18 – Working Memory Limits and Chemistry Instruction

At  https://www.ChemReview.Net/ScienceOfLearning.pdf , Dr. JudithAnn Hartman and Eric Nelson have posted a summary of research on “Working Memory Limits and Chemistry Instruction.”  We hope it may be helpful to all instructors in courses for majors in sciences or engineering.

The intent of this “preprint” is to invite critical feedback — which may be left in the comments section of this post.

This article is an update of our 2014 paper “‘Do We Need to Memorize That?’ Or Cognitive Science for Chemists.”  In the past six years, extensive new research has been published by cognitive neuroscientists on issues in science instruction.

One key finding: According to the consensus of cognitive experts, students in introductory courses for science majors can only reliably solve the kind of problems at the end of the chapter in most textbooks by applying well-memorized algorithms  that apply well-memorized facts.

This is likely not what any of us wanted to hear, but science is not required to heed our preferences.

Cognitive experts say conceptual understanding is the right goal, but it takes quite a bit of thorough memorization to achieve.

And, even if concepts are understood, because of the brain’s working memory limits, to solve problems of any complexity, students still must memorize the fundamental facts and algorithms that their instructors recommend they overlearn — which means memorize so they can be recalled perfectly, repeatedly.

Chem ed journals often deprecate memorization and algorithmic problem solving, but on questions of how the brain works, it may be wise to defer to those whose scientific expertise is the study of how the brain works.

The cognitive experts say, to learn each new topic, students must start with initial and thorough memorization of fundamental vocabulary, facts, and relationships identified by their instructors, followed by problem solving using their new recallable fundamentals to solve problems in a variety of distinctive contexts, including word problems, demonstrations, labs, and simulations.

Again — comments welcome below!


17 – Science Says: Speed in Factual Recall Matters

In a recent online opinion column in Education Week (1/13/2020), math education professor Jo Boaler writes that her paper “Fluency Without Fear … builds a case … on the importance of moving away from speed and memorization toward number sense and conceptual thinking.”

But in fact, what scientists say about memorization and speed is the opposite of what Boaler claims.

The key surprising discovery of cognitive science in recent research:  When solving problems, the human brain has remarkable strengths but also stringent limitations.   Scientists have verified that during the steps of solving a problem,  “working memory” (where your brain solves problems) has an essentially unlimited ability to apply facts and procedures that can be quickly recalled from long-term memory.

However, at each problem step, working memory can generally hold only 3-5 elements of not-well-memorized data and relationships, each for 30 seconds or less (Cowan 2000, 2010; Clark, Sweller, and Kirschner, Spring 2012).

This makes memorization a key part of math (and chem and physics) success. When solving a multi-step problem, if needed relationships must be looked up, calculated on a calculator, or even mentally calculated, storing the answer takes up space that is limited in working memory. This tends to overload working memory, problem data tends to drop out, and confusion tends to result.

Cognitive scientist Susan Gathercole advises  to “avoid working memory overload in structured learning activities.” She explains:

“The capacity of working memory is limited, and the imposition of either excess storage or processing demands in the course of an on-going cognitive activity will lead to catastrophic loss of information from this temporary memory system.” (Working Memory in the Classroom, 2008)

How can the brain work around working memory limits?

In 2008, a U.S. Presidential Commission reported on ways to improve education in fields that relied upon math.  Five leading cognitive scientists  wrote that to get around the “bottleneck” of working memory’s limitations, the “central” strategy is

“the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory.” (Geary et al., . Report of the NMAP Task Force on Learning Processes , page 4-5)

So what student need, say the experts, is fast recall of memorized facts and procedures.  How do students achieve this recall? By exerting effort to memorize. In their task force report, the cognitive scientists noted:

“Verbatim recall of math knowledge is an essential feature of math education, and it requires a great deal of time, effort, and practice.” (p. 4-xii)

Cognitive scientist Daniel Willingham explains:

“Does speed matter? It does. When working a complex problem you not only want to pull simple math facts from memory, you want to do so quickly, so that the other work can proceed apace. Indeed, adults with stronger higher-level math achievement retrieve math facts faster (Hecht, 1999).” (Science and Education blog, 7/2/2017)

Boaler claims “conceptual thinking” can take the place of “speed and memorization.” But scientists tell us that in the brain, to construct conceptual frameworks, small elements of new knowledge first must be stored in the neurons of long-term memory:  “memorized” by effort at recall. Neurons gradually wire together if their knowledge is repeatedly recalled at the same time at steps during problem solving. (Hebb, 1949).

But connections do not grow between empty neurons. Willingham writes:

“A teacher cannot pour concepts directly into students’ heads. Rather, new concepts must build on something students already know.” (American Educator, Winter 2009-10)

Boaler claims students can learn math without the hard work involved in memorizing fundamentals. Perhaps she is “influential” because that’s what we would all prefer to hear.  No one wants Dickensian schools.

But scientists say what she is advocating simply does not work.  Unless teachers ask students to exert the effort needed to achieve quick recall of fundamental facts and procedures, according to science, students will be severely handicapped in their ability to solve math problems above grades K-3, when problems involve more steps. 

Education Week has been publishing a series of articles on “The Science of Reading.”  It’s been incredibly informative for educators seeking to improve student learning. I do not, however, recall any article which covered the consensus of science on how students can and cannot learn math.

In math, they chose to publish views that deny science.  Do such views merit publication?  Would an education journal publish a column by those who argue against vaccination before students are enrolled in school?

As professionals, we as educators have a moral and legal obligation to do our best to follow scientific best practices. If we listen to science on the importance of memorization and speed in recall, student prospects in careers that require math will rise. Poor and minority children will especially benefit.

16 – Online Conference: Help with the Math of Chemistry

NSF data tell us that nationally, about 90% of students who take science-major college chemistry hope to major in biology, health sciences, or engineering.  Those majors require first-year chemistry because molecular behavior is a foundation for all science, but also with the expectation that chemistry will teach calculation skills that are central in scientific disciplines. In scientific calculations, numbers have units attached. In addition, if equations are required, students are expected to determine which data goes where, and in chemistry, students learn to do so.  But most chemistry textbooks optimistically assume that students during K-12 have learned the fundamental rules of arithmetic and algebraic computation. In the U.S. for the current generation, due to circumstances often beyond their control, test data show that computation skills have declined substantially since about 1990.  We also we know from cognitive studies (but did not know until recently) that to solve calculations reliably, students need “very high” rather than “moderate” proficiency in math computation.   And because over-reliance on calculators and a decrease in calculation practice have been a part of K-12 math standards in most states since 1990, current students leaving K-12 are unlikely to have “very high” computational proficiency. These issues are discussed in some detail in the paper

Addressing Math Deficits to Improve Chemistry Success

here: https://confchem.ccce.divched.org/sites/confchem.ccce.divched.org/files/2017FallConfChemP8.pdf This was one of eight papers presented and discussed in the fall of 2017 at an ACS Division of Chemical Education (DivChEd) online conference on “Improving Student Skills in the Mathematics of Chemistry.” Among the topics:
  • Papers 1 and 4 discussed the benefits, procedures, and tips for teaching both first-year and Physical Chemistry without a calculator. These strategies encourage students to use “mental math” to both estimate as a check calculator answers and strengthen the sense of numeracy that helps with conceptual understanding.  Those papers are at:

Paper 1:   https://confchem.ccce.divched.org/2017fallconfchemp1

Paper 4:  https://confchem.ccce.divched.org/content/2017fallconfchemp4

  • A Texas study found that the better first semester general chemistry students did on a test early in the semester of simple “chem math” without a calculator, the better their semester grade tended to be, but the better they did on the same test with a calculator, the worse on average was their semester grade.

These data support the cognitive science prediction that as a basis for placement into “prep chem” vs. “gen chem,” a test of math without a calculator is a better predictor of which students need additional preparation for chemistry.

They also suggest the importance of mental math review as preparation for college chemistry.

Paper 2:  https://confchem.ccce.divched.org/content/2017fallconfchemp2

  • ACS Exam results reported in Paper 5  indicate that students who are given a review of math “just in time” for topics in general chemistry can show substantial gains in achievement, but gains are more likely to be seen for students whose math preparation is “average or above.”  These data suggest that students with below average math scores may need more preparation in math than a “during gen chem class and homework” review can provide.

Paper 5:  https://confchem.ccce.divched.org/content/2017fallconfchemp5

  • Papers 3, 6, and 7 provide additional evidence that a review of pre-requisite math prior to or concurrent with topics in chemistry has a positive impact on chemistry achievement. See:


I believe you will find a wealth of stimulating ideas in these papers for your own experiments to improve student success.

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15 – Needed: A Cognitive Fix for Math Standards

Posted at  www.ChemReview.Net/CCMS.pdf

is an analysis of the alignment of the K-12 Common Core Math Standards (CCMS) (and state math standards similar to the Common Core) with the findings of recent cognitive research.

What does this have to do with Chemistry?  Chemistry is a quantitative science, and the goal of the ChemReview project since 2006 has been to help students with the math needed to succeed in quantitative science courses.  It was hoped that with the adoption in most states of CCMS-type standards, such help for students would no longer be needed, because students would arrive in first-year chemistry with the essential mastery of pre-requisite math fundamentals.  The analysis in the paper above finds this did not happen.  Though the CCMS are superior in some areas to previous math standards in most states, in many key areas the CCMS ask students to solve problems in ways that science says the human brain simply cannot do.

Comments on the paper are most welcome.  If a comment form does not appear below this posting, click on the word “comment” below the title of this post.

14 – Math Computation and Student Success in Science Courses

The paper

Automaticity in Computation and Student Success

in Introductory Physical Science Courses

has been posted at    http://arxiv.org/abs/1608.05006

A PDF may be downloaded from the ArXiv site at no cost.

The article compares US math standards in place in most states until about 2012 (with impact on most current US students) to the recent findings of cognitive science on how the student brain solves problems.  The impact of those standards on student preparation for quantitative science courses is discussed.

Authors Dr. JudithAnn Hartman and Eric Nelson welcome comments, corrections, opposing and/or additional viewpoints.  Click on “Comments” above.

13 — Illustrated Guide: How Your Brain Solves Problems

A break from blogging has been required for work on the second edition of Calculations In Chemistry – An Introduction.  We will be back to blogging when the chapter deadlines have finally passed. Meanwhile, at the link below is posted a presentation that might be titled:

An Illustrated Guide to How the Brain Solves Problems


Why Science and Engineering Major Enrollments Have Fallen

and How We Can Fix Them

at       www.chemreview.net/ChemEd2015Post.pdf

A question from the slides:  For 80,000 Virginia 9th-grade students each year taking the Stanford 9 standardized test, as shown in the graph below, scores in math reasoning went way up.  Why did skills in solving calculations go way down?

Ponder!  Then check the link above to see what science says.

Microsoft Word - WordPlosIowaLand.docx


12 – Help for Students in Mental Math

Do your students know their “times tables” perfectly, without hesitation? Is it important that they do?

In our text Calculations In Chemistry, we included a review of math and algebra fundamentals — just before they were needed for chemistry topics. Working initially with students in engineering chemistry, compared to past semesters, success was pretty spectacular. But in subsequent experiments with “mostly bio-major” general chem and “prep for general chem,” though averages improved notably, more students struggled with quantitative problems than we thought should.

Our diagnostic testing found that strugglers tended to have trouble doing simple math “in their heads.”  Searching the academic literature, we found:

  • In chemistry education journals, others had observed a high correlation between “mental math” skills and general chemistry achievement.
  • Cognitive science emphasized:  Quantitative reasoning depends on  quick, fluent recall of math facts.
  • Math education journals noted that about 10 years ago, when current chemistry students were in 3rd grade, “math standards” in about 40 states required teachers to prepare students to use calculators on 3rd grade state math tests. As a result, many students had essentially never been required to “memorize their times tables.”

For students lacking math fluency, cognitive research suggested that strengthening mental math would help more students succeed in courses in the quantitative sciences.

To assess and sharpen mental math at the start of chemistry:

  1. We have prepared a 15-minute quiz  that will identify students who need help in math-fact recall.  We recommend it be given as early in chemistry as possible, preferably at the start of courses that prepare students for General/AP Chemistry.
  1. For students lacking fluency, we have written homework assignments that practice mental math, with an online quiz added to encourage practice completion.

The activities in the Mental Math packet may be downloaded here:

-For college Gen Chem or Prep Chem courses:  www.ChemReview.Net/GCMentalMath.PDF

-For High School and AP Chemistry:  www.ChemReview.Net/MentalMath.PDF

(click and check your PDF downloads).

In addition, posted at www.ChemReview.Net/WeekOneFiles.pdf  are over 50 pages of free homework tutorials on exponential notation and the metric system that can be used in first-year chemistry at all levels. Editable quizzes may be requested that include  calculations without a calculator.

Researchers say that if during a course, students are given a mix of calculator and “no calculator” problems that keep their mental math sharp, they will better understand quantitative examples and proportional reasoning.

In different populations, skills will vary, but the 15-minute quiz should tell you quickly which individual students will benefit from the “math automaticity” homework and re-quiz.

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11 – “Do we need to memorize that?”

The article

“Do we need to memorize that?” Or Cognitive Science for Chemists

has been published by the journal Foundations of Chemistry. This paper is a summary of recent scientific research on “best practices” for instruction and learning in the physical sciences, engineering, and math.  A pre-print has been posted at


The form below may be used for comments or questions on the article.

In the case of questions from readers, we (article co-authors Dr. JudithAnn R. Hartman and Eric Nelson) hope to be able to compare notes with interested readers on what the cognitive literature has to say.   Opposing and/or additional viewpoints are most welcome!

10 –Willingham’s Columns: Cognitive Science for Educators

In the past two decades, thanks in part to new technologies such as fNMI, PET, and MEG, science’s understanding of how the brain works, learns, and solves problems has increased dramatically. This new knowledge can be of great assistance to instructors and students — after it is translated out of its technical terminology and mined for import to the classroom.

Thankfully, someone has done much of this work — with the aim of helping educators.

Since 2002, University of Virginia cognitive scientist Daniel Willingham has sifted the gold standard, controlled variable, peer reviewed scientific research (as opposed to less rigorous “educational” research). In jargon-free columns in the American Educator, a quarterly professional journal (with free online access provided by the American Federation of Teachers), Willingham  shares the nuggets most useful to instructors.

What science has discovered is different in some areas from what has been claimed by various educational philosophies. Frankly, the science is not necessarily what we wanted to hear. A summary might be: “Evolution has given humans a brain designed to help children learn a language naturally, but other learning is hard work.”

The good news is that science has identified what does work to make study and instruction more effective and efficient. Willingham presents both the findings and the science that supports them. Most of his examples reference K-12 instruction, but for college-level and high school instructors, combining Willingham’s research summaries with the practical tips in the book Make It Stick (see the Read Recs tab above) provides a marvelous summary of the new scientific consensus on learning.

A full listing of Willingham’s articles is available at  http://www.aft.org/newspubs/periodicals/ae/authors5.cfm

Of the 26 columns published to date, below are brief summaries of 9 that I believe are especially relevant to chemistry. Trust me: the summaries below do not do the articles justice. The hope is to entice you to clink the link to the column and explore topics of interest.

To focus on a “science-instruction” perspective, I have “re-titled” the questions addressed by each article as follows:

  1. Making Learning Enjoyable
  2. Why Knowledge In Memory Is Important
  3. The Value of Spaced Study – and Frequent Quizzes
  4. Helping Students Build Effective Study Habits
  5. Why Learning Is Concrete First, then Conceptual
  6. How To Help Students Construct Memory and Understanding
  7. Why Practice Beyond Mastery Is Necessary
  8. Can Critical Thinking Be Taught?
  9. Teaching and Learning Factual, Procedural, and Conceptual Knowledge

I’d suggest: pick a numbered topic above, check the summary below, then try a column or two. For topics in addition to what is in the articles, see Willingham’s book:  Why Don’t Students Like School? available in paperback for under $12.

Article Summaries:

  1. Making Learning Enjoyable

The article is Why Don’t Students Like School? at:


Willingham diagrams the interaction of working and long-term memory. In a human brain which evolved to support the “fluent remembering” required for speech, successful problem solving favors remembering how a similar problem was solved in the past. One welcome finding is that “solving problems brings pleasure” if a problem somewhat challenging but solvable with guidance.

In the age of the internet and calculators, why is it important to memorize? Being a scientist, most of Willingham’s descriptions of research are carefully qualified, so his response here is noteworthy:

“Data from the last 30 years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about. The very processes that teachers care about most—critical thinking processes like reasoning and problem solving—are intimately intertwined with factual knowledge that is in long-term memory (not just in the environment).”

He also cites recent studies indicating “that children do differ in intelligence, but intelligence can be changed through sustained hard work.”

  1. Why Knowledge In Memory Is Important

The article How Knowledge Helps: It Speeds and Strengthens Reading Comprehension, Learning—and Thinking is at: http://www.aft.org/newspubs/periodicals/ae/spring2006/willingham.cfm

In learning, Willingham documents the Matthew Effect: “Those with a rich base of factual knowledge find it easier to learn more—the rich get richer.” When listening or reading, being able to fluently recall knowledge in memory helps in making inferences that improve comprehension. In addition, when students have more background knowledge, space is more likely to be available in working memory to identify and process for long-term memory the conceptual implications: “Whereas novices focus on the surface features of a problem, those with more knowledge focus on the underlying structure of a problem.”

Also noted are studies in science education showing that, to improve students’ problem solving abilities, teaching “problem-solving strategies” was less effective than “improving students’ knowledge base.”

  1. The Value of Spaced Study – and Frequent Quizzes.

The column is Allocating Student Study Time: “Massed” versus “Distributed” Practice  at: http://www.aft.org/newspubs/periodicals/ae/summer2002/willingham.cfm

Is “cramming” a smart way to study? In research comparing students who studied for 8 sessions in one day to those who studied in 2 sessions a day for 4 days, those who spaced their practice were able to remember more than twice as much on a quiz a week later. Other studies showed positive effects of spaced practice on vocabulary retention on tests eight years later.

Willingham suggests:

  • Let students know how “spaced practice” will help on final exams and in recalling information when it is needed in future courses.
  • To encourage spaced study, schedule frequent graded quizzes that require recall from memory.
  • Space topics to include a review of early fundamentals at later points in courses.
  1. Helping Students Build Effective Study Habits

The article is What Will Improve a Student’s Memory? at  http://www.aft.org/pdfs/americaneducator/winter0809/willingham.pdf

What science-based advice can we offer students on how to study? Willingham reviews

  • How memory is the residue of thought: you remember what you think about.
  • How to focus on thought about meaning.
  • Identifying and remembering what is important during self-study.
  • Learning “distinctive cues” that assist in memory retrieval.
  • Why re-copying notes and highlighting is less efficient than “self-quizzing.”
  • The value of mnemonics and visual imagery.
  • The need to “overlearn” fundamentals.

The new book Make It Stick by Brown, Roediger, and McDaniel (see the Read Recs tab) does a more detailed review of study strategies such as self-quizzing (including flashcards), summary sheets, interleaved practice, and elaboration, but Willingham’s explanation of the cognitive principles sets the stage for understanding why those strategies work.

  1. Why Learning Is “Concrete First,” then Concept

Inflexible Knowledge: The First Step to Expertise is at:  http://www.aft.org/newspubs/periodicals/ae/winter2002/willingham.cfm

Willingham notes that much of what is deprecated as “rote learning” is actually “inflexible knowledge:” knowledge with a narrow meaning that is not tied to a deeper conceptual structure. While organizing knowledge by its deeper structure is the goal in learning, he cites extensive evidence that during initial moving of information into memory, “the mind much prefers that new ideas be framed in concrete rather than abstract terms.”

His advice for teachers?

“If we minimize the learning of facts out of fear that they will be absorbed as rote knowledge, we are truly throwing the baby out with the bath water…. What turns the inflexible knowledge of a beginning student into the flexible knowledge of an expert seems to be a lot more knowledge, more examples, and more practice.”

  1. Helping Students Construct Memory and Understanding

Students Remember … What They Think About is at: http://www.aft.org/newspubs/periodicals/ae/summer2003/willingham.cfm

How do we move students from “shallow” learning of facts to seeing the deeper structure that conceptually organizes facts and procedures? The brain tends to remember what it thinks about and elements of the context in which content is encountered.  That context is a key to meaning, and the context is tagged to the knowledge in memory if the context is also thought about.

One consequence of this rule is that while students are more likely to remember what they “discover,” such learning must be done carefully. ”Students will remember incorrect ‘discoveries’ just as well as correct ones.”

Willingham describes how “study guides” for text assignments can be especially helpful if questions steer students toward linkages that construct deeper understanding.

  1. Why Practice Beyond Mastery Is Necessary

The article is Practice Makes Perfect—but Only If You Practice Beyond the Point of Perfection at: http://www.aft.org/newspubs/periodicals/ae/spring2004/willingham.cfm

In order to minimize forgetting what has been learned, cognitive studies recommend spaced overlearning: practice to perfection in recalling new content, repeated over multiple days. “Regular, ongoing review … past the point of mastery” is necessary to gain expertise.

Willingham notes that experts generally attribute their success not to talent, but to extensive practice in a domain, and that experts generally must practice extensively “for at least 10 years” before they make substantive contributions to their field.

In introductory courses, what knowledge and skills are most important to practice? “Core skills and knowledge that will be used again and again.”

  1. Can Critical Thinking Be Taught?

The article is Critical Thinking: Why Is It So Hard To Teach at: http://www.aft.org/pdfs/americaneducator/summer2007/Crit_Thinking.pdf

Willingham’s summary: “Can critical thinking actually be taught? Decades of cognitive research point to a disappointing answer: not really.”

He explains that critical thinking is not, for the most part, a generalized skill. Though students can and should be taught to “look at an issue from multiple perspectives,” or “estimate to check a calculator answer,” to do so requires content knowledge that can be recalled from memory. Most critical thinking strategies are “domain specific:” different, for example, between Pchem and organic synthesis. Though critical thinking strategies can be taught, they nearly always can be explained quickly. In learning, the slow, rate determining step is moving new information into long-term memory, and, via practice, tagging the information with meaning.

In a final section, he describes why scientific thinking in particular depends on scientific knowledge in memory.

  1. Teaching and Learning Factual, Procedural, and Conceptual Knowledge

Is It True That Some People Just Can’t Do Math?  is at http://www.aft.org/pdfs/americaneducator/winter2009/willingham.pdf

Despite the title, this article contains an excellent review of how to structure instruction in both math and the physical sciences. After a few math-specific topics, starting on article page 3, Willingham discusses how math and science instructors can help students learn factual, procedural, and conceptual knowledge.

To start: students need fluent recall of fundamentals. Why? “Complex problems have simpler problems embedded in them…. Students who automatically retrieve the answers to simple… problems keep their working memory (the mental space in which thought occurs) free to focus on the bigger… problem.”

Willingham cites the value of illustrating concepts with plenty of familiar concrete examples when they are available, and familiar analogies when they are not. On the question of what is more important learning: facts, procedures, or concepts? Studies say they are intertwined and reinforce each other.

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Were any of these articles controversial? Surprising? Feel free to express your views in the Comments below.

(We will be taking a break from blogging to do some additional reading and paper work during the next semester but will be back!)

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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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