HINTING BY PARAPHRASTN5 IN AN INSTRUCTION SYS'rEM 
Uladimir PERICLIEU 
Svjato~lav HRAJNOU 
Irina NENOUA 
Department of Mathematical Linguistics 
Institute of Mathematics with Computino Centre, hl.O 
Bulgarian Academ~ OF Sciences 
iI13 Sofia, Bulgaria 
Abs%~act 
Previous work has emphasized the need 
~+or parat\]hrasos as means oF ensurin 0 a 
ffeedbaok with a system. In this paper, we 
diS~LlSS )TDW n paraphrase may he used as a 
hetJuistilz devit~o ,. viz. as a hint, We 
desuFibe an experimental instruction 
system J.*\] mathematics incoFporetins this, 
Featut-e. The system accepts a restricted 
Glass off algebraic storH problems, 
Ecwmulat~d in non-stylized Hulgarian 
lar\]Ouags , arld J.5 capablB oF solving them 
and providi*\]~ one~ Or more "hinting" 
paraphre~e.s, that is, paraphrases 
alley*at kng their Formal*sat*on 
(~'t~rans\].ilti~n into equations\]. 
I. IIWRODUCTION 
PrE\]vious work has emphasized the need 
For paraphrases as means of ensuring a 
Feedback with the system, For example, 
qUL~S t i on'-ansuJsr i no sW sterns, beFolc's 
respond J.n\[J, paraphrase the requests 
fermulatf\]d in natural (Kaplan 1979, 
McKeown :\[9833 or a Formal language (de 
Roeok, I.owdsn 19859 in order that the user 
ascertain that his/her question has reall.y 
been ~or):'ectl N understood. This step is 
necessar!j to avoid (possibly) costly 
searches in the data base For requests 
that hays never bean made. An additional 
reason iq that sometimes the Format of the 
retrieved information also cannot clear up 
a potential misunderstanding CThoma5, 
Gould 1~Z|'75) , 
Howl~ver, there are other 
appli~atLons, diFFerent From Feedback, to 
~hioh ~ paraphrasti~ Facility r.aU 
profitably contribute, lrl thE~ paperj We 
discuss ho~ a paraphrase maw be used as a 
heuristi~\] device, viz. as providing 
hint, in an instruction system in 
mathematics. 
Yho paper is organized in the 
Eollowirpu wag. Sect. ~ is a brief overview 
of the instruction system incorporating 
this \['astute. Seot~. 3 and '{ describe 
r'espsctivelg some general ~equirements 
to ~ hJ,~Itiri~ paraphrase and the measures 
me have taker, to satisfy them in the 
sustomo !~ect, 5 discusses an example, and 
~ect. 5, some implementationdetails. 
2. SYSTEM OVERVIEW 
Me have designed an exporimental 
instruction system in mathematics. The 
system operates in a limited domain: it is 
capable of solving a restricted class of 
stoc~ problems in algebra EOC Ssoondar~ 
Schools in Bulgaria (the so ~alled "number 
problems"). The system accepts non- 
stylized stor'y problems in Bulgarian as 
they can be Found in mathematical 
textbooks of are spontaneously Formulated 
bU the user. It solves the probl~m, and is 
capable of providing either of the 
Following 3 options: 
(a~ Result (resultant number(s) are 
displayed), 
(b9 Equations (the equation(s) to 
which problems translate are displayed). 
(c) Paraphrases (one or more 
"hinting" paraphrases are displayed, 
together with the text of the orioinal 
problem). 
All the three options serve as 
CdiFFecent degrees of) hinting needed in 
case the users (Secondary 5chool pupils) 
have probl~ms with Finding a solution, 
Furtheron, we Focus on problems concerned 
with the hinting paraphrastic Facility oF 
the system. 
3, GENERAL REQUIREMENTS TO A 
HINTING PARAPHRASE 
The profit of using a paraphrase, or 
a "reO'oreulation", oF a problem as a , 
heuristic tool has been emphasized b U 
researchers in heuristics, pedagogy and 
psy~hmlogy of education. Nevertheless, 
such a possibility is usually be qond the 
s~ope OF i~letruction systems (51eeman, 
Brush laB2, Weischedsl st el IS'TO, PLJlrnaN 
19B~), 
l he question still remains as to 
what can count as a hinting pararh£asa 
(HP) (obviously , not ~ paraphrase can 
serve this purpose equally well). Has*no 
ourselves on research in mathematical 
pedagogy and psgchol inguistics (since 
conceptual and linguistic structures in 
this earl, g age are known to be stronoIN 
507 
interdependent\], we derived the followlno 
general requirammnts to a HP: 
i. The HP should ~ the 
original problem (OP) as rsoards the users 
of the system ( by this we msan 
simplification Of OP in both conceptual 
and linguistic aspects with respect to the 
task assigned, viz. to fomalise 
(-translate into equations) th~ OP. 
2. The HP should be ~. From 
the OP (this requirement is self-evident). 
3. The HP should k_eeep close to the OP 
From a conceptual and linguistic 
viewpoi~lts (this is to ensure that the 
USers conceive the "sameness" of HP and 
OF). 
Since the most important task of th~ 
HP is to simplif~ the translation of 
verbally formulated problems into 
equations (solving these equations being 
as a rule wnproblematic for childrHn), ws 
took the Following ~sneral solution 
regardin 0 an "appropriats" HP: An 
"appropriate" HF to a problem is the one 
that can be, somewhat metaphorically 
expressed, literallu translated i~to the 
respmctive equations of this problem. 
Obviously, this would, to the greatest 
extent possible, simplify the OP (in the 
sense in which in the translation from one 
NL to another, the easiest to perform is 
the literal translation\]. This de~ision is 
further supported by the faot that pupils 
usually t~ans\].ate to themselves the OP 
into intermediary languaoe which is most 
close to the equations derivable from this 
problem. 
~. CONSTRAINTS ON 
"APPROPRIATE" HPs 
From what is stated above, a number 
of specific constraints on the content and 
form of the HP can be derived. We briefly 
mention them below in connection with two 
of the major decisions that have to be 
made in a oeneration presses: first, 
makin 0 a decisions as to the 
structure of the HP (i.o. determinin 0 what 
and whsn to sag, or an ordered message to 
be conveyed), and, secondly, making a 
decision as to the y~rbal formulatiom of 
the discourse structure of the HP (i.e. 
determinin 0 how to express this 
in/ormation in Huloarian, what syntantic 
structures to use, what lexemes, etc.). 
At the first staos, we should gain in 
conceptual, and, at the seoond stage, in 
linouistic simplification, thus 
approximating the requirement as to the 
litsralness we have imposed. 
~.i. Discourse structure 
In the light oF our aims, it is clear that 
the discourse structure o~ the HPs should 
be standardized, or ~, This means 
that we need not bs concerned (like most 
scholars workin~ on discourse 
organization, e.g. Mann 198~, McKswon 
508 
IH853 with ~\[W~typss of discourse 
structures of actual texts in the domain 
oE interest, but rather with ~ 
discourse pattern that satisfies the 
discourse ooal. 
Each of the texts in our domain, story 
problems in aloebra For Secondary 5chools, 
is known to be characterized h u Wi!knownf~!, 
(i.e. what is looked for in the problem), 
and ~ (i.e. the equation(s\], 
relating the unknown(s\], or variables, to 
the givenfs), or constants, in the 
problem). Some problems also involve 
~uxiliaru~ (i0e. further 
unknown(s), often mentioned in the 
problem formulation somewhat misleadingly 
(e.g. "...Another number is ~.~"\], 
which have to be manipwlatad~ but are not 
themselves part of the solution\]. 
Yhs discour58 structure of the HPs, 
thsreffore, will have to reflect the basic 
conoeptual constituents of the prohlems: 
I. the unknown(s\] 
2. the auxiliary unknownCs\] 
<optionally> 
3. the condition(s), 
in bb~ partiowl~r order. 
It may be noted that a lot cf 
problems, as they are formulated in 
mathematics textbooks, do not actually 
satisfy this discourse sohsma: the 
unknowns are interspersed in the text, the 
unknownfs) and auxiliary unknown(s\] ar~ 
not sxplioitly discrimisated, the 
conditions precede (auxiliary\] unknown(s), 
etc. 
For instanoe, a typical problem to be 
found in a textbook maw begin as follows: 
"The sum of two numbers is B..." Clearlu, 
5tartin 0 the problem formulation hU a 
condition, instead of with declaring first 
the unknown(s), is misleading. Thus, 
notice that this problem ma~ have quite 
different continuations, among which 
...The first number is 2, 
Which is the second? 
in which we have just ng~q~_ unknown, or 
...Their product is 12. 
What are these numbers? 
a version in which there arm two urlknowns. 
The resolution of this local 
ambiguity requires additional i~itallectual 
effort on the part of the pupilj ~o.- 
readJno, etc., ~ircumstances which ou~ HPs 
should evade. 
In addition to dssoribino the major' 
conceptual constituents of the problems~ 
in the canonical discourse structuma of 
the HPs, the monditions of problems 
themselves, usuall~ compound propositio|is, 
should be brok@n_~o n arts. The 
ordering of these propositions should 
The v~'b~,,J. Formulmt J.~ni off i.h~ I-IP~ 
C.OJ~L~JJh'U\[;'\]_J.L~FK~ tiller ~Ipp~ilh" J%I aLTttlal t~xts 
c " ..... t~: sJ~,;m ~ff the uum ~nd 
Th~'.~ ~, ~x ~i,~ ~ in th~ HI' ~hould 
.~\]ilJ~J\],0, \]lL~ ~ilmissJ, t:l,\].~ ill the HPs, vi~:. tlle~ 
h'ur' l:l\[llAiL~s "1.1~.~ I~l;'SSti'st7 tha~i/l,lith", stP. 
%JtJ.~.~ik t\[7~ l::llt4 V~Jl~'~lT:ll Et}~I3~'Pic-ie,'\[\[LIIIPJ J.li the 
I/j. ldA\]\]. }'i~-i \[tl~ak' \[;I;fJlil ~i}lLt'h iS ~7£tid 
hU~l th~-~ !rJ{P..~ \[~litiL:tJ\[-lll tO ti\]El 7~LIi~'tJiii~Jut~J I\]\[ = 
!A-a~t, 7~J,. 
~IE~LF\]m Id~ \[livtg L~Yl BXalllplB, ~i.ivle~d bL~ 
7g.j.|iliJj, j>f:j.i\]£~l;j.ull.'_i IL'\]~.' tJ'lfJ lip ~l~llisv~irl iti the 
lIP f\['O~" ,=envsi~i~ncl~, th~ |\]P alld thu HP ~:"8 
l;~:-ailslet~d intu EnM\].J.sh). 
TI-i~ OP is: 
(!) It" th~ ~Um Or" aria numbe~ ~iith 
('AT) mhl~h i<<~ mith t:- ~tlIi~\].17t\[" 4\[1}1~ill it 
(3) J.~ lilultlpli£-Id i\]~ ~.i 
('i) ~j~'Ju l, li}_.! L:=lnd "h|ic~ p~-c~duct u17 t'.h~t 
~4~j~l;t~'tlJ i-ilJirih~3~- mit~t thb\] lltllll!lKJ~.' ~\]. 
(;5) b'irl.rI th\[~ t#.!,c!~t llUllrlt:l~?F, 
(i-L) ~li~lJtli~r nlJmhur i5~ \[L!v~tio 
(\['1) ~IJLt th~ tbio IlLliflhEi\['t~. 
({%D~ I\[;' ~\]rJu ii~Ulttpl!_J th~ ~l.lift o|Jtalnsd 
.~slth E~ 
(E~:,' L~otJ bli!\], lfilid tha Ol:'~du~t ~F th~ 
~nd number wlth th~ cumber ~ 5. 
(5) ~' from the ~irst number UOU 
£V) !~c3u will ubt~in the ~ond 
~\[~l \[:umpn~'isei~ ~ith t|lS \[JP, tlis HP 
~xplluh%tSS th~ tt.o rlumb~r5 of the pF\[Jblem 
that mill bs £~Ll~'t\]i~m manipulstsd: Fi~:st, 
t | i~-~ u~ikno ,,~11, mi-ld ~ thatl, th~ aux i i ± arH 
lJrlkrlol~n. In clause (13 of t|iS lip ths 
~p~atioe off ~dditio~ is imp\].icitlu f~iv~n 
b U its ~-~sult ("ths ~um"), mhr~as iH 
clause (39 F~F t\]lO HP th~ same upeF~tio~ J.s 
~laborated h H arl axpli~it msntlorlino ~Jf 
th£~ paFticula~ ~ avithmstic~l npeFation ~Jf 
nddltiun ~ The imhsddsd relative clause 
(2{3 ~)i:" tlls ~P is sxprosssd sepaFat~\]iu \[:ram 
the lii\[~i~ sentence in the HP C(53 arid (7) 
el|? 'h\]'ll:~ \]|P). This pFovidss a puSsJ.bi.LitL4, 
~,"~adi:I\[~ t}1o ~ndition OC the pFoblem |;'rOiil 
Isft ta ciEIht, tn mFite damn, sequsntialJ.u 
~nd :i. ~ldl~pr~Tldsnt 1U, hhs diffff\[~rent 
oquatiuns, in the paraphrase ~ff th.~. 
~-~lativo clause (2) of the \[\]P, the 
z°81ation '"is sili~lla~ than", knumll to 138 
~i\]nEusino ffe~," small child~'en, is ~-splacod 
hi\] its ~ov~sspOild i rllJ operat ioi~ 
~'subtra~tlon", and tl-ls pFOrlLmlir\]a l 
FSffBFSIJUI~ (sXpFSsssd in the Eel/fish test 
\[,~ith "it"3 is avsidsd0 Notice also that 
('i) EFOm t|'is \[JP and (5) if,Fern tile HP ar~ 
~ihvas~d in th~ ~ame ~ta U (thus pFes~Fvin\[~ 
vavtlal Samaesss off the OP and tb~ HP), 
The csoFOallised text of the HP ~an hs 
s~sl\] to Si~lllfioantlu simpliffu th~ DP 
(mhi~h ~ill b~ ps~ti~ulaFlg tFu~ ff~ 
5ec~daFg ~chool children)° 
~, IHPLENENTATION 
Helom ms bristle describe 
~spsots off the implsmsntatiun design. 
Th~ system comprises 3 .iodulss: 
(i) Analwssr 
(ii) Solver 
(iii) Paraphraser. 
~011113 
Yhe Analussr is ~ "traditiunal" 
5~mantiu ~cammam, usis~ hie~a~uhi~allu 
o~osnised ~ The Sulv8~ ~Ives the 
equations obtained as a ~esult of the 
pa~si~S phase fie the sustem is in a 
"~ssult" muds). 
?he gsns~atiun process ~uss thL'ou~h 
ti.~ major phases. The pa~aphra~ti~; 
facilit U of the sustsm has t~o ~ompun~nts, 
vssponsihls ~o~ the tasks at these phaes~: 
the ~, add the ~X. 
In the first phase, the Canonizsr. 
~onstruuts the disoouFs8 stFucture, of thE~ 
canonical fD~m, sff the HP. Ths pFu~sss 
includes the reprsssstatisn sff the 
' dis~ouvsQ st~'uctur'e into a ssqusncs o~" 
slsmsntar.~ pr-upositiens, ±nstaritisted b~ 
509 
the r~sult dmrivsd bg the Analysis module. 
This sequsnom b~gins with the proposition 
describing the unknown(s), and, 
optiooall~, propositions for auxiliar~ 
u~kr~ownfs). In the sequence follow the 
propositions desc~-ibing conditions 
C~squations). 
For example, as a ~'~sult o~ th~ 
analysis of the OF, mentioned in Sscto 5, 
the following sequence As obtained: 
/ 
equal(*C+CX,Y),2),~CV,5)) 
equalC-(X,~),Y) 
unknown(X) 
The Canonizer shifts the last 
proposition unknoen(X) at the begining of 
the sequence of propositions sod adds get 
another proposition auxiliar~ unkno~m(Y)0 
As a result 
unknown(X) 
euxliarg_unknewi~fY) 
equal(m(+(X,Y),2),~CY,5)) 
equal(~(X,~),V) 
is obtained. 
Each compound proposition of the 
latter type is substituted with an 
equivalent ~ ~. In 
order to achieve this, all oonstitusnt 
propositions are substituted bu variablss~ 
after" which the simple proposition 
obtained is unified with the compound 
proposition. 
In the above case, Erom the 
unification of the two compound 
propositions "equal" wlth the simple 
proposition equal(~,B), we obtain: 
equalC~C+CX,V),~),*(Y,5)) - 
-equal(~,B).C~(r,2)/~,~fY,5)/B).C+fX,Y)/~} 
equalC-fX,~),Y)=aqual(~,B).\[-(X,~)/~,Y/B), 
where the expressions in braces are 
substitutions. 
The propositional expression thus 
describes the ~ of obtaining the 
compound proposition in question From 
simple propositions. 
AFter the substitution of Bach 
compound proposition of the equivalent 
propositional expression, the folloeing 
~anonical representation obtains: 
unknown(X) 
auxiliaru unknown(Y) 
BquBI(~,B).C~(r,~)/~,~(Y,S)/B}.\[+fX,Y 
)IF} 
BquBI(-(x,~),Y)-BquaI(~,B).{- 
CX,~)/~,Y/B} 
The canonical representation used is 
easilg seen to have certain advantages. On 
the one hand, it explicates all 
computations nsoessarg ~or construction o~ 
the sostem of equations, and, on the other 
hand, it defines a l~i_~erballzati_o!l, 
to be used b~ the Generator, in which, 
First of all, the simple propositions are 
verbalized, then their verbalizations are 
5\].0 
used in the verbalization of the compound 
propositions at the next higher level of 
hisrav~hu~ and so on0 The text to be 
obtained ~ollowing such a plan of 
verbalization can be literall u translated 
into a system cf equations b U virtue of 
the Fact that the text itself is ~snerat~d 
in inverse order" - From simple to compound 
proposihions, 
In the second phase of the process of 
generation of the HFs, the oanonical Fo~m 
of the HPs is translated into Bulgarian 
text bH the Generator. The 8enerato~ o 
itself is a.~!l~ " . " ~_~ 
m~ (the templates used Fo~ 
generation)° 
Each template describes a sgntacti~ 
construction by means o~ particular 
wordForms, lexi~al classes and variables. 
Some of the templates are used "tO 
propagate anaphorioal relations Cdsfinite 
NPs, or pronominal references). 
As already mentionod, the Generator 
follows the plan For verbalization defined 
bg the canonlcal representation. ~ set of 
s Is" ' governs the choice of 
particular templates, ~L~uniFicatlon 
begins. In case of alternatives as to the 
choice OF a template, the Generator 
consults the derivational historg of the 
analysis, which is kept in a special 
register, and selects the template, and 
the concrete verbal ~o~mulation, used in 
the OP (this 8nsuring partial "sameness" 
of HPs and OPs). 
Yhe system is implemented in PROLO~-2 
and runs on IBM RTs and compatibles. 
7, CONCLUSION 
In the paper, we tried to show hoe a 
paraphrase can be used as a hinting tool 
in an instruction sgstem in mathematics, 
and described a sustee incorporating this 
fsature. In the current implementation, 
the sgstem mag give rsasonablg good 
paraphrases of the original problem, but 
still there is a lot to be desired, even 
abstracting From ang real application ~Or 
educational purposes. It is a rather 
difficult thing to make the "right" 
compromise between the simplification 
needed in such tasks and a nioe verbal 
phrasing of the problems. We shall 
continue the work on the reEinemant of the 
sgstam and on developing an explanation 
Facilitg. 

REFERENCES 

Kaplao S., Cooperative Responses ~rom a 
Portable Natural Language Data Bass 
Ouery, Ph.D. Dissertation, 
University oE Pennsylvania, IS7S. 

Mann W., "~iscours~ structures For text 
generation", COLING8~, Stanford, 
18B½. 

HcKeown I~., "Paraphrasing questions using 
given and new in~ormatlon", RJCL, 9, 
1, I~B3. 

McKeoun K., Text Generation: Using 
Discourse Strategies end Focus 
Constraints to Generate Natural 
LanNuage Text, Cambridge Universitg 
Press, 1585. 

Pulman S., "Limited domain ewstems for 
language tee=hing", In: COLINBBM, 
Stanford, ISB~. 

de Roeck A., B. Lomden, "Generating 
English paraphrases from formal 
relational calculus expressions", 
In: COLINBB8, Bonn, 1986. 

Slesman D., J. Brown fads), Intelligent 
Tutocing SUstsme, N.Y., Academic 
Press, laBS. 

Thomas J., J. 8ould, "A psgchological 
studu OE querg bW example", In: 
Procssdin~s oF NCC, ~, IS75. 

Wsischsdsl R. et el, "An Al approach to 
language instruction", Arti~i=ial 
Intelligenos, i0, 3, IS7B. 
