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How Students Learn Mathematics in the Classroom

Explore how students utilize informal vs. formal mathematical strategies, leveraging preconceptions for deeper learning. Insights from real-world contexts enhance conceptual understanding in the classroom. Discover metacognitive approaches for self-monitoring progress.

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How Students Learn Mathematics in the Classroom

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  1. How Students LearnMathematics in the Classroom

  2. Informal vs. formal mathematical strategies • Brazilian street children could perform mathematics when making sales in the street, but were unable to answer similar problems when presented in a school context. Carraher, 1986; Carraher et al., 1985

  3. Informal vs. formal mathematical strategies • A study of housewives in California uncovered an ability to solve mathematical problems when comparison shopping, even though the women could not solve problems presented abstractly in a classroom that required the same mathematics. Lave 1988; Sternberg 1999

  4. Informal vs. formal mathematical strategies • A similar result was found in a study of a group of Weight Watchers, who used strategies for solving mathematical measurement problems related to dieting that they could not solve when the problems were presented more abstractly. De la Rocha, 1986

  5. Informal vs. formal mathematical strategies • Men who successfully handicapped horse races could not apply the same skill to securities in the stock market. Ceci and Liker, 1986; Ceci, 1996

  6. Principle #1: Teachers Must Engage Students’ Preconceptions • Preconception #1 – Mathematics is about learning to compute. • What approximately is the sum of 8/9 and 12/13? • Conceptually • Procedurally • Sense making

  7. Principle #1: Teachers Must Engage Students’ Preconceptions • Preconception #2 – Mathematics is about “following rules” to guarantee correct answers. • Systematic pattern finding and continuing invention • Tower of Hanoi • Units used to quantify fuel efficiency of a vehicle • Miles per gallon • Miles per gallon per passenger • Compare different procedures for their advantages and disadvantages

  8. Principle #1: Teachers Must Engage Students’ Preconceptions • Preconception #3 – Some people have the ability to “do math” and some don’t. • Amount of effort • Some progress further than others • Some have an easier time • Effort is key variable for success

  9. Key Point to Remember • Without a conceptual understanding of the nature of the problems and strategies for solving them, failure to retrieve learned procedures can leave a student completely at a loss.

  10. How to best engage students’ preconceptions and build on existing knowledge • Allow students to use their own informal problem-solving strategies • Wrong answer (usually partially correct) • Find part that is wrong • Understand why it is wrong • Aids understanding • Promotes metacognitive competencies

  11. How to best engage students’ preconceptions and build on existing knowledge • Encourage math talk • Actively discuss various approaches • Learner focused • Draw out and work with preconceptions • Making student thinking visible • Model the language

  12. How to best engage students’ preconceptions and build on existing knowledge • Design instructional activities that can effectively bridge commonly held conceptions and targeted mathematical understandings • More proactive • Research common preconceptions and points of difficulty • Carefully designed instructional activities

  13. Key Point to Remember • Identifying real-world contexts whose features help direct students’ attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students’ informal experiences and the new formal mathematics they are learning.

  14. Principle #2: Understanding Requires Factual Knowledge and Conceptual Frameworks • MDE HSCE for Mathematics p. 4 • Conceptual Understanding • Procedural Fluency • Effective Organization of Knowledge • Strategy Development • Adaptive Reasoning • How Students Learn Mathematics in the Classroom • A Developmental Model for Learning Functions

  15. A Developmental Model for Learning Functions. Levels of Understanding • 0 – Recognize and Extend Pattern • 1 – Generalize the Pattern and Express it as a function (y = 2x) • 2 – Look at graph and decide if a particular function could model it • 3 – Using Structures from Level 2 to create and understand more complex functions

  16. Principle #3: A Metacognitive Approach Enables Student Self-Monitoring • Learning about oneself as a • Learner • Thinker • Problem solver

  17. Instruction That Supports Metacognition • An emphasis on debugging • Finding where the error is • Why it is an error • Correcting it • Internal and external dialogue as support for metacognition • Help students learn to interact • Model clear descriptions, supportive questioning, helping techniques • Seeking and giving help

  18. How Students Learn Mathematics in the Classroom Careful consideration was taken to the points presented in this powerpoint to insure that the development of our units was guided by the known research

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