PROBLEM SOLVING : HOW CAN WE HELP STUDENTS OVERCOME COGNITIVE DIFFICULTIES

The traditonal approach to teach problem solving usually consists in showing students the solutons of some example-problems and then in asking students to practce individually on solving a certain number of related problems. This approach does not ensure that students learn to solve problems and above all to think about the soluton process in a consistent manner. Topics such as atoms, molecules, and the mole concept are fundamental in chemistry and instructors may think that, for our students, should be easy to learn these concepts and to use them in solving problems, but it is not always so. If teachers do not put emphasis on the logical process during solving problems, students are at risk to become more profcient at applying the formulas rather than to reason. This disappointng result is clear from the outcomes of questonnaires meant to measure the ability to calculate the mass of a sample from the number of atoms and vice versa. A suggeston from the cognitve load theory has proved a useful way to improve students’ skills for this type of problems: the use of worked out examples. The repetton afer two weeks of the Friedel-Maloney test afer the use of worked examples shows that students' skills signifcantly improve. Successful students in all questons jumped from 2 to 64%.


INTRODUCTION
An important and qualifying hallmark of teaching science is the ability to promote problem solving and critcal thinking skills.It is critcal that future citzens have skills in problem-solving to address the range of needs in their life and careers.Problem-solving is an important higher-order cognitve skill."To solve a problem means to search for a way out of a partcular difculty; to fnd a way leading to the pre-set objectve, which is not immediately atainable.Solving problems is a specifc actvity of mind, and mind is specifc only for a human: thus, solving problems is fundamental human actvity" (Polya, 1984, pp. 240).Some scholars connect problemsolving abilites with the defniton of intelligence: "intelligence is by defniton the ability to solve problems."(Nęcka & Orzechowski, 2005, pp. 127).According to John Anderson, "In understanding procedural knowledge we start with problem solving because it seems that all cognitve actvites are fundamentally problem solving in nature."(Anderson, 1995, pp. 237).The Cambridge Handbook of Intelligence reported Wechsler's famous defniton of intelligence: "the aggregate or global capacity of the individual to act purposefully, to think ratonally and to deal efectvely with his environment" (Willis, Dumont & Kaufman, 2011, pp. 50).According to Sternberg, (1988, pp. 72) intelligence "is the purposive adaptaton to and selecton and shaping of real-world environments relevant to one's life."All these defnitons highlight the connecton between the idea of intelligence with the ability to solve problems and to engage in abstract thinking.Established the importance of problem solving, it is necessary to consider what is the nature of a problem in cognitve science and the diference between a problem and an exercise.Problems come in many forms; a general defniton of the term is bound to be abstract.A frst and stll valid defniton comes from Karl Dunker: "A problem arise when a living creature has a goal but does not know how this goal is to be reached" (Dunker, 1945, pp. 1).According to John Hayes, "Whenever there is a gap between where you are now and where you want to be, and you don't know how to fnd a way to cross the gap, you have a problem" (Hayes, 1989, pp. XII).
From this defniton, it follows that not all the 'problems' that are used in educaton are real problems: many are just exercises.Polya calls exercise a routne problem "which can be solved either by putng in special data in already previously solved general problem or so that a certain very shabby patern is followed step by step, without any trace of originality."(Polya, 1984, pp. 241) The "problem" noton is associated with cognitve difcultes and the applicaton of knowledge and skills in logical reasoning in a diferent context.A helpful classifcaton of problem types has been made by Johnstone (2001).He suggested that there are three variables associated with all problems: the data provided, the method to be used, and the goal to be reached.By looking at the two possibilites where each variable is either known or unknown, he came up with eight problem types.The type where each variable is known are not problems, but exercise.According to Jonassen, (2000, pp. 65) a problem is defned by two critcal atributes: it "is an unknown entty in some situaton (the diference between a goal state and a current state).Those situatons vary from algorithmic math problems to vexing and complex social problems, such as violence in the schools.Second, fnding or solving for the unknown must have some social, cultural, or intellectual value.That is, someone believes that it is worth fnding the unknown." As teachers know from their teaching experience, it is not possible to distnguish between problems and exercises just by looking at the problem.The same task can be an exercise for a student and a difcult problem for another student.This is because "Status as a problem is a subtle interacton between the task and the individual struggling to fnd an appropriate answer or soluton" (Bodner, 1987, pp. 513;Bodner & Herron, 2002, pp. 236).
However, the phrase 'problem solving' can be understood in many ways.Problem solving can be seen as the skills associated with solving numerical calculatons, while others might think of practcal skills in the laboratory.It is equally possible to see problems as the real-life problems faced, for example, by many people because of economic difcultes or problems associated with the depleton of the ozone layer or global warming.In this artcle we will restrict our analysis to the numerical calculatons that a freshman meets in a chemistry college course.
The characteristc of the problems that are used in teaching is that they are well-structured and well-defned.Usually the real-world problems are not well structured.(Jonassen, 1997, pp. 68) In the literature there is a fnal distncton between problems: they can be well-defned and ill-defned problems.According to Kahney (1986, pp. 20), in a well-defned problem the solver is provided with several types of informaton: • informaton about the inital state of the problem; • informaton about the goal state; • informaton about legal operators (operatons that are legally allowed); • informaton about operator restrictons which constrain their applicatons.
The well-known Tower of Hanoi problem can be an example of a well-defned problem.(Sapir, 2004, pp. 20) "Given are three pegs and a certain number n of disks of distnct sizes.Initally all disks are stacked on the frst peg (the source) ordered by size, with the smallest at the top and the largest at the botom.The goal is to transfer them to the third peg (the destnaton), while obeying the following rules: • at each step only one disk can be moved; • the moved disk must be a topmost one; • at any moment, a disk cannot reside on a smaller one." We make explicit another restricton of the operator 'move' saying that the discs can only be moved in the 3 pegs.The next problem that the reader will encounter is another example of well-defned problem.

MEANINGFUL PROBLEM-SOLVING
Depending on whether a problem is well-structured or ill-structured, a diferent strategy is required to teach problem-solving skills in instructonal setngs."Requiring learners to justfy (argue for) their positons while solving problems (especially ill-structured problems) should be an essental part of problem-solving instructon."(Shin, Jonassen & McGee, 2003, pp. 28) Solving a new problem requires the producton of new knowledge; "new knowledge is produced through a series of cognitve actvites actng on (external) informaton and prior knowledge."(Taconis, Ferguson-Hessler & Broekkam, 2001, pp. 444).The stoichiometric problems considered in a chemistry course can be described as closed and usually lead to a single 'right' answer.According to Frazer (1982, pp. 173), in closed problems there is 'a single unique soluton', while 'most problems outside the classroom, do not have unique, unambiguously correct solutons -these can be called 'open problems''.
In what follows, well-defned and closed numerical problems will be considered: problems that require numerical calculatons in the context of known theoretcal domains for their soluton.
Problem solving is a complex cognitve task in which critcal thinking and metacognitve actvity play an important role.According to Jonassen (1997, pp. 65-66), "Problem solving, as an actvity, is more complex than the sum of its component parts.… [it involves] domain knowledge, structural knowledge, ampliatve skills, metacognitve skills, motvaton/attudinal components,..and certainly requires knowledge about self".It is necessary to teach the reasoning involved in the problem solutons in ways that are meaningful for students and make sense to them, because "research has made it clear that procedures must take on meaning and make sense or they are unlikely to be used in any situaton that is at all diferent from the exact ones in which they are taught."(Resnick, 1983, pp. 478) In teaching, it is also required to consider the limitatons of human cognitve architecture without unnecessarily overloading the student's working memory.So, "when teaching new content and skills to novices, teachers are more efectve when they provide explicit guidance accompanied by practce and feedback, not when they require students to discover many aspects of what they must learn."(Clark, Kirschner & Sweller, 2012, pp. 6) According to Shavelson (2010, pp. 11), "learning is about change in behaviour" and the purpose of instructon must be to increase the useful knowledge stored in the long term memory, because "If nothing has changed in long-term memory, nothing has been learned."(Sweller, Ayres & Kalyuga, 2011, pp. 24).

OBJECTIVES
The study on recurrent errors and difcultes encountered by students in solving chemical problems has been going on for many years.In this artcle the difcultes that students encounter in the calculatons involving chemical formulas will considered.In this study the informaton in terms of number of moles and molecules, and vice versa, will also be analyzed.According to the Johnstone's triangle (1991) this study will move in the side sub-microscopic -symbolic.For teachers who are experienced, the extensive use of symbols in chemical representatons is a normal occurrence.Instead, the abstract nature of the chemical representatons can mean to the student a signifcant increase of the mental load in the working memory."The reader is invited to see chemistry students as similar to learners working in a second language, where they are expected to be both learning the language and using the language to understand substantve material simultaneously.This is considered more than a metaphor -and highlights the even greater challenge posed to those who are expected to learn this new language of chemistry as they learn the chemistry in the medium of a language that is already not their own."(Taber, 2009, pp. 78) In 1996, students enrolled in a frst year chemistry course in the faculty of Engineering of the Polytechnic University of the Marche were asked to solve this problem, bringing out all the steps: Consider the oxygen contained in 10.00 g of Fe 2 O 3 .How many molecules of oxygen are equivalent to this quantty?(molar mass of Fe 2 O 3 is 159.7 g): The results were quite disappointng (responses' distributon are reported in Table 1): only 15 students out of 87 (17.2%) have given the correct answer ([D]).(Cardellini, 1996).The response distributon shows that students have great difculty in reasoning: Given the confusion between atoms and molecules in the above soluton (molecules O), at the tme this Author suggested that some of the students do not know the diference between atom and molecule.This conclusion may seem excessive, but the word 'know' has diferent meanings in diferent contexts, at least from the operatonal point of view, when the concepts are inserted in stoichiometric calculatons.Faced with the difcultes of the students, responsible teachers should ask themselves why students make mistakes and what we can do to help them to think properly.For teachers, as experts, it is difcult to understand the difcultes of students, also because teachers have forgoten the difcultes, when for the frst tme they solved the same problems.It is worth refectng on the observaton of Keith Taber, (2009, pp. 81) "Where the expert has developed interpretve frameworks that can 'see through' the symbols, the novice may focus on incidental aspects of the formalism used." The objectve of this study is to fnd strategies, instructons, or tps for students to improve their skills in the soluton of these problems.According to Merrill (2002, pp. 47) "Learning is promoted when learners are provided appropriate learner guidance" Students experiencing difcultes in relatonships among the atomic mass, the molar mass, the mass of the substance, the meaning of the subscript, and the diference between atoms and molecules.This aspect has been highlighted many years ago in the literature: "Historically, teachers of beginning chemistry have named the mole concept as the most difcult part of the beginning course.What teachers have in mind is not the concept alone, however: instead they include many relatonships and calculatons that involve the mole."(Dierks, Weninger & Herron, 1985, pp. 839) Because the representaton of the problem is an important step in solving the problem, and the use of a correct representaton is indispensable to avoid errors, it was suggested the use of representaton, associate with the use of proper stoichiometric rato.In general, "A problem representaton is a cognitve structure corresponding to a problem, constructed by a solver on the basis of his domain-related knowledge and its organizaton" (Chi, Feltovich & Glaser, 1981, pp. 121-122).The use of this procedure helped students and 94.4% of them have successfully solved a similar problem (Cardellini, 1996).Visualizing the formulas helps students determine the correct stoichiometric relatonship between the number of atoms and so decreases working memory load.
However, because schools and students have changed through years, it may be interestng to ask whether this strategy is stll valid and sufcient to improve the skills of the majority of the students.To address this queston, a study was designed which engaged students enrolled in the frst year of a chemistry course in a faculty of Engineering.In this study, some cognitve styles and the students' perceptons about the course were explored to see if there was any relatonship between these results and students' abilites in problem solving and the quality of the creatve work resultng from this approach.

METHODOLOGY
In chemistry, as in other situatons, acquiring the knowledge of relevant rules is not sufcient."We can learn the rules of chess in about 30 min and using those rules, we can theoretcally generate every game that has ever been played and that ever will be played.Learning those rules is essental to chess skill but in another sense, it is trivial.Real chess skill comes from acquiring automated schemas."(Sweller et al., 2011, pp. 24) The same thing can be said in stoichiometric calculatons.To know about atoms, molecules, the Avogadro number, molar mass and moles is not sufcient to manage the cognitve skills necessary to master the reasoning that the correct concatenaton of the steps require.It is also necessary to know the theory and do a lot of exercises to become profcient problem solvers.Students enrolled in courses where I teach are requested to solve many problems, to draw concept maps and résumés (write short summaries) using the concepts of the lectures.They are asked also to explain and justfy the various steps of the soluton of a problem and to develop a way for verifying the result.Afer each lesson, students receive e-mails with suggestons about exercises to be solved, the list of concepts presented in class and a copy of the slides for the next lesson.
The study was designed by measuring the students' ability in the calculaton with a test on the frst day of the course and the repetton of the same test afer the interventon.The instrument used was the Friedel-Maloney test, with the additon as the frst queston of the above problem unidentfed.This is also because the results of the four diferent questons in the test are always diferent from each other.Also, the soluton of the calculaton of molecules of oxygen in 10.00 g of Fe 2 O 3 requires conceptual passages similar to the problems that students must solve in writen examinatons.
The Friedel-Maloney test "determine(s) what our students are able to do with the informaton presented in class about moles, molecules, atoms, gram atomic weight, Avogadro's number, and molar mass."(Friedel & Maloney, 1995, pp. 900).The Friedel-Maloney is a paper and pencil test using a four multple-choice optons task and some results in the past have been reported elsewhere (Angawi & Cardellini, 2011).Over the years, the test was repeated, as were the more or less disappointng results.The Friedel-Maloney te st measure the ability to use the relatonship among the atomic mass, the molar mass, the mass of the substance, and Avogadro's number, and calculatons involving atoms and molecules, by means of four problems: • How many oxygen atoms are present in a container with 288 g of O 3 ?
• There are 1.8 x 10 5 atoms in a sample of P 4 .What is the mass of this sample?
• How many atoms of sulfur are in a sample of 963 g of S 6 ?
• There are 2.41 x 10 24 atoms in a sample of S 8 .What is the mass of this sample?
This study reports the data obtained in three diferent courses, in diferent academic years.All relate to students enrolled in the frst year of chemistry at the Faculty of Engineering of the Polytechnic University of the Marche.The frst course in this study had 88 freshmen students enrolled in the frst year of a chemistry course which took place in the second semester of the academic year 2012-13.Of these, 51 were male (58%) and 37 female (42%).The age of students was 19 -21, with 2 students aged 22.The course was held in the second semester of the academic year 2012-2013, for a workload of 6 ECTS credits.In the 48 hours of the course, more than 4,500 solutons of stoichiometric problems were collected (N = 32 Ss; with a mean of 143.1 and standard deviaton of 45.4).From the 32 students that passed the exam, 210 concepts maps and 1,235 résumés were collected.
Three psychological measurements were applied to the group to see if there was any relatonship between these results and the quality of the creatve problem solving resultng from this approach.These were • Formal Operatonal Reasoning, • Disembedding Ability, and The Formal Operatonal Reasoning was measured using the Group Assessment of Logical Thinking (GALT) test (Roadrangka, Yeany & Padilla, 1983).The scores ranged from 8 to 22 (out of 24) with a mean of 16.6 and standard deviaton of 3.6.The disembedding ability was measured by the Field Dependence/Field Independence test devised and calibrated by El-Banna (1987) based upon the original work of Witkin (Witkin, 1974;Witkin & Goodenough, 1981).Out of a possible score of 20, the range achieved was 5-19, with a mean value of 13.7 and a standard deviaton of 3.9.According to Danili and Reid (2004) the extent of feld dependency infuences all academic performance.
The Convergent/Divergent test consisted of 6 mini tests, developed and calibrated at the Centre for Science Educaton in the University of Glasgow, UK, and described elsewhere (Danili & Reid, 2006).In the Italian translaton, the only diference was on test 5: it was adapted to the Italian language and students were asked about the words, which begin with the leter G, and end with the leter A.
The usual modest results were repeated, but the disappointment was even greater considering the results of the second round of the Friedel-Maloney test.The distributon of students' responses are reported in The second queston (Results are reported in Table 3): How many oxygen atoms are present in a container with 288 g of O 3 ?(molar mass of O 3 is 48.0 g): • (a) 3.61 x 10 24 ; • (b) 18.0; • (c) 1.08 x 10 25 ; The third queston (the distributon of students' responses are reported in Table 4): There are 1.8 x 10 5 atoms in a sample of P 4 .What is the mass of this sample?(Molar mass of P 4 is 124 g.): • (a) 9.3 x 10 -18 ; • (b) 3.7 x 10 -17 ; • (c) 5.6 x 10 6 ; The fourth queston (the distributon of students' responses are reported in Table 5): How many atoms of sulfur are in a sample of 963 g of S 6 ?(gram atomic weight of S is 32.1 g): • (a) 3.01 x 10 24 ; • (b) 30.0; • (c) 5.02 x 10 23 ; • Table 5.The distributon of students' responses to the frst and second rounds respectvely, of the problem: How many atoms of sulfur are in a sample of 963 g of S 6 ?
The last queston (the distributon of students' responses are reported in Table 6): There are 2.41 x 10 24 atoms in a sample of S 8 .What is the mass of this sample?(gram atomic weight of S is 32.1 g): Table 6.The distributon of students' responses to the frst and second rounds respectvely, of the problem: There are 2.41 x 10 24 atoms in a sample of S 8 .What is the mass of this sample?
I asked for advice Prof. John Sweller of the University of New South Wales in Sydney, Australia, who originated cognitve load theory.The suggeston was to train students with the use of the worked example method.(Sweller & Cooper, 1985) A worked example provides learners with informaton concerning a problem soluton.
A worked example not only shows the sequence of steps, but also comments on them with explanatons and their motvaton.Five problems, three suggested problems to be solved, and two worked problems, solved with the intenton to "direct atenton appropriately and reduce cognitve load" (Ward & Sweller, 1990, pp. 1), were used in this experiment in two diferent courses.
The second course in this study had 77 freshmen students enrolled in the frst year of a chemistry course which took place in the frst semester of the academic year 2013-14.Of these, 55 were male (71%) and 22 female (29%).The age of students was 19-21, with 2 students aged 23 and 25 years old.The course was held in the frst semester of the academic year 2013-2014, for a workload of 9 ECTS credits.In the 72 hours of the course more than 7,000 solutons of stoichiometric problems were collected (N = 49 Ss; with a mean of 145.3 and standard deviaton of 87.8, from 22 to 373 solutons).In Figure 1 the collected solutons to this and other courses are shown.Figure 2. The correct answer (4) increased from 4% to 69% afer the study and working on the worked example experiment In Figures 3 and 4 are reported the results of the solutons of the second and third questons, respectvely.
Figure 3.The correct answer (3) increased from 42% to 83% Figure 4.The correct answer (1) increased from 19% to 81% In Figures 5 and 6 are reported the results of the solutons of the fourth and ffh questons, respectvely.
Figure 5.The correct answer (4) increased from 46% to 86% Vol. 4(4), 2014, pp 245 Figure 6.The correct answer (3) increased from 50% to 83% As additonal proof I report other results from students enrolled in a course that was held in the frst semester of the academic year 2013-2014, for a workload of 6 ECTS credits.Even if the three courses here described in part difer in content, up to the part of stoichiometric calculatons reported in these studies, the three courses have proceeded in exactly the same way.In the frst test (N = 48), 18 students had wrong answers on all questons, in the second test (N = 39), 22 students solved correctly all 5 questons.The correct answers increased from 8% to 74% in the frst queston, from 13% to 85% in the second, from 11 to 64% in the third queston, from 59% to 82% in the fourth, and from 51% to 77% in the last.

CONCLUSIONS
Knowledge of the diference between atoms and molecules and the stoichiometric relatonships between atoms in chemical formulas are taken for granted.In the university teaching of stoichiometric calculatons, this part of chemistry has very litle tme devoted to it.This is a shame because as these studies demonstrate, there are many conceptual difcultes students encounter in solving problems requiring stoichiometric calculatons.
The results obtained with the use of worked examples are very encouraging.The representaton of the problem is certainly an important aspect in problem solving, but even more important are the logical processes that help solve the problems correctly.The adopton of worked examples really helped the students to improve their skills in this part of stoichiometric calculatons.
It must be remembered that we can't teach people to solve all novel problems beter.What we can do, is turn novel problems into exercises.Worked examples can help in that aim."The purpose of worked examples is precisely to turn problems into exercises.Once sufcient knowledge concerning the problems of an area has been stored in long-term memory, all problems become exercises and high levels of expertse have been atained."(Sweller, 2014) Another educatonal aspect deserves to be considered.During the course my students usually work in cooperatve groups, according to some role, exercising and practcing solving numerous problems.In this way they also become skilled at solving more complex problems.This is an important aspect of meaningful learning because "explaining another person's reasoning, especially a more correct one, raises additonal opportunites for comparing and contrastng the other person's reasoning with one's own.Any conficts observed will naturally elicit more repairs of one's representaton.… exposing a learner to multple perspectves on a problem (or perhaps even multple representaton of a problem soluton), either from a text or from another peer's reasoning, seems to support efectve explaining and thereby learning.(Roy & Chi, 2005, pp. 276-277) In this experiment, the students have solved problems involving atoms, molecules, and formulas individually.We do not know what happens in more complex problems.Because skill is very domain-specifc, probably we will need the use of a greater number of worked examples to prepare students.This study highlights that many students (about 35% of them) contnue to have difculty with this kind of problem.In additon to working in cooperatve groups, less skilled students need more guided instructon."It appears that guided instructon helps less skilled learners by providing task-specifc learning strategies."(Clark et al., 2012, pp. 8).
The distributon of students' responses to the frst and second rounds respectvely, of the problem: There are 1.8 x 10 5 atoms in a sample of P 4 .What is the mass of this sample?

Figure 1 .
Figure 1.Despite the workload required in the course, many students were quite happy to work in this way.In a questonnaire completed by students afer passing the exam, the queston: There was too many problems to be solved scored 3.03, while the queston: I like to solve stoichiometric problems scored 5.50 out of 6 (N = 30 students) The majority of students got the wrong soluton because they used inconsistent relatonships.For example, the typical wrong reasoning is 1 mol O 3 = 1 mol O 2 .The use of this wrong relatonship leads to the response [C] (one-third of the students have made this mistake).Instead, some of the responses [E] have used the procedure: (1.879 x 10 -1 mol O) x (6.022x10 23 molecules O/mol O) = 1.131 x 10 23 molecules O

Table 2 .
The frst queston: Consider the oxygen contained in 10.00 g of Fe 2 O 3 .How many molecules of oxygen are equivalent to this quantty?(molar mass of Fe 2 O 3 is 159.7 g):

Table 2 .
The distributon of students' responses to the frst and second rounds respectvely, of the problem: How many molecules of oxygen are equivalent to 10.00 g of Fe 2 O 3 ?

Table 3 .
The distributon of students' responses to the frst and second rounds respectvely, of the problem: How many oxygen atoms are present in a container with 288 g of O 3 ?