86lA 644 UNDERSTANDING ALGERA EQUATION SOLVING STRATEGIES(U) t/l

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(0 understanding Algebra Equation00 Solving Strategies

p Techrnical Report P00-2

Kurt Vanl-ehn and William Sall

Oeoartrnents of Psychology and Comnouter ScenceCarnegie-Meflon University

Scnenley Park,Pittsburgh, PA 15213 U.S.A.

DEPARTMENTofPSYCHOLOGY

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,ELECTE~ OCT 22 1987~i

Carnegie-Mellon UniversityDts~B~OIN STAI-:fir A'

Approved fol P*..tiCroU c IO04uif 0 ...

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Understanding Algebra EquationSolving Strategies

Technical Report PCG-2

Kurt VanLehn and William Ball

Departments of Psychology and Computer ScienceCarnegie-Mellon University

Schenley Park,Pittsburgh, PA 15213 U.S.A.

6 July 1987

Running head: Algebra strategies

~DTIC~ELECTE '

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• Tnis research was suoDorteC by the Personnel and Training Research Programs, Psychological SciencesDivision, Office of Naval Research, unoer Contract number N00014-86-K-0349. Approved for publicreiease; distrinution unlimited,

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Understanding Algebra Equation Solving Strategies

2 ~RSOA~ u~G(S) Kurt VanLehn & William Ball

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- CCSA7, ::ES 'S Su8,EC- ERMS Conrinuo Ont reverie r mecesiary and4 00er,ty cy 010or mumbOer)

Algebra, equation solving, problem space, problemsolving

S -A CrI'rrnu* on levorle f ftecexzary anal aenriry Oy Otocx 'umoer)

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Abstract

A task analysis of linear algebraic equation solving is presented. The problem space is Shown to have

an elegant mathematical form. Several strategies for searching the problem space are delineated, anc

their orooerties discussed. The forward search strategy, which appears to be the one most commoniy

:augnt :n hign-schooi textbooks, tends to generate non-optimal solution paths. An operator-suogoaling

search strategy -enas to generate shorter Paths. Bundy and Welham's waterfall loop strategy is shown :o

ne a variant of operator-suogoaling that is more amenable for use by humans. This task analysis

s.ggests that *.Ie water'al .oop strategy may be oetter for teaching to nigh-school students :han trie

iorward sea rch straziegy.

04

2

1. IntroductionThis brief note presents an analysis of a small portion of high school algebra, the solving of linear

ec.ations in one variable. The analysis is based on the formal properties of the task, rather than data

fromn human Subjects. Two results are presented. First, the structure of the task domain is uncovered,

and shown to have some elegant propzerties. Second, there is suggestive evidence that forward searcn,

wnich appears to be the strategy most commonly taught in high school algebra, is a less effic:ent searcn

* . strategy than operator subgoalnrg, a strategy based directed on the structure of the task domain, in that it

.er~CS to generate 'onger Solution oaths. This suggests that operator subgoaling may be a better strategy

4a- 'eacr r'g hign school algebra students. However, operator subgoaling seems to require more

cog-"ve resources of line student than forward search. Bundy and Welham's (1,981) waterfal l 000

s:,a-eny NrrC,- s a tyce of ooerator s!uogoaling, is shown to offer reduced requirements 'or cogrnve

-eso ,,ces I: t"Ls cornones tre advartages of short solution paths witn low cognitive loac

A :"oi.gn *'-ese res5,its are suggestive, empirical work is needed in order to build a case 'or clharging

'-e cecagog cai oracces of high school algebra. A tenuous, but still interesting conjecture is that-

.ec g stucents the structure of the solut:on space, as described herein, may lead them to a better

urderstaldifg of th'e process of solving algebra equations.

2. The problem space of simple algebraic equation solvingSo vii-g an algeora equat on can be viewed as search in a problem space (Newell & Simron, 1972) A

oa- e- soace s de':neC by a set of states, a set of olperators 'or moving from one state to another, an

a sla~e arc a descrva on 'or th e des:red final state. For algebra, a state .s just an algebraic ecuat o,

a-': a oce-a~or s , s: an aigeoraic transformation, such as subtract~ng a term from both soes of :he

c-': T-e r,-ai state s :-e given ecuanon, e g.. 6-5ix+.) 7, = S-x. A final state s any equat on

* -as s* o~e occu-e~ce of t"e va- an e a'c Ile var ab e s 'solatec. '.a: s. t stancs a o~e o, -"e e"

*~~ ce a'yse' -e ecua: o"

I"e'- a. s*ates ie c "e' e- ec-a: ors *o oe solvecl eirgercer a "e-ent soec:' c orco e- coaces

-c,,iee a :-e ac:o er- sc aces "s *asK aor-ai n ave *"e sa"'e oas~c 'oocog~cai 'or-' T- s sec, or

s -aCa N. Ve, c-,4 -e -I eracly o' s'a:e vaces s~c- *-a' a state a' : NJ s I'-

a e"0 ~ ~ -a C e3 %' -330 I ' e o. es* s~ate tyce. s*.a~e :yce -csssU"e ac:a

3

equations. Thus, 1+x = 3 and x+1 = 3 are distinct type 1 states. A type 2 state is an equivalence class

of type 1 states that can be reached by the algebraic transformations for communtativity, associativity,

arithmetic combination, reversing the sides of an equation, and simplifying a double unary minus. Thus,

the following equations are all in the same type 2 class:

x+l = 30/3

lax = 30/3

1+x = 10

l+X = -(-10)

10 = l*x

10 = x+1

11-1 = x+l

100-1/: = x+1

Because there are infinitely riany ways to express numbers as arithmetic expressions, there are infinitely

many type 1 states in a type 2 state.

Type 3 states are defined as the equivalence class of type 2 states under the algebraic transformations

that "do the same thing" to both sides of the equations, such as adding the same term to both sides of the

equation. Such equivalance classes have an interesting structure. It is easiest to discuss it with the aid of

several simple examples.

Figure 1 shows eight type 2 states that constitute a type 3 state. They are arranged in a cube in order

to snow the relationsh os among them. The edges that are parallel to the horizontal axis are represent

tre ooerat~ons of mult-oiying or divicing by a. The vertical axis edges represent multiplying or cividing by

b. The remaining edges represent muitiplying or dividing by c. The diagonals of tre c-be (not crawn)

corresoord to inverti g both s:des of the equation.

F gre 2 d:soiays another type 3 state, where the equations are related by accing arc siotracting from

oth s;des. The edges represent adding and subtracting by, respectively, a, b or The diagonals

,eoresent negatirg both sides of the equation.

In general, al equations formed from three atomic subexoressions and an invertible b;rary ooerator wil

-, t'geder a type 3 state that has either type 2 states arranged in a cuoe. However, if *"e s ,oexoress:ors

are ot ato- c, out are hemseives express;ons, then a type 3 state cons ss of wo or -oe cnes ''a*

.%

. . . .. .. -. -. - . .- .- -.- -. - -. -. . , , .. -.11, -. ,- . - .-A. A 2.4, - : : : ; : ; -. .- z ,

a -c.n -c. ao

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ac c c a

a- - C =-.'

Figure 1: Type 3 state for multiplication and division

a - o-co- c: a

a-c 0--c a

.-

. . - 0

~Figure 2: Tyoe 3 s~a'e 'or accit;on ard suotract!on

I

3/30 =1/(x + 2) 1/30 = 1/3(x + 2)

N/3 =30/(x 2) 1 = 30/3(x + 2)

3(x + 2)/30=1 (x +2)/30 =1/3

3(x 2) 30 x + 2 =30/3

x + 2 = 10 x+ 2- 10 = a

x = 10 2 x - 10 = - 2

2 = 10 - x 2 - 10 = - x

0 = 10 - 2 - x -10=- 2 -x

Figure 3: Type 3 state for the equanon 3(x ") 30

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share vertices. Figure 3 illustrates such a state for the equation 3(x+ 1) =0. The uo')er cube treats the

suoexpression (x+1) as atomic, while the bottom cube treats 31-30 as atomic. The multiplication Cube

snares the vertex that stands for the equation x+1 = 3/30 with the addition cube. In general, There are

three vertices in every cube that can be shared. viz, those where a subexciressor that s5 treated as

a~ormic relative to the cube's operations appears isolated on one side of the equal1ty. When a

subexciresSion .s ;solated, its operator is the top operator on one side of the ecuation, and tnus can founld

a "ew "do it to both sides" cube.

1" orr.c:ple, one can ado (or multiply, etc.) any expression to both sides of an equation. Thus, adding

o3' o 0th sides of 2(x+1) = 30 is legal. However, most algebra equations can be solved by applying

cot-s aes operations using only subexpressions that appear in the equation. If we restrict the problem

soace by any allowing the both-sides operators to use subexpressions from the equation, then a tybe 3

s~a,,e always consists of a set of one or more cubes, connected by shared vertices. With this restriction, a

tye3 state hias the following properties:

* All t's constituent type 2 states are equations with exactly the same atomic terms (i.e.,r-ocuio ar'tLnmet';c evaluation, which merges number together).

e Any atom- c *,er- can be isolated by operalios that stay inside the type 3 state.

These orooer-es 'o 'ow directly from the fact that each operation neither destroys terms nor creates

- .*e~s.a~c! th1at a : ocerat ons are invertible, given that the equations are linear equations.

These two orooer*, es :oget'her imply that if any equation in a type 3 state has a single occurrence of the

ia' ac e, -en They al do, and furthermore, that one of the type 2 equations has the variable isolated on

or-e s nc of -e ecozat on TPis, a type 3 state is a "final" state if it contains an equation with a Single

occ:'e-ce o' -~e .... nown For example, the equation 3(x~fl ='10 corresoonos to a finai type 3 state.

cct- s Ces' ccera: o-s are ai -t~a is ecu redl to Convert t Ito The ces rec o.

-~t:-e -.'ooe'/ -o'e generaily, all the ecuations in the tyoe 3 state "ave The same rromer of

c:-e~ce s -.' '-e - .<%wr va-aoler A *yoe 3 state can be ass:gned a leu.,stc value ecual to ,he

- 'ce, o' occ.--e-ces -' e .<-~owr. and ,s value can be used to guid cc Te searcrn among ty e 3

waes -Y aw.ays C'-c00 -q c'oe'at ons medce Ths value Th s strategy S discu.ssed 'ur~her ceow.

V2' N' ~--e set-o ato"- c excress ors n an eci..a*on s to -se so-e 'oir- o' c s" c- c

7 222v7' c-~ -e >s e c eac s% ,i, e .Str at e c 0 Ow

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7

x(ai-b) = C <- - ax-sbx = c

c = (a+b)Ix 4 c = a/x+b/x

*,taxb= C +-- X 5* C

These distribution operations form the type 3 state transitions. Thus, the whole problem space for linear1

algebra equation solving can be reduced dramatically to a problem space of type 3 states connected by

type 3 operators, with the final type 3 state having just a single occurrence of the variable. This idea

forms the basis of th~e operator subgoaiing strategy, whnich is discussed in the next sect:on.

3. Two search strategiesThis secthon ic;sses two search strategies, then compares them. The first strategy derives from the

orociemn soace aralys:s presented aoove. The .dea is to search through the type 3 states using a simple

niil Cimoing strategy -- move to an adjacent type 3 state that minimizes the number of occurrences of ThIe

4 variable -- 'nen searcni thtrough the final type 3 state to find a final type 2 state.

Aithough this sounds cuite simple, t is corolicated by the fact that the combine-term operations, which

are the ooerators used to rove between type 3 states. require that the terms to be Combined are cousirs

n te exoresson tree. Tlat s, wren the expression is viewed as an operator tree (see ligure 4), the twVo

occurrences of the variable must be direct descendents of sibing noces. In the equation 3x+.4x-14

S Possible to coallesce the occurrences of the variable, but in the equation 3x44xr+l) = 7 it is not

oOSSiole apply the combne-terms operation. Thus the strategy requires operator subgoaling, wherein th~e

soiver adopts th~e "ew goal of transforming the equation into a form suitable for applying the comoine-

termrs ooerator. Inti 'ae he subgoal 5s to transform th~e subexpression 4(x+l) into a sumn with x

contained in ore of Is terms. This can be accompiisned by applying the distribution ooerator, wh'tch1

corresoonds to a Yars~tior between two adjacent type 3 states, both of wnicn nave same reurist~c value

In som-e cases, t-e subgoai can cie accom-ol s-ed by stayirg -sice the tyoe 3 state. as .V-e! r -

s t-arsformec to Sxi:- = Sucn s ,ogoa irg can oecorre rath~er comz 'catea. Beca-.se t-e cc-' -a!-:

'orm of act:vt'y s ooerator S,,Ogoai;ng, :h is strategy .s namea operator sutgcaling.

As -ert o-ec ear, er. *ra! states car oe specfied as a corjuncton of two prooert:es. !-:ere so'-e

occ,.,erce of :ii#e variable. ara 2) 't occurs soiated on the left or rgnt side of the equator The ooera*or

'e a-al ,s .ar, orccac y te exte'~d5d lo a ruch arger class of equations Bundy 3'd eharr s , '81 equation so.syse" N r v,.l ie sro.wn aer !o use a scecai case of this strategy can solve rational Poiynomnals containirg tigometroc arl-o

-. -oe'~ o ~o~o%.d%

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8

3 x 4x

Figure 4: Expression tree for the equation 3x+4x+4=7. The .. means multiply.

suogoaling strategy achieves the single-occurrence goal first, then works on the isolation goal. The

searcr strategy that seems to be taught in high school emphasizes the isolation subgoal. In the

extbOOks we have examined, students are taught to clear radicals, fractions, and parentheses first, then

combine terms. (See figure 5 for one popular textbook's description of the strategy it teaches). This

solate-then-combine strategy has the advantage that it is easily implemented as a visually-cued forward

search. The search heuristics are rules such as "If you see some parentheses, then use the distribution

operator to remove them." Because the heuristics are cued by visual features of the equation, they may

ce easier to remember.

* However, the forward search strategy leads to inefficient solutions in some cases:Forward Search Solution Optimal Soluticn_(x+2) = 30 ,x2) = 303x-6 30 x--2 = II)

""x = 24 x =8X =8

T e ootr'al so,.:on patll in this case wou'd be generated by the ooerator subgoaling strategy The

'o'.vatc searcn's strategy s "on-octimal because it moves out of the initial type 3 state, wnich is also a

.a state, a~d to an adlacent type 3 state, which S also a final state. In this case, it is better to stay in

-e al type 3 state. Hee are som,,e "ore cases where the forward search strategy does poorly:

-' -'' . ....

04,~i

iP_- A-.

9

SOLVING AN EQUATION HAVING ONE VARIABLE

1. Clear the equation of fractions, if any, by multiplying both members by the L.C.D. of all

fractions in the equations.

2. Remove parentheses.

3. Clear the equation of decimals. if any, by multiplying both members by the appropriate

power of 10.

If it is a first.degree equation, If it is a second.degree equation,

4. Collect all terms containing the 4. Collect the terms so they are in the

variable so they are in the left left member. The right member

member. Collect all other terms should be zero.

in the right member.

5. Simplify both members. 5. Simplify the left member.

6. Factor the left member.

6. Divide each member by the 7. Set each factor containing the

coefficient of the variable, variable equal to zero and solve

each resulting equation.

7 Check your result by replacing the 8. Check your results by replacing the

variable in the original equation. variable in the original equation.

I"

I"".Figure 5: A procedure from a poloular tigi sciool textbook,

I~iiWelchons etal. (1981), Qg "119

,%%

w.

4b+4V_3x =147 Removing the parentheses is unnecessary

4 b-i~lx+7) -4-16 Shortest path is to divide by lb+4

'9. Y[x+7r+91 102 Shortest path is to divide out the 27 frst

We oerieve :hat it can be shown that the operator subgoalling strategy always p~occes sncr-er

so -tirn oaths than forward search, or a path of equal length. This belief is based mos,,!y on our !-~y to

-c" a counrte rexampie. Further work is needed on this important question.

- 4. Comparing the two strategiesFor '. 'ran sovers, .: s moortant to find Shorter paths, but not so r~cr, cecause itsaves -e, out

,a .- e neca~se it 'ec-ces the chance of error. Excerienced solvers ra;,e rost of teir errors our rg -~e

-2xec,, o' of ooe~at ors, as cl:Dosed to using incorrect or inapproorate ooerat-ons ,Lewis 192T-e

'e.ver :re olce'atlons needed to achieve a Solution, the less chance of error.

However shlortness of bath !s not :he only relevant criterion upon which a searcn strategy should be

:-osen Tne strategy should be easy to use and easy to learn. The operator subgoaling strategy may

ce catcuiary easy :o use, oecause it seemrs to require a goal Stack. That is. the solvers must

* -e-er-roer wnat goal th ey were working on so that they can resume working on it whien they get done withi

:-suogoal Th-e extra memory loao of a rraintaining a goal stack may make the operator suogoaling

s,'a~eqy -e ci"cult to use th an thie forward search strategy, which requires no goal stack because :s

se .3c*ton, ot ooerators s celermineo entirely by the current state.

The Nater'al l oop strategy (Bundy & Wellham, 1981) offers the best of both strategies. It is esser~tal'y

-3- ooerator suogoalnrg strategy, wnich means that it tends to generate opt;(mal solution oaths. but t is

-c~' er-ertec oy several reurstlics tnat are driven almost entirely by the current State, thus -i-izng

=ce- a. -errory 'oad T-e walerla'l cop strategy has tnree meta" ru es

*Isolation: If tlhe equation nas ust one occurrence of tne varaoe, th en acoly a ot-s cesooerator aporoorate to the ari tirmetic ooeratiOn that is the root of 'e exoressocn tree o, t-e

*,sce of 'e eauation contanrig thne variable.

*Combination: If ",he eQuatonr has two occurrences of the varaoe thlat are cous,rs in -'e *.ree.tren comrone t"em,

*Attraction: If tne ecuation has multiple, non-cousin occurrences of the variable, select two"at are nearest in the tree, and apply a transformation that will make them nearer

7-e I s* o' t"ese reta-rules th"at -atcnes the ecuation is fired, then thne .ooo reoeats on t1e new

% -' : - - ~ ' - * . *

equation. Thlat is. control falls through to tlhe rule thiat matcres. *,, coos oacK

Tne waterfall 000p strategy has tre same goal stru'ctre as :-'e oDe-ato' s ,ogoa..ng st',a~egy :so a, oc,

arc com bination are the top level goais of algebra equation solv ng, aio attrac: on s a sulogoa c'

cor-onation. The waterfall looo oiffers from operator suogoanig :n *,a,, it ..ses ro goai staCK as

*e-oorary state for 1!s processing. It is Crve, entirely oy :he appearance of :h~e ecuato-. So *too s a

4su a ly cued, forward search strategy. But its design makes .t a form of operator siogoa. r~g

T-e waterfall 'ooo tr='erefore comoirtes *the best prooertes of ooth 'orward searcn arc ooe-atlc

s-ogoalirg. It tencs to generate olotmrral solution paths, and it is visually cued. Thre ai'fererces ce~wee-

arc t'e usual procedure taught in schools is that its cues concern the number of occurrences of *.e

va' ace a~d te r 'e a*:ve pos ton .n tre equation tree. The forward search procedure .s cued by *-e

c'esence of ;arge 'eatures, s~ci as parentreses and radicals.

Obvious;y, thiese commrertS about ease of use must be viewed as suggestive only. Experimental work,

cerleraoly wi-i ca*m expert ano rov~ces numan solvers, is neeoeol to compare the two strategies.

5. Suggestions for further researchTh !s brief note, al'though sparse on results. opens a number of interesting avenues for researcn. Two

-'ave been mentioned already: a formal demonstration of the optimality of the operator subgoaling

stategy over the forward search strategy, and an empirical demonstration of its superior ease of use.

S, a' y, we reed to experimentally compare the learnability of the two strategies.

A coss bIy ro-e moortant question is whether this new view of algebraic equation solving as s;mor)e

cii-p-g n t:'e t.yce 3 crocblemr soace allows students to truly understand the task comarn Certai v.

6we *eel that we ncerslard a gecraic st'ateg es better for ravrrg LrderStood ! -e str-.cture of te croc e-

sc-ace Per-acs *2's v ew wi l -e c t-e stuce-ts as wei!

To cut th7e su ggest o,' r nc-e corcrete 'arm, sucoose one cuit an aigecra equat'or so~v'rg systemn

aol-g t-e :,es of Ageoraac .BroWr, '983) tnat used a -eru-c-iven irterface to alow 't-e stujcert to

Se_ ect, ocerat o,-s wN, c "e syste ' would tk-en aloy Algebra'a~d Keeos track of th catn th7e stucert

Ic ows arc C sc ays t1- f eStu cert oac~s o. then thre cisolay is a tree, otherwise it :s a oatr, B-own

oa '-s t-at -' s -'ay 'ac tate '-e acc s 'o-~ of rorovec searcr strateges. This claim -s coi-'ssterl ,N2

-esea-c7r y Ace-sor a-c- s c-, eag..es A' e-so-. Boy e & Yost, '985, Ancersor. Boy e & Re se-

%

12

'985), which shows that Similar displays of solution trees seems to help geometry students learn

strategies for finding proofs in plane geometry. The basic message from both sets of researc-'ers s .nat

C sciaying the solution path in a way that emphasizes its tacit stlucture neos students ear- a searc-

st-ategy based on that structure Now, suppose tinat we displayed t~le searcon of an a geora s:-ce":, ' a

-a-ner siriar to :he coes of 'igures 1.2 and 3 We con ec:.re :,rat '-, SC~soay wl: -e D s*:-ce':s comre

.c -- cerstano a'geora strategy in a new, more ber'ef ca way.

13

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456-462.

An'derson, J. R., Boyle, C., & Yost, G. (1985). The geometry tutor In Proceedings of Ninth International

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Brown, J.S. (1983). Process versus product: A perspective on tools 'or communal ario n'or-a

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NY Ecucatonai Broaccas*,ng Comoany

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,ewr *e L.e sets n a'georai C an Duiat on, Artificial Intelligence. 16, ' 89-211

L ew1s. C, (1981) Ski in a geOra. in J, R. Ancerson 1Ec.l, Cognitive Skills and their Acqustllcn

Hilisna,i NJ: Lawrence Er'baum.

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We c-ons A.M , K' C~enoe'-ger, W.R., Pearson, H R., Duffy, A.G., M& MacCaffery, J.M. (19811 Algebra:

Cook ILexirgton, MA Ginn.

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