Research-based MathFor Kindergarten

“Knowing mathematics means being able to use it in purposeful ways. To learn mathematics, students must be engaged in exploring, conjecturing, and thinking rather than only in rote learning of rules and procedures. Mathematics learning is not a spectator sport. When students construct personal knowledge derived from meaningful experiences, they are much more likely to retain and use what they have learned. This fact underlies teachers’ new role in providing experiences that help students make sense of mathematics, to view and use it as a tool for reasoning and problem solving.”

Curriculum and Evaluation Standards for School Mathematics: Executive Summary, National Council of Teachers of Mathematics, (1989)

What Does Good Math Instruction Look Like?It involves good teachers, an effective math environment, and a curriculum that is more than a mile wide and an inch deep.

Research Report, Principal, September/October 2007

Teach for … Conceptual understanding

Developing children’s procedural literacy

Promoting strategic competence through meaningful problem-solving investigations

It is not necessary for teachers to focus first on skill development and then move on to problem-solving. Both can be done together.

What Does Good Math Instruction Look Like?

An Effective Mathematics Environment

Demonstrate acceptance of students’ divergent ideas (1)

Influence learning by posing challenging and interesting questions (2)

Project a positive attitude about mathematics and about students’ ability to ‘do’ mathematics(3)

Students are actively engaged in doing mathematics &

Students are solving challenging problems (4)

What Does Good Math Instruction Look Like?

An Effective Mathematics Environment

Interdisciplinary connections and examples are used to teach mathematics (4)

Students are sharing their mathematical ideas while working in pairs and groups (3)

Students are provided with a variety of opportunities to communicate mathematically (2)

Students are using manipulatives and other tools (1)

“Students learn from the kind of work they do during class, and the tasks they are asked to complete determines the kind of work they do. Work in Mathematics Classes: The Context of Students’ Thinking During Instruction.” Educational Psychologist, Vol.23, Doyle, (1983,1988)

Tasks should encourage reflection and communication Processes through which understanding develops

Tasks should allow students to use tools Skills that have been acquired

Physical materials

Written symbols

Verbal language

Tasks should leave behind important residue Learning that students take with them from solving problems

The single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students. Making Sense Hiebert et al., (1996)

Students solve problems not to apply mathematics but to learn new mathematics

A problem is any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific correct solution method.

The problem must begin where students are.

The problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn.

The problem must require justifications and explanation for answers and methods.

Mathematics is to be taught through problem solving.

A rote procedure with a model is still just that, a rote procedure. Magical hopes: Manipulatives and the reform of math eduation, American Educator, Ball,(1992)

The most widespread error that teachers make with manipulative materials is to structure lessons in such a manner that students are being directed in exactly how to use a model.

Focus should be

On solving problems, not getting answers

Exploring concepts

Construction of understanding

Pocket Problems

Learning Goal: Students will draw pictures to solve problems.

Procedure:

1. Distribute paper and pencils to students.

2. Explain to students that you will be placing some problems in the pocket chart for them to solve. Remind students that they are to sketch a picture to help them solve the problem.

3. After students have had ample time to solve the problem, discuss their solutions. Discuss the similarities and differences among representations. Allow students to explain and justify their solution pictures.

Four features of a productive classroom culture for mathematics..

Making Sense Hiebert et al., (1997)

Ideas are important, no matter whose ideas they are. Students have their own ideas and share them with others and understand that they can also learn from the ideas that others have formulated. Learning mathematics is about coming to understand the ideas of the mathematical community.

Ideas must be shared with others in the class. Respect for the ideas shared by others is critical if real discussion is to take place.

Trust must be established with an understanding that it is okay to make mistakes. Errors are an opportunity for growth as they are uncovered and explained.

Students must come to understand that mathematics makes sense. There is no need for the teacher to provide judgment of student answers.

Children learn math mostly from oral language, rather than textbooks or mathematical symbolism. Teaching English Language learners.

Review of Educational Research, Janzen, (2008)

“Talking Math” is far more than just using math vocabulary Address the language of mathematics, not just the

‘terms’ of mathematics

Relationships and reasoning, not just vocabulary

Incorporate “funds of knowledge” to develop mathematical contexts and understanding

Math used in building, farming, cooking, playing, etc.

Children who have difficulty translating a concept from one representation to another are the same children who have difficulty solving problems and understanding computations.

Problems of representation in the teaching and learning of mathematics, Lesh, Post and Behr, (1987)

Five Representations

As children move between and among these representations for concepts, there is a better chance of a concept being formed correctly and integrated into a rich web of ideas.

Pictures

Written Symbols

Oral Languag

e

Real-world

Situations

Manipulative

Models

The most important content goals for early mathematics are the big ideas of mathematics education—ideas that connect concepts and skills within a domain and across domains.

Engaging Young Children in Mathematics: Standards for early childhood mathematics education, Clements, Sarama,& Dibiase, (2004)

Number Sense is a good intuition about numbers and their 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.

Teaching number sense. Arithmetic Teacher, Howden (1989)

“Is it Counting or is it Adding?”

NCTM

Read the article independently

Big Ideas of Early Number Sense

Subitize – the ability to recognize small quantities without having to count objects

Quick images (dot cards, pictures, dominoes, dice, rekenreks)

Five Frames, Ten Frames

Memory/Concentration

Counting – rote recite numbers in ascending, and descending order

Counting Poems and Finger Plays

Counting Books

Count Around the Circle

Counting Children, snacks,

The Big Ideas of Early Number Sense

One to one correspondence – counting a collection using one and only one number word per object

Counting with Rhythmic movement

Counting Jars/Bags

What’s Hiding

Can you Find it?

Sorting Number sets

Cardinality – understanding that the last counting word represents the total of all the items in the group

Choral Counting of objects

The Big Ideas of Early Number Sense

Hierarchical inclusion – numbers build by exactly one each time--all numbers preceding a number can be or are systematically included in the value of that number (a subset is part of a larger set)

Counting books- building sequential sets

Conservation - the concept that objects remain the same in amount despite perceptual changes.

Sorting Number Sets

Compose numbers- to combine parts or smaller values to form a number and Decompose numbers - to break a number into parts or smaller values

Rekenreks – Guess My Way

Five Frames, Ten Frames

Ten Wand

SWAP MEET

The role of the teacher??

Work with your table groups and create a list of responsibilities of a teacher in a research-based kindergarten classroom.

References AIMS Educational Foundation, Solve It! K-1 Problem Solving Strategies (2007)

Booth, R. S. Siegler, Developmental and individual differences in pure numerical estimation. Developmental Psychology 41, 189 (2006)

Bodovski, G. Farkas, Mathematics growth in early elementary school: The roles of beginning knowledge, student engagement, and instruction. The Elementary School Journal 108, 115 (2007).

Baroody, A.J., & R.T. Coslick. 1998. Fostering children's mathematical power: An investigative approach to K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum.

Bowman, B.T., M.S. Donovan, & M.S. Burns, eds. 2001. Eager to learn: Educating our preschoolers. Washington, DC: National Academy Press.

Bresser & Holtzman, Minilessons for Math Practice (2006)

Campbell, & Silver, Teaching and learning mathematics in poor communities. (National Council of Teachers of Mathematics, Reston, VA, 1999).

Carlyle & Mercado, Teaching Preschool and Kindergarten Math (2012)

Chittenden, E. 1991. Authentic assessment, evaluation, and documentation of student performance. In Expanding student assessment, ed. V. Perrone, 22-31. Alexandria, VA: Association for Supervision and Curriculum Development.

References

Clements, D.H., & Conference Working Group. In press. Part one: Major themes and recommendations. In Engaging young children in mathematics: Findings of the 2000 National Conference on Standards for Preschool and Kindergarten Mathematics Education, eds. D.H. Clements, J. Sarama, & A.-M. DiBiase. Mahwah, NJ: Lawrence Erlbaum Fosnot & Dolk, Young Mathematicians at Work (2001)

Copley, J.V., & Y. Padròn. 1999. Preparing teachers of young learners: Professional development of early childhood teachers in mathematics and science. In Dialogue on early childhood science, mathematics, and technology education, ed. G.D. Nelson, 117-29. Washington, DC: American Association for the Advancement of Science.

Entwisle, &Alexander, Beginning school math competence: Minority and majority comparisons. Child Development 61, 454 (1990).

Griffin, S., Case, R., & Siegler, R. (1994). Rightstart: Providing the central conceptual prerequisites for first formal learning of arithmetic to students at risk for school failure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroom practice (pp. 24–49). Cambridge, MA: MIT Press.

Hiebert, et al., Making Sense (1997)

References Horm-Wingerd, D.M., P.C. Winter, & P. Plofchan. 2000. Primary level assessment for

IASA Title I: A call for discussion. Washington, DC: Council of Chief State School Officials

Janzen, Teaching English language learners. Review of Educational Research 78, 1010 (2008).

Kaplan, Prospective preschool teachers' ideas for developing the number sense of children, Procedia - Social and Behavioral Sciences 46 (2012)

Lee, Racial and ethnic achievement gap trends: Reversing the progress toward equity? Educational Researcher 31, 3 (2002)

Lindquist, M.M., & J.N. Joyner. In press. Moving ahead in support of young children's mathematical learning: Recommendations to conference organizers and participants. In Engaging young children in mathematics: Findings of the 2000 National Conference on Standards for Preschool and Kindergarten Mathematics Education, eds. D.H. Clements, J. Sarama, & A.-M. DiBiase. Mahwah, NJ: Lawrence Erlbaum.

NAEYC, Early Childhood Mathematics: Promoting Good Beginnings A joint position statement of the National association for the Education of Young Children and the National Council of Teachers of Mathematics, (2002)

National Council of Teachers of Mathematics. 2000. Principles and standards for school mathematics. Reston, VA: Author.

References National Council of Teachers of Mathematics, Mathematics in Early Childhood

Learning, (October 2013)

National Research Council, Mathematics in early childhood: Learning paths toward excellence and equity. C. T. Cross, T. A. Woods, H. Schweingruber, Eds., (National Academy Press, Washington, DC, 2009).

Parrish, Number Talks (2010)

Pratt, C. 1948. I learn from children. New York: Simon and Schuster.

Protheroe, What Does Good Math Instruction Look Like? Principal, (Sept/Oct 2007)

Richardson, Developing Math Concepts in Pre-Kindergarten (2008)

Shonkoff, J.P., & D.A. Phillips, eds. 2000. From neurons to neighborhoods: The science of early childhood development. Washington, DC: National Academy Press.

Shumway, Number Sense Routines (2011)

Van de Walle, Teaching Student Centered Mathematics (2006)

Yackel, E., P. Cobb, & T. Wood. 1991. Small group interactions as a source of learning opportunities in second grade mathematics. Journal for Research in Mathematics Education 22: 390-408.