CREATING A CLASSROOM OF MATHEMATICS PROBLEM SOLVERS

Grades PreK-2

BAIN FAMILYThis is our family at the summit of Mt. Ellinor in Olympic National Park. It was a great family adventure. We saw many adults turn around before reaching the summit, but we pressed on and enjoyed the summit with a few mountain goats.

Tricia

Mahreah

Grace

Ben Jonah

Ruby

MY BACKGROUND 1998 graduate of Martin Luther College

with a double major in education and music.

2004 earned post-baccalaureate Wisconsin state teaching license.

Ten-time classroom supervisor of student teachers.

2010 graduate of Martin Luther College with a Master of Science in Education degree—instruction emphasis.

THE IMPORTANCE OF MATHEMATICS& DEFINING PROBLEM SOLVING

TIMSS International

studies; comparing eighth graders

TIMSS = Trends in International Mathematics and Science Study

1995: U.S. ranked 28th out of 41 countries

1999: U.S. ranked 19th out of 34 countries

2003: U.S. ranked 15th out of 45 countries

2007: U.S. ranked 9th out of 47

Why is math important for our country? Why is math important for our church? Why is math important for our

students?

NATIONAL MATHEMATICS ADVISORY PANELFoundations for Success: The Final Report of the

National Mathematics Advisory Panel (2008) “The eminence, safety, and well-being of nations have been entwined for centuries with the ability of their people to deal with sophisticated quantitative ideas. Leading societies have commanded mathematical skills that have brought them advantages in medicine and health, in technology and commerce, in navigation and exploration, in defense and finance, and in the ability to understand past failures and to forecast future developments.” (p. xi)

IS THIS PROBLEM SOLVING? In the numeral 185,

what does the 8 mean?

Jorge has 12 tickets. William has 15 tickets. How many more tickets does William have?

14 – 6 = ______ Which shape has four

sides?

(Students are lined up.) Which student is the tallest?

Erin has 2 cookies. MiKayla has twice as many cookies. How many cookies does MiKayla have?

(There is are several shapes shown at once.) How many triangles are there?

Which animal has more legs, a pig or a chicken?

DEFINITIONS OF PROBLEM SOLVING

NCSM (NATIONAL COUNCIL OF SUPERVISORS OF MATHEMATICS)

NCTM(NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS)

“the process of applying previously acquired knowledge to new and unfamiliar situations”

“problem solving means engaging in a task for which the solution method is not known in advance”

FEATURES OF PROBLEM SOLVING

MEIR BEN-HUR JOAN M. KENNEY

“Problem solving requires analysis, heuristics, and reasoning toward self-defined goals”

a process that involves such actions as modeling, formulating, transforming, manipulating, inferring, and communicating

TEACHING PROBLEM SOLVING

OVERVIEW Key Words Strategies Process by George Pólya Teacher’s Role

KEY WORDS

What words tell a person to add? What words tell a person to subtract? What words do you notice are

particularly troublesome to your students?

AN EXAMPLE OF TEACHING KEY WORDS“There are three bears in the living room. There are two more bears in the kitchen. How many bears are in the house altogether?”

WHAT DOES RESEARCH SAY? Undermines real problem solving Make problem solving a mechanical process

which makes students prone to errors Understanding the language of mathematics is

important.

“There are five bears in the house. Some bears are in the living room. Two more bears are in the kitchen. How many bears are in the living room?”

Ben-Hur, M. (2006). Concept-rich mathematics instruction: Building a strong foundation for reasoning and problem solving. Alexandria, VA: Association for Supervision and Curriculum Development.

Xin, Y. P. (2008). The effect of schema-based instruction in solving mathematics word problems: An emphasis on prealgebra conceptualization of multiplicative relations. Journal for Research in Mathematics Education, 39(5), 526-551.

LITERATURE STRATEGIES FOR MATH WORDS

Word Wall Math Word Dictionary (When they can

write.) Vocabulary Cards (Emphasize examples

& nonexamples. Do together. Use pictures.)

Semantic Feature AnalysisShape Rolls Slides StacksSphere Y N NCone Y Y NCube N Y Y

STRATEGIES INSTRUCTION

WAYS TO INTRODUCE

ONE WAY ANOTHER WAY

Teacher models the strategy.

Students work on problems using that strategy.

Teacher models the strategy.

Students work on problems which may or may not use the modeled strategy.

TYPES OF STRATEGIES Make a Model or

Diagram Make a Table or List Look for Patterns Use an Equation or

Formula

Consider a Simpler Case

Guess and Check/Test

Work Backward Act it Out Others?

ACT IT OUT WORK BACKWARD

Students pretend to be each person. They form a line and check to see if it fits the criteria listed in the problem.

The final statement puts Ashanti as last. Use an A to represent Ashanti. The second to last statement means that Drew cannot be first. That leaves only Chris, so Chris is first.

“Drew, Ashanti, and Chris line up to get off the bus. (Ashanti is the oldest. Drew is the youngest.)The first person is not the youngest.The oldest person is last.Who is first?”

WHAT DOES RESEARCH SAY? Teaching strategies improves mathematics

problem solving abilities. Teaching strategies does not improve

overall math achievement. Teachers need to avoid teaching strategies

as an algorithm.Rickard, A. (2005). Evolution of a teacher’s problem solving instruction: A case study of aligning teaching practice

with reform in middle school mathematics. Research in Middle Level Education Online, 29(1), 1-15.Higgins, K. M. (1997). The effect of year-long instruction in mathematical problem solving on middle-

school students’ attitudes, beliefs, and abilities. Journal of Experimental Education, 66(1).Jitendra, A., DiPipi, C. M., & Perron-Jones, N. (2002). An exploratory study of schema-based word-problem-

solving instruction for middle school students with learning disabilities: An emphasis on conceptual and procedural understanding. Journal of Special Education, 36(1), 23-38.

Mastromatteo, M. (1994). Problem solving in mathematics: A classroom research. Teaching and Change, 1(2), 182-189.

Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145-166.

PROCESS BY GEORGE PÓLYA

THE ELEMENTS OF THE PROCESS Understand the Problem Make a Plan Follow/adjust the Plan Look Back

UNDERSTANDING THE PROBLEM Define important words. Identify necessary and unnecessary information. Stating what is known and unknown. Determine if other information is needed. Decide if calculations need to be made prior to

another calculation. Rephrase the problem. Consider this: pose a problem situation without

a question.

MAKE A PLAN Select a strategy Use what is known to determine how to

find a solution The goal would be that students be

able to explain, with reasons, why they think their strategy could work.

FOLLOW/ADJUST THE PLAN Students carry out the plan they made. Students show the work that they do,

and they may be asked to write explanations.

Students adjust their plan if they notice something isn’t working or they have determined a better way to solve the problem.

LOOK BACK Very valuable! Check that solution fits problem. Consider strategy choices and their

consequences. Create related problem(s) that could be solved

the same way. Offer changes to the problem and infer their

affect on the solution. Connect to other problems already studied. Make generalizations.

THE TEACHER’S ROLE

ADDRESS MISCONCEPTIONSUndergeneralizations Ex.: A rectangle must be horizontal. Ex.: There is only one way, or very few

ways, to represent 3.Overgeneralizations Ex.: Adding two numbers makes a bigger

number Ex.: Misapplication of regrouping Ex.:

THE ENVIRONMENT Foster a classroom environment friendly to asking

questions. Adjust content to meet student needs. Use a wide variety of activities. Allow time for exploration. Organize and represent concepts in different

ways. Pose probing questions to foster meta-cognition. Model meta-cognition. Promote dialogue.

ASSESSMENT/EVALUATION

PROBLEM SOLVING FORM Creates a framework to help students

see the importance of explaining why they are doing what they are doing.

Use in groups or individually. Can be used in portfolios to share with

parents at conferences or for student self-reflection of progress.

ANECDOTAL RECORDS Teacher notes while observing problem solving Listen for evidence the student seeks

information to fully understand the problem. Consider a student’s ability to persevere. Note use of appropriate strategies. Listen to student oral explanations for

misconceptions or proper conceptual understanding.

Look for algorithmic errors.

CHECKLIST OR RUBRIC Much the same as anecdotal records,

but this may be scored from viewing work.

A rubric may be formed using Pólya’s process or specific to the learning goals of the lesson.

Share whatever rubric you use with students and make sure they understand it.

Provide opportunities for self-evaluation.

QUESTIONS OR COMMENTS?