How Children Learn Mathematics
Six Principles that are Fundamental to High Quality Math Education
- The Equity Principle: Excellence in mathematics education requires equity--high expectations and strong supports for all students. (NCTM, 2000, p.12)
- The Curriculum Principle: A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. (NCTM, 2000, p.14)
- The Teaching Principles: Effective mathematics teaching requires understanding what students know and need to learn and then challenging them and supporting them to learn it well. (NCTM, 2000, p.16)
- The Learning Principle: Students must learn mathematics with understanding , actively building new knowledge with experience and prior knowledge. (NCTM, 2000, p.20)
- The Assessment Principle: Assessment should support the learning of important mathematics and furnish useful information to both teachers and students ...Assessment should not merely be done to students; rather, it should also be donefor students, to guide and enhance their learning. (NCTM, 2000, p.22)
- The Technology Principle: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances student learning. (NCTM, 2000, p.24)
Research shows that children learn mathematics best through classrooms that are structured around interactive activities, engagement, and investment in students' awareness and sense.
Teaching the Standards
The Five Process Standards
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The Five Content Standards
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The Standards for Mathematical Practice from the Common Core State Standards
- Make sense of problems and preserve in solving them
- Reason abstractly and quantitatively
- Construct valuable arguments and critique the reasoning of others
- Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make use of structure
- Look for and express regularity in repeated reasoning
The Learning Trajectories for Primary Grade Mathematics:
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A Problem Centered Classroom
Before a Lesson
After the Lesson
Why teach in this format?
Teaching Through Problem Solving helps students in the following ways: 1) Focuses students' attention on ideas and sense making 2) Develops mathematical processes 3) Develops student confidence and identities 4) Provides a context to help students build meaning for the concept 5) Allows an entry and exit point for a wide range of students 6) Allows for extensions and elaborations 7) Is interactive and engaging Problem Based Lesson Plan Format:
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What to Teach and How to Teach It (K-8)
Teaching Fractions, Decimals and Percents (K-8)
Hot Topics for Fractions
Hot Topics for Fractions
- For students to really understand fractions, they must experience fractions across many constructs, including part of a whole, ratios, and division.
- Three categories of models exist for working with fractions--area, length, and set or quantity.
- Partitioning and iterating are ways for students to understand the meaning of fractions, especially numerators and denominators.
- Students need many experiences estimating with fractions.
- Understanding equivalent fractions is critical. Two equivalent fractions are two ways of describing the same amount by using different-sized fractional parts.
Hot Topics for Decimals and Percents
- The base-ten place-value system extends infinitely in two direction: to tiny values as well as to large values. Between any two place values, the 10-to-1 ratio remains the same.
- The decimal point is a convention that has been developed to indicate the units position. The position to the left of the decimal point is the unit that is being counted as singles or ones.
- Decimal fractions are simply another way of writing fractions. Both notations have value. Maximum flexibility is gained by understanding how the two symbol systems are related.
- Percents are simply hundredths and as such are a third way of writing both fractions and decimals.
- Addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like positions values -- a simple extension from whole numbers.
- Multiplication and division of two numbers will produce the same digits, regardless of the positions of the decimal point. As a result, the computations can be performed as whole numbers with the decimal placed by way of estimation.
Fractions, Decimals, and Percent Activities
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Teaching Algebra K-8
Kaput's Three Strands of Algebraic Reasoning
Kaput's Three Strands of Algebraic Reasoning
- Study of structures in the number system, including those arising in arithmetic (algebra as generalized arithmetic)
- Study of patterns relations and functions
- Process of mathematical modeling, including the meaningful use of symbols
Developmentally Appropriate Tasks
Algebra Activities
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Iterative Rule
Five Representations of Functions
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Teaching Geometry K-8
Van Heile's Levels of Geometric Thought
Van Heile's Levels of Geometric Thought
- Level 0: Visualization (the objects of thought at level 0 are shapes and what they "look like")
- Level 1: Analysis (The objects of thought at level 1 are classes of shapes rather than individual shapes)
- Level 2: Informal Deduction (The objects of thought at level 2 are the properties of shapes)
- Level 3: Deduction (The objects of thought at level 3 are relationships between properties of geometric objects)
- Level 4: Rigor (The objects of thought at level 4 are deductive axiomatic systems for geometry)
Characteristics of the van Hiele Levels
- The products of thought at each level are the same as the objects of thought at the next level. The objects (ideas) must be created at one level so that relationships between these objects of thought can become the focus of the next level.
- The levels are not age dependent. A third grader or a high school student could be at a level o.
- Advancement through the levels requires geometric experiences. Students should explore, talk about, and interact with content at the next level while increasing experiences at their current level.
- When instruction or language is at a higher level than that of the student, students will not be able to understand the concept being developed. They may memorize a fact but not construct the actual relationship of the properties of a square.
Spatial Sense
- Intuition about shapes and the relationships between shapes and is considered a core area of mathematical study, like number. Spatial sense includes the ability to mentally visualize objects and spatial relationships - to turn things around in your mind. It includes a comfort with geometric descriptions of objects and position. People with well-developed spatial sense appreciate geometric form in art., nature, and architecture and they use geometric ideas to describe and analyze their world.
Geometry Activities
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Integrating Technology
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Technology can be used to teach, learn, and assess student learning in Mathematics
To Learn Mathematics:
Interactive white boards, such as a SMART Board, for students to practice problems on the board, also can provide games that the whole class can participate in
Digital video cameras can be used for students to make mathematical presentations to demonstrate their understanding of different math concepts
Calculator
- Use as a study aid in order to relate what the information they are learning to the calculator procedure
- Can practice geometry and trigonometry with different short cuts the calculator provides
- Can practice graphing data and functions used in mathematics
- Use to compare combinations and values of different numbers
- Use to create different problems they are learning about
Computer Software
- Geometry drawing programs can be used for students to practice finding number of sides, angles, and measure different geometry elements
- Spreadsheets can be used with measurement to compare and record data
- Word processing can be used to reflect on concepts that have been learned and make graphs
- Presentation programs, like PowerPoint, can be used by students to present information they found to the class
To Teach Mathematics:
Calculators
- Study aids help teach students the calculator procedure along with the new information at the same time
- Can be used to teach about different concepts relating to geometry and trigonometry
- Can be used to teach about graphing and how different values affect the result a graph produces
- Can be used to show entire class examples of how different problems are formulated
Computer Software
- Geometry drawing programs can be used to show students different elements apart of geometry
- Word processing can be used to create rubrics, worksheets, or other important elements needed within a classroom
- Presentation programs, like PowerPoint, can be used to present information to an entire class
SMART Board
- Use to present new information to class
- Use to clarify information to the whole class
- Use to project problems so students can see the teacher work through different problems and concepts
Internet
- Use to research math concepts and for projects
- Create custom mathematical web search on Google
- Use websites, for example YouTube to watch informational videos
To Assess Student Learning:
To Learn Mathematics:
Interactive white boards, such as a SMART Board, for students to practice problems on the board, also can provide games that the whole class can participate in
Digital video cameras can be used for students to make mathematical presentations to demonstrate their understanding of different math concepts
Calculator
- Use as a study aid in order to relate what the information they are learning to the calculator procedure
- Can practice geometry and trigonometry with different short cuts the calculator provides
- Can practice graphing data and functions used in mathematics
- Use to compare combinations and values of different numbers
- Use to create different problems they are learning about
Computer Software
- Geometry drawing programs can be used for students to practice finding number of sides, angles, and measure different geometry elements
- Spreadsheets can be used with measurement to compare and record data
- Word processing can be used to reflect on concepts that have been learned and make graphs
- Presentation programs, like PowerPoint, can be used by students to present information they found to the class
To Teach Mathematics:
Calculators
- Study aids help teach students the calculator procedure along with the new information at the same time
- Can be used to teach about different concepts relating to geometry and trigonometry
- Can be used to teach about graphing and how different values affect the result a graph produces
- Can be used to show entire class examples of how different problems are formulated
Computer Software
- Geometry drawing programs can be used to show students different elements apart of geometry
- Word processing can be used to create rubrics, worksheets, or other important elements needed within a classroom
- Presentation programs, like PowerPoint, can be used to present information to an entire class
SMART Board
- Use to present new information to class
- Use to clarify information to the whole class
- Use to project problems so students can see the teacher work through different problems and concepts
Internet
- Use to research math concepts and for projects
- Create custom mathematical web search on Google
- Use websites, for example YouTube to watch informational videos
To Assess Student Learning:
- All of the resources under learning and teaching mathematics can be used to assess student learning. Each resource provides ways for students to show their understanding of mathematical concepts.
Integrating Children's Literature
Reasons
One innovative way to get students excited and engaged in mathematics is to incorporate literature that they love. The wonderful thing about math is that it is everywhere. The same with children's literature: you can read it everywhere and for any reason or purpose. Below are some rich texts that are already made for the purpose of math. On the other hand, anyone can pick an authentic piece of literature and pull mathematics from it. Also provided are a few read aloud videos that show how reading and math can be tied together in the classroom. Genres and Types of Children's Literature Picture books Realistic fiction Poetry Historical fiction Traditional Literature Modern Fantasy Biography Graphic Novels Informational Literature |
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Work Cited:
Van de Walle, J.A., & Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th Ed.), Boston: Pearson.
Van de Walle, J.A., & Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th Ed.), Boston: Pearson.