A Semantic-Linguistic Method of Solving Verbal Problems Franklin R. Hoggard Southwest Missouri State University, Springfield,MO 65804 Solving verbal problems is difficult for many students. This method provides a linguistic algorithm for solving generic chemistry prohlems that is partly adopted from the artificial intelligence languages LISP' and PROLOGZ. While using this algorithm, the attention of the student is directed to processing information by using a small set of rules. The basic mental concepts of translation, rotation, mirror image symmetry, superpositioning, disjoining, and conjoining are employed. The method can be expanded to solve other more complex prohlems than those shown below. Analysis 01 Problems Problems are divided into linguistic clauses. A linguistic clause is not necessarily the same as a grammar clause. Punctuation marks and connecting words such as "hut", "if', and "then", symbols such as "M" and " % , and verbs with the suffix "ing", mark linguistic clauses. T o reduce anxiety and information overload, each clause is considered and dealt with separately. Each clause is set off with parentheses; an opening parenthesis signals the beginning of a clause and a closing parenthesis marks the end of a clause. (How many centimeters are in 14.7 in.?)
(1)
The entire sentence in (1) is a clause. However, a clause can he emhedded in a clause, which is analogous to the embedding of functions in math. (8+ (5 + (3 + (5 + 7))))
(2)
The evaluation in (2) must he suspended repeatedly until the "bottom" is reached. Then, the evaluation is carried "up"level by level to the "top" where "8 20 = 28". However, in linguistic analysis, i t is more useful to evaluate emhedded clauses from "top to bottom".
+
(What volume of aluminum, (density 2.70 g/cm3,)is found in 54 g?)
Arguments and Predicators In linguistics, the words in a clause are divided into arguments and predicators. These terms are analogous to "elements" and "operators" in mathematics. Arguments name something. Predicators connect arguments. Language usually follows the pattern, argument predicator argument predicator argument. . .
In (6), "compound", "carbon", and "nitrogen", are arguments; they name something. The predicator is "contains". There are four kinds of arguments. In order of precedence they are (1)discrete objects, (2) substances, (3) concepts, and (4) units of measure. The argument with the highest precedence is the headword and the arguments with a lower precedence are the completiues. In (4), "car" is the headword, and the substance "gasoline" and the units are completive~. (How many meters in 0.565 mi?) (The ore is (11.0%)iron.) (Methaneburns to form carbon dioxide.)
(9)
(10)
(3) L = 0 What
L
(4)
-I
volume o f aluminum
is found I n 54 Q? denslt
Figure 1. A sentence with a clause at level 0 and at level 1. Tap level (L = 0 ) clauses have the least amount of deleted information.
The clause pattern in (4) is identical to that of (3) (A rock (containing (14.0%(mlm)) magnesium) has a mass of 1.30 kg.)
(8)
The arguments in (8-10) have the same precedence; hence, the predicators must determine which argument is the headword. There are five kinds of predicators: (1) marks of location, such as "in" and "contains", (2) possessive ("of', "has", etc.), (3) temporal-action, such as "reacts to give", (4) copulas or intransitives. and (5) . . comoarisons such as " ~ e r "and "more than". In (a), both arguments are units of measure. However, "in", a mark of location, indicates that the unit "mi1es"is the headword argument. An argument following a copula is the completive. In (9),"is" identifies "iron" as the completive. In (lo), there is a temporal predicator. The argument
In (3), there is aclause a t the "top level" (L = O), anda clause a t a deeper level (L = 1). See Figure 1. (A car (moving 60 kmh) used 18.25 L of gasoline.)
(7)
(5)
There are four opening and closing parentheses in (5). The clauses are emhedded from L = 0 toL = 3 as shown in Figure 2. ((A compound contains (46%)carbon) and (54%)nitrogen.) (6)
"And", in (6), allows for a partial superpositioning of two top-level clauses. Both top-level clauses contain an emhedded clause (see Fig. 3).
' Winston, P. H.; Horn, 6. H. LISP; Addison-Wesley: Reading, MA. 1981.
Clocksin, W. F.; Meilish, C. S. Programming h P R O L W Springer-Verlag: New York. 1981.
-0 -1 L - 2 L - 3 L
L
has a mass o f 1.30 k ~ .
R IOC~
containln
ma~ne~lum
Figure 2. A sentence with four clauses. eachat adifferent level. Theamaunt of deleted information in the clauses increases at deeper levels.
L
-0
L - I
A C O ~ P O UCOntalnS ~ ~
~ a l 0 0 n . il c~mpliuna contalni
a&
nltrooen.
a
Figure 3. The partially superpositioned clauses in (6)have been disembedded into two top-level clauses ( L = 0). Each clause contains en L = 1 clause.
Volume 64 Number 1 January 1987
49
Discrete Object
= Discrete Object
t1
subitance
tr
Concept
II
Untt
Count (1) the number of units, (2) the number of concepts measured, (3) the number of substances or objects, and (4) determine if the clause is "open" (asks a question) or "closed". This information is used to build a list. The wellformed clause in (15) gives the list,
It
substake ConceDt
Obtalnlng Information from Well-Formed Clauses
It It Unit
(two units, one concept measured, one substance, open.) (19)
Figure 4. The clause template. All clauses are mapped onto this template.
(twounits, two concepts measured, one substance, closed.) (20)
(carbon dioxide) occuring last in time is the completive. (125 mL of (5.00%)alcohol solution.. .)
(11)
"Alcohol" modifies "solution" in (11). "Alcohol" is the completive. Language is the most complex structure known. Deletion, embedding, and recursion are employed so that it can operate a t maximum efficiency and productivity. Predicators point arguments to arguments, thus, all clauses, even sparsely worded ones, can be mapped onto the template in Figure 4. (12)
This becomes
(4.0 g of CHI reacts to give 11.0 g of C02.)
(21)
(twounits, one concept measured, two substances, closed.) (22) A list containing two operative substances/objects is a correspondence element. There are no other kinds of well-formed clauses. RehabllltatlngBadly Formed Clauses There are three ways to rehabilitate badly formed clauses:
(1) A unit may he inferred from a concept. For example,
(How many inches of length of a substance of an ohjeet that is the same object of the substance of length 14.7 cm?) (13) Upon readina- (13) it is apparent why deletion is used with .. language. An obiect mav to itself as shown in (13). Also, an . point . object can serve as its own unit and concept: (16 test tubes have a mass of 324 g.)
(14)
maps exactly onto the template in Figure 4. (2, h clause aith only one unit maps partially u n t u the template; thus, i t is bod/) Ivrrnud. Aclauseu~iththreeunits ihalsu bdly fcmlcd sinre i t mapsunlo one templatcand part of another. ( 3 , A stip~lott~eclause contains m, units of measure. It 81x0 maps partidly mlc, the template. (15)
The clause in (15) is well formed; it contains two units. The clause maps exactly onto the template in Figure 4 by inferring (1) the concept "energy", (2) a "substance", and (3) a "portion" or "sample". This gives statement (16): How many calories of energy are releasedlahsorhed by the substance of a portion that is the same portion ofthe same substance releasinglabsorbing energy of 45.0 J? (16) The top-level clause in (17) is badly formed: (What volume of aluminum (density 2.70 g1mL) is found in 35.0 g?)
There are no units in (18). The clause is a stipulation. Journal of Chemical Education
(How many joules are released when 3.0 g of methane burns?) (24)
results. List (25) contains an intensive element (2) Two hadly formed clauses can he conjoined.
(A car (moving 60 kmlh) burned 18.25 L of gasoline.) (How many kilometers did it go) if (it used 6.4 Lib?) (26) The top-level clause, "A ear burned 18.25 L of gasoline," is badly formed. The elause, "Haw many kilometers did it go .. . ,"is also badly formed. Conjoining the clauses produces the single well-farmed clause, (How many kilometers did go itlear burned 18.25 L of gasoline?) (27) From (27),the list, (two units, two concepts measured, two substances, open) (28) is obtained. List (28) identifies a correspondence element. (3) A clause with three or more units, must he disjoined further. (16 chickens can eat 21 lbs of feed in a week.)
(17)
If a unit of volume were present, it would map exactly onto the template, thus becoming well formed. The embedded clause a t L = 1is well formed. (Zinc reacts with sulfur to form zinc sulfide.)
Example (23) is badly formed because it has only one unit of measure. Infer an energy unit for "heat":
(two units, two concepts measured, one substance, open) (25)
(1) A clause with two units of measure is well formed because it
(How many calories in 45.0 J?)
(Haw much heat is released when 3.0 g of methane burns?) (23)
Now, (24) is well formed, and the list,
Classlfylng Clauses Clauses are classified into three types:
50
A list containing one substance or object and two concepts measured is an intensive element.
The clause in (21) is well formed and gives the list,
Arguments Point to Arguments
(How many inches are in 14.7 em?)
If no substance or object is mentioned, one is inferred since "nothing" cannot be measured. A list containing one substance or object and one concept measured is a conversion element. The embedded and well-formed clause in (17) gives the list,
(18)
(29)
Clause (29) is badly formed and must be disjoined. "Chickens" is the headword. (16 chickens can eat 21 lbs of feed.) (16 chickens eat for 1week.) List (32)taken from (30) identifies a correspondence element: (twounits, two concepts,two substanceslobjects,closed.) (32)
The Problem-Solving Protocols
In (311,there is an intensive element: (two units, two concepts, one object, closed).
(33)
A chicken is its own concept and unit
Webs The units in well-formed clauses areused to build webs3.A web is analogous to a directed graph in math (Fig. 5) with one important difference: A directed graph is made up of ordered pairs while the units in a web are paired but not ordered. Webs are used to provide the rules for solviugproblems and to discover "silent" clauses. A web is constructed for (34) below.
(How many meterdkilograms are there in a uniform wire,) (if 14.0cm of the wire has a mass of 16.5g?) (34) The clauses have been set off with parentheses. Both are a t the top level (L = O), and they are well formed. Each clause contains an intensive element. The first is open (how many meters/kilogram) and the other has closure. This leads to Protocol E in the table. Protocols D and F are valid for other problems. After a solver constructs several webs for problems, it becomes evident that problem elements have the following order of precedence: (1) correspondence, (2) intensive. and (3) conversion. To begin a web, the closed element having the highest precedence is placed in the same position as the ordered pair (a, b) in Figure 5. The corresponding open element is placed a t (c, d). Accordingly, the closed intensive element in the first clause in (34) is placed as a ratio in the (a, b) position in Figure 6. A web does not order the units; thus, the ratio could have been inverted. The open intensive element from the second clause in (34) has been placed in the (c, d) position in Figure 6. Open parentheses areused for the open units, "how many meters/kilograms." The directed graph in Figure 5 shows that only like arguments can be connected between ordered pairs. This is also true for webs (see Protocol G in the table). In Figure 6, "meters" cannot be pointed to "centimeters", nor can "grams" be pointed to "kilograms". The incomplete web reveals two silent clauses. Centimeter-meter and kilogramgram "bridges" must be built into the web (see Protocol I and then Figure 7). Both conversion elements have a lower precedence than the intensive element. In problems containing wrrmpondence elements, both intensive and nmveriion elemenrs rake on the subordinate bridging role. The web in Fizure "~ 7 is closed and the oroblem is solved when the factor label matrix is used to order the units. Because of personal r reference the matrix annears reversed comoared to those h u n d in textbooks. But t i e directed graph in'Figure 5 shows that a clockwise or counterclockwise rotation is exactlv equivalent. ~
~~~
~
Protocol A.
Units must be paired.
Proto~olB.
The number of closed correspondence elements will be one less than the total number of operative substances and objects.
Protrnol C.
If the number of canesoondence elements for Protocol B is
0, then and only then, there will be one less closed intensive elements than the total number of concepts measured. Proto~olD. Protocol E. Protocol F.
If a conversion element is desired, a closed conversion element must be provided. If an intensive element is desired. a closed intensive element must be provided. if a correspondence element is desired, a closed wrrerponden- element must be provided.
hotocol G.
A unit in a problem element can only be pointed to the same unit in another problem element.
Protocol H.
Units m a y be pointed to different units only through closed problem elements. If the units beween elements are different. "bridging" el* ments are needed.
Protocol I. Protocol J.
If a capacity factor is desired, a capacity factor must be pravided.
sample, (2) potassium, and (3) K~Cr04.Hence by Protocol B, t h e r e will be two closed correspondence ele(b,~ ments in the problem. By ( d, a ) Protocol F, there must also be an open correspondence element. By Protocol H, a problem asking for a capacity factor (how m a w grams Figure 5. A directed graph ordered potassium) must also pro- pairs. The behavior of this sequence is vide one (45.0 g potassium pertectly predictable aner just one obchromate). ~ewation.
2
(c,d) 1
~
Figure 6. (len) An Incomplete web. "Meters" cannot be pointed to (mapped onto) "centimeters", and "grams" cannot be painted to "kilograms".
Counting Substances
A useful oreliminaw activitv is to count the number of ~~erationaisubstances and objkcts found in aprohlem. If no substance or object is found, one is assumed. An operational substance is easy to identify; it has a unit of measure pointing to it. Other substances such as "excess acid", or "all the sodium chloride". have existential.,null..or universal auantifiers. They are nominal substances/objects and are not counted. Isomorphic substances and objects are also found in problems. For exam~le.in "a 6.5-a samole of NaCl", count either "sample" or "N'aCi" but notlhoth.. Protocols B and C in the table provide advance information about the problem web.
Figure 7.(right) A completed web. The conversionelemems assume a bridging function because they have a lower precedence Man intensive elements.
Highly Deleted Clauses
An embedded clause may he deleted to the point that only a symbol is left to indicate it. Examples are "%" for percent, "M" for molarity, and "ppm" for parts per million. To restore deleted clauses, templates in the form of lists are provided. For ''percent", the template is (#,unit, completive argument, per 100,unit, headword argument).
(36)
(How many grams of potassium are found in 45.0 g of a sample) (35) (that is (18.0%)K2Cr04?) There are three operational substances/objects in (35); (1)
Hanf, M. J. Reading 1971, 14(14), 225.
Volume 64 Number 1
January 1987
51
)
78.2 o K 1 18.0 g K2Cr04 191.2 o KzCrOd [ IOO Q sample
1
45.0 Q sample
I
I
- 3,26
Figure 8. A web wim two correspondence elements having equal precedence. Thus, they are placed side by side.
K - 2 mol x p&g
w
=
5.00 L 011. 501. I I rnL con. IOI.
Figure 10. The two cwrespondence elements occupy the position of highest precedence. The molar mass and density elements take the subordinate role of bridges. As each element is added to the web in the appropriate place, the the problem taking on a form, (2) the gaps in the web. (3) student can see: (I) what elements are missing, and (4) when closure is accomplished.
78.2 9
= 52.0
As students become familiar with this method, more complex language patterns can be learned. For example, Figure 9. A formula web containing many embedded and highly deleted wellformed clauses.
The template for "molarity" is (#,moles, completive, per liter, headword).
(37)
Other templates can be easily prepared from concept definitions. In (35), the top level clause contains (two units, one concept measured, two substances, open) (38) which is the open correspondence element predicted above. The copula "is" in the L = 1clause, "that is (part) K2CrO4", identifies potassium chromate as the completive argument. Recursing hack to the L = 0 clause reveals that "sample" is the headword. Now, the percent template in (36) can be completed to give the list, (18.0, grams, potassium chromate, per 100,grams, sample). (39) List (39) is well formed and i t contains (twounits, one concept measured, two substances, closed) (40) One of the two correspondence elements in (35) has been found. I t is entered as a ratio in Figure 8 along with the open correspondence element from (35). The other closed correspondence element is embedded in the chemical formula. Chemical Formulas
A chemical formula represents many embedded and highly deleted clauses. Because it operates on more than one clause level, students are encouraged to place formulas in brackets (i.e., [KzCr04])as a reminder to use i t as a "name" at L = 0 and to use it to form a matrix a t a deeper level. The matrix in Figure 9 has closure. Any pair of units selected from it is well formed because the pair can he mapped exactIv onto the temvlate in Fieure 4. ' I!sc ~n,ton,l'lfrom the'tahle to determine that 78.2 g K1 194.2 e KKrO, should he entered in Fieure 8 as the second corresiondenck element. The factor label matrix orders the units and provides the answer. Inverse Relatlonshlps
Concepts such as "pressure" and "speed" are frequently measured with units unique to that concept, for example, "nascal" and "knots". Actuallv. these units are intensive eiements (force per unit area ar;d length per unit time). The use of such units eives rise to inverse relationshins which must he entered into a web in a multiplicative form (i.e., 102 kPa X 1.2 L).
-
Words that Operate on Different Clause Levels
Words such as "density", "pressure", "concentration", and "speed", are placed in brackets because they operate on more than one clause level. 52
Journal of Chemical Education
(The [density] of aluminum is 2.70 glcm3).
(41)
In (41),the copula "is" acts as a node dividing the clause into two parts. The first part is stipulative and the second part is well formed. However, the well-formed part is dangerously deleted. The copula maps i t back onto the stipulative part. In turn, "density" maps "2.70 gIcm3" onto its deeper level template (mass, substance, object, same object, same substance, volume), to give "2.70 g aluminum sample per 1cm3 aluminum sample". Complex problems contain words like "pH" that have a deeper level template. Often, the template will contain a word that has its own still deeper template. This has the effect of forming a "tree" or "vine" as used in computer programming. The solver must have a knowledge of these words and the templates they produce. Texts give definitions of these words, but in this method definitions are provided as lists to be completed. Problems with additive functions cause the protocols in the table to break down. Three higher order protocols are needed: (1)the law of addition, (2) the law of remainders, and (3) the law of relative complement. Examples of how complex problems and problems with additive functions are solved with this method are beyond the scope of this paper. Solving a Problem
(5.00 L of dilute (0.100 M) [KzCrOJ solution must be prepared from a (30.0%(mlm)) concentrated [K2CrOJ solution.) (How many milliliters of the concentrated solution are needed) if (it has a [den(42) sity] of 1.28 gImL?) (1) Count the number of substances and objects: (dilute solution, concentrated solution, potassium chromate). (2) Eliminate the isomorphic and the nominal substances and objects. There are none. (3) By Protocol B, there will he two correspondence elements. (4) . . Bv. Protocol F. there will be an oven corresoondence element. (5) Break up the clauses using parentheses as shown in (42). Conjoin the two badly formed clauses to give, "How many milliliters of concentrated solution are needed to prepare 5.00 L of dilute solution?" This is well formed and it contains an open correspondence element. I t is entered into the web in Figure 10 a t the (c, d) position as shown in Figure 5. Use of the percent template in (36) t o obtain the list, (30.0, g, KzCrOa,per 100,g, concentrated solution).
(43)
This is the first correspondence element. Use the template in (37) to fill out the molarity clause: (0.100 mol K2Cr04,per 1L, dilute solution).
(44)
This is the second correspondence element. Add to the web in Figure 10 by placing the elements found
i n ( 4 3 4 4 ) side by side (they have equal precedence) a t the (a, h) position shown in Figure 5. Use the density element as a bridge in t h e web. Intensive elements take a lower precedence than correspondence elements. A silent clause is present because t h e web still does not have closure. Use the molar mass for KzCrOa from Figure 9 (see Protocol I). Use a factor label matrix t o find t h e answer. Concluslon This method is taught t o 250 college students of average deductive and inferential ability each semester. Students who can already solve problems are left alone t o d o i t their way. T h e method has been taught t o a group of junior a n d senior high science teachers using a C B H E economic securit y grant. As p a r t of t h e grant study, t h e participants will teach this method a t t h e appropriate prohlem-solving level t o their students during t h e 1 9 8 6 1 9 8 7 school year. T h e results will he evaluated during t h e spring of 1987.
I sincerely appreciate t h e suggestions a n d help provided h v D. Brooks, Universitv of Nebraska, a n d J. Northrip, ~ b u t h w e s~t i s s o u rsi t a t e University. Glossary completive argument. An argument in a clause taking a lower preeedenee. Completives coupled with the headword (subject) and the oredicator (thematic). eives a meanineful clause. conjoin. The act of combining a sentence, clause, or word with
another sentence, clause, or word to form one sentence, clause or word. disjoin. The act of breaking a sentence, clause, or word up into more than one sentence, clause or word. headword argument. The argument in a clause that takes the highest precedence (see Figure 4). If the arguments have the same precedence such as a substance paired with a substance, or a concept with a concept, the predicator will identify the headword. For example, in "12 g of salt in 120 g solution", the predicator "in", a mark of location, identifies "solution" as the headword. isomorphic substance. In languagc two idcnriral nqumcnt.i may br identified with different nilmcq. "AI~ohol''and "sample" arc ~ w m c r n h rin~ "A I? 0-r sample of nlcnhd war burned." C'nlcirc and caicium carbonate is another example. nominal argument. Often, arguments are placed in a problem just to provide a realistic framework. These arguments have a null, universal, or existential quantifier. For example, "use all the acid," or "place several drops of water in the glass." operational argument. An argument with a specific numerical value as in "use 14 mL of water." "points to". Predicators "operate" on arguments, thus pointing them into a meaningful sequence by identifying relative location, temporality, relative value, and property. This helps the reader to assign meaning to the language. "The dirt shoveled the man," is incorrect because the predicator points one argument to another in the wrong sequence. aredieator. A word or eroun .~ .. . of words that establishes a theme or rrlntimnl .spttmg fur the nrgulnrnrs in n clause. Fm rxamplr, "nre iound in" .n ' I 4 8 sah arc found in 1FOg solution." Alro, " d ' n n d "reacted" i n "1.1 ml. oi hydrogen reacted " silent clause. A clause not mrntioned in a pmhlem. hlost often, it is nrlausc cmtsining aconwmion element as in theproblrm,"How mmv gmms in U 51 Ibs?" The cmverswn hrtwcrn pounds and grams is not given, hence a silent clause.
Volume 64
Number 1 January 1987
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