FOR SOLVING MULTI-STEP PROBLEMS IN GENERAL CHEMISTRY

The learning of solving mult-step problems is a relevant aim in chemical educaton for engineering students. In these questons, afer analyzing inital data, a complex reasoning and an elaborated mathematcal procedure is needed to achieve the correct numerical answer. However, many students are able to efectvely use algorithms even with a lack of meaningful understanding of involved chemical concepts. This paper reports the applicaton of some collaboratve actons in order to induce a complete acquisiton of problem solving skills. The studied approaches, performed inside and outside the classroom, are classifed in low-collaboratve and highcollaboratve actvites, depending on the relatve partcipaton of instructor/students in their development. The critcal descripton of the proposed methodology and the produced outcomes are exposed. The contributon of each major reasoning mode (model, rule or case based) employed in these actvites is analyzed. Also, the percepton of students is evaluated on the basis on the data provided by direct observaton and a specifc survey with Likert-scale and open-ended questons. The results indicate that the changes of teaching to a more conceptual orientaton lead a deeper understanding, minimizing misconceptons or a rote learning.


INTRODUCTION
The learning of solving numerical problems mainly involves both conceptual and algorithmic components, together with a heuristc approach for a global strategy. The frst is related to the understanding of underlying concepts from the basic theories of science, while the second is related to mathematcal procedures for applying specifc equatons. So, the general algebraic system is at the heart of advanced modern calculaton techniques in engineering (Lee, 2003). During the teaching and assessment of chemistry, as other sciences, the instructors must design and prepare actvites, materials, and other educatonal resources for guaranteeing that the engineering students should be able to learn both contributons.
Several authors have demonstrated that students might exhibit diferent behaviors answering chemistry questons (Bodner, 2003;Talanquer, 2006;Taber & Bricheno, 2009;Taber & García-Franco, 2010). This variety of alternatve conceptons is a result of applying a set of more or less integrated cognitve resources that guide but also constrain their answers. In a linear diagram of cognitve capabilites, the students can be classifed from conceptual learners to algorithm learners. The students at the frst end fail in the algebraic manipulaton of mathematcal formulae, although they understand the theoretcal background of the queston. The last end corresponds to students that are able to successfully apply algorithms even in the absence of meaningful conceptual understanding.
The involved levels of cognitve processing and the common questons proposed in chemistry subjects have been examined by several educatonal researchers. A relevant study is the investgaton of Zoller, Dori and Lubezky (2002), classifying chemistry questons in lower-order cognitve skills (LOCS) and higher-order cognitve skills (HOCS). Smith, Nakhleh and Bretz (2010) reviewed the published categorizatons of chemistry questons. This study also suggested that a useful framework for conceptualizing existng results is the revised Bloom's taxonomy.
Diferent models of reasoning are triggered when a student is solving questons (Kraf, Strickland & Bhatacharyya, 2010;Christan & Talanquer, 2012). Major modes are model-based reasoning (MBR), case-based reasoning (CBR), and rule-based reasoning (RBR). MBR is an abstracton to understand mater at diferent scales following conceptual/mathematcal approaches. CBR may imply using old solutons or experiences to interpret or explain new situatons and solve a less familiar task. RBR involves inducing paterns of behavior from direct experience, and then the questons are solved based on generally forward-working schemes associated to each specifc framework.
Several studies in chemical educaton have been focused on looking for teaching methods (Dori & Hameriri, 2003). The descripton of algorithm to get the correct answers is the main objectve in conventonal teaching. In this sense, some teachers ofen prefer expository methods -presentaton of a nearly closed strategy -, rather than those related to the use of an elaborated judgment by students. The teacher can control the reasoning mode and direct the students through the algorithm. The students' atenton can be focused on the key points to solve the queston. But, actve learning methods appear to produce an improved learning environment leading to increased student satsfacton, and even, beter academic performance (Kovac, 1999). Then, the revised conceptons of problem solving need a transformaton from routne tasks to an atractve and valuable assignment. Changes from some instructonal practces based on teacher-controlled actvites to ones grounded in student-controlled actvites are required. The applicaton of various methodological resources is especially recommended for novel engineering students, because that should provide a solid background for more specialized subjects. A shif of chemistry instructon is needed from an algorithmic orientaton to a more conceptual orientaton.
The central goal of this study was to investgate the integraton of collaboratve actvites in teaching in order to induce a deeper learning and increasing the success in solving chemical questons. In partcular, the investgaton was guided by how to modify instructon to beter support students' reasoning inside and outside the classroom. The proposed approach was based on mult-step questons that involved both algorithmic and conceptual orientatons requiring more advanced levels of cognitve processing. Also, a collaboratve learning was a key strategy for facing to complex mult-step questons. Two or more peers working together on learning actvites was the best tool for partcipants actvely involved in conceiving and internalizing the solving knowledge and skills.

DESIGN/METHODOLOGY/APPROACH 2.1 Partcipants
This study was conducted in Universitat Politecnica de Valéncia, a public technological university in the western Spain. Partcipants were Agriculture Engineering students enrolled in General Chemistry (frst year course, frst semester). The methodological changes were performed during two consecutve academic courses. The number of students enrolled in the subject was 184 (2012-2013) and 170 (2013-2014) divided in three groups supervised by a teacher each one. Only a group partcipated in the entre innovatve experience, the number of students being 64 (2012-2013) and 44 (2013-2014).

Actvites
The actons focused on the learning of solving problems were integrated in the development of the subject. These actvites corresponded to about 40 % of course tme. Designed strategies for the interactve solving of relevant problems were divided in low-collaboratve and high-collaboratve actvites. The frst category included complete group to self-learning or peer-learning actons. The second category is composed by actvites performed in pairs or small groups for the solving of proposed questons. The main resources were textbook, blackboard, multmedia slides, and multmedia material obtained from the Internet. The textbook was composed by 450 mult-step questons and 200 short questons. Each unit contained solved problems and unsolved problems with solutons . Table 1 summarizes the applied actvites with tme distributon, partcipaton, and resources used. Most actvites were performed for all students enrolled in the subject. Nevertheless, an increment of highcollaboratve actvites (about 20 %) -decreasing the number of low-collaboratve actons -was promoted in the partcipant group. Also, inquisitory style was the predominant in the lectures, compared to expository style of the other groups.

Survey
The percepton of innovatve experience was collected by Likert-scale and open-ended questons (see Fig 1). The survey was anonymously flled up in the last lecture of the course, afer a short explanaton performed by instructor. First queston was related to the cognitve capabilites and students had to classify themselves the kind of learner, from conceptual learner to algorithm learner. In the following questons, the students evaluated the contributon of diferent methodologies available to learn how solving mult-step problems.
Finally, an open-ended queston allowed them a comment about teaching quality and development of subject.

Analysis of reasoning models and common errors
The frst step was the analysis of the rubrics generated from the paper based evidences collected in the last years. The reasoning strategies and errors performed by students during the solving of chemical questons were determined.
In the frst part of rubrics, the contributon of major modes of reasoning applied to solving quanttatve chemistry problems (MBR, CBR, and RBR) was studied. The results indicated that a mixture of CBR and RBR was the mode of reasoning commonly displayed in the observed study groups. Students applied normatve, empirical, or theoretcal rules extracted from some examples as the main approach to solve a problem. The process started with the search of keywords in the statement, for instance the students underlined some relevant words or data. These chemical terms, equatons, units or numeric data allowed students to establish a patern for comparing with similar mult-step questons. A linear, forward-working, and ratonal manner was applied following empirical generalizaton. An example was that students included words as "step 1", "step 2", etc. These rules served as heuristcs to detect relevant informaton, and make quick predictons and decisions. This type of reasoning was quite efcient in generatng satsfactory answers with low cognitve efort and processing tme. Unfortunately, the indiscriminate applicaton of rules across many situatons yielded systematc errors and reasoning biases. For instance, novices ofen failed to recognize new situatons that justfed the use of other approach and tended to overextend the applicaton scope of a given rule.
In the second part of rubrics, the types of mistakes students made were analyzed. General limitatons were identfed allowing the categorizaton of common difcultes or mistakes (mathematcal errors, scientfc errors, chemical errors, and handling of units). These errors refected the sources of difcultes that students encounter in chemical questons. An obvious correlaton found was that the success rate of the in solving multstep problems decreased as the problem difculty increased. Data showed that the most frequent errors were related to required mathematcal skills and the lack of a deep understanding of concepts. For instance, the problem was not completed because they did not solve a second order equaton. Also, an incorrect relaton among concepts and their quanttatve aspects was observed. Hence, some students were unable to correctly solve due to their limitatons in a general level (scientfc) or a specifc level (chemical).
As it has been described in several research studies, the quanttatve aspect of chemistry is ofen an obstacle for freshmen in an introductory college course of chemistry (Dori & Hameiri, 2003). The performance of student achievement was signifcantly dependent on the mathematcs level. Mathematcal skills required throughout a chemistry course mainly involve simple algebraic methods. But, students showed important problems to rearrange an algebraic equaton to solve for an unknown variable or incognita. The following category was composed by defciencies in general skills related to scientfc tools or methods, such graphical data, conversion within metric system, or scientfc notaton. In the area chemistry skills, the most frequent student faults agreed with those previously described (Lee, 2003). They included from incorrect names for quanttes to procedure skills, such chemical language or stoichiometric relatonships. Students in their frst year of University had a hard tme fnding the connecton between macroscopic, sub-microscopic, and symbolic levels, as it was observed for several researchers (Dori & Hameriri, 2003). Finally, students ofen found difcultes in the handling of units, the common errors being noninclusion, incorrect conversions, and dimension heterogeneity.
In conclusion, student defciencies in numerical and scientfc skills detected in this study were the same as those widely recognized among the engineering disciplines.
The analysis of reasoning models, levels of cognitve processing and common errors allowed us to design a more productve instructon, in terms of teaching successful strategies to solve questons.

Studied strategies for teaching how solving mult-step chemical questons
Diferent strategies were introduced in this General Chemistry course to promote the collaboratve acquisiton of efectve skills for solving mult-step questons. The goal was to get the student to solve problems in a ratonal manner and not only mechanically.

Collaboratve lectures
An increment of student partcipaton was applied. Expository and inquisitory presentatons were selected as model of teaching to engage students in discovering rules and relatonships of the subject area. The role of the teacher was shifed from presentng the algorithm for solving problems to actvatng conversaton with the students to encourage them to discover the answers. This approach led to a beter context, drawing upon basic knowledge, correctng common errors, and giving feedback. The students discussed about what they were doing, or how to apply the problem-solving process.
First, a general solving strategy for each kind of mult-step queston was provided indicatng relevant informaton: equatons, stoichiometric transformatons, and conceptual-algorithmic components. The appropriate steps were proposed for solving as routne exercises (linear, forward-working mode) as novel problems (fexible mode). This acton yielded a progressive change of cognitve process dimensions, in other words, a transiton from conceptual knowledge to procedural knowledge. Secondly, blackboard and slides supported the teaching of selected mult-step examples. Diferent reasoning model (MBR, CBR or RBR) were used. The selecton of the solving strategy drastcally infuenced on the future approach selected by students, because most of them directly repeated the reasoning procedure that the instructor shows in the classroom. Thirdly, the solving procedure was applied inducing the partcipaton. The students collaborated sharing conceptual understanding, analyzing partal results, and discussing alternatve strategies. The correct handling of chemical formulae, variable dimensions, and units was partcularly studied. Also, intermediate results were tested to see whether any progress toward an answer was achieved. It allowed focusing on the diferences between successful and unsuccessful solutons. In conclusion, a discussion scenario with actve contributon of students was induced in the classroom and diferent reasoning models were exposed to increase the successful transfer of queston solving skills (conceptual and algorithmic, LOCS and HOCS).

Of-class actvites
Actvites outside of classroom have taken on special importance in the new context of undergraduate educaton. Learning outcome, student satsfacton, and used tme depends on both the type of actvity and the format (paper documents or computer assisted). So, the use of computer provided best results for openquestons or actvites of informaton seeking, but paper-documents were more useful for problem solving. Then, a design of instructonal strategies, mainly supported by paper documents, was elaborated for the ofclass learning.
End-of-chapter questons and problems for selected general chemistry textbooks is a common strategy outside of classroom (Dávila & Talanquer, 2010). Recently, our educatonal research group has elaborated a specifc textbook for learning about mult-step problems and opened-questons, containing examples of applied solving strategies, illustratng diferent approaches and difculty grades. Also, some notes are included for understanding conceptual terms involved in the problem and emphasizing the algorithm contributon. This collecton of questons allowed from self-learning (individual student) to collaboratve learning (self-initated study groups).

Of-class group actvites
Induced tasks and collaboratve actvites were performed outside of classroom. In the frst acton, students were engaged in pairs into some out-class actvites for solving of practce questons. Two peers working together forced them to maintain some agreement and to reach eventually a shared soluton. Students controlled their own learning process and partcipated in the learning process of other students (peer-learning).
In the second opton, small study groups were formed without partcipaton of instructors for studying purposes. The group size, group roles, actvites, resources, and temporal distributon were self-regulated. The students had to think how to resolve the problem by actve learning techniques and they were all are colearners.
In absence of instructors, the potental relatonships between modes of reasoning and levels of cognitve processing can change. For instance, solving strategies were strongly dependent on students' prior knowledge (experience of what it was important for success). This fact afected both their comprehension of core chemistry concepts and the development of mental models to make predictons and build explanatons related Vol. 4(4), 2014, pp 255 to the selecton of solving strategy. Christan and Talanquer (2012) identfed dominant modes of reasoning expressed by college chemistry students while working in self-initated study groups. The group talk was largely focused on issues invoking rule-based reasoning. Also, there were some discussions about chemical reactons heavily relied on case-based reasoning, but model-based reasoning was minimally applied.

Group tutorials and specifc seminars
Two kind of small group actvites were developed controlled by instructor. The frst acton was based on classical tutorials where a group of students with similar difcultes partcipated in an adapted teachinglearning session. These actvites were performed in a small room or the ofce of instructor. The second acton was the development of specifc seminars designed with the aim of discussing about the best procedure to solve, in planned sessions. The number of students in these classes is lower than 25 and organized in small groups (2-4 students). In previous courses, the teachers had an important role controlling the discussion dynamics, helping them in the group reasoning, and fnally, solving the queston. This adaptaton of problembased learning has been well accepted by most of the students enrolled in this subject from course 2002. However, some students had a passive attude waitng untl the fnal solving.
The innovaton approach proposed in this study was based on an increase of collaboraton between students. The discussion and the applicaton of algorithm to reach by consensus were directed by students. They expressed more freely their reasoning (MBR, CBR, and RBR) and the capacity to implement their knowledge for solving the queston from the understanding statement untl the achievement of the fnal result. The teacher just partcipated for emphasizing theoretcal concepts or practcal skills related with the problem (conceptual or algorithm) or for the detecton of misconceptons. These sessions were more educatonal than informatve.

Impact of the actons
Learning achievements were analyzed by comparing those obtained by the partcipants (higher contributon of high-level collaboratve actvites) and by the rest of students enrolled in the subject. First, the learning of the skills related to solve problems was observed during seminars. On average, the number and the complexity of questons to the teacher as well as the tme needed to solve a problem were reduced. Second, the collected material during the course (exams, actvites, seminars) was analyzed by the evaluaton rubric. No signifcant diferences were observed respect to the reasoning models performed by students during the solving of chemical questons. However, the total number of errors was reduced respect to the previous courses (up to 10 %). The higher decrease corresponded to the chemical errors, indicatng that a deeper knowledge was induced. Third, the marks for mult-step questons of the exams were analyzed. On average, partcipants in this study had slightly higher grade point averages (a mean increment about 10%) than the average for all of the students in the course. From the teacher's point of view, the proposed change involved similar eforts for preparing materials. The innovaton was centered in the teaching style, more inquisitory lectures and more actvites controlled by the students (collaboratve actvites). Hence, the general conclusion was that the teaching adaptaton to high-level collaboratve actons between students showed positve consequences in two year-study. However, more data are required to assure the statstcal signifcance of these results, discriminatng the efect of teaching-learning acton and the inter-variaton of student groups. For instance, the improvement in student achievement was signifcantly dependent on the previous chemical and mathematcs level and motvaton. Additonally, the previous learning experiences conditoned the proposed teaching acton. It was difcult to change the reasoning strategy, because those students were used to solve questons following a predetermined mode (efectve or not).

Student percepton
Afer a short explanaton of the educatonal research concepts, the student opinion survey based on a questonnaire was performed, including some Likert-scale questons and an open-ended queston. The main results are shown Figure 2. In the frst queston, the students were auto-classifed on cognitve development spectrum solving the mult-step problems. They had to decide from a learner focused on the mathematcal Vol. 4(4), 2014, pp 256 solving to a learner focused on the conceptual aspects. An important part of asked students (44%) considered themselves as high or medium algorithmic learners. Also, the percentage of high or medium conceptual learners was relevant (24%). Despite of the teaching eforts, these results confrmed the importance of a shif of chemistry instructon to a more conceptual orientaton in order to balance the learning of chemistry, but teachers cannot forget that some students show defciencies in algorithmic components of this subject. In the following questons, the incidence of diferent actvites or resources available in the teaching-learning scenario was evaluated. According to the opinion of partcipants, low-collaboratve actvites helped them to learning solving strategies. The survey data surprisingly indicated than inquisitory lectures (14.5%) had a comparatve contributon than expository lectures (14.4%), even with an increment of student partcipaton and an induced peer-learning. On the other hand, the textbook specifcally designed for self-learning and collaboratve actons (13.8%) showed a higher impact than other textbooks or internet resources (4.4% and 7.1%, respectvely). With respect to high-collaboratve actvites, induced tasks (8.4%) were apparently less efcient than collaboratve actons carried out by self-initated study groups (15.2%). Nevertheless, innovatve specifc seminars (14.4%) were beter accepted by students than classical group tutorials (7.9%). In the open-ended queston, the students positvely evaluated the methodology used for problem solving and they also proposed changes, for example that the teacher should provide more solved problems. Chemistry teachers utlize diferent tools to build and communicate the knowledge and the skills required for engineering students. Mult-step questons are an excellent platorm for showing how chemical concepts are useful for solving scientfc-technical problems. Also, these questons induce more robust reasoning models and, consequently, are able to be applied in novel situatons. For that, the algorithmic-type learning is a common practce. Unfortunately, such memorizaton techniques promote an approach that hinders meaningful learning and true understanding. The results of this paper have showed that collaboratve actvites are a positve acton to support deep knowledge and high-level thinking skills. A modifcaton of instructon has been analyzed indicatng a mode to do it, inside and outside of classroom, and the expected outcomes and difcultes. It is worth mentoning that the proposed changes involve a small adaptaton of common lectures, seminars, and resources. However, the actve partcipaton of students leads to a more efectve teaching and peer-learning, correctng misconceptons, and reinforcing theoretcal concepts. Collaboratve actvites make easy the problem-solving process because students discuss about the relevant informaton from the problem statement, the analysis of chemical concepts, and the selecton of the best algorithm toward the answer. These steps are partcularly important in determining the success or failure to obtain the correct numerical answer, including its units. Data collected by observaton and survey (questonnaire) indicated that the applied methodology was satsfactorily accepted by students.