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MATH PROBLEM SOLVING

MATH PROBLEM SOLVING. FOUR STAGES. SOLUTION EXECUTION. PROBLEM TRANSLATION. Calculations Fast & Accurate. Basic Facts Presented Math Knowledge. SOLUTION PLANNING & MONITORING. PROBLEM INTEGRATION. Break into Subgoals Step by Step Plan Where am I in my plan?.

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MATH PROBLEM SOLVING

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  1. MATH PROBLEM SOLVING FOUR STAGES SOLUTION EXECUTION PROBLEM TRANSLATION Calculations Fast & Accurate Basic Facts Presented Math Knowledge SOLUTION PLANNING & MONITORING PROBLEM INTEGRATION Break into Subgoals Step by Step Plan Where am I in my plan? Recognize Problem Type Build Coherent Representation

  2. SAMPLE PROBLEM Floor tiles are sold in squares 30 cm on each side. How much would it cost to tile a rectangular room 7.2 meters long and 5.4 meters wide if the tile cost $ .72 each? ONE POSSIBLE SOLUTION PLAN Step 1: change width & length into number of tiles 540/30 = 18 tiles 720/30 = 24 tiles Step 2: determine how many square tiles cover the floor 18 times 24 = 432 tiles Step 3: determine the cost of the 432 tiles 432 times $ .72 = $ 311.04

  3. WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM? Step 1: Problem Translation Linguistic Knowledge - need to be able to understand English sentences in order to recognize the facts of the problem (What are the givens? What is the problem goal?) For the tile problem: - room is rectangle, 7.2 by 5.4 meters - each tile costs $ .72 - goal is to find total cost of tiling the room Translation process also requires factual knowledge about mathematics - one meter equals 100 cm

  4. WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM? Step 2: Problem Integration Schematic Knowledge - need to integrate information into a coherent representation, need to recognize problem type For the tile problem: - this is a rectangle problem - need to use the rectangle area formula to solve the problem Area = length x width *** Problem Integration involves more than statement by statement translation

  5. PROBLEM INTEGRATION ENABLES YOU TO RECOGNIZE INCONSISTENCIES “The number of quarters a man has is seven times the number of dimes he has. The value of dimes exceeds the value of the quarters by $2.50. How many has he of each coin? WRITE THE EQUATIONS YOU NEED TO SOLVE THE PROBLEM AND THEN SOLVE! Q = 7 D D (.10) = 2.50 + Q (.25) Anything wrong?

  6. PROBLEM INTEGRATION ENABLES YOU TO CONSTRUCT A SITUATION MODEL Can you give an example of a concrete situation that corresponds to: 3 4 1 2 1 That is, create a simple word problem that could be solved by the above equation. Problems of this type were given to elementary school teachers in the U.S. and China (try to do this yourself, sample answers - next slide)

  7. STUDY ON SITUATION MODELS U.S. vs. CHINESE TEACHERS Incorrect model: “If you have one pie and 3/4 of another pie to be divided equally by two people, how much pie will each person get?” Correct model: “If a team of workers construct 1/2 kilometer of road per day, how many days will it take them to construct a road 1 and 3/4 kilometers long?” Results: 96% of the U.S. teachers either could not describe an appropriate concrete situation or produced an incorrect model. 90% of the Chinese teachers produced correct models. YIKES!!!!!

  8. WHICH TWO PROBLEMS BELONG TOGETHER? 1. A personnel expert wishes to determine whether experienced typists are able to type faster than inexperienced typists. 20 expert typists (5 yr or more experience) and 20 inexperienced typists (less than 5 yrs) are given a typing test. Each typist’s average number of words per minute is recorded. 2. A personnel expert wished to determine whether typing experience goes with faster typing speeds. 40 typists are asked to report their years of experience as typists and are given a typing test to determine their average number of words per minute. 3. After examining weather data for the last 50 years, a meteorologist claims that the annual precipitation varies with average temperature. For each of 50 years, she notes the annual rainfall and average temperature. Experienced math problem solvers pick 2 and 3 Inexperienced math problem solvers pick 1 and 2.

  9. CAN THESE PROBLEMS BE SOLVED? ANY IRRELEVANT INFORMATION? 1. A rectangular lawn is 12 meters long and 5 meters wide. Calculate the area of a path 1.75 meters wide around the lawn. 2. The length of a rectangular park is 6 meters more than its width. A walkway 3 meters wide surrounds the park. Find the dimensions of the park if it has an area of 432 square meters. 3. The lengths of the sides of a blackboard are in a 2:3 ratio. What is the perimeter (in meters) of the blackboard? *** Most high school students make mistakes on more than half of problems like the ones shown above.

  10. IMPLICATIONS FOR INSTRUCTION: TEACHING PROBLEM INTEGRATION SKILLS • use varied presentation to encourage students to discriminate among problem types • encourage students to draw diagrams • practice sorting problems into categories • practice identifying relevant and irrelevant information

  11. WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM? Step 3: Solution Planning and Monitoring Strategic Knowledge - need general strategies that can be used to devise and monitor a solution plan For the tile problem: - draw a picture - work backwards from goal: goal is to find total cost of tiling floor, so you need to know the # of tiles that cover the floor - divide into subgoals: change dimensions into # of tiles, then determine how many tiles cover the floor, then determine the cost of all the tiles *** general strategies are italicized 7.2 m 5.4 m

  12. WHAT DO YOU NEED TO KNOW TO SOLVE THE TILE PROBLEM? Step 4: Solution Execution Procedural Knowledge - computational procedures from simple procedures (e.g., single digit addition or subtraction) to more complex procedures (e.g., subtraction of multiple digit numbers) For the tile problem: 540/30 = 18 tiles 720/30 = 24 tiles 18 x 24 = 432 tiles 432 times $ .72 = $ 311.04 *** Key Point: able to do computations with no difficulty, fast and accurate (achieve automaticity, direct retrieval from long-term memory)

  13. FAULTY BELIEFS ABOUT MATH THAT UNDERMINE EFFECTIVE PROBLEM SOLVING 1) Ordinary students cannot expect to understand math, they have to memorize it, and just apply what they have learned mechanically and without understanding. 2) All story problems can be solved by applying operations suggested by key words in the story (in all suggests addition, left suggests subtraction, share suggests division - 3rd graders) 3) Any assigned problem should be solved within five minutes or less. (High school students estimated the typical problem should take about 2 minutes) 4) Math is not particularly useful or sensible. Math is mostly a set of rules and mathematics learning means memorizing the rules (54% of 4th graders and 40% of eighth graders; females’ attitudes toward math more negative).

  14. ATTRIBTUTION STYLE UNDERMINES EFFECTIVE MATH PROBLEM SOLVING Researchers gave 10 year old children a questionnaire asking about their likely reactions to hypothetical failures. They identified two attribution styles: Mastery-oriented: likely to think they should work harder in the face of failure/difficulty Helpless: likely to respond to difficulty with negative attributions about ability *** There were no IQ differences between these two groups. *** Many more girls were categorized as “helpless.”

  15. ATTRIBTUTION STYLE UNDERMINES EFFECTIVE MATH PROBLEM SOLVING - CONTINUED Researchers next gave the children a series of confusing math problems (difficult to solve), and then a batch of easy math problems (that all children should be able to solve). What happened? Mastery-oriented children: These children were able to recoup from the negative experience and solved the easy problems with ease. Helpless: These children were thrown by the confusing problems and didn’t try very hard on the easy problems, getting many of them wrong.

  16. ATTRIBTUTION STYLE UNDERMINES EFFECTIVE MATH PROBLEM SOLVING - CONTINUED Researchers wanted to know why girls were more likely to adopt a “helpless” attribution style. What happens in the classroom? Boys and girls receive the same amount of negative comments. But the nature of these comments differ. Boys: Criticisms sometimes focus on intellectual quality, sometimes on neatness, conduct, or effort. Boys and girls both think teachers like girls better. Girls: Teacher criticisms focus consistently on the intellectual quality of the work. End Result: Boys attribute failure to any number of factors, girls are left with negative attributions concerning their ability.

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