# What is Wrong with Mathematics Instruction

Share Cognitive Science Implications for Teaching Mathematics
•
•
•
•
•

What is Wrong with Mathematics Instruction

We do not teach definitions.

• Without definitions there can be no communication.  Without communication there can be no teaching or instruction.  Without teaching or instruction there can be no learning.
• Students cannot use definitions to properly identify objects.
• Students cannot use definitions to answer questions.

We do not teach an appreciation for distinction between stipulative and lexical definitions.

•  Lexical definitions report common usage.
• Stipulative definitions announce (stipulate) what will be meant by a word (or term) in the present work.
• Mathematics uses only stipulative definitions.
• Mathematics definitions are very precise.
•  Stipulative definitions are poorly learned by usage.
•  Learning stipulative definitions must begin with memorization.

We teach to “get the answer” as opposed to teaching for understanding.

•   Mathematics is not about answers, it’s about processes.
•   Many students and teachers erroneously believe that knowing the steps is understanding.
• 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, relations, unary operations, binary operations and mathematical objects such as equations, inequalities, algebraic fractions, polynomials, and functions.

We do not teach students that in mathematics we solve ONLY equations and inequalities.

• The action verb “to solve” should not be the first thing a student thinks of when presented with a mathematics statement.

We do not teach students to use concepts to analyze problems and answer questions.

• For example, students should be taught to use The Transitive Property of Equality to analyze “word” problems.
• For example, students should be taught to use The Law of Trichotomy and a minimum of computation to consider and solve all three of an equation and its corresponding inequalities. To emphasize the close relation between an equality and the corresponding inequalities, I have begun to refer to them as siblings.
• For example, students should be taught to perform simple derivations such as using the Pythagorean Theorem to derive the formula for a circle with center (h, k) and radius r.  Most of my Community College students in recent years could not perform such derivations.
• During the last ten years or so of my teaching career I drew special attention to “The Big Four” (Distributive Property, Transitive Property, Zero Factor Property, and the Law of Trichotomy).  I diligently and regularly illustrated how and where deductive reasoning could utilize these properties to analyze questions.  I am working on a page for this blog which will expand on “The Big Four.”

We do not teach coherent structures which answer a range of questions.  Instead we train students to follow specific steps in each of thousands of insipid problems.

• For example, most of my Community College students have learned five distinct processes to solve the following five equation types:
• 1)  Solve 3x = 7
• 2)  Solve
• 3)  Solve
• 4)  Solve (3 + 4i)x = 7 – 5i
• 5)  Solve AX = B where A, X, and B are matrices with appropriate dimensions.
• If they had received proper instruction they would use the same procedure to solve all five of the equations.  Each of the five equations would be solved by multiplying both sides of the equation by the multiplicative inverse of the coefficient of the variable.
• If they had received proper instruction they would be confident in each of the five cases that the solution set they obtained was correct.  They would know it was correct because they would know and understand that such multiplication yielded an equivalent equation whose solution set is obvious.  Of course they would know that two equations are equivalent if and only if they have the same solution set.
• Observe that if they had received the proper instruction they would become independent of their math teacher; there would be no need to remember what the teacher wanted them to write.
• Many of my Community College students do not realize that to solve an equation means to find all those values which make the statement true when substituted for the variable.  To most of them it simply means to repeat the steps their teacher insisted on.

We do not show students how the act of generalizing reduces many seemingly new concepts to nothing more than a reexamination of an old familiar concept.

• For example, in early childhood a student learns that the absolute value of a number is the number with neither a + or – attached.  That works fine for some time but eventually it becomes desirable to replace that definition with the statement that the absolute value of a number is its distance from zero on the number line.  Currently common practice is to present this new definition of absolute value simply as a replacement of the old with no attempt to explain it as a generalization of the previous definition.  Then in algebra it becomes necessary to replace that definition with the following algebraically more useful definition:  .  Currently common practice is to present this new definition of absolute value simply as a replacement of the old with no attempt to explain it as a generalization of the previous definition.
• It is important for students to realize that these generalizations are not new and different from the previously learned topics.  They must realize that the generalization is simple an examination of a familiar concept in a different context.  The concept does not change. Understanding of the concept expands.

We do not teach students to use deductive reasoning to justify each step in a mathematical process or to produce any other logical argument.

• The definition of graph of a function f is the set of points of the form (a, f(a)) where a is a domain element and f(a) is the corresponding unique range element.  Students should learn that if (3, 8) is on the graph of a function f, then as a result of deductive reasoning, f(3) = 8.  This must be deductive reasoning on the part of the student not recall of a procedure illustrated by a teacher.  Teaching students to replace memorized procedures with deductive reasoning is a very difficult teaching mandate.

Our teaching does not illuminate the structural aspects of mathematics.

• For example, students entering college algebra should be able to identify each “thing” as a relation, a unary operation, a binary operation, or a mathematical object.
• For example, students should be shown that there is a division algorithm for the Natural Numbers, the Integers, polynomials with integer coefficients, but not for Rational Numbers, Irrational Numbers, or Complex Numbers.

We do not provide students with appropriate organizing schemes or teach them how to abstract relevant principles from what they are learning.

Many students fail to identify polynomials or their classifications.

• For example, generating a sequence of equivalent equations which terminates in a simplest equation is an elegant general approach to solving equations. We fail to teach students this elegant general approach to solving equations and inequalities.  We fail to teach students to devise appropriate modifications when necessary.

We have not changed the curriculum to match the needs of the 21st century.

• Mathematics required in the workplace is different than in previous centuries.
• Technology has assumed most computational details.

Gagne identified five learning outcomes: Intellectual Skill, Cognitive Strategy, Verbal Information, Motor Skill, and Attitude.  The desired outcomes in mathematics instruction are Intellectual Skill, Cognitive Strategy, and Verbal Information.  However, instruction in most early math classes presents mathematics as a Motor Skill.

Two additional sections are required to complete this essay.  I will discuss the causes and the cures. These sections have not been completed.  Outlines (subject to change) of these two sections are present below.

What are the Causes

Societal lack of interest and motivation.

Common conception that mathematics is about computation.

An outdated curriculum

• A mile wide and an inch deep.
• The “spiral approach” has become mere repetition.
• Successful curricular reform requires informed staff development and engagement of teachers.

Bad outdated textbooks

• Textbooks stress computation, algorithmic procedure, and artificial story problems.
• Textbooks promote topics techniques which may be temporarily and narrowly effective and efficient but give the wrong mathematical lesson and therefore ill-serve the student. An example of this would be the FOIL method.

Teacher’s lack of mathematics knowledge.

• Nearly half of math and science teachers do not possess adequate subject-matter training.
• “School boards do not understand what math is, neither do educators, textbook authors, publishing companies, and sadly, neither do most of our math teachers. The scope of the problem is so enormous I hardly know where to begin.”
• Paul Lockhart – A Mathematician’s Lament – 2002

Teacher’s lack of knowledge of cognitive science.

Teachers tend to teach the way they were taught.

Teachers’ lack of autonomy in the classroom.

Student’s attitude toward learning.

• Student’s level of curiosity has declined over the last two decades.
• Recently one of my algebra students told me; “The reason Asian students do so much better is they work very hard. Personally I am not willing to work that hard for anything”.

Minimal and improper mathematics content required for teaching degree.

• Teacher educators should focus first on the education of mathematics specialists at the elementary and secondary levels.
• Education professors’ lack of understanding of mathematics.

Teacher Recruitment

• The quality of an educational system cannot exceed the quality of its teachers. South Korea therefore recruits from the top 5% of graduating college students. America, in contrast, tends to recruit from the bottom third.

What are the Cures

Drastic improvement in teacher education.

Localize education.

Remove DOE, unions, giant publishers, and political groups from the process of determining classroom activity

Empower teachers in the classroom.

Adopt instructional materials developed by subject matter experts well versed in the latest cognitive science as applied to mathematics.

Modernize the curriculum to reflect needs of the 21st century.

Infuse the latest cognitive science into mathematics teaching and learning.