1. Pursuing Wisdom in the Mathematics Classroom RUSMP Spring 2011 Networking Conference by Arthur C. Howard Mathematics Teacher Providence Classical School

2.   Loosely define mathematical Wisdom.  Discuss some practices that may limit or prevent the growth of wisdom in student thinking, along with …  Some proposals that are likely to foster its growth.  Some resources to help you develop wise students. Pursuing Wisdom

3.   Know how to do necessary calculations?  Memorize multiplication facts?  Know how to solve equations? Mathematical Wisdom

4.   Means different things at different ages.  Involves seeing the “big picture.”  Discovering transcendent patterns in mathematics. Mathematical Wisdom

5.   Primary Grades  Realize that collections of things can be categorized in different ways.  Understand how “2 + 1 = 3” can be used to find “12 + 1 = 13” and “20 + 10 = 30”  Defend “8 + 5 = 13” by regrouping Base Ten blocks.  Recognize that addition and subtraction are related to each other. Mathematical Wisdom

6.   Intermediate Grades  Understand the relational nature of arithmetic.  Determine what operation best fits a verbal situation.  Defend an answer as reasonable. Mathematical Wisdom

7.   Middle School Grades  Understand the relational nature of Arithmetic.  Determine if an answer is reasonable and if it answers the question that was asked.  Gives an example of infinity. Mathematical Wisdom

8.   High School Grades  Recognize linear and non-linear behavior in words and data.  Understand the relational nature of functions.  Ask the right questions.  Know where to start.  Understand the inherent risk of extrapolation. Mathematical Wisdom

9.   Can be fostered.  Classrooms that welcome thinking.  Classrooms that welcome questioning.  Classrooms that foster conjecturing and defending. Mathematical Wisdom

10.   Unquestioning acceptance of any answer.  Unreasonable answers.  Incomplete answers.  Answers with the wrong label, or no label at all.  Having the necessary tools, but not knowing how to use them, or that they even should be used.  Data used in inappropriate places.  Not knowing how to use data to construct a model.  Inability to solve a problem presented “out of sequence” or in a different context. Evidence of a Problem

11.   Not understanding basic properties of mathematics.  Misuse of Distributive Property and laws of exponents.  Confusing division of 0 and division by 0.  Poor understanding of integers and absolute value.  Confusing multiplication and division of fractions. Evidence of a Problem

12.   Procedural thinking VS Logical thinking Evidence of a Problem

13.   “Top times top and bottom times bottom.”  Change the wording.  “Three-fourths of two-thirds.” Teaching tricks

14.   Ours is not to reason why. Just invert and multiply!  Change the wording.  “How many times is 4 contained in 12?”  How many times is contained in 1?  How many ones do I have? Teaching Tricks 2 5 = 10

15.   How many are there in 1?  How many ones? Teaching Tricks 12

16.   Cross-multiply Teaching Tricks

17.   Cross-multiply Teaching Tricks

18.   Cross-multiply Teaching Tricks

19.   Cross-multiply Teaching Tricks

20.   Addition and subtraction of fractions are done in the same way as addition and subtraction of whole numbers. Disconnected Math

21.   Teach that dividing by three is the same as taking of a number.  What is of 20?  will be 3 times as much. Disconnected Math 5 15

22.   Teach the WHY, not just the WHAT. Some More Proposals

23. 

24.   Teach the WHY, not just the WHAT. Some More Proposals indeterminate

25.   Model good problem solving.  Do Think-Alouds with Commentary. Some More Proposals The length of a rectangle is 5 cm longer than the width. If the perimeter is 38 cm, what are the measures of the length and the width? ww w + 5

26.   I. Hand-Holding Phase  Solve good problems.  Work the problem with them, showing every step.  II. Independent Phase  Give good problems.  Monitor as they work.  Insist they use the techniques you modeled.  III. No-Holds-Barred Phase  Give good problems.  Solve any way possible.  Use algebra as a last resort. Problem-Solving Steps

27.   Let students struggle with novel problems. More Proposals

28.   Let students struggle with novel problems. More Proposals

29.   Make connections to other disciplines.  Review often.  Provide opportunities for mental math.  Give “Challenge” problems.  Give students time to THINK. More Proposals

30.  Those who are wise think out of the boxquestion Those who innovatecreateleadmake a difference

31.   Teach for understanding.  Encourage, praise and reward questioning and conjecturing.  Provide for problem-solving experiences not tied to the curriculum. Putting it all together:

32.   Do something with your students starting Monday.  Convince colleagues by example.  Spread the movement to every grade in your school.  Dialogue with your feeder pattern schools.  Creating mathematically wise adults is a K – 12 effort. Make it happen!