Post on 03-Jan-2016
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
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.)
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.
One Solution(from The New Yorker)
Interactive Mathematicsby Academic Systems
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.
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.
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!
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.
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).
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!
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!
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.
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Program B
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Stored solution never clears the fractions. It helps neither studentfind his or her sign error!
Note: Sign Error
Note: Sign Error
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)
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.
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?
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.
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.
By an Israeli Math Professor
Program Y
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!
xyAlgebraA Completely Free Program By John C. Miller
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.
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.
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.