The importance of the statement in addition and subtraction word problems.

The importance of the statement in addition and subtraction word problems. 1. IntroductionThe research carried out on simple addition and subtraction subtraction,fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number a−b is that number (called the difference) which when added to b (the subtractor) equals wordproblems (solved by x + y = z or x - y = z) has been very extensive.Ample bibliographical bibliographicalpertaining to the literature of a subject.bibliographical toolsthe ways in which a bibliography can be approached or managed. details on this subject may be found in researchsurveys by Fuson (1992) and Verschaffel and De Corte Corte (Corsican Corti) in is a town and a commune in the Haute-Corse d��partement in central Corsica, in France. It is the fourth-largest commune in Corsica (after Ajaccio, Bastia, and Porto-Vecchio), with a 1999 census population of 6,329 inhabitants. (1996).From experience, and the results of research, we know that eachstudent has a varying degree of problem-solving problem-solvingn → resoluci��n f de problemas;problem-solving skills → t��cnicas de resoluci��n de problemasproblem-solvingn → success with differentproblems and also that different students have different levels ofsuccess in each problem. These facts are explained through differentproblem characteristics. Several classes of additive additiveIn foods, any of various chemical substances added to produce desirable effects. Additives include such substances as artificial or natural colourings and flavourings; stabilizers, emulsifiers, and thickeners; preservatives and humectants (moisture-retainers); and problems are wellknown: Combine, Change, Compare and Equalize e��qual��ize?v. e��qual��ized, e��qual��iz��ing, e��qual��iz��esv.tr.1. To make equal: equalized the responsibilities of the staff members.2. To make uniform. (Carpenter and Moser Moser is a family name shared by the following individuals, companies and works: Alphabetical listingAnnemarie Moser-Pr?ll (born 1953), Austrian skier Dietz-R��diger Moser (born 1939), German academic Edda Moser (born 1938), German soprano Edward W. , 1982;Riley et al., 1983). In this paper we are particularly interested incertain aspects of the problems that we now summarize sum��ma��rize?intr. & tr.v. sum��ma��rized, sum��ma��riz��ing, sum��ma��riz��esTo make a summary or make a summary of.sum .In certain numerical numericalexpressed in numbers, i.e. Arabic numerals of 0 to 9 inclusive.numerical nomenclaturea numerical code is used to indicate the words, or other alphabetical signals, intended. situations two states are compared: a smallstate ("Juan Juan (IPA: [xwan]) is a Spanish form of the given name John (q.v.). It was the 55th most popular name in the United States as of 2003. has 2 pesetas") and a big state ("Pedro Pedro.For Spanish and Portuguese rulers thus named, use Peter.Pedroin marrying former mistress of enemy. [Ger. Opera: d’Albert, Tief land, Westerman, 371–374]See : Innocence has5 pesetas"). We can use the scheme s + d = b, where s and b arestatic situations and d is the difference. There are two ways in whichthe difference may be expressed. In Compare problems, the difference isexpressed as "more than" ("Pedro has 3 pesetas more thanJuan") or "less than" ("Juan has 3 pesetas less thanPedro"). In Equalize problems, the expression would be "howmuch" the small state must increase to equalize the big state("If Juan earns 3 pesetas, then he has the same as Pedro") or"how much" the big state must decrease to equalize the smallstate ("If Pedro loses 3 pesetas, then he has the same asJuan").In other situations we have a start state ("Before, Juan had 2pesetas"), a variation ("then he earned 3 pesetas") andan end state ("Juan has now 5 pesetas"). These problems havethe scheme s + v = e and are associated with dynamic situations. Thereare two types of expression for the variation: in Change problems, thevariation is expressed in a simple way ("Juan has earned ..."or "Juan has lost ..."). In Change-Compare problems thevariation is expressed as more than or less than, in a similar way toCompare problems ("Now, Juan has 3 pesetas more than he hadbefore"). We don't know Don't know (DK, DKed)"Don't know the trade." A Street expression used whenever one party lacks knowledge of a trade or receives conflicting instructions from the other party. of any research study that coversChange-Compare problems. These classes of problems are shown in Table 1.The above distinction between scheme and expression is not usual.To sum up, in an additive situation where three numbers are involved: a+ b = c, the scheme refers to the numerical situation and the expressionrefers to the manner of saying (or writing) the variation and thedifference.Fuson and Willis Wil��lis, Thomas 1621-1675.English anatomist and physician known for his studies of the nervous system and the brain. He discovered the circle of Willis at the base of the brain. (1986) noted that Compare and Equalize problemshave different problem-solving difficulty levels: Compare problems aregenerally more difficult to solve than Equalize problems; the expressionof the difference therefore influences the level of difficulty. In thepresent research, we show that the expression of the variation, inChange problems and in Change-Compare problems, is of greatsignificance. Several researchers have showed the importance of otherexpressions in the statement of problems (De Corte and Verschaffel,1991; Teubal and Nesher This article is about the city. For the fighter aircraft, see IAI Nesher. Nesher (Hebrew: נשר‎) is a city in the Haifa District in Israel. , 1991).In our research, Combine problems (where the addition of twopartial states equals the total state) are not considered, because ourinterest is really focussed on the contrasts between differentexpressions in a same numerical situation. Of course, other classes ofproblems have certainly been described in the literature on this subject(Bruno and Martinon, 1996, 1997).There are many contexts within which it is possible to stateadditive problems, such as: temperature, chronology chronology,n the arrangement of events in a time sequence, usually from the beginning to the end of an event. , length, etc. In ourresearch, however, we have preferred to analyze an��a��lyzev.1. To examine methodically by separating into parts and studying their interrelations.2. To separate a chemical substance into its constituent elements to determine their nature or proportions.3. all of them within thesame "having money" context ("Juan has 2 pesetas","Juan has earned 2 pesetas", etc.), so as to fix the contextvariable as a standard and also because we consider students aregenerally more familiar with this approach. Please note that in order tosimplify, we will now use "pta" instead of "pesetas"and the initials J, P, E, T for persons' names.In each simple additive situation there are three problems,depending on the unknown (these are described in the next section). Theunknown also has an important influence on the problem-solving result(Carpenter and Moser, 1982; Riley et al., 1983).In our research involving students in Third, Fourth, Fifth andSixth Year of Primary Education in Spain The framework of Education in Spain is described in this article. State Education in Spain is free and compulsory from 6 to 16 years. The current education system is called LOGSE (Ley de Ordenaci��n General del Sistema Educativo). (aged 8-12), we have analyzed an��a��lyze?tr.v. an��a��lyzed, an��a��lyz��ing, an��a��lyz��es1. To examine methodically by separating into parts and studying their interrelations.2. Chemistry To make a chemical analysis of.3. the levels of problem-solving difficulty of the four classes of problemsand have evaluated the results in accordance Accordance is Bible Study Software for Macintosh developed by OakTree Software, Inc.[]As well as a standalone program, it is the base software packaged by Zondervan in their Bible Study suites for Macintosh. with certaincharacteristics which we consider certainly influence the problems. Thecharacteristics related to the expression of the difference and thevariation, which will be described in more detail in the followingsection, are identified as follows: "I" = use of"inconsistent Reciprocally contradictory or repugnant.Things are said to be inconsistent when they are contrary to each other to the extent that one implies the negation of the other. language" and "R" = the referent ref��er��ent?n.A person or thing to which a linguistic expression refers.Noun 1. referent - something referred to; the object of a reference isthe unknown. The combination of these two characteristics (IR) mayappear in a strong form (s) or in a weak form (w), according to according toprep.1. As stated or indicated by; on the authority of: according to historians.2. In keeping with: according to instructions.3. theorder of the data in the statement. The problems identified with thecharacteristic IR(s) are in fact those with a lower problem-solvingsuccess rate and appear in Change Compare and Compare problems.Change-Compare problems, therefore, have certain factors thatdistinguish them from Change problems and, certainly, from Compare andEqualize problems.Student problem-solving success rates, for the problems that wereset, certainly confirm our belief that they are influenced by the way inwhich the variation and the difference are expressed and indeed theorder of the data itself. In our opinion, the reason is that there arecertain ways of expressing variation and difference, or the sequence ofthe data as such, that create greater difficulty for the student whentrying to understand the statement of the problem, or to visualize thenumerical situation involved, really making a solution very difficult.Our explanation is therefore based (in the case of these simple additiveproblems with positive numbers) on the fact that levels ofproblem-solving success are directly related to the degree to which thestatement of the problem is clearly understood.It should be noted that in this paper we have "forced"the usual syntactic Dealing with language rules (syntax). See syntax. order in English 1. English - (Obsolete) The source code for a program, which may be in any language, as opposed to the linkable or executable binary produced from it by a compiler. The idea behind the term is that to a real hacker, a program written in his favourite programming language is in order to maintain the syntacticpattern used in the statements of the problems is Spanish Spanish,river, c.150 mi (240 km) long, issuing from Spanish Lake, S Ont., Canada, NW of Sudbury, and flowing generally S through Biskotasi and Agnew lakes to Lake Huron opposite Manitoulin island. There are several hydroelectric stations on the river. : "AhoraPedro tiene 5 pesetas. Ahora tiene 4 pesetas menos que antes an��te?n.1. Games The stake that each poker player must put into the pool before receiving a hand or before receiving new cards. See Synonyms at bet.2. . ?Cuantaspesetas tenia tenia/te��nia/ (te��ne-ah) pl. te��niae ? taenia. te��ni��an.Variant of taenia.teniapl. teniae [L.] a flat band or strip of soft tissue. antes?" ("Pedro has now 5 pesetas. He has now 4pesetas less than he had before. How many pesetas did he have before?)2. Types of problemsIn this section, we introduce the terminology The terminology used in the computer and telecommunications field adds tremendous confusion not only for the lay person, but for the technicians themselves. What many do not realize is that terms are made up by anybody and everybody in a nonchalant, casual manner without any regard or used, describe thetypes of additive problems that were set for the students and highlightthe characteristics of the problems we consider more relevant.Before we refer to such additive problems, we will discuss theadditive story (Rudnisky et al., 1995) or additive situation, in which asituation involving the addition or subtraction of two numbers isdescribed. In our research, simple additive stories with positivenumbers are considered, which are those with the addition x + y = z orthe subtraction x - y = z, where x, y, z are all positive. We considerfour classes of additive stories and, consequently, four classes ofadditive problems: Compare, Equalize, Change and Change-Compare, asdescribed in the following paragraphs.We can have three types of problems associated with a story of thiskind, depending on the unknown. The story "J has 2 pta and P has 5pta, so P has 3 pta more than J," gives rise to the following threeproblems:* J has 2 pta and P has 5 pta. How many pta more than J does Phave?* J has 2 pta P has 3 pta more than J. How many pta does P have?* P has 3 pta more than J. P has.5 pta. How many pta does J have?2.1 CompareWe consider two states s ("J has 2 pta") and b ("Phas 5 pta"). Both of these interrelate in��ter��re��late?tr. & intr.v. in��ter��re��lat��ed, in��ter��re��lat��ing, in��ter��re��latesTo place in or come into mutual relationship.in through difference d = b - s("P has 3 pta more than J"). We can therefore say this type ofstory does have the scheme s + d = b, where, to avoid confusion ofclassification, we will take it that d > 0; that is, s is the smallstate and b is the big state (s < b). We say that an additive history(and its associated problems) is a Compare history if its scheme is s +d = b and the difference d is expressed in some of the following ways:* More: when it is said how much the big state is "morethan" the small state* Less: when it is said how much the small state is "lessthan" the big stateExample: "J has 2 pta and P has 5 pta"; the comparisonmay be expressed in the following ways: "P has 3 pta more thanJ" (More) and "J has 3 pta less than P" (Less). There arethree problems according to the unknown: s (small state), d (difference)or b (big state). For example: "J has 3 pta less than P. P has 5pta. How many pta does J have?"2.2 EqualizeIn the stories with scheme s + d = b the difference d may also beexpressed in the following ways:* Add on: when it is said how much the small state must"increase" to equalize the big state.* Take away: when it is said how much the big state must"decrease" to equalize the small state.Example: "J has 2 pta and P has 5 pta". Then, "If Jearns 3 pta, then he will have the same as P" (Add on) and "IfP loses 3 pta, then he will have the same as J" (Take away). Thestories of this type are referred to as Equalize. Also, in this case,there are three problems associated with each story. For example:"If P earns 2 pta, then he will have the same as R. R has 6 pta.How many pta does P have?"2.3 ChangeNow we consider a start state s ("this morning J had 2pta"), time goes by and we have the end state e ("J has now 5pta"), and there is a variation v = e - s ("J has earned 3pta"). We may say this type of story has scheme s + v = e. If thevariation v is expressed in simple form ("J has earned","J has lost"), then we say that this is a Change story.Consider the following example: "In the morning, J had 2 pta and inthe evening he has 5 pta". Then we tell: "J earned 3 ptathroughout the day" (Simple). According to the sign of v, we willrefer to this as a Change (Increase) when v > 0 ("J has earned 3pta") or as a Change (Decrease) when v < 0 ("J has lost 3pta"). In this class of problems, the unknown may be s (startstate), v (variation) or e (end state). Example: "J had 5 ptabefore and he has now 2 pta. How many pta did he lose?"2.4 Change CompareA story is said to be of class Change-Compare if it is an additivesituation with scheme s + v = e and the variation v is expressed usingthe words more than or less than, which are used in the expression ofthe comparison in the stories of the Compare type. Example: "In theevening, J has 3 pta more than he had in the morning" (More) and"In the morning, J had 3 pta less than he has in the evening"(Less). Of course, there are three problems associated with each story,depending on the unknown (start state, variation, end state). Now wealso distinguish between classes Change-Compare (Increase) andChange-Compare (Decrease).2.5 Order of presentation of numbers in the statementIn the problems with scheme x + y = z, the data in the statementcan be presented following the x, y, z order (x,y; x,z; y,z) or theopposite order (y,x; z,x; z,y). We have identified the problems with an'opposite order' with an asterisk (1) See Asterisk PBX.(2) In programming, the asterisk or "star" symbol (*) means multiplication. For example, 10 * 7 means 10 multiplied by 7. The * is also a key on computer keypads for entering expressions using multiplication. *. For instance, thefollowing problems have a different data order:* P had 6 pta before and he has now 4 pta. How many pta more thanhe has now did he have before?,* P has now 4 pta and he had 6 pta before. How many pta more thanhe has now did he have before?2.6 Types of problemsEach problem will be classified according to the followingcategories:* Type of story: Compare, Equalize, Change, and Change-Compare* Type of expression: more, less, add on, take away.* Unknown: start state, variation, end state, small state,difference, big state.* Order of the data: scheme order, opposite order (*).The 39 problems considered in our research study are listed in theAppendix appendix,small, worm-shaped blind tube, about 3 in. (7.6 cm) long and 1-4 in. to 1 in. (.64–2.54 cm) thick, projecting from the cecum (part of the large intestine) on the right side of the lower abdominal cavity. and the terminology used is contained in Table 2. Of the 39problems that we have considered, 9 are of the class Change, 6 ofEqualize, 6 of Compare and 18 of Change-Compare.In the following sections we analyze some characteristics of theproblems that we consider relevant in the solution of problems bystudents.2.7 Characteristic 1: Inconsistent languageIn the literature on this subject, it is usually stated that aproblem has inconsistent language (1) when the "key words"used in the statement might be considered to suggest a differentcalculation to that which really applies. For instance, in the problem:* Compare 2: T has 2 pta. T has 3 pta less than E. How many ptadoes E have? We must add 2 + 3, but the expression "less than"may lead certain students to subtract A relational DBMS operation that generates a third file from all the records in one file that are not in a second file. . Similarly, in the problem:* Change 5: J has earned 3 pta and he has now 5 pta. How many ptadid he have before?We must subtract 5 - 3, even though the term "earned" maysuggest addition. That is to say: a problem is worded with inconsistentlanguage when addition is required and expressions such as "lessthan" or "to lose" are used, or if subtraction isnecessary and the expressions "more than" or "toearn" are used. (Problems in which inconsistent language is usedare shown in table 2).2.8 Characteristic R: the referent is the unknownIn the Compare problems and the Change-Compare problems therelationship between the two states is expressed by taking one of themas the referent. For instance, in the Compare problem:* Compare 1: T has 2 pta. M has 5 pta more than T. How many ptadoes M have? The referent is what T has. If the comparison is expressedwith "less", as in:* Compare 2: T has 2 pta. T has 3 pta less than M. How many ptadoes M have? The referent is now what M has, which coincides with theunknown. In the cases in which the referent coincides with the unknownwe note by R. In Change-Compare problems there is also a referent. Inthe problem:* Change-Compare 1.2: Before, A had 3 pta. Before he had 5 pta lessthan he has now. How many pta does he have now?the referent is the number of pta he has now. In the Changeproblems, we have not considered a referent to exist. The R-problems ofour research are shown in Table 2.2.9 Characteristic IR: Inconsistent language and the unknown is thereferentWe apply the symbol "IR" when the I and R characteristicsare both present. It should be noted that this only appears in certainChange-Compare and Compare problems (see Table 3). In these problems,and only in them, the referee A judicial officer who presides over civil hearings but usually does not have the authority or power to render judgment.Referees are usually appointed by a judge in the district in which the judge presides. is the known state, whereas the unknownstate is the referent. In the example Change-Compare 5.5, the referee iswhat L has now and the referent is what L had before:* Change-Compare 5.5: Now, L has 3 pta more than he had before.Now, he has 5 pta. How many pta did he have before?Let us consider the above and the following problems:* Change-Compare 2.1: Before, P had 8 pta. Before he had 3 pta morethan he has now. How many pta does he have now?* Compare 2: T has 2 pta. T has 5 pta less than E. How many ptadoes E have?* Compare 5: T has 5 pta more than M. T has 7 pta. How many ptadoes M have?In the Change-Compare 2.1 and Compare 2 problems, the abovecondition is expressed in a strong from (s) because of the"repetitive" expressions "Before ... Before ... now ...now" and "T ... T ... E ... E", whereas in theChange-Compare 5.5 and Compare 5 problems it appears in a weak form (w)"Now ... before. Now ... before" and "T ... M ... T ...M". We believe the distinction between the strong and weak forms tobe of importance because we also consider that strong form statementsare more difficult for students to understand than weak form statements.It should be noted that if the order of the numbers presented inthe statement of Change-Compare 2.1 [IR(s)] is changed, the problembecomes Change-Compare 2.1* [IR(w)]:* Change-Compare 2.1*. Before R had 3 pta more than he has now.Before, he had 8 pta. How many pta does he have now?If we change the order of presentation of numbers in the statement,then a problem with characteristic IR(s) is converted into a problemwith characteristic IR(w) and vice versa VICE VERSA. On the contrary; on opposite sides. .We should mention that some authors (e.g., Verschaffel, 1994) saythat a problem is of "inconsistent language" when it has theIR-characteristic, which is yet a stricter use of that terminology.3. The research studyA research study was carried out among several groups of studentsto contrast Change-Compare problems with Change, Equalize and Compareproblems, and to establish the influence of the expression of thevariation and the difference, as well as the effect of the problemcharacteristics. The results of the research study are described in thispaper.Several groups of students of Third, Fourth, Fifth and Sixth Yearof Primary Education in Spain (aged 8-12) were asked to completedifferent written tests that contained the 39 additive problems referredto in the previous section. The groups of students were selectedstarting from the third level because the students had previously workedwith these problems (on the first and second level).A total of 267 students of two State Schools (S1 and S2) located inthe suburbs of Santa Cruz de Tenerife Santa Cruz de Tenerife(săn`tə krz dā tānārē`fā), city (1990 pop. 222,892), capital of Santa Cruz de Tenerife prov. (Canary Islands Canary Islands,Span. Islas Canarias, group of seven islands (1990 pop. 1,589,403), 2,808 sq mi (7,273 sq km), autonomous region of Spain, in the Atlantic Ocean off Western Sahara. They constitute two provinces of Spain. Santa Cruz de Tenerife (1990 pop. , Spain Spain,Span. España (āspä`nyä), officially Kingdom of Spain, constitutional monarchy (2005 est. pop. 40,341,000), 194,884 sq mi (504,750 sq km), including the Balearic and Canary islands, SW Europe. ), took partin our experiment, and formed 15 different groups (G1, G2, ... G15).The large number of problems considered in this experiencesuggested to distribute them into different tests with an acceptablenumber of problems on each one, in such a way that the students wereable to answer them during one session. The research study covered 10different tests (T1, T2, T3, ... T10) of six problems each, details ofwhich are given in Table 3. The tests were randomly distributed in eachgroup and the students were asked only to answer one test each.Table 4 shows the contents of the tests. Please note that the T1 toT6 tests do not contain *-problems and that this type was only includedin the T7 to T10 tests. It was intended that a test should containproblems to allow diverse aspects to be examined simultaneously si��mul��ta��ne��ous?adj.1. Happening, existing, or done at the same time. See Synonyms at contemporary.2. Mathematics .For instance, Test 1 included the following problemsT1: Change 5, Change-Compare 2.1, Change-Compare 5.5,Change-Compare 5.6, Equalize 5, Compare 5.Problems Change 5, Change-Compare 5.5 and Change-Compare 5.6 onlydiffer in the expression of the Variation; analogously a��nal��o��gous?adj.1. Similar or alike in such a way as to permit the drawing of an analogy.2. Biology Similar in function but not in structure and evolutionary origin. , with Equalize 5and Compare 5 problems. On the other hand, the Equalize 5 andChange-Compare 5.5 problems only differ in their scheme s + d = b or s +v = e. We also wished to examine the contrast between the Change 2 andthe Change 5 problems. In paragraph 4.7, advantage was taken of the factthat certain sets of problems appeared in the same test.In order to avoid possible influences through the order in whichproblems are set, several different formats were used for eachindividual test, changing the order of problems.The students' usual answer was either to write an operation(addition, subtraction, multiplication multiplication,fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. or division), or simply to writethe solution to the problem without giving details of their operations.Their answers were classified as right, wrong or blank Lacking something essential to fulfillment or completeness; unrestricted or open. A space left empty for the insertion of one or more words or marks in a written document that will effectuate its meaning or make it legally operative. .4. Results and discussionTable 5 shows the percentage of general success achieved for eachof the problems, the number of students to whom it was put and thecharacteristics of such problems, with students of all levels and groupsbeing evaluated together.It may be readily appreciated that the results are of a verydiverse nature and a detailed study, based on problem characteristics,is required.4.1 General resultsIt is evident that we cannot establish a strict level of difficultybetween the four classes of problems under consideration. Therelationship of the I-, R- and IR-characteristics, especially thelatter, with the greatest problem-solving difficulty, is evident inTable 5. It can be appreciated that IR(s)-problems are generally thosewith a lower success percentage, particularly the Change-Compareproblems; particularly, Change-Compare 1.2 (37%) and Change-Compare 2.1(29%) problems. In the IR(w)-problems, the success rates are higher:Change-Compare 5.5 (65%) and Change-Compare 6.6 (47%). The greaterinfluence of the strong form on the results may be appreciated moreclearly by reference to the (*)-problems: Change-Compare 2.1 [IR(s)] hasa 29% success percentage and Change-Compare 2.1 [IR(w)] has a 45%;similarly, problem Change-Compare 6.6* [IR(s)], with 21% as against 47%in the Change-Compare 6.6 [IR(w)] problem. The influence of theIR-characteristic on the Compare problems is not so marked as that inthe Change-Compare problems.4.2 Data orderWith regard to the (*)-problems, we can establish whether the orderin which data is presented has any influence on the success rates. Aquick reference to Table 5 clearly shows that this is important with theChange-Compare problems but of little or no consequence where Changeproblems are concerned. We have already pointed out that the order inwhich the data is expressed in IR problems causes the statement to adopta strong or weak form of expression, with a direct repercussion on thesuccess rates. Such results lead us to believe that the Change-Compare1.2* and Compare 2* problems (which, we repeat, have not been studied inthe present work), both IR(w), are more easily solved by students thantheir IR(s) counterparts: the Change-Compare 1.2 and Compare 2 problems,respectively. Similarly, we think that the Change-Compare 5.5* andCompare 5* problems, both IR(s), are found to be more difficult thantheir IR(w) counterparts: Change-Compare 5.5 and Compare 5,respectively.4.3 ChangeProblems under this heading are clearly those with the best problemsolving problem solvingProcess involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. success rates, all between 75% and 100% (see Table 5). It may bethat, because more attention is given at school to this type of problemand to those of the "states' combination" type, suchsuccess rates are so positively influenced. The lower values in the75%-100% interval interval,in music, the difference in pitch between two tones. Intervals may be measured acoustically in terms of their vibration numbers. They are more generally named according to the number of steps they contain in the diatonic scale of the piano; e.g. are related to 1-characteristic problems, actuallybetween 75% and 81%, while the rest are between 91% and 100%. Pleasenote that problems Change 6* (81%), Change 4* (94%) and Change 2 (94%)figured exclusively in Test 10, which corroborates our opinion thatproblems worded with inconsistent language "I", are thosewhere students encounter the greatest difficulty. In the comparison ofour results with those of Martinez Martinez(märtē`nəs), city (1990 pop. 31,808), seat of Contra Costa co., W Calif., on Carquinez Strait between San Pablo and Suisun bays, in a farm area; inc. 1884. Its major industry is petroleum refining. and Aguilar Aguilar refers to: PeopleAndrew Aguilar Actor Antonio Aguilar (1919–2007), Mexican singer Ant��nio Aguilar, Portuguese rugby player Baron Diego Pereira D' Aguilar (1699–1759), Spanish Marrano (1996), theirinvestigation shows that the worst results were also related to problemscontaining the 1-characteristic. It should be noted, however, thatproblems with the IR or simply R characteristics do not exist in theclass of the Change problems.4.4 Change CompareAs may be seen in Table 5, three of the problems in this categoryshow success rates that are clearly inferior INFERIOR. One who in relation to another has less power and is below him; one who is bound to obey another. He who makes the law is the superior; he who is bound to obey it, the inferior. 1 Bouv. Inst. n. 8. to all the others in thisresearch study: Change Compare 1.2 (37%), Change-Compare 2.1 (29%) andChange Compare 6.6* (21%), which are the three IR(s)-problems. Lowsuccess rates also affect three other problems with theIR(w)-characteristic: Change-Compare 5.5 (65%), Change-Compare 6.6 (47%)and Change-Compare 2.1* (45%).As to the problem-solving difficulty levels of the Change-Compare(Increase) problems, we have: the IR(s) with a 37% success percentage,the IR(w) with 65%, followed by the one with I-characteristic,Change-Compare 3.3, with 78%, and lastly the problems without a specificcharacteristic, between 80% and 93%. The Change-Compare (Decrease)problems, however, show a different pattern, the IR(s) type proving tobe the most difficult, with success results of 21% and 29%, followed bythe IR(w), with results of 45% and 47%. Nevertheless, there are twoI-problems, Change-Compare 4.3 (95%) and Change-Compare 4.3* (64%),which, of all problems without the IR characteristic, have the lowestand highest success percentage figures, respectively, something wecannot clearly explain.4.5 CompareThe two problems with the lowest success rates do possess theIR-characteristic: Compare 5 [IR(w)] 63% and Compare 2 [IR(s)] 59%. Theyalso represent by far the lowest two success rates in the study ofMartinez and Aguilar (1996). Of the two problems of unknown Diff, thelowest percentage was in respect of Compare 3 (74%), an I-problem (SeeTable 5).4.6 EqualizeAs shown in Table 5, the two problems, Equalize 6 and Equalize 1,with the unknown as referent, do not have low success rates. The lowestpercentages are shown against Equalize 2 (50%), Equalize 5 (57%) andEqualize 3 (75%), which are the three problems with theI-characteristic, the same result having been obtained by Martinez andAguilar (1996). In this class of problems, therefore, theI-characteristic makes a problem more difficult to solve than theR-characteristic.4.7 Similar problemsFor the problems Change-Compare 5.6 (80%), Change-Compare 2.2*(79%) and Compare 6 (80%), not only were similar success rates achievedbut their respective statements may be obtained from each other byinterchanging their words. Thus, if in the case of Change-Compare 5.6:* Change-Compare 5.6. Before E had 4 pta less than he has now. Now,he has 9 pta. How many pta did he have before?we replace "before" and "now" by"now" and "before", respectively, and verbal VERBAL. Parol; by word of mouth; as verbal agreement; verbal evidence. Not in writing. formsare appropriately modified mod��i��fy?v. mod��i��fied, mod��i��fy��ing, mod��i��fiesv.tr.1. To change in form or character; alter.2. , Change-Compare 5.6 becomes Change-Compare2.2** Change-Compare 2.2*. Now J has 2 pta less than he had before.Before, he had 6 pta. How many pta does he have now?If with Change-Compare 2.2*, "now" and "before"are replaced by "R" and "L", respectively, itbecomes Compare 6:* Compare 6. R has 3 pta less than L. L has 5 pta. How many ptadoes R have?Similar possibilities exist with other sets of problems, as shownin Table 6, although the success percentage values are not so close toeach other as in the previous example. Nevertheless, each of thedifferent sets-of-three problems was included with the same individualtest and the results confirm the coincidence Coincidence is the noteworthy alignment of two or more events or circumstances without obvious causal connection. The word is derived from the Latin co- ("in", "with", "together") and incidere ("to fall on"). .Considering (*)-problems of Change (Increase) and of Compare (notthe subject of the present research), we can take it for granted thatthe sets-of-three problems shown in Table 7 below would have similarsuccess percentage values.Also, the similar results obtained in the Change-Compare 6.6 (47%)and Change-Compare 2.1* (45%) problems, as also in the Change-Compare6.6* (21%) and Change-Compare 2.1 (29%) problems, are certainlyremarkable. In the following statements:* Change-Compare 6.6. Now, M has 4 pta less than he had before.Now, he has 5 pta. How many pta did he have before?* Change-Compare 2.1*. Before R had 3 pta more than he has now.Before, he had 8 pta. How many pta does he have now?if we replace the words "now" by "before" and"less" by "more", one type of problem is convertedto the other. The same occurs in the following example:* Change-Compare 6.6*. Now, P has 5 pta. Now, he has 4 pta lessthan he had before. How many pta did he have before?* Change-Compare 2.1. Before, P had 8 pta. Before he had 3 pta morethan he has now. How many pta does he have now?Similar comments may also be made about other sets of problems. Itseems, therefore, that all problems with similar statements willnormally be solved with near-equal success results.5. ConclusionsThe main object of the research presented in this paper is tohighlight that the manner in which certain problem statements areexpressed certainly influences problem-solving success results.A simple additive situation can be expressed in different ways. Ifthe situation is associated to a scheme s + v = e (start state +variation = end state), there are three ways of expression of thevariation (simple, more than, less than); the results of our researchshow that the kind of expression used can affect the level of success.Analogously, if the scheme is s + d = b (small state + difference = bigstate), then there are four ways of expression of the difference (morethan, less than, take away, add on), and the expression used in thestatement is related with the difficulty of the problem.In order to confirm our idea concerning the importance of theexpression, we have taken four classes of additive problems withpositive numbers, three of which (Change, Compare and Equalize) werepreviously quite well-known well-knownadj.1. Widely known; familiar or famous: a well-known performer.2. Fully known: well-known facts. , and the fourth class (Change-Compare) hasbeen investigated in our research for the very first time. The Changeand Change-Compare problems have the same scheme s + v = e, but theexpression of the variation is different; also the Compare problems andthe Equalize problems have the scheme s + d = b, but the difference isexpressed in different ways. Moreover, the expressions for the variationand the difference are similar in the Change and Equalize problems, andalso in Change-Compare and Compare problems.With the object of investigating what influence the manner in whicha problem is expressed may have, certain problem characteristics wereconsidered. Characteristic I is used to indicate "inconsistentlanguage" in the expression of a problem that might suggestaddition when in fact the correct action to be taken is to subtract, orto subtract when addition should apply. Characteristic R will also applywhen the referent is the unknown. The results of our investigation provethat the problem-solving results are adversely affected by both thecharacteristics, although when both coincide (IR) the difficulty createdis much greater. This is particularly true when the order in which thedata is expressed "forces" the characteristic IR to adopt thestrong form: IR(s).After the detailed analysis carried out, we can establish thatChange-Compare problems have a particular condition of their own. On theone hand, Change problems do not have the IR-characteristic, whereas itcertainly appears in Change-Compare problems, therefore, making themmore difficult to solve. On the other hand, Change-Compare problemsdiffer greatly from Compare ones, because Change-Compare problems withthe IR-characteristic are among those with the lowest problem-solvingsuccess rates. There are others, however, with the I-characteristic,having similar success rates for both classes. So, it can be said thatthe IR-characteristic is a particularly negative condition withChange-Compare problems but of less consequence in Compare or Equalizeproblems.The fact that some of the problems show a low percent of successdoes not imply that they should be excluded from the teaching. On thecontrary, they should have a careful didactical di��dac��tic? also di��dac��ti��caladj.1. Intended to instruct.2. Morally instructive.3. Inclined to teach or moralize excessively. treatment at the schoolthrough methodologies giving advantage to the understanding of thestatement. In the context of the problems to which we have referred inthe present investigation, we are sure an understanding of the numericalsituation expressed in the statement of the problem, providing thestudent with some kind of mental picture, is of great importance forachievement of the correct solution. Therefore, we believe the poorersuccess rates that correspond to certain problems simply result fromstatements with numerical information that are difficult to grasp; sothe student in his/her reply, if any, just "processes" thedata in what he believes to be a rational way (using key words). Thisfact had already been established with regard to the Compare problemsbut our research has nevertheless allowed us to extend the results tothe Change-Compare problems.It is really necessary to complement our investigation with certaincognitive cog��ni��tiveadj.1. Of, characterized by, involving, or relating to cognition.2. Having a basis in or reducible to empirical factual knowledge. aspects of student in the problem-solution process.Specifically, it will be interesting to ascertain what conceptual con��cep��tu��aladj.Relating to concepts or the the formation of concepts. schemeis used by students and whether they in fact utilize the same scheme forthe solution of Change-Compare as for the Change problems, or,contrarily, whether they just use the same as for the Compare problems.Clarification ClarificationThe removal of small amounts of fine, particulate solids from liquids. The purpose is almost invariably to improve the quality of the liquid, and the removed solids often are discarded. of this particular point would be vital in establishingexactly whether the dominant influence in the solution of a problem isthe structure (the scheme) of the problem, or the way in which theproblem statement is actually expressed.It is possible that a "compatibility hypothesis An assumption or theory.During a criminal trial, a hypothesis is a theory set forth by either the prosecution or the defense for the purpose of explaining the facts in evidence. ", similarto that of Lewis and Mayer, in the solution of Compare problems(Verschaffel, 1994), may be considered applicable to some students, whowould evidently prefer the statement of a Change-Compare problem to beexpressed without the IR-characteristic. When it is present, however,they could "mentally reorganize re��or��gan��ize?v. re��or��gan��ized, re��or��gan��iz��ing, re��or��gan��iz��esv.tr.To organize again or anew.v.intr.To undergo or effect changes in organization. " the problem statement toexclude the characteristic in question and exchange a More with a Lessone. A further research is required to confirm this hypothesis, and weare already planning to carry that out.Our investigation has produced certain results that should favoreducation. From what we have stated, it is evident that it is essentialthat the student clearly understands the statement of a problem so thathe/she is able to create a mental picture of the numerical situationsinvolved. In this respect, it has been showed in several investigationsthat the use of diagrams and graphics, or mathematics manipulative ma��nip��u��la��tive?adj.Serving, tending, or having the power to manipulate.n.Any of various objects designed to be moved or arranged by hand as a means of developing motor skills or understanding abstractions, especially in teaching material, can be of invaluable assistance in the comprehension comprehensionAct of or capacity for grasping with the intellect. The term is most often used in connection with tests of reading skills and language abilities, though other abilities (e.g., mathematical reasoning) may also be examined. of problems (De Corte, 1993; Riley et al., 1983). It is also beneficialfor students to get directly involved with the statements of problems,either by changing statements previously given to them in their ownwords (Verschaffel, 1994), or just making up completely fresh problemsthemselves (Rudnitsky et al., 1995).Several authors (such as Rudnitsky et al., 1995) consider itconvenient that the knowledge of the different classes of problems formpart of the normal school education curriculum, and that students shouldbe able to identify them as Change, Equalize or Compare. If we acceptthis fact, it is thought that working with students about the fourclasses of problems we have discussed here will improve theirunderstanding of problematic situations and, therefore, improve theirproblem-solving ability. We are also sure the distinction we have madebetween the scheme of the problem (s + v = e; s + d = b) and the ways inwhich the relationship between the two states (Variation and Difference)may be expressed, will both contribute to achieve this purpose.Finally, the conclusions we have drawn from our work actuallystrengthen those already described in various publications on theimportance of the particular attention that textbook textbookInformatics A treatise on a particular subject. See Bible. authors should payto the expressions employed in the enunciation enunciation(inun´sēā´shn),n an auxiliary function of teeth, particularly those in the anterior sector of the dental arch; the formation of sounds of problems. Schoolteachers should also be conscious of the relevance of the expression inproblem enunciation, not only when the problem is being solved but alsoduring subsequent analysis of the students' results.Appendix: Problem StatementsNote: In this paper, we have forced the usual syntactic order inEnglish in order to show the syntactic pattern used in the statements ofthe problems in Spanish.Change* Change 1: Before, A had 4 pta. Then he earned 3 pta. How many ptadoes he have now?* Change 2: Before, J had 9 pta. Then he lost 4 pta. How many ptadoes he have now?* Change 2*: Before losing 4 pta, J had 9 pta. How many pta does hehave now?* Change 3: J had 2 pta before and he has now 5 pta. How many ptadid he earn?* Change 4: J had 5 pta before and he has now 2 pta. How many ptadid he lose?* Change 4*: E has now 2 pta and he had 5 pta before. How many ptadid he lose?* Change 5: J has earned 3 pta and he has now 5 pta. How many ptadid he have before?* Change 6: After losing 3 pta, R has now 4 pta. How many pta didhe have before?* Change 6*: R has now 4 pta, after losing 3 pta. How many pta didhe have before?Change-Compare* Change-Compare 1.1: Before, P had 5 pta. Now, he has 4 pta morethan he had before. How many pta does he have now?* Change-Compare 1.2: Before, A had 3 pta. Before, he had 5 ptaless than he has now. How many pta does he have now?* Change-Compare 2.1: Before, P had 8 pta. Before, he had 3 ptamore than he has now. How many pta does he have now?* Change-Compare 2.1*: Before, R had 3 pta more than he has now.Before, he had 8 pta. How many pta does he have now?* Change-Compare 2.2: Before, A had 6 pta. Now, he has 2 pta lessthan he had before. How many pta does he have now?* Change-Compare 2.2*: Now, J has 2 pta less than he had before.Before, he had 6 pta. How many pta does he have now?* Change-Compare 3.3: L had 4 pta before and now he has 7 pta. Howmany pta more than he had before does he have now?* Change-Compare 3.4: L had 5 pta before and he has now 9 pta. Howmany pta less than he has now did he have before?* Change-Compare 4.3: P had 6 pta before and he has now 4 pta. Howmany pta more than he has now did he have before?* Change-Compare 4.3*: P has now 4 pta and he had 6 pta before. Howmany pta more than he has now did he have before?* Change-Compare 4.4: M had 9 pta before and he has now 5 pta. Howmany pta less than he had before does M have now?* Change-Compare 4.4*: M has now 5 pta and he had 9 pta before. Howmany pta less than he had before does M have now?* Change-Compare 5.5: Now, L has 3 pta more than he had before.Now, he has 5 pta. How many pta did he have before?* Change-Compare 5.6: Before, E had 4 pta less than he has now.Now, he has 9 pta. How many pta did he have before?* Change-Compare 6.5: Before, L had 3 pta more than he has now.Now, he has 2 pta. How many pta did he have before?* Change-Compare 6.5*: Now, J has 2 pta. Before, he had 3 pta morethan he has now. How many pta did he have before?* Change-Compare 6.6: Now, M has 4 pta less than he had before.Now, he has 5 pta. How many pta did he have before?* Change-Compare 6.6*: Now, P has 5 pta. Now, he has 4 pta lessthan he had before. How many pta did he have before?Equalize* Equalize 1: R has 4 pta. If R earns 2 pta, then he will have thesame as P. How many pta does P have?* Equalize 2: A has 4 pta. If C loses 2 pta, then he will have thesame as A. How many pta does C have?* Equalize 3: T has 3 pta and M has 8 pta. How many pta does T haveto earn to have the same as M?* Equalize 4: C has 2 pta and A has 7 pta. How many pta does A haveto lose to have the same as C?* Equalize 5: If P earns 2 pta, then he will have the same as R. Rhas 6 pta. How many pta does P have?* Equalize 6: If C loses 5 pta, then he will have the same as A. Chas 7 pta. How many pta does A have?Compare* Compare 1: T has 2 pta. M has 5 pta more than T. How many ptadoes M have?* Compare 2: T has 2 pta. T has 5 pta less than E. How many ptadoes E have?* Compare 3: R has 2 pta and A has 7 pta. How many pta more than Rdoes A have?* Compare 4: P has 4 pta and A has 6 pta. How many pta less than Adoes P have?* Compare 5: T has 5 pta more than M. T has 7 pta. How many ptadoes M have?* Compare 6: R has 3 pta less than L. L has 5 pta. How many ptadoes R have?Table 1. The four classes of problems analyzed in our research study(pta = pesetas) Scheme: s + v = e Scheme: s + d = bExpression: Change EqualizeSimple Before Juan had 4 pta. Juan has 4 pta. Then he earned 3 pta. If Juan earns 2 pta, How many pta does he have now? then he will have the same as Pedro. How many pta does Pedro have?Expression: Change-Compare CompareMore/Less Before Juan had 4 pta. Juan has 4 pta. Now he has 3 pta more than he had Pedro has 5 pta more before. than Juan. How many pta does he have now? How many pta does Pedro have?Table 2. Types of problems considered in this research studyType of problems Expression Unknown Sign of Characteristic variationChangeChange 1 Add on End IncreaseChange 2 Take away End DecreaseChange 2*Change 3 Add on Variation Increase IChange 4 Take away Variation DecreaseChange 4*Change 5 Add on Start Increase IChange 6 Take away Start Decrease IChange 6* ICompareCompare 1 More BigCompare 2 Less Big IR(s)Compare 3 More Difference ICompare 4 Less DifferenceCompare 5 More Small IR(w)Compare 6 Less SmallEqualizeEqualize 1 Add on Big REqualize 2 Take away Big IEqualize 3 Add on Difference IEqualize 4 Take away DifferenceEqualize 5 Add on Small IEqualize 6 Take away Small RChange-CompareChange-Compare 1.1 More End IncreaseChange-Compare 1.2 Less End Increase IR(s)Change-Compare 2.1 More End Decrease IR(s)Change-Compare 2.1* IR(w)Change-Compare 2.2 Less End DecreaseChange-Compare 2.2*Change-Compare 3.3 More Variation Increase IChange-Compare 3.4 Less Variation IncreaseChange-Compare 4.3 More Variation Decrease IChange-Compare 4.3* IChange-Compare 4.4 Less Variation DecreaseChange-Compare 4.4*Change-Compare 5.5 More Start Increase IR(w)Change-Compare 5.6 Less Start IncreaseChange-Compare 6.5 More Start DecreaseChange-Compare 6.5*Change-Compare 6.6 Less Start Decrease IR(w)Change-Compare 6.6* IR(s)Table 3. Distribution of students by levels, groups, schools and testsproposedLevel Age S G N T1 T2 T3 T4 T5 T6 T73[degrees]-6[degrees] 8-12 267 30 32 32 32 32 30 223[degrees] 8-9 70 4 8 8 8 8 8 8 1 G1 23 3 4 4 4 4 4 1 G2 21 1 4 4 4 4 4 2 G3 15 5 2 G4 11 34[degrees] 9-10 59 8 8 8 9 8 10 2 1 G5 17 4 4 3 4 1 1 1 G6 24 4 4 4 4 3 5 1 G7 10 - - 1 1 4 4 2 G8 8 25[degrees] 10-11 66 8 8 8 7 8 6 6 2 G9 23 4 4 4 4 4 3 2 G10 22 4 4 4 3 4 3 2 G11 21 66[degrees] 11-12 72 10 8 8 8 8 6 6 2 G12 18 4 4 4 4 2 - 2 G13 15 4 4 3 - 2 2 2 G14 15 2 - 1 4 4 4 2 G15 24 6Level Age S G N T8 T9 T103[degrees]-6[degrees] 8-12 267 19 22 163[degrees] 8-9 70 6 6 6 1 G1 23 1 G2 21 2 G3 15 2 4 4 2 G4 11 4 2 24[degrees] 9-10 59 1 5 - 1 G5 17 1 G6 24 1 G7 10 2 G8 8 1 5 -5[degrees] 10-11 66 6 5 4 2 G9 23 2 G10 22 2 G11 21 6 5 46[degrees] 11-12 72 6 6 6 2 G12 18 2 G13 15 2 G14 15 2 G15 24 6 6 6S = School; G = Group; N = Number of studentsTable 4. Contents of the testsTest Types of problems 1 Change 5 Change-Compare 2.1 Change-Compare 5.5 Change-Compare 5.6 Equalize 5 Compare 5 2 Change 1 Change-Compare 6.5 Change-Compare 1.1 Change-Compare 1.2 Equalize 1 Compare 1 3 Change 2 Change-Compare 5.6 Change-Compare 2.2 Change-Compare 2.1 Equalize 6 Compare 6 4 Change 6 Change-Compare 1.2 Change-Compare 6.6 Change-Compare 6.5 Equalize 2 Compare 2 5 Change 3 Change-Compare 4.3 Change-Compare 3.3 Change-Compare 3.4 Equalize 3 Compare 3 6 Change 4 Change-Compare 3.4 Change-Compare 3.4 Change-Compare 3.3 Equalize 4 Compare 4 7 Change-Compare 6.5* Change-Compare 5.5 Change-Compare 2.1* Change-Compare 1.1 Compare 5 Compare 1 8 Change-Compare 2.2* Change-Compare 5.6 Change-Compare 6.6* Change-Compare 1.2 Compare 6 Compare 2 9 Change-Compare 4.4* Change-Compare 3.3 Change-Compare 4.3* Change-Compare 3.4 Compare 3 Compare 410 Change 6* Change 4* Change 2* Equalize 6 Equalize 4 Equalize 2Table 5. Success rates, number of students and characteristics of theproblems % N Char % N Char % N CharChange Change 1 Change 3 Change 5 94 32 75 32 I 77 30 I Change 2 Change 4 Change 6 91 32 100 30 78 32 I* problem 94 16 94 16 81 16 IChange- Change-Compare 1.1 Change-Compare 3.3 Change-Compare 5.5Compare 93 54 78 54 I 65 52 IR(w) Change-Compare 1.2 Change-Compare 3.4 Change-Compare 5.6 37 83 IR(s) 90 84 80 81 Change-Compare 2.1 Change-Compare 4.3 Change-Compare 6.5 29 62 IR(s) 95 62 I 70 64* problem 45 22 IR(w) 64 22 I 91 22 Change-Compare 2.2 Change-Compare 4.4 Change-Compare 6.6 84 32 94 30 47 32 IR(w)* problem 79 19 82 22 21 19 IR(s)Equalize Equalize 1 Equalize 3 Equalize 5 91 32 R 75 32 I 57 30 I Equalize 2 Equalize 4 Equalize 6 50 48 I 98 46 85 48 RCompare Compare 1 Compare 3 Compare 5 80 54 74 54 I 63 52 IR(w) Compare 2 Compare 4 Compare 6 59 51 IR(s) 89 52 80 51% = Success; N = Number of students; Char = CharacteristicTable 6. Sets-of-three similar problems A T A T A TChange-Compare 5.5 65 55 Change-Compare 2.1* 45 45 Compare 5 63 68Change-Compare 3.3 78 73 Change-Compare 4.3* 64 64 Compare 3 74 68Change-Compare 1.1 93 95 Change-Compare 6.5* 91 91 Compare 1 80 95Change-Compare 5.6 80 79 Change-Compare 2.2* 79 79 Compare 6 80 74Change-Compare 3.4 90 91 Change-Compare 4.4* 82 82 Compare 4 89 86Change-Compare 1.2 37 32 Change-Compare 6.6* 21 21 Compare 2 59 42A = Average for all tests;T = Average for individual tests with the sets-of-three problems in thesame rowTable 7. Possible sets-of-three similar problemsChange-Compare 5.5* Change-Compare 2.1 Compare 5*Change-Compare 3.3* Change-Compare 4.3 Compare 3*Change-Compare 1.1* Change-Compare 6.5 Compare 1*Change-Compare 5.6* Change-Compare 2.2 Compare 6*Change-Compare 3.4* Change-Compare 4.4 Compare 4*Change-Compare 1.2* Change-Compare 6.6 Compare 2*REFERENCESBruno, A. and Martinon, A. 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Lit.: The Tempest]See : Treachery MartinonUniversity of La Laguna The University of La Laguna is situated in San Crist��bal de La Laguna, on the island of Tenerife. It is the oldest university in the Canary Islands, and has the highest student population of any university in these islands. , SpainFidela VelazquezSecondary Institut San Hermenegildo, Spain