Chapter 5 The Domain in MIRA: Algebra Word Problems

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Chapter 5. The Domain in MIRA: Algebra Word Problems 902. Read the problem description and identify the givens of the problem. Create a variable foreach given and assign the known value to this variable.3. Identify the unknowns that are explicitly described in the problem statement and definevariables to represent them.4. Select from the problem description the sentences that define relationships between two ormore elements.5. Rewrite each relation substituting the parts of the sentence that refer to the components bytheir variables names. If no previously identified component matches any of the componentsin the relation, identify its type (givens, goals, or unknowns) and create an appropriatevariable for it.6. Analyse the mathematical meaning of the relation and decide which arithmetic operatorsshould be applied (+, -, * , /, =).7. Translate the relation into an equation, using the variables and the operators organized inthe form that best matches the meaning of the original sentence.8. Read each equation produced (in more than one way if possible) to verify that it translatesback to the original sentence.9. Check carefully if any equation has been omitted and if all variables previously identifiedhave been used in the equations.5.6 Designing a Tool to Support Word Problems TranslationIn order to enable students to use the heuristic proposed, we have designed a tool to support thetranslation of word problems into equations. As explained in the previous sections, one of thehardest problems in translating algebra problems into equations is comprehending the situationand constructing representations of it. We would like to develop an instructional approach thatgives students control over their process while helping them to build this representation.The PAL Tool was designed to meet this requirement. It is the tool in MIRA that helps studentsto organize and represent the problem into its smaller components (the givens, the goals,the unknowns and relations). The PAL Tool implements a specific strategy for translating algebraword problems. This strategy is in line with strategies proposed by Polya (1945), Mayer (1999)and many algebra courses (e.g. the Math Forum 2 ). It motivates the students to look for specificelements in an algebra problem, labelling these elements as givens, goals, and unknowns. Also, ithelps in making the connections between the relations and the elements previously identified. Itprovides an interface for describing the components of the problem, creating variable names, identifyingthe relations and the components that are involved in each relation. It focuses exclusivelyon the translation stage of algebra problem solving and is intended to provide limited help on thetask, letting students figure out the equations on their own.5.6.1 Requirements of the PAL ToolThe conditions or requirements the PAL tool had to meet in its design to achieve the goals were:1. Help students to isolate and categorize parts of the problem;2 The Math Forum is an Internet web site for mathematics. http://mathforum.org/
Chapter 5. The Domain in MIRA: Algebra Word Problems 912. Make students think about implicit information (sub-goals, unknowns, etc.);3. Help students to think about the relations in the problem and their conditions;4. Assist students in making explicit the relations identified;5. Help students to create a basic notation for thinking about the problem;6. Provide input information to the metacognitive engine that is used to infer about students’understanding of the problem;5.7 A Pilot Study with the PAL ToolPrior to the implementation of the PAL Tool a low fidelity prototype built on paper (Preece et al.,2002) with the features being considered for the final software was designed to inform the finaldesign. A summary of the pilot study using this prototype and its results were published in Gama(2002).The conceptual design of the prototype was simple, with some tables in which each line stoodfor an element of the problem. Figure 5.2 shows this initial mock-up of the tool.The buttons on the paper served to trigger help on the task. There were two types of help:Show Hint and Give me One. The hint consisted of a multiple choice question with three possibleanswers about the givens, goals, unknowns, or parts related. When students asked for a hint, aquestion with the possible answers was shown, but no feedback was provided regarding whichanswer was the correct one. They had to figure it out by themselves. The give me one button, onthe other hand, provided complete information about one part of the problem (givens, goals, etc.).For example, if the student placed a rubber (this was the way to indicate the action of “clicking on abutton”) on top of the give me one button next to the “goal table” the experimenter would producea piece of paper containing one goal description, together with an identifier and the correspondingunit (i.e. a complete row of the goal table).The “fade-out” function was provided as a pen that students used to cross out irrelevant information.Figure 5.3 shows a participant using the paper version of the Pal tool - notice the rubber,pen for “fade-out” functionality and pieces of paper with hint messages.The conceptual model of the low-fidelity prototype aimed at helping learners to perform thefollowing activities:1. find and label relevant elements of the problem.2. classify the elements in three distinct categories: givens, goals, and unknowns.3. write down the relationships identified in the problem, with the parts related (givens, goals,unknowns).4. write down any assumption they made to solve the problem.On a separate piece of paper the students had to write down the equations and solve the problem.Some students were asked to try to write down the equations before using the Pal Tool. Aftera few minutes, they were given the PAL Tool sheet to organize the problem and then go back tothe equations. Others were asked to first organize the problem using the Pal Tool sheet and thentry to write down the equations.
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Chapter 5 The Domain in MIRA: Algebra Word Problems

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