Measurable Objectives • Identify multiple problem solving strategies for a given • problem • Identify skills needed as a problem solver • Engage in problem solving • Identify problem solving processes applied in personal and • student problem solving • Identify curriculum resources associated with problem • solving instruction
“We can't solve problems by using the same kind of thinking we used when we created them. ” ~Einstein, 1949 What skills are needed as a problem solver?
Reflecting on the Processes Find the problem card at your table. Take a few minutes on your own to think about the solution. • How did you solve this problem? • What skills did you need to solve this problem? • How did you learn how to solve problems like this? • What factors influence differences to approach solution-finding?
Why Problem Solving? “The single best way to grow a better brain is to engage in challenging problem solving.” ~Jensen (1998)
“It doesn’t matter to our brains whether we come up with the right answer or not: the neural growth happens because of the process, not because we have found the correct answer.” ~Jensen (1998)
What is in your textbook/curriculum related to problem solving instruction? Take five minutes to reflect in your journal about the resources that you have available for problem solving instruction.
"Textbooks commonly… • depict problem solving as a linear process. • present problem solving as a series of steps. • imply that solving mathematics problems is a procedure to be memorized, practiced, and habituated. • lead to an emphasis on answer getting.” ~Polya, 1945
Instructional programs from Pre-K through grade 12 should enable all students to… • build new mathematical knowledge through problem solving; • solve problems that arise in mathematics and in other contexts; • apply and adapt a variety of appropriate strategies to solve problems; • monitor and reflect on the process of mathematical problem solving. Taken from: NCTM, 2000. Principles & Standards for School Mathematics.
“If problem solving is treated as ‘apply the procedure,’ then the students try to follow the rules in subsequent problems. If you teach problem solving as an approach, where you must think and can apply anything that works, then students are likely to be less rigid.” ~ Suydam, 1987
The Mangoes Problem One night the King couldn't sleep, so he went down into the Royal kitchen, where he found a bowl full of mangoes. Being hungry, he took 1/6 of the mangoes. Later that same night, the Queen was hungry and couldn't sleep. She, too, found the mangoes and took 1/5 of what the King had left. Still later, the first Prince awoke, went to the kitchen, and ate 1/4 of the remaining mangoes. Even later, his brother, the second Prince, ate 1/3 of what was then left. Finally, the third Prince ate 1/2 of what was left, leaving only three mangoes for the servants. How many mangoes were originally in the bowl? ~NCTM, Illuminations, 2008
Guess and Check 14 - 1/2 - 1/3 - 1/4 - 1/5 - 1/6 ~NCTM, Illuminations, 2008
Draw a Picture ~NCTM, Illuminations, 2008
Work Backward "Six represents two-thirds of something, so one-third must be three. So to get three-thirds, you must add the six (for two-thirds) to three (for one-third) and you have nine mangoes." Then, going the next-backward step, he said, "Nine needs one-fourth" (his words, meaning that since nine is three-fourths of the previous amount, it "needs" another fourth of this amount added to it), "so nine is three-fourths: divide by three (i.e., 9/3) and add this to nine, obtaining twelve." ~NCTM, Illuminations, 2008
Write an Equation • Since the King removed (1/6)x, then x - (1/6)x mangoes are left after his removal. Thus, (5/6)x mangoes are left. • The Queen removed one-fifth of (5/6)x, so (5/6)x - (1/5)(5/6)x, or (4/6)x, mangoes are left after her removal. • The first Prince removed one-fourth of (4/6)x mangoes, so (4/6x - (1/4)(4/6)x, or (3/6)x, mangoes are left after the first Prince's removal. • The second Prince removed one-third of (3/6)x, so (3/6)x - (1/3) (3/6)x, or (2/6)x, mangoes are left. • Finally, the third Prince removed one-half of (2/6)x, leaving 3 mangoes, so (2/6)x - (1/2)(2/6)x = 1/6x = 3. Solving 1/6x = 3 results in x = 18. ~NCTM, Illuminations, 2008
George Polya (1887-1985) George Polya was one of the most famous mathematics educators of the 20th century. He strongly believed that the skill of problem solving could and should be taught—it is not something that you are born with. He identified four principles that form the basis for problem solving: • Understand the problem • Devise a plan • Carry out the plan • Look back (reflect)
Getting Acquainted With A Problem • Where should I start? • Start from the statement of the problem. • What can I do? • Visualize the problem as a whole as clearly and as vividly as you can. Do not concern yourself with details for the moment. • What can I gain by doing so? • You should understand the problem, familiarize yourself with it, impress its purpose on your mind. The attention bestowed on the problem may also stimulate your memory and prepare for the recollection of relevant points. ~Polya, 1945
Introduce New Problem: Applying the Process The houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? ~Reardon, 2001
Area 1: Understand the Problem • What are you asked to find out or show? • Can you draw a picture or diagram to help you understand the problem? • Can you restate the problem in your own words? • Can you work out some numerical examples that would help make the problem more clear?
Critical Questions • Do you understand all the words used in stating the problem? • What are you asked to find or show? • Can you restate the problem in your own words? • Can you think of a picture or diagram that would help you understand the problem? • Is there enough information to enable you to find a solution?
What if students cannot understand the problem? • What can you do?
Area 2: Devise a Plan to Solve the Problem A partial list of problem solving strategies include: • Solve the simpler problem • Experiment • Act it out • Work backwards • Use deduction • Change your point of view • Guess and check • Make an organized list • Draw a picture or diagram • Look for a pattern • Make a table • Use a variable
Critical Questions • Have you seen it before? Or, have you seen the same problem in a slightly different form? • Do you know a related problem? Do you know a theorem that could be useful? • Look a the unknown, and try to think of a familiar problem having the same or a similar unknown. • Could you restate the problem? Go back to definitions
What if students cannot understand the problem? • What can you do?
Area 3: Implementing a Solution Plan • Carrying out the plan is usually easier than devising the plan • Be patient—most problems are not solved quickly nor on the first attempt • If a plan does not work immediately, be persistent • Do not let yourself get discouraged • If one strategy isn’t working, try a different one
Experiment with different plans • Allow for mistakes • Work collaboratively • Check How do students implement? • Draw it • Use calculator • Use computation skills
Area 4: Reflecting on the Problem: Looking Back • Does your answer make sense? Did you answer all of the questions? • What did you learn by doing this? • Could you have done this problem another way—maybe even an easier way? • ~Reardon Problem Solving Gifts, Inc. (2001)
“Students need to view themselves as capable of using their growing mathematical knowledge to make sense of new problem situations in the world around them.” ~Newell & Simon, 1972
Reflection • What problem solving processes do you most commonly use or encourage students to use? • What strategy have you learned or been reminded to use more often?
Welcome Back ! To Day 2 of Keys of Problem Solving (KoPS)
Review of Day 1 • Legislation steering mathematics • Recommendations from research • Next Generation Standards • RtI • Problem solving process
Day 2 Focus will be on instruction and intervention based on assessment results 1-accurate problem identification 2-problem analysis 3-design a plan and implement 4-evaluate effect
Polya hypothesized that problem solving is not an innate skill, but rather something that can be developed. He explains, “Solving problems is a practical skill, let us say, like swimming…. Trying to solve problems, you have to observe and imitate what other people do when solving problems, and, finally, you learn to solve problems by doing them.” ~Polya, 1945, p. 5
Example Three darts hit this dart board and each scores a 1, 5, or 10. The total score is the sum of the scores for the three darts. There could be three 1’s, two 1’s and a 5, one 5 and two 10’s, and so on. How many different possible total scores could a person get with three darts?
Additional ways to reflect on the problem… • Reflect on the plan • Justify and explain your answers
How Does the Process Relate to Student Learning? My Instruction?
Holistic Rubrics ~ Florida Department of Education. FCAT Mathematics Sunshine State Standards Test Book, Released: Fall 2007.
Problem Solving Process/Holistic Rubric Using this one source of student data, what does this student sample reflect? Florida Department of Education. FCAT Mathematics Sunshine State Standards Test Book, Released: Fall 2007.
Mathematical Thinking “Providing a challenging investigation to small groups of students facilitates ongoing reasoning, argument, and assessment throughout the problem solving process.” ~NCTM, (2005)
Problem Solving is a dynamic process 1-accurate problem identification 2-problem analysis 3-design a plan and implement 4-evaluate effect
Growing Giant Sequoias • Goals: • Use given data to decide which growing condition fosters better growth of tree seedlings • Develop a decision-making procedure based on the data for use in future experiments • ~NCTM (2005)
Advance Organizer • Read about sequoias from newspaper article • Answer readiness questions • Learn about professor’s work regarding growing conditions • Analyze professor’s data ~NCTM (2005)
Connections to Problem Solving • Build new mathematical knowledge through problem solving • Solve problems that arise in mathematics and other contexts • Apply and adapt a variety of appropriate strategies to solve problems • Monitor and reflect on the process of mathematical problem solving • ~NCTM (2005)