Everyday Math: The Method Behind the Madness Dover Area School District June 16, 2009 Rina Iati Professional Development Specialist LIU #12

The Final Report of the National Mathematics Advisory Panel 2008 U.S. Department of Education

Math in the US During most of 20 th century, the US possessed peerless mathematical prowess But without substantial and sustained changes to its educational system, the US will relinquish its leadership in the 21st century. The safety of the nation and the quality of life- not just the prosperity of the nation-are at issue

Math in the US 26% of science/engineering degree holders are age 50 or older Demand for employees in this sector expected to outpace overall job growth The combination of retirements and increased demand will stress nation’s ability to sustain adequate workforce

National Assessment of Educational Progress (NAEP) There are positive trends of scores at Grades 4 and 8, which have reached historic highs. However… 32% of our students are at or above the proficient level in Grade 8 23% are proficient at Grade 12 Vast and growing demand for remedial math education for college freshman.

NAEP Percent At or Above Proficiency in PA: Grade 4: 47 in in in 2003 Grade 8 38 in in in 90

Concerns… There are large persistent disparities in math achievement related to race and income. Sharp falloff in math achievement begins as students reach late middle school…where algebra begins Research shows that students who complete Algeba II are more than twice as likely to graduate from college

Startling Statistics 78% of adults cannot explain how to compute interest paid on a loan 71% cannot calculate miles per gallon on a trip 58% cannot calculate a 10% tip for a lunch bill. A broad range of students and adults have difficulties with fractions

Startling Statistics 27% of eight graders could not correctly shade 1/3 of a rectangle 45% could not solve a word problem that required dividing fractions 10% of US fourth graders scored at the advanced level in TIMMS compared to 41% of 4 th graders in Singapore. Close to half of all 17 year olds cannot read or do math at the level needed to get a job at a modern automobile plant. (Murnane &Levy, 1996) National Math Panel report

PreK – 8 curriculum should be streamlined. Use what we know from research –Strong start, develop conceptual understanding, procedural fluency, recall of facts, effort counts. Mathematically knowledgeable classroom teachers Put First Things First Recommendations of the Panel

Practice should be informed by high quality research NAEP and other state assessments should be improved Nation must continue to build capacity for more rigorous research Put First Things First

Recommendations Conceptual Knowledge and Skills (Curricular Content) Learning Processes Teachers and Teacher Education Instructional Practices Assessment Learning As We Go Along…

Major Topics of School Algebra: Table 2 Curricular Content Recommendations

Readiness for Learning Recommendation Importance of early mathematical knowledge Broaden instruction in estimation Curriculum should allow for sufficient time to ensure conceptual understanding of fractions Contrary to Piaget, spatial visualization skills for geometry have already begun on young children Change student beliefs from a focus on ability to a focus on effort

Teachers must know in detail the mathematical content they are responsible for teaching The Mathematics preparation of elementary and middle school teachers must be strengthened Research be conducted on the use of full- time math teachers in elementary schools Teachers and Teacher Education Recommendations

Instruction should not be entirely “student centered” or “teacher directed.” Regular use of formative assessment, particularly in elementary grades The use of Computer-assisted instruction (CAI) for math facts and tutorials Faster pace for mathematically gifted Instructional Practices Recommendations

Publishers must ensure mathematical accuracy of their materials More compact and coherent texts States and districts should strive for greater agreement regarding topics to be emphasized. Instructional Materials Recommendations

NAEP and state tests for students through grade 8 should focus on Panel’s critical Foundations of Algebra Improve NAEP and state tests Assessment of Learning Recommendations

We need more research Research Policies and Mechanisms Recommendations

Standards Aligned System and the Math Big Ideas

The Big Ideas igned_system/4228/mathematics/440539

The 5 Content Standards Number and Operations Algebra Geometry Measurement Data Analysis and Probability

The 5 Process Standards The mathematical process through which students should acquire and use mathematical knowledge Problem Solving Reasoning and Proof Communication Connections Representation

Problem Solving All students should build new mathematical knowledge through problem solving Problem solving is the vehicle through which children develop mathematical ideas. Learning and doing mathematics as you solve problems

Reasoning and Proof Reasoning is the logical thinking that helps us decide if and why our answers make sense Providing an argument or rationale as an integral part in every answer Justifying answers is a process that enhances conceptual understanding

Communication The importance of being able to talk about, write about, describe, and explain mathematical ideas. No better way to wrestle with an idea than to articulate it to others

Connections Connections within and among mathematical ideas. i.e. Fractions, decimals, and percents Connections to the real world, and frequently integrated with other discipline areas

Representations Symbols, charts, graphs, and diagrams, and physical manipulatives should be understood by students as ways of communicating mathematical ideas to other people. Moving from one representation to another is an important way to add understanding to an idea.

learning should build upon knowledge that a student already knows; this prior knowledge is called a schema. learning is more effective when a student is actively engaged in the construction of knowledge rather than passively receiving it. children learn best when they construct a personal understanding based on experiencing things and reflecting on those experiences. Constructivism: A New Perspective

Learning new concepts reflects a cognitive process This process involves reflective thinking that is greatly facilitated through mediated learning. New concepts are not simply facts to be memorized and later recalled, but knowledge that reflects structural cognitive changes. Constructivism: A New Perspective

New concepts permanently change the way students think and learn Balance procedural skills with discussion of concepts Constructivism

1.Discovery Phase 2.Concept Introduction 3.Concept Application 4.Learning Structures 5.Student Assessment 6.Student reflection 7.Authentic Assessment Constructivism

The Math Wars Traditional Reformed

Developing Number Concepts and Number Sense

Make a list of all the things children should know about the number 8 by the time they finish first grade….

Number Grid Puzzles

Common uses of Computation

Algorithm An Algorithm is a systematic, step-by-step procedure used to find an answer, usually to a computation problem.

More or less than 100?

43 x 26 More or less than 1000? 43 x 26

Do the following problems mentally, jotting down intermediate steps if necessary. Share your methods with your group. Your group should be prepared to share at least 2 different methods with the class – x ÷ 17

Developing Number and Operation Sense Look at numbers as whole quantities Look at problems as a whole before choosing a strategy Think from left to right about numbers Use landmarks in the number system Reason from number relationships that you know 5 x 8 = 40, so 7 x 8 = = = 12, so = 120 Use any operation that makes sense for the situation

Students learn new algorithms faster when they first experience an algorithm through alternative visual models and discuss their logic, than when the teacher tells about the algorithm and then drills students on its application Students who are skillful with a particular procedure are very reluctant to attach meanings to it after the fact. Algorithms

The traditional, rote approach to teaching algorithms fosters beliefs such as: Math consists mostly of symbols on paper Following the “rules” of math is most important Math is mostly memorizing facts and rules Speed and accuracy are more important than understanding There is one right way to solve any problem Math symbols and rules have little to do with common sense, intuition or the real world.

our advanced calculus and AP physics students rate last in the world. (Wm. Schmidt, MSU, The Widening Achievement Gap) Our 8th grade advanced students rank in the middle of the pack in the world.(See TIMS and TIMSS-R 1996 and In one study only 60% of 10 year olds achieved mastery of subtraction using the standard “borrowing” algorithm. A Japanese study found that only 56% of third graders and 74% of fifth graders achieved mastery of this algorithm. Traditional math instruction isn’t working as well as we might think.

Research tells us that between 3rd and 6th grade the equivalent of one year is spent teaching long division. Yet, when 17 year olds were tested for division with a 2 digit divisor, fewer than 50% answered it correctly. Children learn to perform the steps (rules) of the algorithm without thinking about the math behind it

900 ÷30 = __ 30 ) =___

Standard Algorithm Partial Sums Column Addition Lattice Addition Opposite Change

Problems can, and should, be solved in more than one way Studying several algorithms for an operation can help students understand the operation Providing several alternative algorithms for an operation affords flexibility Presenting several alternatives gives the message that mathematics is a creative field.

If you consider the traditional instructional model to be handing the kid a shovel and making him expert in using the shovel, then the 21 st century approach is handing the kid a fully-stocked tool shed and training him to select the most appropriate tool from the shed to apply to the math task at hand. The kid who only knows the shovel can only respond if the task is digging, but the kid who owns the tool shed can respond flexibly to any task. Bruce Harrison

Standard Algorithm Trade-First Partial Differences Counting Up Same Change

Standard Algorithm Partial Products Lattice Russian Peasant Same Change

Long Division Friendly Numbers Partial Quotients Column Division