Math+Resources

MATHEMATICALLY POWERFUL STUDENTS —

• Understand the power of mathematics as a tool for making sense of situations, information and events in their world; • Are persistent in their search for solutions to complex, “messy” or “ill defined” tasks; • Enjoy doing mathematics and find the pursuit of solutions to complex problems both challenging and engaging; • Understand mathematics, not just arithmetic; • Make connections within and among mathematical ideas and domains; • Have a disposition to search for patterns and relationships; • Frequently make conjectures and investigate their validity and their implications; • Have “number sense” and are able to make sense of numerical information; • Use algorithmic thinking and are able to estimate and compute mentally; • Are able to work both independently and collaboratively as problem posers and problem solvers; • Are able to communicate and justify their thinking and ideas both orally and in writing; • Use tools available to them to solve problems and to examine mathematical ideas.

The goal of mathematics education should be to produce learners who are both mathematically competent and confident. Mathematical competence does not come from simply memorizing rules and procedures. It comes from understanding mathematical relationships so that you can recognize those relationships and use them to make sense of information, situations, and problems that you encounter. Mathematical confidence does not come from practicing rote procedures until you can do then automatically. Mathematical confidence comes from knowing that you understand mathematics and its beauty and utility, and from knowing that with persistence you can make sense of information and situations you encounter, and that you can solve even complex and messy or ill-defined problems.

WHAT DO WE DO ABOUT ARITHMETIC?

• Develop a sense of number.

• Emphasize mental computation and algorithmic reasoning.

• Recognize that children must be able to work with numbers with accuracy, efficiency, and flexibility. Computational proficiency is still important.

• Recognize that, contrary to what we all grew up believing, the teaching of standard paper-and-pencil algorithms for computation can actually interfere with the development of children’s numerical reasoning … especially when those algorithms are taught prematurely.

• Recognize that mathematics is about sense-making. Instead of teaching mathematics as a collection of isolated procedures to be memorized and followed, encourage children to use numerical relationships to make sense of situations.

• Build many kinds of fluency with small numbers.

• Develop an understanding of our number system.

• Encourage children to talk about their work and ideas.

• Provide meaningful and engaging contexts for drill with number relationships.

• Provide a safe environment for risk-taking.

BUILDING FACILITY WITH NUMBERS: GRADE-LEVEL-APPROPRIATE EXPECTATIONS FOR MENTAL COMPUTATION

Facility with a variety of algorithms for mental computation is an important part of mathematical power. Much of the information that we get about the world comes to us mathematically in the form of statistics and other numerical information. We often receive this information in an auditory mode (listening to the news, talk shows, or advertising while driving a car or watching television) or in a visual mode (reading the newspaper, books and magazines). When we receive numerical information, it is not often while we are sitting with paper and pencil or even a calculator ready to compute in order to determine if the information being presented is reasonable or accurate.

The ability to successfully use mental computation and estimation is essential to making sense of numerical information that surrounds us on a daily basis. In addition to number skills needed to be an informed citizen, it is important to recognize that the needs of today’s workplace demand that we redefine what is important with regard to number or arithmetic. In today’s workplace, most tedious computations are done exclusively by machines. We need workers who understand mathematical concepts well enough to tell the machines what to do, and who have sufficient mental facility with numbers to translate the output of machines and determine whether the results are reasonable. Most of us who learned computation primarily through drill with paper-and-pencil algorithms find that we struggle when it comes to having mental facility with numbers. Children who have had experiences with diverse algorithms or ways of solving number problems, on the other hand, often have a much easier time dealing with numbers and numerical information.

A prevalent misconception is the belief that emphasizing harder computational skills earlier in the grade levels raises standards and results in more academic rigor. In reality, teaching abstract skills prematurely often results in a failure to build the real foundations—the understandings on which children’s mathematical futures will depend. Building fluency with small numbers is all too often under-emphasized as teachers and parents push to get children to the harder concepts. It is fluency and understanding of small numbers that is the foundation of success with larger numbers.

This fluency with small numbers is sometimes referred to as basic facts. I would rather rename them basic relationships, for it is an understanding of the relationships that leads to facility with mental computation. A child who has learned that 7 x 8 is 56 by practicing with flash cards and timed tests will not necessarily have the understandings that help with mental computation when encountering 7 x 8 in the context of a larger problem. A child who has had opportunities to understand relationships of multiplication also sees 7 x 8 as 5 x 8 (40) and 2 x 8 (16) or 40 and 16 (56) or as 7 x 7 (49) + 7 (56). It is the taking apart of numbers that is essential when we encounter larger problems such as 28 x 27. Many of us who learned multiplication through rote drill find that we cannot easily solve such a problem mentally. You might want to stop reading for a moment and try to mentally solve the problem 28 x 27 before reading on. A child who understands relationships has a variety of ways to solve the problem mentally, such as 25 x 28 is 1/4 of 2800 or (700) plus 2 more 28s (56) resulting in 756. Fluency with addition and subtraction of small numbers is equally important to mental computation with larger numbers. Once again, it is the understanding of relationships that leads to proficiency with mental computation. Yes, we want children to know their “basic facts.” It is important that we help children and parents understand the difference between knowing or understanding and simply committing something to memory without understanding the relationships involved. Perhaps an illustration would be useful here. I have memorized e=mc2. I can even tell you that it is related to Einstein’s Theory of Relativity. But because I have only memorized e=mc2, I am unable to recognize its relevance when it might be useful for solving problems, and even unable to use it to solve problems if told that it would be helpful. This information is not useful to me because I have only memorized the information and have no understanding of the relationships involved. Einstein, on the other hand, knew (or understood) e=mc2. He would have been able (and, indeed, was able) to use the information to solve problems.

How do we build the foundation for success with numbers or computation? By identifying those skills or relationships that are appropriate to most children at a given grade level, and by working to gain fluency with smaller numbers before moving to larger computational problems. What follows is my initial attempt to identify grade level appropriate skills that should provide the focus for frequent drill with mental computation. You are invited to help identify those skills that seem most appropriate to your children. I should first define my use of two terms: fluency and ease. When I refer to fluency, I am suggesting that children should be able to take apart and put back together the numbers with little thinking involved. They should have internalized those relationships. When I refer to ease, I am suggesting that children can use number relationships to get accurate results without much of a struggle. They can compute accurately, flexibly and efficiently.

A Focus on Grade Level Expectations………

Please note that I have provided just a few examples at each level. You will want to add further examples so that children get enough practice at each level to develop both confidence and competence in their understanding of the relationships involved, and so that they have a foundation on which to build. Also note that children who have not experienced mental computation on a regular basis or who have only experienced numbers at the symbolic level may not have fluency with small numbers. It will be important at each grade level to develop fluency with smaller numbers.

Two cautions seem in order. First, although we want to provide mental computation practice with smaller problems in order to build a foundation, this in no way implies that children cannot work with larger numbers or solve big problems. The can and should. We want to be sure, however, that we also provide adequate practice building fluency with smaller numbers.

Second, not all children in a given classroom will be ready to move on at the same time. It is important that we find ways to keep all children working to push the boundaries of their understanding. We need to find ways to give children adequate practice at levels appropriate to their individual needs. One way I was able to accomplish this was to work on mental computation with small groups during “menu time” (see About Teaching Mathematics, by Marilyn Burns and Mathematical Power: Lessons from A Classroom, by Ruth Parker). Since “menu” is a learning environment where children are surrounded with a concept and work independently on a variety of problems over a period of several days, I spent some of my time, during menu, working with children who needed extra help with mental computation or who needed challenges that the class as a whole might not have been ready for. If you are not using menu in your classroom, it will be important to find some time when children are working independently so that you can work on mental computation with smaller groups of children at levels appropriate to their needs.


 * Remember that the focus throughout should be on diverse approaches to dealing with numbers. You will want to ask the question, “Who thought of it differently?” repeatedly.

THIRD GRADE Children who have had practice with mental computation and NCTM Standards based instruction at both 1st and 2nd grade levels, should be fluent with numbers to 10 before 3rd grade. If however, your students have experienced only conventional mathematics instruction, you cannot anticipate that they will have this fluency. If they do not, then developing fluency with addition and subtraction of numbers to 10 will be important.

• Fluency with doubles and neighbors –

3 5 6 6 7 9 4 8 9 etc. +4 +4 +6 +7 +8 +8 +5 +8 +10

• Fluency with addition and subtraction of number to 19 –

7 7 8 13 16 16 16 18 19 etc. +9 +8 +6 -7 -5 -7 -9 -9 -8

• Fluency in adding and subtracting 9 or 8 from any number –

14 23 31 42 16 54 68 75 82 etc. -9 -9 -8 -8 -9 -8 -9 -8 -9

• Ease with doubling any number to 15 – Double this number.

7 8 9 13 14 6 15 12 11 6 etc.

• Ease with addition and subtraction of two digit plus-or-minus one digit numbers –

23 46 63 47 62 84 38 76 53 etc. +8 +7 -7 -8 +7 -5 +6 +5 -6

• Ease with addition and subtraction of 10 and multiples of 10 –

9 17 14 27 32 36 63 47 54 87 etc. +10 +10 +10 +10 +20 +40 -10 -27 -30 -50

• Ease with addition and subtraction of two digit plus-or-minus two digit numbers –

23 64 52 63 72 37 37 81 93 48 etc. +57 +57 -18 -26 -27 +37 -28 +35 -56 +66

• Ease with multiplication relationships to 5 x 5 –

3 3 5 2 5 4 2 2 5 etc. x 3 x 4 x 3 x 4 x 5 x 5 x 3 x 5 x 3

FOURTH GRADE All of third grade skills plus the following:

• Fluency in addition and subtraction using any multiples of 100 –

200 247 676 527 676 731 423 423 etc. +300 +300 -400 -300 +397 -298 -107 +197

• Ease with addition and subtraction using landmark numbers such as 25, 50, 75 –

25 125 125 125 125 150 125 26 51 +50 +75 +50 -50 -75 -75 -75 +75 +75

• Ease with addition and subtraction of three digit plus-or-minus three digit numbers –

278 621 547 603 600 541 541 358 etc. +357 -256 +396 -289 -286 +397 +285 -169

• Fluency in multiplication of numbers to 6 x 6 –

3 4 5 3 6 3 3 6 6 6 etc. x 6 x 5 x 6 x 4 x 2 x 5 x 3 x 4 x 3 x 6

• Ease with one digit by one digit multiplication –

5 8 4 7 7 7 8 7 6 9 etc. x 9 x 5 x 8 x 4 x 9 x 6 x 9 x 8 x 8 x 3

• Fluency in multiplying a number by 10 and 100 –

13 17 15 23 64 86 8 15 100 37 etc. x 10 x 10 x 10 x 10 x 10 x 10 x 100 x 100 x 21 x 100

• Fluency in dividing multiples of 10 by 10 –

etc.

• Fluency in doubling any number to 20 – Double this number.

8 12 7 15 13 14 18 20 17 19 etc.

• Ease doubling any number to 50 –

21 24 30 25 23 32 43 33 41 27 etc.

• Beginning practice multiplying two digit numbers by one digit numbers –

11 13 16 15 14 14 14 18 17 etc. x 6 x 5 x 4 x 6 x 7 x 3 x 6 x 8 x 3

FIFTH GRADE All of third and fourth grade skill plus the following:

• Fluency in halving any number to 10 – What is half this number?

4 3 5 1 6 9 8 7 2 10 etc.

• Fluency in halving any multiple of 10 – What is half this number?

40 60 30 70 90 80 110 50 20 120 etc.

• Ease in halving any number to 20 – What is half this number?

14 11 12 15 16 20 18 17 19 13 etc.

• Ease in finding a quarter of any number to 10 – What is one-fourth of this number?

1 2 4 6 8 3 5 7 9 10 etc.

• Ease in finding a quarter of any number to 20 – What is one-fourth of this number?

12 11 16 14 15 18 17 19 20 13 etc.

• Ease in doubling any number to 100 – Double this number.

22 25 35 23 46 28 56 65 69 71 etc.

• Ease with combinations that make 100 – Given this number, what will it take to make 100?

32 56 24 27 61 58 43 78 82 46 etc.

• Ease in multiplying two digit by one digit numbers –

23 54 86 91 47 74 35 34 46 37 etc. x 6 x 7 x 6 x 5 x 6 x 7 x 8 x 6 x 4 x 9

• Ease in multiplying two digit by two digit numbers (keeping track with a pencil might be necessary for some of these problems) –

48 22 24 53 65 48 44 51 76 53 etc. x 25 x 31 x 46 x 48 x 27 x 26 x 43 x 34 x 52 x 84

• Ease with two digit by one digit division problems –

etc.

• Ease with three digit by one digit division problems (keeping track with a pencil might be necessary for some of these problems) –

etc.

• Practice with division problems using landmark numbers 10s, 25, 50, 75 –

etc.

• Practice using fractions, decimals, and percents – Is it closer to 0, 1/2, or 1?

1/4 2/3 .4 .17 .63 4/5 11/13 1/8 2/5

Estimate the answer to:

12/13 + 8/9 3/6 + 7/8 .5 + 1.4 3/4 + 1/2 2/5 + 4/5

4/3 + 2/6 3 – 1.4 .8 – 1 1/2 + 5/8 etc.

MATERIALS REFERENCED IN RUTH PARKER’S TALK

Math By All Means, a series of books for grades 2 - 4, Marilyn Burns. These are multi-week units of study in the areas of geometry, place value, multiplication, division, probability, time and money. Available through ETA Cuisenaire (800) 445-5985 www.etacuisenaire.com

A Collection of Math Lessons, from grades 3 – 6, Marilyn Burns. This is the source of the multiplication lessons with rectangles and patterns in the multiplication tables. For information see above.

Developing Number Concepts, Kathy Richardson. This series of books and the Richardson reference that follows focus on building fluency with number with young children. Available through ETA Cuisenaire (800) 445-5985 www.etacuisenaire.com

Math Time: The Learning Environment, Kathy Richardson. Available through Teaching Resource Center, (800) 833-3389. www.trcabc.com

Mathematical Power: Lessons from a Classroom, Ruth Parker. This book is the result of a year spent as a researcher in a fifth grade classroom. It addresses the need for complex changes in the teaching of mathematics and follows a teacher as she works to make those changes in her classroom. Available at Heinemann Press, (800) 225-5800 http://www.heinemann.com/

Seeing Fractions, Susan Jo Russell and Rebecca Corwin. A 6-8 week unit on fractions appropriate for grades 4–6. This is a unit that you saw children’s work from during the parent session. A good illustration of what good mathematics curriculum can look like. Out of print; may be available through http://www.amazon.com. Originally published by Calif. Dept. of Education Press, 800-995-4099, http://www.cde.ca.gov/cdepress. The unit is now incorporated into the K – 5 textbook series, Investigations in Number, Data & Space, published by Dale Seymour, (800) 872-1100.

Why Numbers Count: Quantitative Literacy for Tomorrow’s America, Lynn Arthur Steen, The College Board, 1997, ISBN 0-87447-577-5.

Some Quality Mathematics Resources for Parents………..

GRADES K – 8 About Teaching Mathematics, Marilyn Burns. This book is a rich resource for great problems across the strands of math. It’s usually the first resource I use when looking for good math problems for all my nieces and nephews. Available at ETA Cuisenaire (800) 445-5985 www.etacuisenaire.com . Beyond Facts & Flashcards: Exploring Math With Your Kids, Jan Mokros. This is a book that suggests many practical, everyday ways of exploring mathematics as a family. Available at (800) 225-5800 http://www.heinemann.com/ ISBN 0-435-08375-9.

Family Math, Lawrence Hall of Science, Berkeley. Available through ETA Cuisenaire (800) 445-5985 www.etacuisenaire.com

The I Hate Math Book, Marilyn Burns. Full of fun math problems to work on. Available through ETA Cuisenaire (800) 445-5985 www.etacuisenaire.com