icme12 understanding math usiskin

What Does It Mean to Understand Some Mathematics?

ICME-12Seoul, KoreaJuly 10, 2012

Zalman Usiskin

[email protected]

Hans Freudenthal, Didactical Phenomenology of Mathematical

Structures (1983)

my perspective

James Hiebert and Thomas Carpenter, “Learning and Teaching with Understanding”

in Handbook of Research on Mathematics Learning and Teaching (1992)

mathematical

classroom

Conflicting views of doing and understanding

“I hear and I forget, I see and I remember,I do and I understand.” – Confucius (6th century B.C.)

“They can do it but they don’t understand what they are doing” – teachers (21st century A.D.)

Behaviorists and Cognitivists

Most teachers are both behaviorists and cognitivists.

Behaviorists want students to answer questions correctly and do not necessarily care how they got the answer.

Cognitivists want to know what students are thinking as they work and typically ask them to show their work.

Is skill a type of understanding?

Richard Skemp (1976) wrote, regarding instrumental (procedural) understanding and relational (conceptual) understanding:

“I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’.” (emphasis his)

What is mathematics?

Mathematics is an activity involving objects and the relations among them; the objects may be abstract or abstractions from real objects.

The activity consists of concepts and problems or questions. Mathematicians employ and invent concepts to understand questions and problems; they pose questions and problems to delineate concepts.

The full understanding of mathematics in schools requires a view of

understanding from...

• the standpoint of the learner

• the standpoint of the education policy maker

• the standpoint of the teacher

• the standpoint of the mathematician.




• the standpoint of the education policy maker


• the standpoint of the research mathematician.

The range of school mathematics in the U.S. Common Core State Standards (CCSSM, 2010)

• Statistics – Yes• Physics – No• Formal logic – No• Applied mathematics – Yes• Telling time – Yes• Reading tables – Yes• Locating places on maps – No• Sudoku Puzzles – No• Using a calculator – Mixed message




• the standpoint of educational policy


• the standpoint of the research mathematician.

Classics displaying the mathematician’s perspective on understanding

• Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945)

• G. H. Hardy, A Mathematician’s Apology (1940)

• George Polya, Mathematical Discovery I: On Understanding, Learning, and Teaching Problem Solving (1962) and How to Solve It (1957)

Understanding the lives of mathematicians• Reuben Hersh and Vera John-Steiner, Loving and Hating

Mathematics: Challenging the Myths of Mathematical Life (2010)

Polya’s four phases in problem-solving (from How To Solve It)

1. Understanding the problem

2. Devising a plan

3. Carrying out the plan

4. Looking back

Understanding the problem(from Polya’s How To Solve It)

What does the problem say – can it be stated in your own words? What is the unknown? What are the data [givens]? What is the condition [or conditions]? Is it possible to satisfy the condition[s]? Will a figure help? Is it necessary or helpful to introduce suitable notation?

A person does not understand some mathematics when that person

acts blindly to the prompts in the situation,

or

acts incorrectly to the prompts.

Understanding

multiplication of fractions

Meanings for “fraction” – an essential part of understanding fractions

• a number between 0 and 1 (or between 0 and -1?)(“He earns only a fraction of what she earns.”)

• a (positive?) number that is not an integer(“I want an answer that is a fraction.”)

• division indicated with a fraction bar or a slash (or some other symbol?)

(“…an expression of the form a/b.”)

• a number expressible in the form a/b where a is a whole number and b is a positive whole number.

(the meaning in the U.S. Common Core)

Different algorithms for multiplying fractions depending on the numbers involved

Dimension 1: Skill-algorithm understanding of the multiplication of fractions

…requires making decisions

…is choosing an appropriate algorithm depending on the numbers in the fractions

…is being able to check an answer using a different method than was employed to get the answer

Dimension 2: Property-proof understanding of the multiplication of fractions

We wish to justify:

Property-proof understanding of the multiplication of fractions

Variants of the algorithms for property-proof understanding

Dimension 3: Use-application understanding of the multiplication of fractions

1. Area: A farm is rectangular and 2/3 km by 4/5 km . What is its area?

2. Rate factor: An animal travels at an average rate of 2 km in 3 hours (that is, 2/3 km per hour) for 48 minutes (that is, 4/5 hours), how far will it have traveled?

3. Probability: If independent events have probabilities 2/3 and 4/5, what is the probability of both happening?

4. Size change: If a segment on a sheet of paper is 4/5 unit long and is put into a copy machine to be copied at 2/3 its original length, what will be the length of the segment on the copy?

5. Two size changes: If something is on sale at 1/3 off (you pay 2/3) and you get a 20% discount (to 4/5 the sale price) for opening a charge account, your cost is what part of the original price?

Use-application understanding of the multiplication of fractions

1. Area: A farm is rectangular and 2/3 km by 4/5 km . What is its area? measure X measure

2. Rate factor: An animal travels at an average rate of 2 km in 3 hours (that is, 2/3 km per hour) for 48 minutes (that is, 4/5 hours), how far will it have traveled? rate X measure

3. Probability: If independent events have probabilities 2/3 and 4/5, what is the probability of both happening? scalar X scalar

4. Size change: If a segment on a sheet of paper is 4/5 unit long and is put into a copy machine to be copied at 2/3 its original length, what will be the length of the segment on the copy? scalar X measure

5. Two size changes: If something is on sale at 1/3 off (you pay 2/3) and you get a 20% discount (to 4/5 the sale price) for opening a charge account, your cost is what part of the original price? scalar X scalar (X measure)

Dimension 4: Representation-metaphor understanding of the multiplication of fractions

4/52/3 • 4/5

C

Understanding

congruence in geometry

General definition of congruence (dynamic)(informal)(after Euclid)

Two figures are congruent* if and only if one can be placed on top of the other.

* Euclid called such figures “equal”.

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Specific definitions of congruence(static)(formal)(after Hilbert)

Two segments are congruent if and only if they have the same length.

Two angles are congruent if and only if they have the same measure.

Two triangles are congruent if and only if there is a correspondence between their vertices with corresponding sides having the same length and corresponding angles having the same measure.

Two circles are congruent if and only if they have the same radius.

General definition of congruence (using transformations)(formal)(after Klein)

Two figures are congruent if and only if one can be reflected, rotated, and/or translated onto the position of the other. (dynamic)

α β

Two figures α and β are congruent if and only if there is a distance-preserving transformation that maps α onto β. (static)

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Dimension 1: Skill-algorithm understanding of congruence includes…

Drawing, constructing, and visualizing congruent figures, including images under isometries.

Identifying the transformation that maps one figure onto a congruent figure.

Recognizing congruent and non-congruent figures.

Cube without perspective Cube with perspective

Dimension 2: Property-proof understanding of congruence includes…

Conditions that cause two triangles or other figures to be congruent (SAS, SSS, etc.).

Properties of each of the isometries and relationships among the isometries.

Derivations using congruence of the basic formulas for the areas of triangles and other polygons.

Relations between congruence and similarity.

Dimension 3: Use-application understanding of congruence includes…

Congruence in structures and mass-produced parts.Congruence and symmetry in nature.Applications of isometries (rotations and turns, translations and slides, reflections and mirrors, glide reflections and walks).Tessellations and packing problems.Applications of measurements of angles, lengths, areas, and volumes

Dimension 4: Representation-metaphor understanding of congruence includes…

Placement of congruent figures on geoboards.

Algebraic formulas for images of figures in the coordinate plane under isometries.

Matrices for isometries.

Descriptions of isometries using complex numbers.

Skill-algorithm understandingfrom the rote application of an algorithm through the selection and comparison of algorithms

to the invention of new algorithms (calculators and computers included)

Property-proof understandingfrom the rote justification of a property through the derivation of properties to the proofs of

new properties

Use-application understandingfrom the rote application of mathematics in the real world through the use of mathematical

models to the invention of new models

Representation-metaphor understandingfrom the rote representations of mathematical ideas through the analysis of such

representations to the invention of new representations

Dimensions of mathematical understanding

Skill-algorithm understandingfrom the rote application of an algorithm through the selection and comparison of algorithms

to the invention of new algorithms (calculators and computers included)

Property-proof understandingfrom the rote justification of a property through the derivation of properties to the proofs of

new properties

Use-application understandingfrom the rote application of mathematics in the real world through the use of mathematical

models to the invention of new models

Representation-metaphor understandingfrom the rote representations of mathematical ideas through the analysis of such

representations to the invention of new representations

History-culture understandingfrom rote facts through the analysis and comparison of mathematics in cultures to

the discovery of new connections or historical themes.

A fifth dimension of mathematical understanding for learners

Dimension 5: History-culture understanding includes …

The invention or discovery of mathematics.

The growth of mathematics and its applications over time.

The differences among cultures of formal and informal mathematics, including ethnomathematics.

The notion of relative truth vs. absolute truth and relations with philosophy.

Recreational mathematics and other mathematics done “just for fun”.

Applications of the multi-dimensional view of understanding

• clarifies the meaning of “concept”

• broadens options for developing concepts

• opposes the one-dimensional models of item difficulty

• helps to develop richer curricula

• provides a tool for analyzing curricula (ideal, implemented, or learned)

A CCSSM Fraction Standard (Grade 5)

• Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

• Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

A CCSSM Fraction Standard (Grade 5)

• Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

• Interpret the product (a/b) × q as a parts of a partition of q into b equal parts (Vocabulary); equivalently, as the result of a sequence of operations a × q ÷ b (Property-Proof). For example, use a visual fraction model to show (2/3) × 4 = 8/3(Representation-Metaphor), and create a story context for this equation (Use-Application). Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) (Skill-Algorithm)

A CCSSM Congruence Standard (Grade 8)Understand congruence using physical models, transparencies, or geometry software.•1. Verify experimentally the properties of rotations, reflections, and translations (Skill-Algorithm and Property-Proof): a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines.•2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations (Property-Proof); given two congruent figures, describe a sequence that exhibits the congruence between them (Skill-Algorithm).•3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates (Representation-Metaphor).

Frequencies of dimensions of understanding in the CCSSM standards

Totals K-5 6-8 9-12 9-12 STEM

Vocabulary 51 21 18 11 1Skill-Algorithm

186 88 37 38 23

Property-Proof

196 67 47 61 21

Use-Application

147 55 47 32 13

Representation-Metaphor

118 46 30 32 10

History-Culture

0 0 0 0 0

Problem Solving

13 7 1 2 3

MAXIMUM POSSIBLE

385 148 81 113 43

Totals 385 284 180 176 71

Realms of understandings a teacher needs to have

• Pedagogical Content Knowledge

• Concept Analysis

• Problem Analysis

• Connections to Other Mathematics

Pedagogical Content Knowledgeincludes…

• Designing and preparing for a lesson• Explaining and representing ideas new to students• Finding and analyzing student errors • Responding to questions that learners have

about what they are learning

Concept Analysisincludes…

• Engaging students in justifying their actions and thinking

• Considering alternate definitions and their consequences

• Using representations to clarify concepts• Recognizing and employing alternate algorithms • Explaining why concepts arose and how they have

changed over time• Dealing with the wide range of applications of the

mathematical ideas being taught

Problem Analysisincludes…

• Engaging students in problem solving

• Discussing alternate ways of approaching problems with and without calculator and computer technology

• Examining different student solution methods• Offering extensions and generalizations of

problems

Connections to Other Mathematicsinclude…

• Extending and generalizing properties and mathematical arguments

• Explaining how ideas studied in school relate to ideas students may encounter or have encountered in other mathematics study

• Realizing the implications for student learning of spending too little or too much time on a given topic

• Comparing different resources’ treatments of a mathematical procedure or topic

Summary

• Understanding mathematics is different for the policy maker, the mathematician, the teacher and the student.

• Mathematics consists of concepts and problems; this talk has mainly been about understanding concepts. (See Polya for understanding problems and problem-solving.)

• There are at least five dimensions to the understanding of mathematical concepts:

Skill-algorithm, Property-proof, Use-application, Representation-metaphor, and History-culture.

Thank you!

For a copy of the ppt, e-mail

[email protected]

icme12 understanding math usiskin

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