9 – The Limits of Reasoning

When students try to solve problems by “thinking like a scientist,” what happens?   Let’s try an experiment (adapted from a suggestion by University of Virginia cognitive scientist Daniel Willingham).


You will need a sheet of paper and pencil or pen.


  1.  Try multiplying 68 x 87 “in your head.” Do not use fingers, toes, pencil, paper, computer, or calculator. Set a 2-3 minute limit. Pick up the pencil only when you are ready to write your 4-digit answer.
  1. Now multiply 68 x 87 using pencil and paper (but no calculator).

Discussion Questions:

On each of these, spend 2 minutes max to jot down answers.

  1. Were you able, without the pencil, to successfully reason the answer? What happened mentally when you tried?
  1. How difficult was solving with pencil and paper?   Did you use a procedure memorized long ago? Why was “using pencil and paper” easier than “solving in your head?”
  1. Given 3 ways to solve a simple task (2 digits times 2 digits): reasoning, applying a memorized algorithm, or using a calculator/computer – for you, which one measurably does not work?
  1. For fundamental tasks, if students must “use a computer” to solve, will they be prepared for higher level courses in the sciences?

What Studies of the Brain Predict:

  1. You will have considerable difficulty solving “in your head.” Even when your “times tables” are very well memorized, trying to keep track of the problem goal, your strategy, where you are in the process, and middle step numeric answers that are not well-memorized will likely exceed the “3-5 chunk limit” of your working memory.

The confusion you encountered is similar to what students experience when they try to solve a science problem by reasoning with information that is not well-memorized.

  1. Allowed the pencil, you were able to apply an algorithm (a step-by-step procedure) recalled from your long-term memory. The algorithm broke the problem into “one at a time” steps to avoid overloading working memory at any step.
  1. Nearly always, we solve problems with more than two steps by fluent, intuitive recall of memorized algorithms or procedures.
  1. Computers can solve some problems, but for probems with several steps, if students do not have in memory the facts and algorithms that solve problems when the numbers are simple, they will not have the skills in memory needed to choose the right programs to use or keep straight the order in which to push buttons on the calculator..

If instructors cannot use “reasoning without memorized algorithms” to solve 2 digits times 2 digits, is it fair to expect our students to use that kind of reasoning to solve problems of any complexity?

No single experiment proves a theory, but any scientific theory on reasoning should be able to explain the results of experiments such as the one above.

In our own scientific specialty, when judging proposed theories, we are taught to set aside our preferences and beliefs and to ask dispassionately:  “what do the data say?”

Outside of our specialty, we judge theories by asking, “What do the experts in that sub-discipline agree the data say?” As chemists, we say “evolution is correct” not because we dig up fossils or sequence species DNA. In science, we accept that “science” is what the experts in that sub-discipline agree it is.

But science must also allow for change. In 1920, chemists taught the Bohr model as our best explanation for atomic structure. When measurements found some predictions of the Bohr model to be incorrect, experts including Schrödinger and Heisenberg proposed an improved model — which other scientists accepted because the model predicted the data observed.

The model for cognition proposed by Swiss psychologist Jean Piaget (1896-1980) was widely-accepted for a time, but some predictions of Piaget’s model are not in agreement with recently measured limits on reasoning (see post #2 references). In response to this anomalous but verified data, the model accepted by cognitive science for how the brain solves problems has changed.

What experts in cognition are now telling us (in detail in posts #2 and #11) is this:

When trying to solve a problem, if even small amounts of needed information cannot be automatically recalled from long-term memory, the brain is likely to become confused, because working memory has very little space for knowledge that has not been previously moved into, and linked within, long-term memory (LTM).

For this reason, in science courses prior to graduate school, a primary focus for students must be learning to recall from LTM the core facts and algorithms of a discipline.

If we guide our students based on improved scientific understanding of how the brain works, are they likely to be more successful in STEM courses and careers?

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