“When parents and teachers alike believethat hard work pays off and whenmathematics is taught and learned byusing all the strands of proficiencymathematics performance improves forall students.”Helping Children Learn Mathematics(2002) Kilpatrick and Swafford
At The Crossroads—Meeting the Challenges of a Changing World — ISM 2012
The world we know ischanging 75 % of jobs will be in STEM Not just STEM careers, it is STEM in every job Technology as a “global knowledge economy” is the future, and it requires different skills. Business and industry want employees with these skills! OECD ISM 2012
Our Challenge The top 10 in-demand jobs in 2010 did not exist in 2004. Estimated that todays learner will have 10-14 jobs by the age of 38. The amount of new technical information is doubling every 2 years For students starting a 4 year technical degree this means that half of what they learn in their first year of study will be outdated by their third year of study. ISM 2012
21st Century Learning“We are responsible for preparing students to address problems we cannot foresee with knowledge that has not yet been developed using technology not yet invented.” APEC ISM 2012
Scenario My child doesn’t know his number facts. Why aren’t we teaching the basics and teach REAL math.ISM 2012
Myths and Facts Why Has Mathematics Instruction Changed? ISM 2012
Myth Fact Some people Every adult and just can’t do every child can math. do mathematics.ISM 2012
Myth Fact Literacy is Literacy is far reading and broader, includin writing. g an understanding of graphs, tables, s tatistics, probabil ity...ISM 2012
Myth Fact Students learn Students learn math best math best when through they are active teacher- participants in directed the learning lessons. process.ISM 2012
Myth Fact The use of When used calculators appropriately, cal will have a culators improve negative computation impact on skills. students’ computation skills.ISM 2012
Myth Fact Manipulatives Manipulatives are for young help connect abstract thought children. to something tangible --- for learners of ALL ages.ISM 2012
Myth Fact The school Rigor does not equate to program is not difficulty. rigorous.ISM 2012
What is mathematics? Mathematics is a way of thinking about, understanding, explaining, and expressing phenomena. Mathematics is about inquiry and insight. Computation is (usually) a means to an end.SM 2012 I
How fluent are you inmathematics?Add 346 and 484 62- 381½ x 2¼ 2 ÷⅜43 X 37 196 ÷ 18 ISM 2012
Reading Fluency Efficiency FLUENCYProsody AccuracyFluency is the ability to read with sufficient ease and accuracy thatone can focus attention on the meaning and message of text.” Adams, 2002 ISM 2012
MATH FLUENCY (Russell, 1999) Efficiency: Accuracy: Student does not get A working knowledgebogged down into too many steps of number facts, combinations,or lose track of logic or strategy. and other important number (WORKING MEMORY) relationships. (AUTOMATIC RETRIEVAL) FLUENCY Flexibility: Knowledge of more than one approach to problem solve. Allows student to choose appropriate strategy and to double check work. (EXECUTIVE FUNCTIONING) ISM 2012
What is your number sense?Add 346 and 484 62- 381½ x 2¼ 2 ÷⅜43 X 37 196 ÷ 18 ISM 2012
What is Number Sense?A “good intuition about numbers andtheir relationships.It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989). ISM 2012
Learning The knowledge base for classroom assessment is closely tied to contemporary theories of learning -----about how knowledge is organized in the mind and about how participation in communities of practice shapes understanding. How People Learn Bransford, Brown, & Cocking, 1999, NRCISM 2012
Learning Progressions describe skills, understandings, and knowledge in the sequence in which they typically develop: a picture of what it means to ‗improve‘ in an area of learning.‖ (Masters & Forster, 1996).ISM 2012
Relevant Research “When students compute with strategies… they choose because they are meaningful …they are able to remember and apply their knowledge.” Baroody, Arthur J. “The Development of Basic Counting, Number, and Arithmetic Knowledge among Children Classified as Mentally Handicapped.” International Review of Research in Mental Retardation, edited by LaraineMasters Glidden. pp.51-103. New York: Academic Press, 1999.ISM 2012
How Students Learn “Can engage in instructional activities but teaching has not occurred until student learning has occurred“ “…covering the material and explaining it well is NOT the same as the student learning it.” NRCISM 2012
Learning Mathematics For all students to become mathematically proficient, major changes must be made in instruction, assessments, teacher education, and the broader educational system. ◦ Adding It Up (NRC)ISM 2012
Building Mathematical Concepts Concrete Pictorial AbstractManipulatives Representation Symbols IIII 4+4=8 2x4=8 IIII ISM 2012
The Bridge To Understanding Representation/Pictorial ―SEEING‖ StageConcrete Abstract―DOING‖ Stage ―SYMBOLIC‖ Stage ISM 2012
According to Piaget (1949, 1958), children learn primarily by manipulating objects until the age of 12. If children are not taught math with hands-on methods, between years 1 and 12, their ability to acquire math knowledge is disturbed at the point when hands-on explorations were abandoned in favor of abstractions. This clearly sets them up for mathematical disabilities in the next developmental period of formal propositional operations. Center for Teaching/Learning Mathematics, 1986ISM 2012
Mathematical thinking . . .A gateway to higher mathematics? OR A wall blocking path for students? ISM 2012
Knowing Understanding“Understanding is the key to remembering what is learned and being able to use it flexibly.” - Hiebert, in Lester & Charles, Teaching Mathematics through Problem Solving, 2004. ISM 2012
Students Can Do Basics, ... 347 + 453 90% 864 – 38 73% … But Students Cannot Solve Problems Ms. Yost’s class has read 174 books, and Mr. Smith’s class has read 90 books.33% How many more books do they need to read to reach the goal of reading 575 books? Source: NAEP 2009 ISM 2012
Explain your thinking• A furniture maker was putting stools and chairs together. The stools were the three- legged kind and the chairs had four legs. If the seats were the same for the stools and chairs, how many of each could he make if he had 21 seats and 72 legs?
Explain your thinking• 2/3 of my students play the piano.• 1/2 of the students who play the piano also play the violin.• 1/4 of those students also play the flute.• What fraction of my students play all three instruments? ISM 2012
Math Literacy = Just as knowing the definitions of words does not make a person literate, knowing the rules and algorithms to solve mathematics problems does not make a person mathematically literate. ISM 2012
Conceptual Understanding Procedural Problem Fluency SolvingISM 2012
ConnectingProcedural Knowledge with Conceptual Understanding • What misconception(s) is the student holding? Measurement • What key concept is the Place Value student missing? • How might this misconception have Operation Sense developed? and Algebra Decimal • What types of activities Fractions and conversations help Common students construct correct Fractions ISM 2012 concepts?
A Learning Progression Model: Learning Progression: A learning progression is a sequenced set of subskills and enabling knowledge that, it is believed, students must master en route to mastering a more remote curricular aim. (Popham 2008)ISM 2012
Scenario Parent: How can I help my child with math? It is so different from when I went to school.ISM 2012
Changes• Textbooks and learning materials• Types of questions posed• Types of responses needed• Assessment and grading• Uses of class time
Yes, I can …Talk about math in positive ways I’m sure you will I could never do understand if the math either. you… Don’t worry about Let’s figure it out the math. together.
Yes, you can …Support your child through homeworkchallenges by listening and askingquestions
Yes, I can …• Allow my child to struggle through the process of problem solving.• Discuss mistakes as learning opportunities.
Questions to Ask …• What do you need to find out?• Tell me what you know?• Show me what you started?• What can you try first?• Can you make a drawing or picture?• Will a list or table help?
Prompts …• Ask your child to show you how he/she is finding the answer.• Share your method.• Explain to each other why your different methods are successful. Providing shortcuts for getting the answer might hinder deeper student understanding.
Manipulatives and Technologies Their Role in Teaching and Learning MathematicsISM 2012
Research … “Manipulatives support the conceptual development of important mathematical ideas for tactile and visual learners.”ISM 2012
Research … • Students learn in the presence of technology. • Technology should influence the mathematics content taught and how it is taught.ISM 2012