What is Literacy? • Literacy is the ability to identify, understand, interpret, create, communicate and compute, using printed and written materials associated with varying contexts. Literacy involves a continuum of learning in enabling individuals to achieve their goals, to develop their knowledge and potential, and to participate fully in their community and wider society. • United Nations Educational, Scientific, and Cultural Organization (UNESCO)
Problems and Problem Solving “Most, if not all, important mathematics concepts and procedures can best be taught through problem solving.” --John Van de Walle
What is Problem Solving? • “Problem solving means engaging in a task for which the solution method is not known in advance.” --Principles and Standards for School Mathematics • It encompasses exploring, reasoning, strategizing, estimating, conjecturing, testing, explaining, and proving.
What is a Problem? • A problem is a task that requires the learner to reason through a situation that will be challenging but not impossible. • Most often, the learner is working with a group of other students to meet the challenge.
Problem or Exercise? • An exercise is a set of number sentences intended for practice in the development of a skill. • A problem is what we commonly refer to as a “word problem.” • But beware! Problems can become exercises!!
Common Characteristics of a Good Problem • It should be challenging to the learner. • It should hold the learner’s interest. • The learner should be able to connect the problem to her life and/or to other math problems or subjects. • It should contain a range of challenges. • It should be able to be solved in several ways.
What Does It Mean to Be Successful at Problem Solving? • Having success means that the child has discovered a way of thinking about mathematics that he had not experienced before he came upon this problem. • Success will involve the process of problem solving as well as understanding the content presented.
Success with “How Many Rectangles” • Do the students resolve the question about whether to include the squares in their count of rectangles? • Do they understand that squares meet all the criteria to be considered a rectangle? • Do they recognize that there are many different sizes of rectangles in the drawing?
Success with “How Many Rectangles” (continued) • Have the students devised a way of counting the rectangles they find? • Do they find patterns in the number of different-sized rectangles? • Do they think about the concepts embedded in the problem differently than before?
The Teacher’s Role in Problem Solving “The more regularly that teachers make it part of the curriculum, the more opportunities students will have to become successful problem solvers.” --Children Are Mathematical Problem Solvers
Choosing Problem-Solving Tasks • The problem must be meaningful to the students. • The teacher must sometimes adapt the problem to make it more meaningful. • The teacher must work the problem to anticipate mathematical ideas and possible questions that problem might raise.
Presenting the Problem • It must be interesting and engaging. • It must be presented so that all children believe that it’s possible to solve the problem, but that they will be challenged. • The teacher has to decide whether students will work individually or in groups.
Group Work or Individual Work? • In groups, students don’t give up as quickly. • Students have greater confidence in their abilities to solve problems when working in groups. • When in a group, students hear a broad range of strategies from others. • Kids enjoy working in groups! • Students remember what they learn better when they assist each other. • If students are less productive, arrangements can be made for them to work alone. • There will be a heightened noise level—but conversation is an important part of the learning process.
Once the Kids Are Working… • Allow students to “wrestle” with the problem without just telling them the answer! • If we are just telling them what to do, the students are not engaged in the process. • Finally, teachers have to determine how to assess what the students are learning and what they need to learn next. • There are several ways to do this…
Assessing Understanding • Listen to and record the students’ conversations as they solve the problem. • Have students explain their solutions in writing. • Give them another problem that requires them to use what they learned in the first problem.
Learning Mathematics through Problem Solving • Students learn to apply the mathematics as they are learning it. • They can make connections within mathematics and to other areas of the curriculum. • Students can understand what they have learned.
Expressions and Problem Solving “Math Expressions was developed to meet the national need for a balanced program that could expand the types of word problems to those solved by other countries and use an algebraic approach to word problem solving.”
In Kindergarten… • Students act out family experiences about meals they might eat at home. • “Tom sets the table. He puts down 3 plates and then 1 more. How many plates are on the table?” • Using paper plates, each child can act successfully solve a story problem.
In First Grade… • Students might solve the following problem and then explain their solutions at the board: • “I took 4 rides on the roller coaster. My sister took 5 rides. How many roller coaster rides did we take in all?” • Students could use any way that makes sense to them to solve this problem.
In Second Grade… • Using Solve and Discuss, students might solve the following problem: • “Last year our school had 5 computers in the library. They bought some more over the summer. Now there are 12. How many computers did they buy over the summer?” • Two or three children might show their solutions on the board. Students at their desks should be encouraged to ask questions:
5 + 7 = 12 C buy now • How did you get 7 more? • Why did you start with 5? • How did you know 7 was a partner? • How did you know 12 was the total? • Where did you get 5 + 2? 5 + 5 + 2 = 12 C buy now xxxxx xxxxxxx 12 5c 7 buy now
In Third Grade… • Students might solve the following problem and record their answers in several ways: • “Chris picked 8 apples. His mother picked 6 more. How many apples do they have now?” • Children might show their solutions in several ways:
Math Mountain: Count All: 14 now xxxxxxxx xxxxxx Chris Mom now T 14 8 6 P P Chris Mom Count On: Equation: 8 + 6 = 14 P P T 8 xxxxxx 14 had count on now 6 more
In Fourth Grade… • As the problem become more complex, students may rely less on pictures and more on ways to represent the steps in the problem: • “In the morning, 19 students were working on a science project. In the afternoon, 3 students left and 7 more students came to work on the project. How many students were working on the project at the end of the day?” • Manipulatives and drawing paper should still be available for those students who would like to use it. Following are abstract ways to represent this problem:
Tommy’s Method Write an equation for each step. Lucy’s Method Write an equation for the whole problem. Find the total number of students who worked on the project. 19 + 7 = 26 Subtract the number of students who left in the afternoon. 26 – 3 = 23 Let n = the number of students working on the project at the end of the day. Students who left Students who arrived in the afternoon. in the afternoon. 19 – 3 + 7 = n 23 = n
In Fifth Grade… • Students are still encouraged to solve problems any way that works for them. • “A right triangle has sides of 4 feet, 5 feet, and 1 yard. What is its perimeter in inches?” • In this case, students may well want to draw a picture to assist them in solving this problem.
Expressions: Inquiry + Fluency • Using Expressions, students balance deep understanding with essential skills and problem solving. • Students invent, question, discover, learn, and practice important math strategies. • Students explain their methods daily.
Sources • Children Are Mathematical Problem Solvers • by Lynae E. Sakshaug, Melfried Olson, and Judith Olson • Math Expressions • developed by The Children’s Math Worlds Research Project; Dr. Karen C. Fuson, Project Director and Author