Grades 3-8. Creating a Classroom of Mathematics Problem Solvers. Grace. Ben. Jonah. Tricia. Mahreah. Ruby. Bain Family.
This 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.
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.My Background
Why is math important for our country? in education and music.
Why is math important for our church?
Why is math important for our students?
Foundations 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)National Mathematics Advisory Panel
NCSM Mathematics Advisory Panel
(National Council of Supervisors of Mathematics)
(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”Definitions of problem solving
Meir Ben- Mathematics Advisory PanelHur
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 communicatingFeatures of Problem Solving
Key Words Mathematics Advisory Panel
Process by George Pólya
What words tell a person to multiply? Mathematics Advisory Panel
What words tell a person to subtract?
What words do you notice are particularly troublesome to your students?
“Two flags are similar. One flag is three times as long as the other flag. The length of the smaller flag is 8 in. What is the length of the larger flag?”An Example of Teaching Key words
Undermines real problem solving the other flag. The length of the smaller flag is 8 in. What is the length of the larger flag?”
Make problem solving a mechanical process which makes students prone to errors
Understanding the language of mathematics is important.
“Two flags are similar. One flag is three times as long as the other flag. The length of the larger flag is 8 in. What is the length of the smaller flag?”
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.What does research say?
Word Wall the other flag. The length of the smaller flag is 8 in. What is the length of the larger flag?”
Math Word Dictionary
Semantic Feature AnalysisLiterature Strategies for Math Words
One Way the other flag. The length of the smaller flag is 8 in. What is the length of the larger flag?”
Using a Formula the other flag. The length of the smaller flag is 8 in. What is the length of the larger flag?”
Make a Model or Diagram
Perimeter of a square : P = 4S.
Using this formula, students could determine the side lengths for each of the squares as 10 inches and 9 inches.
Area of a square: A = S2.
Larger square = 100 in.2
Smaller square = 81 in.2
The difference between the areas of the two squares is found by subtracting the smaller area from the larger area.
Use graph paper to draw one square inside the other.
Count the squares to find the difference.
“One square has a perimeter of 40 inches. A second square has a perimeter of 36 inches. What is the positive difference in the areas of the two squares?”
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.What does Research Say?
Understand the Problem abilities.
Make a Plan
Follow/adjust the Plan
Look BackThe Elements of the Process
Define important words. abilities.
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.Understanding the Problem
Select a strategy abilities.
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.Make a Plan
Students carry out the plan they made. abilities.
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.Follow/adjust the Plan
Very valuable! abilities.
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.Look Back
Ex.: 3:4 and ¾ are different things
Ex.: “=“ only means perform operations to find answer
Ex.: Multiplying two numbers makes a bigger number
Ex.: Misapplication of regrouping
Ex.: Finding common denominators when multiplying fractions.Address Misconceptions
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.
Promote dialogue.The Environment
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.Problem Solving Form
Teacher notes while observing problem solving explaining why they are doing what they are doing.
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.Anecdotal Records
Much the same as anecdotal records, but this may be scored from viewing written 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.Checklist or Rubric