To solve word problems, students must read the problem, decide what to do, solve the problem, and check that the answer is reasonable. The SOLVE IT! strategy is designed to help students “understand the mathematical problems, analyze the infor-mation presented, develop logical plans to solve problems, and evaluate their solu-tions” (Montague et al., 2000, p. 111). SOLVE IT! is a structured strategy that uses explicit instruction in problem solving steps. SOLVE IT! features a structured series of steps, each of which incorporates self-instructions and self-monitoring (i.e., Say, Ask, Check). Students are taught to carefully read problems, paraphrase the problem, ana-lyze the information, form a plan, solve the problem, and assess their solution. Figure 10.5 shows the steps in SOLVE IT! and the associated instructions and self-monitoring.
Schema-Based Strategies
The wide variety of possible word problems is one factor that makes them more diffi-cult for students. The ability to organize word problems into a small number of groups
FIGURE 10.4. Examples of computation strategies. Subtraction and addition strategies adapted from Frank, A. R., & Brown, D. (1992). Self-monitoring strategies in arithmetic. Teaching Excep-tional Children, 24(2), 52–53. Copyright 1992 by The Council for ExcepExcep-tional Children. Adapted by permission. Multiplication strategy reprinted from Reid, R. (1992). A brief multiplication strat-egy. Unpublished manuscript, University of Nebraska–Lincoln.
Subtraction Strategy
SUBTRACT. Remember the 4 B’s:
Begin? In the 1’s column.
Bigger? Which number is bigger?
Borrow? If bottom number is bigger I must borrow.
Basic Facts? Remember them. Use Touch Math if needed.
- Begin - Begin - Begin
- - - Bigger - - - Bigger - - - Bigger
- - - Borrow - - - Borrow - - - Borrow
- - - Basic Facts - - - Basic Facts - - - Basic Facts
8 7 6 6 2 3 5 6
– 3 9 8 – 1 5 – 3 5
Addition Strategy ADD. Remember SASH:
Start in the 1s column.
Add together the numerals in each column.
Should I carry a numeral?
Have I carried the correct numeral?
- Start - Start - Start
- - - Add - - - Add - - - Add
- - - Should I carry - - - Should I carry - - - Should I carry - - - Have I carried - - - Have I carried - - - Have I carried
1 2 7 3 5 7 6 4
+ 3 9 8 + 1 2 + 1 7
Multiplication Strategy MULTIPLY. Remember MAMA:
Multiply the 1’s column.
Across Do I need to go across to the 10’s?
Multiply the bottom 1’s digit with the top 10’s digit.
Add any number that was carried in Step 2.
- Multiply - Multiply - Multiply
- - - Across - - - Across - - - Across
- - - Multiply - - - Multiply - - - Multiply
- - - Add - - - Add - - - Add
1 7 3 5 6 4
× 8 × 3 × 7
FIGURE 10.5. The SOLVE IT! strategy. From Montague, M., Warger, C., & Morgan, T. (2000).
Solve It! Strategy instruction to improve mathematical problem solving. Learning Disabilities Research and Practice, 15, 110–116. Copyright 2000 by Blackwell Publishing Ltd. Reprinted by per-mission.
Read (for understanding)
Say: Read the problem. If I don’t understand, read it again.
Ask: Have I read and understood the problem?
Check: For understanding as I solve the problem.
Paraphrase (your own words)
Say: Underline the important information. Put the problem in my own words.
Ask: Have I underlined the important information? What is the question? What am I looking for?
Check: That the information goes with the question.
Visualize (a picture of a diagram)
Say: Make a drawing or a diagram.
Ask: Does the picture fit the problem?
Check: The picture against the problem information.
Hypothesize (a plan to solve the problem)
Say: Decide how many steps and operations are needed. Write the operation symbols (+, –, x, ÷).
Ask: If I do , what will I get? If I do , then what do I need to do next? How many steps are needed?
Check: That the plan makes sense.
Estimate (predict the answer)
Say: Round the numbers, do the problem in my head, and write the estimate.
Ask: Did I round up and down? Did I write the estimate?
Check: That I used important information.
Compute (do the arithmetic)
Say: Do the operations in the right order.
Ask: How does my answer compare with my estimate? Does my answer make sense? Are the decimals or money signs in the right place?
Check: That all the operations were done in the right order.
Check (make sure everything is right) Say: Check the computation.
Ask: Have I checked every step? Have I checked the computation? Is my answer right?
Check: That everything is right. If not, go back. Then ask for help if I need it.
with common characteristics that can then be represented and solved simplifies the difficulty of word problems greatly. Schema-based strategies approach word prob-lems from this perspective. Schemas are representations of word-problem structures.
Schemas “capture both the patterns of relationships as well as their linkages to opera-tions” (Marshall, 1995, p. 67). Thus, schema based approaches allow the student to both understand how to represent problems and identify the correct operations for solving them (Jitendra, DiPipi, & Perron-Jones, 2002). An advantage of schema-based ap-proaches is that when one piece of information is retrieved other information that is linked to it will also be activated (Jitendra et al., 2002; Marshall, 1995). Major types of problem schemas are “change, equalize, combine, compare, vary, and restate.” These are the most typical types of word problems in elementary and middle schools (Riley, Greeno, & Heller, 1983; Van de Walle, 1998). Figures 10.6 and 10.7 show examples of problem types for each schema.
Jitendra and her colleagues (Jitendra & Hoff, 1996; Jitendra et al., 1998, 1999, 2002) have developed and validated a schema-based approach to word-problem solving. The strategy requires students to learn the types of schemas to mastery and to match each schema with the appropriate diagram (developed by Marshall, 1998). The diagram serves to remind the student to record the important information and to cue the appro-priate arithmetic operation. The steps in the strategy are:
• Identify problem schemas: Students are taught the types of schemas to mastery and how to differentiate between them through the use of several examples.
• Create an appropriate diagram: Students are then taught how to appropriately dia-gram the different types of schemas (Figure 10.8). The diadia-grams serve as graphic orga-nizers that help students organize and remember important information.
• Flag the missing element with a question mark: The missing element, or the answer that the problem is requesting, is then flagged with a question mark. The question mark lets students know that they must use a mathematical operation to figure out the number to go in that box or circle (Figure 10.8).
• Apply the appropriate operation to solve the problem: The type of schema and dia-gram will dictate the operation to be used. Students will need to be taught which oper-ation goes with which type of schema and diagram (Figure 10.8).
• Ask if the answer made sense: Once students have solved the problems, they are to check to see if the answers make sense (e.g., if the operation is addition then the answer should be greater than both the addends). Students could use estimating to determine if their answers are reasonable.
• Check the work: Students should be taught to “work the problem backwards.”
Working a problem backwards requires students to do the opposite operation to deter-mine whether or not the answer is correct (e.g., subtraction→ addition, and multipli-cation→ division).
Note that all of these steps must be taught to a high degree of mastery. Figure 10.8 shows examples of how a graphic organizer could be used with the different types of story problems.
180 Results unknown
Dr. Gerber has 6 golf balls. Dr. Lloyd gave him 8 more. How many golf balls does Dr. Gerber have?
Sue has 21 cats. She gave 11 to John. How many cats does Sue have left?
Change unknown
Reese had 7 baseballs. Chris gave him some of her baseballs. Now Reese has 23 baseballs. How many baseballs did Chris give Reese?
John has 12 slices of pizza. He gave some pizza to Stan. Now John has 3 slices of pizza. How many slices did John give to Stan?
Start unknown
Torri had some hamburgers. Then Wendy gave her 7 hamburgers. Now Torri has 18 hamburgers.
How many hamburgers did Torri have at the beginning?
Trevor had some cows. He gave 6 cows to Maci. Now Trevor has 22 cows. How many cows did Trevor have before he gave some to Maci?
Equalize
Mike has 7 dollars. Ron has 14 dollars. How many dollars does Mike need to have as many as Ron?
Javon has 25 trading cards. Fred has 11 trading cards. How many cards would Javon have to give away to have as many as Fred?
Combine
Total set unknown
Melody has 5 flowers. Emma has 9 flowers. How many flowers do they have in total?
Subset unknown
Emma and Leigh have 28 rabbits altogether. Leigh has 13 rabbits. How many rabbits does Emma have?
Compare
Difference unknown
Alex has 11 books. Joe has 5 books. Alex has how many more books than Joe?
Laura has 17 pens. Ross has 11 pens. Ross has how many fewer pens than Laura?
Compared quantity unknown
Andy has 4 computers. Matt has 8 more computers than Andy. How many computers does Matt have?
Laura has 23 books. Ross has 6 fewer books than Laura. How many books does Ross have?
Referent unknown
Jorge has 10 DVDs. He has 3 more DVDs than Cindy. How many DVDs does Cindy have?
Nirbhay has 17 dollars. He has 9 dollars less than Alan. How many dollars does Alan have?
FIGURE 10.6. Addition and subtraction word-problem types.
From Robert Reid and Torri Ortiz Lienemann (2006). Copyright by The Guilford Press. Permission to photo-copy this figure is granted to purchasers of this book for personal use only (see photo-copyright page for details).
IMPLEMENTATION PLANS
In this section, we provide partial examples of implementation plans for the math strat-egies previously discussed.
Stage 2 for Computation Strategies: Discussing the Strategy
This is the first stage in “initiating” the strategy. In this stage it is important to stress the relevance of the strategy. During an initial conference the teacher will want to discuss the students’ current performance. It is also important for the teacher to stress the value
Vary
Size of groups unknown
In Steve’s basketball camp there are 5 balls for 25 players. How many players must share each ball?
Whole unknown
George worked picking up bottles for 6 days. He earned 54 dollars for each day he picked up bottles. How much did George earn?
Compare
Referent unknown (compared is part of referent)
Mike and Ron bought some cherries. Mike bought 4 pounds of cherries. Ron bought one-third as many cherries as Mike. How many cherries did Ron buy?
Compared unknown (compared is part of referent)
Stan and Michalla both got speeding tickets. The amount that Stan had to pay was one-half the amount that Michalla had to pay. Stan had to pay 40 dollars.
How much did Michalla have to pay?
Compared unknown (compared is multiple of referent)
John has 20 doughnuts. Evie has 4 times as many doughnuts as John. How many doughnuts does Evie have?
Restate
Susan and Lynette took a walk. Lynette walked half as far as Susan. If Susan walked 18 miles, how far did Lynette walk?
FIGURE 10.7. Multiplication and division story problem types.
From Robert Reid and Torri Ortiz Lienemann (2006). Copyright by The Guilford Press. Permission to photocopy this figure is granted to purchasers of this book for personal use only (see copyright page for details).
of the strategy. Brainstorm with the students on situations where using this strategy or completing the given task accurately is important. For example, the following might be appropriate brainstorming ideas in response to the question “When would it be impor-tant for you to make accurate math calculations?”
• Balancing your checkbook
• Trying to figure out if you have enough money to buy what you want
• Following a recipe Change Problem
Frank has 8 seashells. Edward gave him 8 more. How many sea shells does Frank have now?
Vary Problem
In Steve’s basketball camp there are 5 balls for 25 players. How many players must share each ball?
FIGURE 10.8. Examples of graphic organizers for word problems.
From Robert Reid and Torri Ortiz Lienemann (2006). Copyright by The Guilford Press. Permission to photocopy this figure is granted to purchasers of this book for personal use only (see copyright page for details).