Mathematics Education – Facing the Problems

Share Cognitive Science Implications for Teaching Mathematics

7 NOV 2015   Revised 18 Aug 2018

As long as I have been teaching I have heard professors say: “The really good students will succeed in spite of us.”  I fear there is more truth in that aphorism than society is willing to acknowledge.  The fact that the good student succeeds does not excuse the education community for its general failure to educate.

By every measure I have seen in the past 10 or more years, US math students fall far behind many countries of the world and fall behind our own achievements of past decades.

Part of the problem is the fact that as a society we “preach” about the importance of education, but our actions do not match our words.  A cynical friend recently remarked: “If we could elevate math education to the same level of importance as football, we could achieve wondrous things in the math classroom.”  Fixing this is a monumental task.  I offer no further comment.

Part of the problem is that we do not treat teachers as professionals and they in turn do not act professionally.  We treat them as hourly workers with lots of vacation.  We do not promote professional development.  Fixing this is a monumental task. I offer no further comment.

My opinions are shaped by current experiences against a backdrop of many years of university teaching.  My opinions are based on personal experiences with mathematics students as they begin college.  My opinions are further shaped by extensive studying (for more than 30 years) especially in cognitive science research.

I will begin by discussing (A) the decline in education success and secondly, I will address (B) the knowledge retention issue.

Part A: Decline in Education Success

Part A: Cause #1 – Incorrect Decision Makers

Decisions about subject matter content, teaching methods, instructional material, and assessments are being made by politicians, union leaders, and giant publishing companies.  These entities are not qualified to make decisions about education.  The result is one foolish program after another (all with cute names) is introduced only to be replaced by another fad a few years later.

Remedy – Correct Decision Makers

Decisions about subject matter content and instructional material should be made by unbiased subject matter experts with expertise in education along with education experts who are able, and willing, to evaluate and implement cognitive science research.

Decisions about teaching methods should be made by unbiased education experts whose expertise derives from a knowledge, honest evaluation, and implementation of current cognitive science and classroom practices (not restricted to anecdotal evidence).

Decisions about assessment should be made by unbiased experts in the field of assessment.

Contrary to the current practice, all these decisions should be pushed as far down the organizational chart as possible.

  • This requires that education of our teachers will include rigorous study of subject matter, a rigorous study of cognitive science, a solid foundation in assessment, and construction of assessment tools. Education of our teachers must be of such high quality that we can trust individual teachers to make good decisions about content and methods to match their individual classroom.
  • This requires that instructional material be written by individual subject matter experts rather than minimally qualified teams assembled by a giant publishing company. It is very important that these authors be unbiased and have expertise in education including current cognitive science research.
  • This requires that we abandon the ridiculous idea of uniformity for: all states, all counties and parishes, across all school districts, across all schools, across all classrooms, for every student. Learning is an intensely individual activity.  We should recognize that simple fact and permit well-trained teachers to react to that individuality.
  • A natural outcome of pushing all decisions down the organizational chart will be the elimination of the Department of Education, creation of competition between newly created publishers of education material, as well as removal of politicians and party politics from education.

Part A: Cause #2 – Ignoring Science

Education and cognitive science are largely separate worlds – Benedict Cary

As early as 1875 Hermann Ebbinghaus (of forgetting curve fame) firmly established the disadvantage of massed practice and the advantage of spaced practice.  Yet all math textbooks and other study materials are based on massed practice with virtually no spaced practice. It is my prediction that the education community will continue to steadfastly ignore the amazing advances by cognitive scientists in the last 15 years.

Educators fail in their responsibility to teach students how to study and learn.

The fallacy in thinking that repetitive exposure builds memory has been well established through a series of investigations going back to the mid -1960s, when the psychologist Endel Tulving at the University of Toronto began testing people on their ability to remember lists of common English nouns.

However, rereading text and massed practice are still the preferred study strategies of most learners and teachers, but rereading and massed practice are also among the least productive study strategies.

Additionally, teachers do not dispel the myth of multitasking.

It is a proven fact: That part of the brain which is involved with learning is NOT capable of multitasking.

Learners must understand that they cannot study effectively while attending to TV, social media, texting, phone calls, and other distractions.  To be an effective learner a student  must avoid distractions while studying.

Remedy – Listen to Science

Professors in university Departments of Education must teach methods based on current cognitive science.  Teachers need to teach students how to study more effectively as determined by cognitive science.

Learning means to acquire knowledge and skills and having them readily available from memory for use in the future. Learning requires memory.  Forgetting is the major impediment to learning.  The learner must be taught study techniques which mitigate forgetting and then the learner must implement those techniques.

Three kinds of memory are involved in learning: Sensory Memory, Short-Term or Working Memory, and Long-Term Memory.  Forgetting occurs in (or from) each of these memories.  Recent cognitive science research aided with neuroimaging has firmly established:

  • Forgetting from Sensory Memory is mitigated by learner paying attention.
  • Forgetting from Short-Term Memory is mitigated by motivation, mental repetition, and association with prior knowledge.
  • Forgetting from Long-Term Memory is mitigated by
  • Retrieval practice such as self tests (flash cards) and low-stakes quizzes
  • Interleaved retrieval practice with related but different topics or completely different topics
  • Spaced retrieval practice to interrupt the forgetting curve
  • Elaboration – linking to prior knowledge, examples, images
  • Reflection – reviewing recently learned topics and asking questions about it
  • Generation – solve a problem without being shown a method
  • Calibration – use an objective instrument to evaluate your understanding
    • flash cards
    • self-tests
    • practice tests
  • Mnemonics – translate information into a form that the brain can retain better than its original form

These points are summarized in a diagram latter in this essay.

A very important fact: Retrieval practice (recalling facts or concepts or events from memory) is a more effective learning strategy than review by rereading.

A proven fact: That part of the brain involved with learning is NOT capable of multitasking.

Learners must understand that they cannot study effectively while attending to TV, social media, texting, phone calls, and other distractions.  To be effective, learners must avoid distractions while studying.

Part A: Cause #3 – Incorrect Goal

The ultimate goal of our educational system should be to produce educated persons.  An educated person is one who can intelligently contribute to society while enjoying the societal fruits produced by the arts, science, and technology. A person who graduates from our education system should be able to communicate effectively, appreciate the arts, understand as well as produce logical arguments, use critical thinking to solve problems, and be a lifelong learner.

The goal of our education system seems to be to stuff the student full of facts whose recall will assure passing the next test.  All too often in mathematics classrooms education has been replaced with training.

In mathematics we teach students that the goal in a math class is to “get the answer.”  I, and many other math professors, admonish our students that their primary goal is not to “get the answer” but rather to understand concepts.

Each of the following are ignored in most teaching materials and by most teachers:

  • concepts
  • structure
  • deductive reasoning
  • problem solving techniques
  • critical thinking
  • abstraction
  • generalization
  • constructing a valid argument
  • writing a valid argument

The above skills should receive strong emphasis because they have application far beyond the mathematics classroom or the next test.  These are precisely the skills we should be teaching, and our students should be learning, but they are ignored in most classrooms.

Remedy – Correct Goals

The purpose of early college level algebra courses is to introduce the student to the use of abstraction, generalization, problem solving techniques, deductive reasoning, and critical thinking while exploring the structure, patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions.

Textbooks and other learning materials should be reorganized to emphasize structure, deductive reasoning, interrelations between concepts, generalizations, abstraction, and problem-solving techniques.  These new materials should minimize manual computation and emphasize concepts.

All learning material should discourage the belief that the purpose of an exercise is to “get the answer.” All learning material should instill the conviction that an understanding of pertinent concepts coupled with good reasoning skills enables one to solve a variety of problems.

Part A: Cause #4 – Inattention to Technology

Learners frequently use technology as a means of distraction.

Teachers do not teach students to effectively use technology.

Teachers do not effectively use technology to teach understanding of concepts.

Neither teachers nor students use technology as a powerful research tool.


Remedy – Embrace Technology

It is a proven fact: That part of the brain involved with learning is NOT capable of multitasking.  Technology can serve a valuable purpose, but it also can produce ineffectiveness and fatigue because it causes the brain to continually switch attention from one task to another.


Technology in the classroom should be used to enhance understanding of concepts.  Let the computer do the drudge work and let humans soar with the eagles amongst concepts.


Technology and the Internet provide amazing research capabilities (Facebook is not one) and the capabilities will continue to expand at an increasing rate.  Our classes should teach students:

  • what resources are available,
  • what resources are expected to become available,
  • how to stay abreast of new resources when they come online,
  • how to judge new resources,
  • how to use those resources,
  • how to evaluate the information obtained form those resources.


Part A: Cause #5 – Inattention to Changing Needs

Every report I have read in recent years indicates that employers want to hire persons with the following traits:

  • ability to communicate effectively,
  • ability to solve problems,
  • ability to work individually as well as a team member,
  • ability to use critical thinking to analyze a problem,
  • ability to use deductive reasoning to solve a problem,
  • ability to learn and understand abstract concepts.

Remedy – Adjust to Current Needs

An employer does not want employees spending time manually doing lengthy mathematics computations.  Therefore, we should not unnecessarily burden the student with such computations. We should teach them to use technology to be an efficient employee.

An employer would rather that an employee be able to devise a method and then use technology to perform the computational drudgery.  Such an employee must be able to use critical thinking and deductive reasoning, be able to think in generalities and abstractions, and then must be able to write and otherwise communicate his method and conclusions.

Our courses should educate students to match those needs.  An educated student would use The Rational Zeros Theorem to find all possible rational zeros of a polynomial function, and then use a graphing tool (rather than synthetic division) to determine a list of probable rational zeros.  My favorite example has 24 possible rational zeros and only 4 probable rational zeros.

When I develop a comprehensive mental image of the elementary algebra courses I see one giant example of the desirable traits listed above.  We should teach that comprehensive image, rather than convincing students that math consists of thousands of disparate problems to remember.

Part B: Lack of Retention

Part B: The Cause

Education and cognitive science are largely separate worlds – Benedict Cary

There are many reasons why persons forget what they have learned.  I will address a few of the more obvious causes.

Forgetting is a natural and very predictable action of the brain. In 1885 Hermann Ebbinghaus discovered and quantified the forgetting curve. After 133 years most of us are still completely ignorant of how to mitigate forgetting.

If a learner does not attach importance to a piece of information, the probability is quite high that the information will be forgotten. There is a widespread, unfounded, and completely erroneous attitude that mathematics is not important. Consequently, there is a high probability that mathematics will be forgotten.

For many students the “cram and dump” mentality dominates their approach to learning.  Inherent in this approach to learning is a deliberate desire to forget.

An awful and unnatural compartmentalization, mostly disconnected, has been imposed on our courses (especially math) with corresponding compartmentalized assessment tools.  The impact of this compartmentalization is that even if the learner senses an overarching logical structure, it is evaluated as not important and therefore is forgotten.

An unreasonable and irresponsible pressure to “pass the next test” combined with very shallow tests have trained students to temporarily memorize procedures to mimic on the test.  Temporary memorization does not enhance long-term memory and therefore the information is forgotten.

By time I meet a student in beginning algebra classes all intellectual curiosity and fascination with learning has been drummed out of them.  When intellectual curiosity and fascination with learning are absent it is impossible to be attentive and interested in an academic subject. Without attention and interest the information is rapidly forgotten.

Some Remedies – Current Cognitive Science Research

Forgetting is the major impediment to learning.  The learner must be taught study techniques which mitigate forgetting and then the learner must implement those techniques.  The following schematic summarizes major aspects of memory, learning, forgetting, and mitigation of forgetting. Please study this diagram carefully.  This diagram shows how information is stored in long-term memory, the mechanisms of forgetting, and techniques to mitigate forgetting. 

Every day I observe forgetting from each of sensory memory, short-term memory, and long-term memory.

In my class an inattentive student is not able to repeat even the simplest fact immediately after hearing it. That is an example of forgetting from sensory memory. Due to inattention the student forgets the information almost at the same instant it is perceived.

In my class a student can repeat a fact one day and not be able to recall that same fact the very next day.  That is an example of forgetting from short-term memory.  Because the student lacks motivation, fails to perform mental repetition, or fails to associate the new information with prior knowledge, the fact is forgotten from short-term memory.

In my own experience, my long-term memory contains advanced mathematics related to research performed around 1972, but I am not able to recall that mathematics without some related study.  That is an example of forgetting from long-term memory.  Moreover, my experience illustrates that forgetting from long-term memory is due to the weakening of a retrieval path.  When I study those topics, I strengthen the retrieval paths and that segment of my long-term memory is restored.  Had I continued doing mathematics research, those retrieval paths would have grown stronger not weaker.