Developing Math Sense for the 21 st Century June 11, 2008 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

TIMMS 4th Grade In 2003, U.S. fourth-grade students exceeded the international averages in both mathematics and science. In mathematics, U.S. fourth-graders outperformed their peers in 13 of the other 24 participating countries, and, in science, outperformed their peers in 16 countries.

In 2003, U.S. eighth-graders exceeded the international average in mathematics and science. U.S. eighth-graders outperformed their peers in 25 countries in mathematics and 32 countries in science. TIMSS 8 th Grade

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 7% of US fourth graders scored at the advanced level in TIMMS compared to 38% 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)

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 1 Curricular Content

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

Standards Aligned System and the Math Big Ideas

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.

The Big Ideas

Grade Level Competencies

Fractions lesson

How many 12s are in 36? 36 ÷ 12 = 4 How many 40s are in 360? 360 ÷ 40 = 9 Draw a picture to show how many 5s there are in 15.

How many ½ s fit in a whole? 1 ÷ ½ = 2

How many ¾ fit in a whole?

How many 5/6 fit in a whole?

1 ÷ ½ = ____ 1 ÷ 3/4 = ____ 1 ÷ 5/6 = ____ What is consistent in these cases? Do you think it is always true? How does it happen to be so?

Write a mathematical rule for dividing one by a fraction.

Reciprocal What is the question (math problem) that the reciprocal provides the answer to? 1 ÷ a/b = b/a

Reciprocal 1 ÷ ¾ = 4/3 2 ÷ ¾ = double(1 ÷ ¾) 2 ÷ ¾ = double(4/3) 2 ÷ ¾ = 2 x 4/3

2 ÷ ¾ = 2 x 4/3 =8/6

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….

What is Number Sense? A person's ability to use and understand numbers: * knowing their relative values, * how to use them to make judgments, * how to use them in flexible ways when adding, subtracting, multiplying or dividing * how to develop useful strategies when counting, measuring or estimating.

Spend a few minutes discussing the information on your handout, particularly as it relates to your own experiences. Design a 5 minute presentation for the whole group. Developing Number Sense through Counting

Spatial Relationships Children can learn to recognize sets of objects in patterned arrangements and tell how many without counting. For most numbers there are several common patterns. Patterns can also be made up of two or more easier patterns for smaller numbers. Dot Plate Flash Learning Patterns Dominoes

One and Two More, One and Two less Involves more than just the ability to count on or count back 2. Make a Two-More-Than Set More or Less Calculator

Anchoring Numbers to 5 and 10 Since 10 plays such a large role in our numeration system and because 2 fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10. Five Frame Tell About Crazy Mixed-up Numbers Ten-Frame Flash

Part-Part-Whole Relationships To conceptualize that a number is made up of two or more parts. Build it in parts Two out of Three Covered Parts Missing Card Parts I wish I had

An increased facility with a strategy but only with a strategy already learned A focus on a singular method and an exclusion of flexible alternatives A rule-oriented view of what mathematics is all about. A false appearance of understanding Drill will NOT provide any new strategies or skills. Drill only focuses on what is already known. Repetitive, non-problem based exercises designed to improve skills or procedures already acquired

An increased opportunity to develop conceptual ideas an more elaborate and useful connections An opportunity to develop alternative and flexible strategies A greater chance for all students to understand, not just a few A clear message that math is about figuring things out and making sense Different problem based tasks or experiences, spread over numerous class periods, each addressing the same basic ideas.

Take a few minutes to examine the student journal/workbook/ text. Are they drill, practice, or both?

Mastering the Basic Facts

One-More-Than, Two- More- Than

When Tommy was at the circus he saw 8 clowns come out of a little car. Then two more clowns came out on bicycles. How many clowns did Tommy see in all? One-/Two-More-Than Dice One-/Two-More-Than Match One-More-Than, Two- More- Than

Facts with Zero What’s Alike? Zero Facts

Doubles Double Images Calculator Doubles

Near-Doubles Double Dice Plus One

Make-Ten Facts The Ten Frame

The Last Six Facts

Doubles plus Two, or Two apart facts Make-Ten extended Counting On ????? The Last Six Facts

The 36 “Hard” Subtraction Facts

Doubles Multiplication

Fives Facts

Zeros and Ones

Nifty Nines

Helping Facts

When and How to Drill? Do not subject any student to fact drills unless the student has developed an efficient strategy for the facts included in the drill. Drill in the absence of strategies has proven to be ineffective.

Timed Tests? Cannot promote reasoned approaches to fact mastery Will produce few long-lasting results Reward few (the goal –oriented, under pressure) Punish many Should generally be avoided Can be used to diagnose which combinations are mastered, but not often.