Problem Solving K-2 “Learning mathematics should make Sense!” “Real understanding comes from solving problems.”

Objectives for today… To learn the 11 different problem types. To be able to write your own problems of any type. To know and use the progression of teaching the problem types to your students. To be able to distinguish helpful resources vs. misleading resources

Building the background…

Problem Solving Approach Small-group work Whole- class discussions where children explain their thinking Justify their solutions Question each other NCTM-Reasoning/Sense Making and Communication

Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

Teacher’s Role Asks questions Challenge children’s ideas Offer guidance HOWEVER – Teacher refrains from traditional practice of showing children a single procedure for solving a certain type of problem

Why do Problem Solving? Educators need to know how and why mathematics work. Not the rote following of set steps and procedures. Become more confident and comfortable Richer understanding And may even enjoy math!!!! (Cobb, Wood, Yackel 1991)(NCTM Standards)

Why do Problem Solving (continued)… Construct mathematics relationships Helps children develop their own strategies Using key words fails to engage children in inquiry process Suzie had 8 apples. Altogether Suzie and Johnny had 19 apples. How many apples did Johnny have? (Why is 27 a typical answer to this question?) Following facts and rules doesn’t serve them well to apply math to solve problems outside of school or learn more advanced mathematics It should be a sense-making activity.

Children’s Beliefs about Solving Problems (Ask yourself…How do my beliefs compare?) There is one right way to solve a problem. Mathematics is a set of rules and procedures. Learning mathematics is mostly memorizing. Elementary school math is computation. Mathematics problems should be solved quickly. The goal of math is to obtain “right answers” The teacher and the textbook are the mathematical authority. Take a minute to share your thoughts with a neighbor

Most Textbooks Understand the Problem Devise a Plan Carry Out the Plan Look Back

Problem Solving Suggestions STUCK, Good! RELAX and ENJOY it! Sort out What you KNOW and What you WANT, Try some strategy you may have tried before, Make a prediction, Find someone to whom to explain why you are stuck.

Problem Solving Strategies Guess and Check Make a List or Table (“Brian’s Way”) Write an Equation Work Backwards Break into Smaller Parts Draw a Picture Act it out Look for a pattern Do Something Take a Break and Try Again!

Using Strategies as Rules IS not True Problem Solving Strategies should be tools Textbooks offer hints on what strategy to use Children ignore other strategies Who tells you what strategy to use in your life? Choosing a strategy = significant step in process Hints hinder development in selecting strategies Our goal = provide opportunities for children to learn and feel comfortable with a variety of tools.

Mathematical Proficiency Conceptual understanding – comprehension of mathematical concepts, operations, and relations Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic competence – ability to formulate, represent, and solve mathematical problems Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy (NRC, 2001, p. 5) Summary: Skill in basic arithmetic is no longer a sufficient mathematics background for most adults

Problem Solving Types Jigsaw at your table (Ten Minutes) Join (ACES) Separate (TWOS) Part-Part-Whole (THREES) Compare (FOURS) DEFINITION, EXAMPLES. What does it take to be in your problem type classification?

Classification of Word Problems Change: (Join or Separate) There is an initial quantity and a direct or implied action causes an increase or decrease in that quantity. “Jose had 18 baseball cards. His friend Juan gave him 6 more baseball cards. How many baseball cards does Jose have now?” Combine/Group/Part-Part-Whole: Two distinct groups or subsets combine to form a new group or set. “There are 12 students in a school play. 4 are boys and the rest are girls. How many girls are there?” Compare: The comparison of two distinct, disjoint sets (compared and referent). “Andy has 14 music CDs and Lisa has 9 music CDs. How many fewer CDs does Lisa have than Andy?” (Carpenter & Moser, 1983)

Classify these problems… All on your own, put the number of the problem in the box you think it should go. When finished, verify your answers with your group or a partner at your table. When the timer goes off, we will reveal the correct answers.

Write your own problems… IN GRADE LEVEL GROUPS, choose 2 themes that students in your grade level would be interested in (holiday, boy vs. girl themes, science, social studies, health, etc.) Write 11 problems on each theme, using all the different problem types. Share out 2 problems from each theme.

What should the progression of introducing the different problem types be? Suggested Word Problem Progression of Teaching 1. Join – Result Unknown Separate – Result Unknown Part-Part-Whole or Combine – Whole Unknown 2. Join – Change Unknown Separate – Change Unknown 3. Compare – All types Part-Part-Whole or Combine – Part Unknown 4. Join – Start Unknown Separate – Start Unknown (The numbers above are not grade level specific)

Where can you find resources? Are all the resources you find going to represent all the problem types? What does the textbook do with word problems that we should try to modify?

Resources Textbook, google, standards, problem of the day items, ADD 2’s, Math Stretches, morning meetings, Study Island, Math journal

What were our objectives for today?

Objectives for today… To learn the 11 different problem types. To be able to write your own problems of any type. To know and use the progression of teaching the problem types to your students. To be able to distinguish helpful resources vs. misleading resources

Easels with sticky paper to share out jigsaw, blank classification chart, markers, packets, suggested progression Classify advanced problems Get a new group (by card number) and write K-2 appropriate problems for your problem type (use laptops and share quickly, to Corey) Progression—what do you think? How do you fit in problem solving to your day?