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Problem Solving: How Can We Do It Better? September 20, 2012 Presented by Lois Williams, Ed.D. Tidewater Team College of William and Mary What do we know about problem solving? Research Students’ “ability to solve word problems falls far below their ability to compute.”(Burns, 2000) Research • This discrepancy is not because children have poor computation or reading skills, but because children “do not know how to choose the correct operation to apply to the problem.” (Burns, 2000). Research • The more problems you solve, the better the problem solver you become. • Students construct their own rules for solving problems. Example Megan had $45 on Monday. On Tuesday she earned $16 babysitting and on Thursday she earned $25 cutting grass. If she did not earn any more money that week, how much money did Megan have by Friday? Megej khj $45 on mosnfn. Onjbf kishfbj knnb $16 njdfh skf ,mkldfhjkn jnjkjf m sjhe jkdhh $25 jhdfif ufhhjf . Jgdffjhdfb kfhdfkjhdfn ufhkjdhj fjkdhf ldjfhjhgnjfcnfi;zpfj iur eiy ? Example • Mhe kjh ckikh kjh kj khk 96, 79 and 87 oj injhk kihe kjnj kjhej. Iohh lj ;pwe rfh kkkke h? • Megan received a 96, 79 and 87 on her most recent math quizzes. What is her average on these quizzes? Research Students more accurately solve problems with diagrams than without. Research • About one-fifth of students disagreed with the statement that a mathematical problem can be solved in different ways (Lindquist, 1989). Research • Good problem solvers monitor their thinking regularly and automatically. This is called metacognition. (Schoenfeld, 1992) • There is evidence that metacognitive behavior can be learned. (Garofalo, 1987; Thomas, 2006) Strategies Key Word Strategy What is it? Teaching students to scan a problem for specific words that suggest an operation such as “in all” means “add”. Disadvantages of Key Word Strategy 1. Key words are often misleading. (Drake and Barlow, 2007) There are three boxes of chicken nuggets on the table. Each box contains six chicken nuggets. How many chicken nuggets are there in all? Disadvantages of Key Word Strategy 2. Many problems have no key words. The rope is 25-feet long. How many 7-foot jump ropes can be made? Disadvantages of Key Word Strategy 3. The key word strategy sends the wrong message about doing mathematics. (van de Walle, 2010) What message are we trying to send? Working with Diagrams Working with Diagrams THURS. FRIDAY SATURDAY SUNDAY SUPERMAN 234 250 275 200 SPIDERMAN 250 255 260 225 BATMAN 240 275 250 200 Two-Step Problems Tony bought three dozen eggs for 89 cents a dozen. How much was the bill? How much change did Tony get back from $5? Tony bought three dozen eggs for 89 cents a dozen. How much change did Tony get back from $5? Two-Step Problems 1. Give students a one step problem and have them solve it. 2. Before discussing the answer, have each child or group use the answer to the first problem to create a second problem. 3. Collect the second problems. 4. Discuss answer to first. 5. As a class, start answering the second problems. Two-Step Problems 7. Select one first and second question. Put them on the board and show the students how to make a “hidden” question. 8. Have students do it. 9. Share the “hidden question” problems with the class. Ask students to solve AND identify the “hidden questions”. Singapore Bars • 1995 TIMSS placed the United States and Singapore near the bottom of the list of countries in mathematics and science. • The next TIMSS places Singapore in first place in mathematics. What Happened? Singapore: Join/Separate (part/whole) At the Virginia Museum of Art there are 135 European artists represented and 119 American artists represented. How many European and American artists are represented? 135 119 Singapore: Difference (part/whole) There are 253 artists represented in the Virginia Art Museum. 134 are European artists. The rest are Virginia artists. How many are Virginia artists? ? 134 253 Singapore: Add/Subtract Comparison 154 women took part in an art competition. 25 fewer men than women took part. How many men took part in the competition? 154 women ? men 25 Singapore: Add/Subtract (comparison model) 154 women and 127 men took part in an art competition. How many more women than men took part in the competition? 154 women 127 men ? Singapore: x/÷ (part and whole model) Lois saved $80 a week for 6 weeks. How much did she save altogether? $80 Singapore: x/÷ (part/whole model) Lois saved $120 in 6 weeks. How much did she save each week? $120 Singapore: x/÷ (part /whole model) Lois saves $20 each week, How many weeks will it take her to save $120? $120 $20 Singapore: x/÷ (comparison model) • There are 9 green apples. There are 3 times as many red apples as green apples. How many red apples are there? 9 green 9 9 9 Singapore: x/÷ (comparison model) There are 27 red apples. There are 3 times as many red apples as green apples. How many green apples are there? 27 red apples ? Singapore: x/÷ (comparison model) There are 27 red apples and 9 green apples. How many times as many red apples as green apples are there? 9 27 Singapore: Fractions Lauren bought 24 flowers. 3/4 of them were white. How many white flowers did Lauren buy? 24 ? Try One! Alex spent 4/5 of his money on a storybook. The storybook cost $20. How much money did he have to begin? Lois Williams [email protected]