echoes of working on a problem at discovery’s edge here?

Consider this:

As someone pursuing discovery you have almost undoubtedly been taught ways to solve known kinds of problems in your area of interest (or you may have taught yourself well-known methods by doing a lot of studying using the internet).  So, essentially, you are like students in the first theme — you have a set of schemas to solve certain kinds of problems.  These are problems with well-known answers that you already know can be solved.  And these are methods you already know work.

At discovery’s edge you now come to a problem that you don’t know how to solve (or you are trying to identify a previously unrecognized problem and point out that it needs solving).  You still have tried and true methods, but now you have no idea how to get those to work on your new problem.  Aspects of the new problem may or may not resemble your old problem.  And for a scientific discovery scale problem, you (or someone else) will have already tried all the known methods and shown they don’t work.

So you’re stuck in the second theme, the schema-broadening problem.  How do you get the methods you know to apply to a problem that’s new?

 

Schema is Just a Fancy Word for Category

 

I loved Powell’s article because I realized that a problem students with learning disabilities may have is the same one discoverers might face.  I mean that these two situations,  in spirit, are similar (and in no way mean to trivialize the nuances or differences between the two situations).

Once I realized this, it put into perspective why I myself had felt the need, when I first started working on how to improve the techniques of scientific discovery at the individual skill level, to jump in and start generating categories.  I had phases of scientific discovery (a way to put a process into categories); I was trying to compile strategies (categories for solving discovery obstacles); and I even spent a lot of time trying to find out what scholars were saying about the types of scientific discoveries (categories of discovery).

But then I second-guessed this approach, because I wasn’t quite sure why I thought it was so valuable.  Was it just habit?

In science, especially certain topics like geology, paleontology, and particle physics, we are prone to “reductionism,” the tendency to want to break everything down into the smallest parts and to assume the behavior of the whole can be precisely determined from knowledge of the parts.  But this is not true in many natural phenomena (known as “emergent” phenomena), where the behavior that results from the interactions of the smallest parts is highly sensitive to many factors, and cannot be reduced in this simple toy-model kind of way.  Nonetheless, reductionism tends to be a mental trap and blind spot to which many scientists fall prey (myself included).

But this idea of schemas, and our ability to call them up based on our mental association between a problem type and a particular schema, sort of summed up the implicit philosophy I was following:  If I could come up with types of problems related to achieving scientific discovery, and even types of scientific discoveries, then maybe I could identify a set of schemas to overcome those problems, and those schemas might be teachable.

In fact schemas are themselves just more categories, ways to put mental processes and beliefs into categories that we can use and implement at will.

Schema-broadening then is the crux of the problem as to why we don’t yet know how to “teach the skill of scientific discovery.”  We haven’t spent enough time thinking explicitly about why the schemas we have don’t apply to novel problems, or why we fail to recognize that a known schema can in fact solve a novel problem.  If we put more emphasis there, on studying how we transfer schemas from one problem to another, then maybe we can boost our ability to discover the undiscovered.

 

Building the Big Picture

 

The image that came to mind was of a vast and complicated mosaic.  This mosaic not only creates one large picture, but also contains within it many smaller pictures, set pieces within the larger world of the whole mosaic.  The information we gain through observation and experimentation are like the tiny tiles which need to be placed within the mosaic.  Our theories and hard earned insights are like the set pieces.  Nature herself is like the whole mosaic.  But schemas are like the unseen outlines that tell us where the tiles should be placed in order for the mosaic to be a reflection of the real world, instead of a fantastical mirage.

It’s that intangible scheme, lurking behind the finished whole, that deserves our attention as much as the finished mosaic itself.

 

Interesting Stuff Related to This Post

 

1. Sarah R. Powell, “Solving Word Problems using Schemas: A Review of the Literature,” Learning Disabilities Research & Practice 26(2), pps. 94-108 (2011).  Open access version available here.
2. George Polya, How to Solve It (1945).
3. Liane Gabora, “Toward a Quantum Model of Humor”, Psychology Today online blog, Mindbloggling, April 6, 2017, https://www.psychologytoday.com/us/blog/mindbloggling/201704/toward-quantum-model-humor.

 

How to cite this post in a reference list:

 

Bernadette K. Cogswell, “An Intangible Scheme”, The Insightful Scientist Blog, August 25, 2019, https://insightfulscientist.com/blog/2019/an-intangible-scheme.

 

[Page feature photo: Mosaic commemorating the death of Beatles band member John Lennon in New York City’s Strawberry Fields, Central Park.  Photo by Jeremy Beck on Unsplash.]