Helping Children with Problem Solving Presentation by: Tabatha Leak, Sara Hoagland, Devon Pashia, & Vickie Burke
What is a problem? • Something a person needs to figure out. • Something where the solution is not immediately obvious.
What is problem solving? • Requires creative effort and higher-level thinking. • Skill in solving problems comes through experiences with solving many problems of many different kinds.
As a teacher, you must be sure not to shield children from problem-solving challenges by assigning “problems” that really are just exercises. -An example is: 3194 5346 +8877 -Another example: 7809 people watched TV on Monday. 9060 people watched TV on Tuesday. 9924 people watched Wednesday. How many people watched in these three days? This is a routine problem or exercise-task that can be solved by applying a mathematical procedure in much the same way as it was learned.
Example of a problem: Begin with the digits 1 2 3 4 5 6 7 8 9 Use each digit at least once, and form three four-digit numbers with the sum of 9636. • This is a non-routine problem because it requires thinking. This is because the procedure to solve it is not immediately obvious.
Teaching Mathematics through Problem Solving The Primary goal of school mathematics instruction should be to ensure that students make sense of the mathematics they are learning.
Tips for teaching Mathematics through Problem solving • Allow Math to be problematic for students • Do NOT give extra work. • Just allow students to solve their own problems, without your step-by-step help. • Focus on the methods used to solve problems • Learning takes place when a student is challenged to make sense of a problem. • Learning is extended when they share their approaches. • Tell the right things at the right time • Do NOT help to early it eliminates the challenge.
Factors for Success in Problem Solving • Teachers must see the importance of: • Knowledge • Students must learn to make connections between new and old problems • Beliefs and Affects • Teachers should show students everyone is a good problem solver • Control • Students need to learn to monitor their own thinking about problem solving • Sociocultural Factors • Classroom environment must encourage students
Choosing Appropriate Problems • Teaching through problem solving requires planning and coordinating the problems so students have a variety of problem types to analyze.
Problems…. • that ask students to represent a mathematical idea in various ways • that ask students to investigate a numeric or geometric concept • that ask students to estimate or to decide on the degree of accuracy required or to apply mathematics to practical situations • that ask students to conceptualize very large or small numbers • that ask students to use logic, to reason, to strategize, to test conjectures, or to gauge to reasonableness of information. • that ask students to perform multiple steps or use more than one strategy
Finding Problems • Resources: • Articles and books • Websites • Write your own • Attending workshops • Have children write their own problems • Use children's literature as a connection
Using Calculators • Consider letting students use calculators when: • They let children solve more complex problems or problems with realistic data • They eliminate time consuming computations and relieve anxiety • Their special functions can help children explore mathematical objects, concepts, and operations
Classic Four-Stage Model of Problem Solving Understand the problem. Devise a plan for solving it. Carry out your plan. Look back to examine your solution.
Useful Problem-Solving Strategies Act It Out Making a Drawing or Diagram Look for a Pattern Construct a Table Guess & Check Work Backwards Solve a Simpler or Similar Problem
Act It Out Acting out a problem helps children visualize what is involved in a problem. When using this strategy children can use themselves to perform the actions described in the problem or manipulate objects.
Problem 1 Two students are standing at their seats. Three students join them. How many students are standing at their seats?
Making a Drawing or Diagram This strategy lets students depict the relationships among the different pieces of information in a problem in a way that makes those apparent.
Problem 2 Aunt Katrina wants to hang 6 decorative plates on her dining room wall, in a straight line, spaced evenly apart (from each other and the edges of the wall) . Each plate is 8 inches in diameter and the wall is 104 inches long. How far apart should the plates be hung?
Look for a Pattern In problem-solving, children look for patterns in more active ways- for example, by constructing a table that might help them see a pattern. Understanding “patterns, relations, and functions” is one of the major goals of the Algebra Standard (K-12) in NCTM’s Principle and Standards for School Mathematics.
Problem 3 A pattern of dots is shown below. Determine the number of dots in the 20th step. Step One Step Two Step Three
Construct a Table Organizing information into a table often helps children discover a pattern and identify missing information. Constructing a table is an efficient way to classify and order large amounts of information and it also provides a record of what has been tried.
Problem 4 Suppose somebody offers you a job for 15 days. They offer you your choice of how you will be paid. You can start at 1 cent a day, get 2 cents the next day, 4 cents the next day and continue doubling the amount every day. Or you can start with $1 on the first day , get $2 the next day, $3 dollars the next day and continue adding a dollar a day. Which option would you choose? Why?
Guess & Check Guessing can be a useful strategy if students incorporate what they know into their guesses (educational guesses, not random guesses.)
Problem 5 Suppose it costs 35 cents to mail a postcard and 37 cents for a letter. Bill wrote to 12 friends and spent $3.46 for postage. How many letters and postcards did he send?
Work Backward Students must work backward to solve a problem if the problem states a result or an endpoint and the students have to figure out the initial conditions or the beginning. (Many mazes are worked by solving from end to the beginning.)
Problem 6 Sue baked some cookies. She put half of them away for the next day. Then she divided the remaining cookies evenly among her two sisters and herself, so each got four cookies. How many cookies did she bake?
Solve a Similar but Simpler Problem Students who know how to solve a problem can usually solve a second problem that is somewhat similar, even if the second problem is somewhat more difficult. The insight and understanding they gain from solving easier problems , where relationships are more apparent, carry through and let them solve harder problems.
When students do not understand a problem, you can try asking them to restate the problem in their own words. This can help you identify what the child did not understand or it can help the child figure out what the problem is asking.
Why look back? • Looking back at the problem. -This helps students generalize the problem. This means they can relate the problem to other problems in the future. • Looking back at the answer. -Does the answer make sense? • Looking back at the solution process. -Using different techniques may help students. • Looking back at one’s own thinking. -An effective teaching technique is to model the type of questions students should ask themselves when thinking about their own thinking. -Example: Was this problem like any I had even seen before? Was this problem easier (or harder) than I expected?
Helping All Students with Problem Solving How does a teacher structure problem solving sessions for a full class of diverse students? Vickie Burke
Managing Time • Students need time to think about the problems. • Students need to be encouraged. • A teacher needs to make time to help students. • Managing Classroom Routines • Large-group instruction – Teaching the whole class • Small-group instruction – Group students by problem-solving ability and interests. • Having children work in pairs – Students can teach each other. • Having children solve problems individually – Also necessary, so students can progress at their own pace and use strategies they find most comfortable.
Managing Student Needs • Compensatory strategies – Specific strategies that individual students can use to deal with their own specific learning needs. • Using an approach that fits their way of learning. • Using their strengths to communicate their thinking and keep a record of how they solved a problem. • Rephrasing a problem in their own words in order to understand the problem better. • For students with special needs: • Help them get started or have another student explain the problem. • Help them create a journal or a card file of “types of math problem I can solve.” • Allow extra time for students with abstract reasoning or reading difficulties to break down problems by making summarizing notes.
Modifying instruction to help all students • Assign different types of problems. • Call on students to re-phrase problems. • Have pairs or small groups to share their solutions. • Allow students to write on chalkboard/whiteboard. • Allow students to work in quiet places. • Students with learning disabilities • Accustomed to instruction that focuses on learning through imitation. • Students may need time to adjust to new approach. • Gifted students • Prepare problem extensions (general problem with larger numbers).
Cultural Connections Trends in International Mathematics and Science Study (TIMSS) found that Japanese students attained highest overall test scores while American students’ scores were among the lowest. Vickie Burke
American lessons generally had two phases and involved students doing many “problems” but were really just exercises instead of actual problems. • Teacher demonstrated or explained sample problems. • Students worked on problems (really exercises) on their own while the teacher helped students that were having problems. • Japanese lessons lasted 45 minutes and consisted of four or five phases focuses on one of two genuine problems. • Teacher posed a complex problem. • Students worked on the problem on their own or in pairs. • Whole class discussed different approaches to the problem. • Teacher summed up. • Depending on time, students could work on extensions of the original problem.
Key difference between American and Japanese teaching is the role of the teacher while children are working on the initial problem. • Research shows (Shimizu 2003): • Suggestion 1 - Label students methods • Suggestion 2 - Use the chalkboard effectively. • Suggestion 3 – Use the whole-class discussion to polish the students’ ideas. • Suggestion 4 – Choose the numbers in, and the context of, the problem carefully. • Suggestion 5 – Consider how to encourage a variety of solution methods. • Japanese understand the importance of thinking deeply about the relationship between the mathematical content to be taught and the problems they assign. Anticipating student responses is a crucial aspect of lesson planning.
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