Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus...

25
Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented at AMATYC 2011 on November 12, 2011, in Austin, Texas

Transcript of Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus...

Page 1: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Instructors Assess Every StepSo Why Can’t Software?

By John C. MillerProfessor EmeritusDepartment of MathematicsThe City College of C.U.N.Y.

Presented at AMATYC 2011on November 12, 2011, in Austin, Texas

Page 2: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Math Software Syllogism

• Math software should emulategood math instructors.

• Good math instructors always askto see all the students’ steps.

• So … . (We will return to this syllogism later.)

Page 3: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

A Brief History of Math Software

• In the 1980s, multiple-choice problems were universal in math software.

• Eventually, more and more faculty objectedto the multiple-choice format atsoftware presentations and elsewhere.

• This inspired publishers to move frommultiple-choice to short answer problems.

Page 4: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

One Solution(from The New Yorker)

Page 5: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Interactive Mathematicsby Academic Systems

Page 6: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Interactive Mathematicsby Academic Systems

• The Academic Systems software, released in 1993, enjoyed stunning sales due to:• Superior graphics.

• Very aggressive promotion.

• Short answers, not multiple choice.

Page 7: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Interactive Mathematicsby Academic Systems

• The following actual problem fromAcademic Systems shows both:• The obvious superiority of short answers

as compared to multiple choice.

• The ultimate shortcoming of short answers.

• Note that the second line of the student’s upcoming solution is clever, and unexpected!But there’s a sign error in her last line.

Page 8: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

2

YOUR ANSWER IS INCORRECT.

Explain

Apply

Explore

Evaluate

Homework

Overview

Leo is now 4 years old.Let Leo’s age now = x.Then Steve’s age now = 2x - 1.Three years ago, Steve’s age was (2x - 1) - 3.At that time, Steve was as old as Leo is now.Steve’s age three years ago = Leo’s age now. (2x - 1) - 3 = x 2x - 4 = x x - 4 = 0 x = 4So Leo is now 4 years old.

Let a = Leo’s age now.

a + 3 = Steve’s age now.

a + 3 = 2a – 1.

2 = a.

Steve’s age now is 1 year less than twice Leo’s age now.Three years ago, Steve was as old as Leo is now.How old is Leo now? Years old.

Done

Interactive Mathematicsby Academic Systems

Response implies that student’s solution is TOTALLY WRONG!

Clever use of 2nd sentence!

Sign error!

Page 9: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Interactive Mathematicsby Academic Systems

• In response to Academic Systems’ success, most publishers produced clones that:

• Used short answer problem formats.

• Showed a stored solution after any error.

• Used “Algorithmic problem generation,”i.e., randomly varied coefficients.

• Academic Systems went public in 1999, but fell prey to the dot-com bust, and vanished.

Page 10: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

The Bad News• For the last 15 years:

• Good instructors have continuedinsisting on seeing all student steps.

• Major publishers’ software has continued accepting only short answers.

• We show two recent examples from major publishers (anonymously, in accord with AMATYC policies).

Page 11: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Program A

•In this problem, involving a system of two linear equations:

•The student begins by solving the first equation …

•… but makes a sign error in substituting.

•Program A’s stored solution begins by solving the second equation!

Page 12: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Program A

y = 10-3x

3x-10 = -y

2x-y = 0{2x-10-3x = 0

-10 = x

y = 10-3(-10)

y = 40

Response is no help to student. Worse, it impliesthat the student should always begin by solvingthe second equation, since beginning withthe first equation results in an incorrect solution!

Solved 1st equation.

Solved 2nd equation.Note: sign error!

Page 13: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Program B

•In this problem, involving a fractional equation:

•One student clears fractions by multiplying both sides by the L.C.M of the denominators.

•Another student uses “cross multiplication.”

•Each student makes a sign error when transposing.

Page 14: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

9

10

109

142499

7

3

4

2

1

x

x

x

x

9

10

109

142499

7

3

4

2

1

x

x

x

x

Program B

9

10

9

10

109

142499

7

3

4

2

1

x

x

x

x

9

10

109

142499

7

3

4

2

1

x

x

x

x

9

10

109

142499

7

3

4

2

1

x

x

x

x

9

10

3027

4272279

7

6

839

7

3

4

2

1

x

x

x

x

x

9

10

3027

4272279

7

6

839

7

3

4

2

1

x

x

x

x

x

9

10

3027

4272279

7

6

839

7

3

4

2

1

x

x

x

x

x

9

10

3027

4272279

7

6

839

7

3

4

2

1

x

x

x

x

x

9

10

3027

4272279

7

6

839

7

3

4

2

1

x

x

x

x

x

Stored solution never clears the fractions. It helps neither studentfind his or her sign error!

Note: Sign Error

Note: Sign Error

Page 15: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Math Software Syllogism

• Math software should emulategood math instructors.

• Good math instructors always askto see all the students’ steps.

• But let’s rethink our premisesand propose a higher standard… .

(Revisited)

Page 16: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Math Software Syllogism

• Math software should emulate the aspirations of good math instructors.

• Good math instructors aspire to be thereto help each student with each step.

• So math software should:• Accept step-by-step solutions.

• Flag any incorrect step immediately

• Review as appropriate after an incorrect step.

• Suggest a reasonable next step at any stage.

Page 17: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

So How Difficult Can It Be …

• To Write an Equivalence Checkerto determine if an arbitrary student stepis equivalent to the preceding step?

• To Write a Next Step Generatorto suggest a reasonable next step afterany correct partial solution to any problem?

Page 18: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Two More Programs

• Both of which:• Were written by individual math professors.

• Were self-published via the Internet.

• Have been available for more than 15 years.

• Include both equivalence checking andsuggested next step generation capability.

• Appear to have made virtually no impact.

Page 19: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

By an Israeli Math Professor

Program Y

• This program poses a system of linear equations.• Its hint implicitly suggests the addition method.• The student begins with the substitution method.• The program accommodates to this preference.• It evidently contains both an equivalence checker and a next step generator.

Page 20: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

By an Israeli Math Professor

Program Y

Page 21: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

xyAlgebraA Completely Free Program By John C. Miller

• An xyAlgebra student solves a fractional equation.

• The program initially suggests clearing fractions.

• Instead, the student “collects” the fractions.

• After detecting an error, xyAlgebra applies its solution generator to the student’s last correct step, and suggests a correct next step appropriate to the student’s solution method!

Page 22: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

xyAlgebraA Completely Free Program By John C. Miller

Page 23: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

How Can We GetBetter Math Practice Software?

• The responsibility lies with our profession.So why aren’t we insisting on better materials?

• We don’t care about our students.

• We can’t figure out what’s needed.

• We’re too polite to make a fuss.

• We’re too busy or can’t be bothered.

• We feel threatened by technology.

Page 24: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

How Can We GetBetter Math Practice Software?

• Let’s demand better math practice softwarewith intelligent step-by-step help:

• At commercial software presentations.

• At commercial software exhibits.

• From publishers’ sales representatives.

• By enlisting colleagues.

Page 25: Instructors Assess Every Step So Why Can’t Software? By John C. Miller Professor Emeritus Department of Mathematics The City College of C.U.N.Y. Presented.

Instructors Assess Every StepSo Why Can’t Software?

John C. MillerProfessor EmeritusDepartment of MathematicsThe City College of C.U.N.Y.

The entire xyAlgebra package is available on the“Free Downloading” page at www.xyalgebra.org.Thank you very much for your interest.