A. Introduction
All teachers are likely to agree that they not only want their students to understand information that is presented to them but also want them to understand what they are doing when they are solving problems or conducting an experiment. In helping the students develop any real understanding of the subject taught, the teachers will have to teach them how to think, how to make decision, and how to learn from their experiences both in and out of school. To do this, the teachers need to select the teaching strategies carefully so that they suit both their students and the subject matter they are teaching.
Jones & Idol (1990) stated that such strategies or approaches have in common an understanding that learners construct knowledge by making connections between their prior knowledge and new information, and analyze new learning activities to determine the most effective approach to achieve learning goals. Cognitive instruction seeks to facilitate learning by making students aware of their own mental processes and by providing direct instruction in thinking and learning strategies.
Nevertheless, the teachers may not find it easy to decide what teaching strategy to use for a particular lesson because the number of factors that should be considered is quite large. Having content knowledge is simply not enough to enable the teachers to teach effectively. The teachers must know how to engage the students in the learning process and know how to organize information so that others can learn it. How students learn is as important as what students learn because their learning experiences directly influence their willingness to engage in future learning and their future learning strategies. As a result, the requirements of the learners, the learning context, and the content must guide the teachers` selection of teaching strategies (Killen: vii-x).
In this paper, we, in group work, will present one of the very common teaching strategies, that is, problem solving. The discussion of this paper comprises introduction, theoretical framework of the strategy, the anticipation of the strategy, the strengths and weaknesses, and the implementation of problem solving as a teaching strategy in classroom activity.
B. Theoretical Frameworks
Many current views of problem solving, such as described in Keith Holyoak and Robert Morrison's Cambridge Handbook of Thinking and Reasoning (2005) or Marsha Lovett's 2002 review of research on problem solving, have their roots in Gestalt theory or information processing theory.
Gestalt Theory. The Gestalt theory of problem solving, described by Karl Duncker (1945) and Max Wertheimer (1959), holds that problem solving occurs with a flash of insight. Richard Mayer (1995) noted that insight occurs when a problem solver moves from a state of not knowing how to solve a problem to knowing how to solve a problem. During insight, problem solvers devise a way of representing the problem that enables solution. Gestalt psychologists offered several ways of conceptualizing what happens during insight: insight involves building a schema in which all the parts fit together, insight involves suddenly reorganizing the visual information so it fits together to solve the problem, insight involves restating a problem's givens or problem goal in a new way that makes the problem easier to solve, insight involves removing mental blocks, and insight involves finding a problem analog (i.e., a similar problem that the problem solver already knows how to solve). Gestalt theory informs educational programs aimed at teaching students how to represent problems.
Information Processing Theory. The information processing theory of problem solving, as described by Allen Newell and Herbert Simon (1972), is based on a human computer metaphor in which problem solving involves carrying out a series of mental computations on mental representations. The key components in the theory are as follows: the idea that a problem can be represented as a problem space—a representation of the initial state, goal state, and all possible intervening states—and search heuristics—a strategy for moving through the problem space from one state of the problem to the next. The problem begins in the given state; the problem solver applies an operator that generates a new state, and so on until the goal state is reached. For example, a common search heuristic is means-ends analysis, in which the problem solver seeks to apply an operator that will satisfy the problem-solver's current goal; if there is a constraint that blocks the application of the operator, and then a goal is set to remove the constraint, and so on. Information processing theory informs educational programs aimed at teaching strategies for solving problems.
1. The Definition of Problem and Problem Solving
A problem can be defined as any situation in which some information is known and other information is needed. The problem might be something that gives rise to doubt or uncertainty, or something that is hard to understand, or a difficult task or question, or inquiry that start from given condition to investigate facts or principles. Awhile, problem solving can be considered as the process of applying existing knowledge to a new or unfamiliar situation in order to gain new knowledge. According to Mayer and Wittrock, problem solving is “cognitive processing directed at achieving a goal when no solution method is obvious to the problem solver” (2006, p. 287). Thus problems solving is a form of inquiry learning. When used as a deliberate teaching strategy, problem solving can help students to release that the knowledge they have already gained can be applied to a new situations, and that this process can lead them to gain new knowledge.
2. The Distinction between Teaching Problem Solving and Problem Solving as a Teaching Strategy
It is important to distinguish between teaching problem solving and using problem solving as a teaching strategy. Teaching problem solving is exactly that-teaching students how to solve problems. Meanwhile, problem solving as a teaching strategy is a technique in which problems are used deliberately as a means of helping students to understand or gain insight into the subject they are learning. So, in the first thing the emphasis is on solving problems and the latter the emphasis is on learning about the subject through solving problems.
Quite clearly, the two concepts are closely related, and it is probably best to think of problem solving as a teaching strategy as being the broader concept that includes teaching problem solving. When problem solving is used as a teaching strategy, the emphasis should be on students learning about the subject, rather than simply learning to solve problems. The reason why this point is being stressed is that if we, teachers, simply teach students how to solve problems they may learn very little other than the sequence of steps they need to follow in order to solve a particular type of problem. There is considerable research evidence that students who are taught to solve problem with a “means-ends” approach can do so “with very little knowledge acquisition” (Owen&Sweller, !985, p.273).
There is some evidence to suggest that problem solving depends very heavily on domain-specific knowledge. It means that the ability to solve problems in a particular area is so dependent on specific knowledge relevant to that area that one would expect minimal, if any, transfer of problem-solving ability from one area to another. (Schuell, 1990)
In many classrooms, especially those of developing countries, the problem solving process may not be feasible because of factors beyond teachers’ control. However, this does not mean that learners in such countries should be disadvantaged from learning or acquiring skills that the problem-solving process offers. Learners can still attain such skills if teachers employ the problem-solving teaching approach. In this kind of teaching, teachers’ techniques, especially their questioning styles, can help learners develop critical and logical thinking, as well as the ability to generalize.
Helping students develop problem solving skills is a frequently cited goal of science educators. The National Science Teachers Association (NSTA), in its 1980 position statement, advocated that science teachers help students learn and think logically, specifying that "...high school laboratory and field activities should emphasize not only the acquisition of knowledge, but also problem solving and decision making" (1985:48).
Problem solving means many things to many people. For some, it includes an attitude or predisposition toward inquiry as well as the actual processes by which individuals attempt to gain knowledge. Usually, when teachers discuss problem solving on the part of pupils, they anticipate pupils will become involved with the thinking operations of analysis, synthesis, and evaluation (considered as higher-level thinking skills). The American College Testing program has redesigned its college admissions test with a new emphasis on assessing higher-order thinking skills (EdLine,1989).
3. When Problem Solving as a Teaching Strategy
Teaching and learning strategies are the purposeful actions and thoughts learners engage in for understanding, storing, and remembering new information and skills. Some learning strategies are observable, as in note-taking or writing a plan for problem solution. Many learning strategies, however, are non observable because they are purely mental processes. Examples of non-observable strategies are monitoring comprehension or activating prior knowledge. Since learning strategies can be used with any learning task, including mathematics and language tasks, they have considerable potential for enhancing the academic achievement of linguistic minority students.
Three types of learning strategies are commonly discussed in the literature: metacognitive strategies, or the executive strategies that individuals use to plan for, monitor, or evaluate learning; cognitive strategies, the actual manipulation of learning materials by reorganization and grouping, elaboration or relating one new idea to another and relating new ideas to existing knowledge; and social-affective strategies, in which the learner calls on another person for assistance or works cooperatively with others on a common task (Pressley & Harris, 1990).
An important reason for having students solve problems is to help them gain insight that will enable them to understand the subject matter better and to look at it from different perspectives and so problem solving is an ideal strategy to use in relation to group work. To achieve the benefits of group work, the teachers need to encourage students to discuss with one another what they understand the problem to be, how they feel about the problem, and the various steps in their attempts to solve the problem. In short, the most appropriate times for the teachers to use problem solving as a teaching strategy are as follows:
• When the teachers want their students to gain a deep understanding of the subject matter, rather than just remembering bits of it.
• When the teachers want to develop their students` thinking and reasoning skills-that is, their ability to analyze situations, to apply their existing knowledge to new situations etc.
• When the teachers want to develop their students` problem solving skill- that is, their ability to assess and respond to new situations.
• When the teachers want to challenge their students intellectually.
• When the teachers want to encourage their students to take greater responsibility for their own learning.
• When the teachers want students to understand the relationship between what they are studying and the real world (the relationship between theory and practice)
• When the teachers want students` learning experiences to be varied and interesting.
C. Anticipation of problem
To use problem solving effectively as a teaching strategy you will need to develop certain teaching skills, just as the students will need to develop the thinking skills described previously. Fullerton (1992) suggests that you will also need to be able to engage in the following activities.
1. Formulating the problem
You must help the students to examine the problem from a number of perspectives so that they will understand exactly what the problem is.
2. Analyzing the problem
Before students try to solve a problem, you need to get them to break down the problem into its various components and assess the importance of each component.
3. Generating Ideas
When you present students with open-ended problems (rather than problems that have a specific answer), one of your major tasks will be to help students generate ideas or data that can be used for solving the problem. To do this you might :
• Help students to relate the problem situation to real situation
• Help students to develop or explore analogies that will lead them to creative solutions for the problem.
4. Evaluating ideas
When solving open-ended problems, students may have difficulty deciding on the relative merits of various ways of approaching the problem and/or the merits of various suggested solutions. You can help the students to select techniques for solving a problem, and to judge the merits of alternative solutions, by giving them specific criteria for making judgments. You could :
• Have students list all the available alternative, examining the strengths and weaknesses of each, and then selecting the one that fulfils the problems objectives best.
• Have students generate a list of ideas and evaluate each idea against explicit criteria, with all students having equal input into the selection of the final idea.
In solving any problems, it helps to have a working procedure. We might want to consider this four-step procedure.
Understand : before you can solve a problem you must first understand it. Read and re-read the problem carefully to find all the clues and determine what the question is asking you to find.
What is the known ?
What are the data ?
What is the condition ?
Plan : one you understand the question and the clues, it’s time to use your previous experience with similar problems to look for strategies and tools to answer the question.
Do you know a related problem ?
Try it : after deciding on a plan, you should try it and see what answer you come up with.
Can you see clearly that the step is correct.
Look back (reflect ) : once you tried it and found an answer.
Check result, argument, result differently and a glance.
When we use problem solving as the principal teaching strategy in a lesson, we are passing to students some of the responsibility for their own learning. If we have planned carefully, you will have created a situation in which the students can learn about the subject by solving the problem(s) that you have posed.
If the students work on problems individually, the teacher should expect them to:
Satisfy the students that they know what the real problem is before they try to solve it
Ask question if they are not clear about any points
Develop a plan for solving the problem before they jump into calculations or experimentation or research
Keep a written record of how they attempted to solve the problem and the result they obtained
Make brief notes of what they learned by solving the problem
If the students work in groups or pairs to solve problems, we should expect that they will do the things listed above and it addition:
Explain to one another what they understand about the problem and how it might be solved;
Co-operate with one another to complete the tasks and maximize their learning;
Explain their solutions to one another and show interest is one another’s explanations; and
Persist until they agree that they have solved the problem satisfactory.
If the students work on the problem as a whole-class activity, you should expect that they will:
Explain what they understand by the problem and how they attempted to solve it.
Listen to and try to understand, the ideas and explanations of other students;
Pose questions and ask for clarifications;
Comment on the ideas and solution offered by other students; and
Help to draw conclusions from the problem-solving experience.
D. Strengths and Weaknesses:
1. Strengths
When used effectively, problem solving has a new number of advantages over direct instruction. In summary, these advantages are as follows:
Problem solving provides a challenge for students, and they can derive great satisfactions from discovering new knowledge for themselves.
Problem solving engages students actively in learning.
Problem solving helps students to develop new knowledge for themselves and feel responsible for their own learning.
Problem solving teach students that their solutions should be explainable and just justifiable.
Problem solving can show students that our subject is a way of thinking and doing things that should make sense to students, not just to the teacher or the textbook author.
Problem solving can be fun and rewarding learning experience.
Problem solving can develop students` critical thinking skills and their ability to adapt to new learning situations.
Problem solving can help to keep alive students` natural curiosity.
Problem solving encourages students to talk about the concepts they are trying to understand.
Problem solving develops students` ability to make informed judgements.
Problem solving provides students with an opportunity to apply their knowledge and to see that their knowledge has some real world applications.
Real-life problems can help students to integrate the knowledge they gain fro studying several different subjects.
Problem solving can help students to develop qualities such as resourcefulness, independence, patience and tenacity.
Problem solving can show students the value of exploring alternative approaches, in contrast to finding the answer as quickly as possible.
Appropriately challenging problems can engage students in learning long after the formal lesson is over.
Problem solving helps increase retention and provides a sound foundation from which students can transfer their knowledge to other situations.
Problem solving can create three conditions that assist in subsequent retrieval and appropriate use of new information: activation of prior knowledge, similarity between the contexts in which information is learned and later applied, and opportunity to elaborate on that information.
Problem solving can help students to develop the confidence to try things even when they think they might make a mistake, and to keep trying when they do make mistakes.
Problem solving can give the teachers a better understanding of the abilities and special talents of their students.
2. Weaknesses
There are many occasions when we are teaching maybe more effective or efficient if we use techniques other than problem solving because it has the following limitation:
Unless the problems are motivating, students may see them simply as busywork.
Unless students are interested and believe that they can solve the problem, they may be reluctant to try
Appropriate problems take time to develop since each problem needs to be carefully structured to produce specific student learning outcomes.
Unless our students understand why they are attempting to solve a particular problem, they may not learn what we want them to learn.
E. Implementation
The implementation of problem solving strategy can be seen from using it through studying mathematics. At this studying, students should be able to solve problems encountered in the real world and to reason and talk about their solutions. In this approach to mathematics instruction, language plays a much larger role than has been the case in traditional computation-based programs. For example, students need good comprehension skills (both oral and reading) in order to understand a problem, and they also need good speaking skills in order to discuss the problem and explain their path to solution. Writing skills are also needed in mathematics if students are to write about their problem solutions (Dossey, 1989). The strategic approach generally recommended for mathematics problem solving is based on Polya’s model, which consists of understanding the problem, devising a plan, carrying out the plan, and looking back (Polya, 1957, 1973 in Chamot, A.U., & O'Malley, J.M. 1993). In addition to language skills, mathematics problem solving also requires a strategic approach to understanding and representing the problem, and making and carrying out a plan for its solution.
The important of following the problem solving steps of the Polya model in sequence is a highly effective approach to solving word problems (Pressley & associates, 1990). Specific learning strategies appropriate for each step of the model include elaboration of prior knowledge, selective attention or focusing on important information, evaluating the plan, and representing the problem pictorially (Chamot & O’Malley, 1993). Cooperation, or solving problems in small groups, is another strategy that has a positive effect on problem solving and on helping students develop metacognitive awareness of their own mathematical thinking. Benefits of solving problems cooperatively include sharing strategies, communicating mathematically, and developing skills needed for independent learning.
In addition to following specific problem solving steps, effective problem solvers maintain a reflective view of their own problem solving processes. They analyze related information, look for possible solutions, and check the accuracy of alternative solutions.
Such metacognitive knowledge and executive control over problem solving provides the student with flexible and autonomous control over the learning process. In addition to metacognitive control, an effective problem solver will brainstorm a variety of alternative plans or solution strategies, activate what has already been learned, try the plan out with the current problem, and evaluate its application to the solution. The student’s evaluation of the plan must be performed with respect to the original problem representation rather than to the simple computational procedures used to find the answer (Noddings, Gilbert-MacMillan, &Leitz, 1983).
The purpose of the representation step is to help students organize the data, define the problem and identify key issues. In this phase, we might ask students to:
frame the problem in their own words
define key terms and concepts
determine statements that accurately represent the givens of a problem
identify analogous problems
determine what information is needed to solve the problem
In the solution phase, one develops and then implements a coherent plan for solving the problem. As we help students with this phase, we might ask them to:
identify the general model or procedure they have in mind for solving the problem
set sub-goals for solving the problem
identify necessary operations and steps
draw conclusions
carry out necessary operations
In all cases, the more we get the students to articulate their own understandings of the problem and potential solutions, the more we can help them develop their expertise in approaching problems in our discipline.
A few different ways we can introduce a problem to students include:
demonstrate a problem solution by systematically explaining each step and its rationale
ask the students how they would approach solving the problem
ask the students to help we solve the problem by posing questions at key points in the process
have the students work together in small groups (3 to 5 students) to solve the problem, and then have the solution presented to the rest of the class (either by us or by a student in the group)
Students in problem-solving classes indicate that they appreciate:
clear step-by-step explanations
questions posed at key points in the solution
overviews and summaries of problem-solving approaches
opportunities to ask questions about problems and solutions
explicit links to similar problems
opportunities to interact with each other and with the instructor while working towards solving problems.
To make the students to become better at problem solving, the teacher needs to give them appropriate practice by doing the following activities:
a) Deciding what the teacher wants students to learn.
b) Selecting or developing appropriate problems.
c) Presenting the problems to students in a stimulating way so that they are encouraged to try to solve them.
d) Monitoring the students` progress and encouraging them to think about their problem-solving processes.
e) Helping students to understand how problem solving is helping them to learn about the subject they are studying.
Teachers` Activities Developed during Problem Solving Lesson
To use problem solving effectively as a teaching strategy the teacher will need to develop certain teaching skills by engaging in the following activities ( Killen, :
a) Formulating the problem by highlighting the parameters of the problem and helping the students view the problem from new perspectives.
b) Analyzing the problem by assisting students to identify the major elements in a problem and assisting students to identify how the components of a problem interact.
c) Generating ideas by helping students to relate the problem situation to real situations with which they are familiar and helping students to develop or explore analogies that will lead them to creative solutions for the problem.
d) Evaluating ideas by having students list all the available alternatives and having students generate a list of ideas and evaluate each idea against explicit criteria.
F. Conclusion
A problem can be defined as any situation in which some information is known and other information is needed.
Problem solving can be considered as the process of applying existing knowledge to a new or unfamiliar situation in order to gain new knowledge.
Thus problems solving is a form of inquiry learning
Learners can still attain such skills if teachers employ the problem-solving teaching approach
In this kind of teaching, teachers’ techniques, especially their questioning styles, can help learners develop critical and logical thinking, as well as the ability to generalize.
When teachers discuss problem solving on the part of pupils, they anticipate pupils will become involved with the thinking operations of analysis, synthesis, and evaluation
The American College Testing program has redesigned its college admissions test with a new emphasis on assessing higher-order thinking skills (EdLine, 1989)
The purpose of the representation step is to help students organize the data, define the problem and identify key issues
In all cases, the more we get the students to articulate their own understandings of the problem and potential solutions, the more we can help them develop their expertise in approaching problems in our discipline.
When we use problem solving as the principle teaching strategy in a lesson, we are passing to students some of the responsibility for their own learning.
When used effectively, problem solving has a new number of advantages over direct instruction
Appropriate problems take time to develop since each problem needs to be carefully structured to produce specific student learning outcomes.
References
Chamot, A.U., & O'Malley, J.M. 1993. The CALLA handbook: How to implement the Cognitive Academic Language Learning Approach. Reading, MA: Addison-Wesley.
Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5), Whole No. 270.
Jones, B.F., & Idol, L. 1990 Introduction. In B.F. Jones & L. Idol (Eds.), Dimensions of thinking and cognitive instruction (pp. 1-13). Hillsdale, NJ: Lawrence Erlbaum.
Holyoak,, K. J., & Morrison, R. G. (Eds.). (2005). The Cambridge handbook of thinking and reasoning. New York: Cambridge University Press.
Mayer, R. E., & Wittrock, R. C. (2006). Problem solving. In P. A. Alexander & P. H. Winne (Eds.), Handbook of educational psychology(2nd ed., pp. 287–304). Mahwah, NJ: Erlbaum.
Newell, A., & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice Hall.
Noddings, N., Gilbert-MacMillan, K., & Leitz, 5. 1983, April. What do individuals gain in small group mathematical problem solving? Paper presented at the annual meetings of educational Research Association, Montreal.
Pressley, M. & Associates. 1990. Cognitive strategy instruction that really improves children’s academic performance. Cambridge, MA: Brookline Books.
APPENDIX A
WORD PROBLEM PROCEDURE
The CALLA Mathematics curriculum has been supplemented with a Word Problem Procedure (WPP) which Spanos (Arlington Public Schools,, 1991) developed for use in his high school CALLA math class. This procedure is in the form of a one-page worksheet and contains the following 11 steps:
1. Choose a partner or partners. Write your names above.
2. Choose a problem. Write the problem in the space below.
3. Select one student to read the problem aloud. Re-read the problem if necessary. Together, discuss the vocabulary in the problem and circle any words you don’t understand. Write the words below.
4. Find the meanings of the words by using a bilingual dictionary, asking your partner(s), or asking your teacher.
5. Write what the problem asks you to find below.
6. Write the operation(s) you need to solve the problem below:
Add? Subtract? Multiply? Divide?
7. Solve the problem in the space below.
8. Check your answer.
9. Explain your answer to your partner(s). Write your explanation below.
10. Explain your answer to the class.
11. Write a similar problem on the back of this page.
Steps 1-4 correspond to Step 1 of Polya’s method (Understanding the Problem), Steps 5-6 correspond to Step 2 of Polya’s method (Devising a Plan), Step 7 to Polyas Step 3 (Carrying out the Plan), and Steps 8-11 to Polyas Step 4 (Looking Back). The steps in the procedure are analyzed in the curriculum guide in terms of the learning strategies that are involved, e.g.,
Steps 1-6 are planning steps and therefore invite metacognitive strategies, while Steps 7 and 8 require students to manipulate the mathematical content and therefore involve cognitive strategies. The entire Word Problem Procedure calls on students to use social affective strategies by asking for clarification and working cooperatively on the problem solution. Finally, students use academic language related to mathematics as they read, discuss, write individual explanations of how the problem was solved, and explain to their partners and the class how they arrived at the answer. In this way, the WPP embodies the CALLA philosophy of integrating content, academic language, and learning strategies.
APPENDIX B
INTERVIEW GUIDE FOR
THINK-ALOUD AND RESTROSPECTIVE
INTERVIEWS
STUDENT INTERVIEW GUIDE
Part I: Warm-Up - Background Information (2 minutes)
1. Introduce self and ask student’s name.
2. What is your native language?
3. What country are you from?
4. How long have you been in the U.S.?
5. Did you go to school in your country? What grade did you finish?
6. Explain purpose of interview.
Part II: Think-Aloud Warm-up (3 minutes)
1. Here’s an easy problem. Can you tell me what you are thinking while you solve it?
Student is given a sheet of paper with this problem:
53
-28
2. (If student needs more practice with thinking aloud). Do this problem, too:
15
x3
If student falls silent while working, say TELL ME WHAT YOU’RE THINKING.
Part III: Word Problem Think-Aloud (5 minutes)
1. Start with Word Problem 1. If the student is completely unable to handle it, use Word Problem 2. Say, “Okay, why don’t we try this problem instead.”
2. Script: “I’d like you to try to solve this problem.” (Hand Word Problem 1 to student.) “Here’s a pencil and a calculator, if you want to use it. You can write or put anything on this paper to help you solve the problem. Why don’t you read the problem aloud first, and then talk aloud while you solve the problem.”
Say TELL ME WHAT YOU’RE THINKING. WHAT STRATEGIES ARE YOU GOING TO USE? If student doesn’t understand, say JUST TELL ME WHAT YOU’RE THINKING. WHAT ARE YOU GOING TO DO?
If, in spite of prompting, student does not want to talk while solving the problem, ask after he or she has completed the problem: “How did you get that answer? Can you tell me what you did?” Have the student go back through each step and describe what he or she did.
WORD PROBLEM 1
Carlos and Gloria work at McDonald’s at 4238 Wilson Boulevard. Carlos works 8 hours per week and gets $4.50 per hour. Gloria works 6 hours per week and gets $6.15 per hour. Who works more? Who gets more money per week? How much more?
WORD PROBLEM 2 (ALTERNATE)
In 1989, there were 36 students in the HILT math class. In 1990, there were 27 students. Which year had more students? How many more students were there in that year?
Part IV: Learning Strategy Discussion (5 minutes)
Say WHAT STRATEGIES DID YOU USE? if student does not understand, ask WHAT DID YOU DO TO HELP YOU SOLVE THE PROBLEM? if student is not able to generate answers, ask the following questions in sequence:
1. “How did you feel about solving the problem?” (If necessary, ‘Were you nervous? Interested?)
2. “Have you solved other problems like this one?” (if yes, “Did you remember how you did another problem to help with this one?”)
3. “Did you understand the problem right away?” (If no, “What did you do about it?”)
4. “Did you make a plan of what to do?” (If yes, “Tell me about your plan.”)
5. “Did you look for important words to solve the problem?” (If yes, “What were they?”)
6. “Were there any words you didn’t understand?” (If yes,
“What were they? Could you solve the problem anyway without those words?”)
7. “How did you decide which numbers to use?”
8. “Did you cross out, or not use, information that you didn’t need?” (If yes, “What was it? What didn’t you need?”)
9. “How did you decide which operation to use?”
10. “Did you make a picture in your head or draw a picture or table?” (If yes, “Can you show/tell me about it?”)
11. “Did you check your answer? (If yes, “How did you do that?”)
THANK YOU FOR YOUR WORK. YOU DID A REALLY GOOD JOB.
APPENDIX C
CALLA TEACHER QUESTIONNAIRE
Teacher Name ____________________Date____________________
1. Check which of the following math strands you have covered this year:
_______ Whole Numbers
_______ Fractions
_______ Time & Money
_______ Ratio & Percent
_______ Geometry
_______ Decimals
_______ Graphs, Charts, Statistics, &
_______ Probability
_______ Other (Describe_______________________________ )
2. Check the average amount of time spent on word problems:
_____ once a week _____ 4-5 times a week
_____ 2-3 times a week _____ part of every period
3. Do you think it is necessary to teach learning strategies directly (i.e., to name the strategy and tell why it is important)? __ Yes __ No
Why or why not?
_________________________________________________________
_________________________________________________________
4. Which learning strategies do you teach directly (D) or indirectly (I)?
__ D __ I Elaborating prior knowledge
__ D __ I Cooperation
__ D __ I Graphic Organizers
__ D __ I Classifying/grouping
__ D __ I Making inferences/predicting
__ D __ I Summarizing
__ D __ I Using images/visualizing
5. Do you teach problem-solving strategies? __ Yes __ No
6. If yes, check which strategies you use*:
__ Finding needed information
__ Finding extra information
__ Cooperative learning
__ Guessing & checking
__ Choosing operations
__ Making organized lists/tables
__ Drawing pictures/diagrams
__ Finding patterns
__ Writing simple problems
__ Solving simpler problems
__ Using logical reasoning
__ Working backward
__ Writing number sentences
7. Additional comments _____________________________________
________________________________________________________

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