Difference between pages "Carraher & Schliemann (2007)" and "Hiebert & Grouws (2007)"

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imported>Raymond Johnson
(Created page with "{{Title|Early Algebra and Algebraic Reasoning}} * Authors: David W. Carraher and Analúcia D. Schliemann * Book: Second Handbook...")
 
imported>Raymond Johnson
(Created page with "{{Title|The Effects of Classroom Mathematics Teaching on Students' Learning}} * Authors: James Hiebert and Douglas A. Grouws * Book: Second Handbook...")
 
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{{Title|Early Algebra and Algebraic Reasoning}}
{{Title|The Effects of Classroom Mathematics Teaching on Students' Learning}}


* Authors: [[David Carraher|David W. Carraher]] and [[Analúcia Schliemann|Analúcia D. Schliemann]]
* Authors: [[James Hiebert]] and [[Douglas Grouws|Douglas A. Grouws]]
* Book: [[Second Handbook of Research on Mathematics Teaching and Learning]]
* Book: [[Second Handbook of Research on Mathematics Teaching and Learning]]
* Year: 2007
* Year: 2007
Line 7: Line 7:


==Outline of Headings==
==Outline of Headings==
* Why Algebraic Reasoning?
* Teaching Matters...But ''How'' Is Not Easy to Document
* Rationale and Structure of Chapter
** A Claim That Appears Obvious is Strikingly Difficult to Specify
** The Recent Focus on Algebraic Reasoning in the Early Grades
** Useful Theories of Teaching Are in Short Supply
** A Decisive Moment
*** Challenge 1: Different Teaching Methods Might Be Effective for Different Learning Goals
*** Event 1: NCTM's Endorsement
*** Challenge 2: Teaching is a System of Interacting Features
*** Event 2: The RAND Mathematics Study Panel Report
*** Challenge 3: The Influence of Mediating Variables
** Five Key Issues
** Documenting the Effects of Teaching on Learning is Methodologically Difficult
*** Issue 1: The Relations Between Arithmetic and Algebra
*** Accounting for Relevant Factors
*** Issue 2: Process versus Object
*** Creating Appropriate Measures
*** Issue 3: The Referential Role of Algebra
** Teaching Often Gets Interpreted as Teachers
*** Issue 4: Symbolic Representation (narrowly defined)
** Despite the Difficulties, Research-Based Claims Are Emerging
*** Issue 5: Symbolic Representation (broadly defined)
* Opportunity to Learn: Still the Key Enabling Condition
* A Traveler's Guide to Early Algebra
* Two Patterns: Teaching for Skill Efficiency and Teaching for Conceptual Understanding
** School Algebra and EA
** Becoming Efficient in Executing Skills
** EA Versus Pre-Algebra
** Developing Conceptual Understanding
*** Pre-Algebra Approaches
** Teaching That Promotes Conceptual Development: Two Key Features
*** EA Approaches
*** Feature 1: Teachers and Students Attend Explicitly to Concepts
*** On the Possibilities of EA
*** Feature 2: Students Struggle with Important Mathematics
*** Parsing (Early) Algebra
*** Teaching Features that Promote Conceptual Understanding also Promote Skill Fluency
** Algebra Is Latent in the Existing Early Mathematics Curriculum
*** Summary
* Arithmetic and Numerical Reasoning as an Entry Point Into EA
** Absence of Features That Support Conceptual Development from U.S. Mathematics Teaching
** The Field Axioms and Other Properties of Numbers
* Future Directions for Research That Connects Teaching and Learning
** Studies That Introduce Algebra Through Generalizations About Numbers
** Be Explicit About Learning Goals
** Quasi-Variables
** Build and Use Theories
** Summary: Numerical Reasoning and EA
** Set Realistic Expectations for What a Knowledge Base Can Do
* Arithmetic and Quantitative Reasoning as an Entry Point Into EA
** Account for the Costs and Benefits of Different Research Approaches
** Quantities, Measures, and Magnitudes
** Comparing the Effects of Different Instructional Methods
** Quantitative Thinking and Number Lines
** Correlating Features of Teaching with Students' Learning
** Can Students Apply Other Properties of Arithmetic to Magnitudes?
** Balancing the Benefits of Small-Scale Qualitative Studies and Large-Scale Quantitative Studies
** Referent-Transforming Properties
* Final Thoughts
** EA Studies that Focus on Magnitudes and Measures
** Why Quantitative Thinking Is Unavoidable in EA
** The Davydov Approach to EA
** The Measure Up Project
** Summary: Quantitative Reasoning and EA
* Arithmetic and Functions as an Entry Point Into EA
** Can Young Students Reason with Functions?
** Functions As Rules for Generating Collections of Figures
** Functions Expressed Through Multiple Representations: The Early Algebra, Early Arithmetic Project
** Summary: What Can Young Student Learn About Functions?
* Concluding Thoughts
** What Kinds of Representations Express Algebraic Ideas?
** Patterns and Functions
*** Issues Common to Patterns and Tables
** Is a Scalar Approach Valid?
** What Goals Are Achievable in the Short Term, Mid Term, and Long Term (for Students and Teachers)?


==Corrolary==
==Corrolary==
;APA
;APA
: Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), ''Second handbook of research on mathematics teaching and learning'' (pp. 669–705). Charlotte, NC: Information Age.
: Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester (Ed.), ''Second handbook of research on mathematics teaching and learning'' (pp. 371–404). Charlotte, NC: Information Age.
;BibTeX
;BibTeX
<pre>
<pre>
@incollection{Carraher2007,
@incollection{Hiebert2007,
address = {Charlotte, NC},
address = {Charlotte, NC},
author = {Carraher, David W. and Schliemann, Anal\'{u}cia D.},
author = {Hiebert, James and Grouws, Douglas A.},
booktitle = {Second handbook of research on mathematics teaching and learning},
booktitle = {Second handbook of research on mathematics teaching and learning},
chapter = {15},
chapter = {9},
editor = {Lester, Frank K.},
editor = {Lester, Frank K.},
pages = {669--705},
pages = {371--404},
publisher = {Information Age},
publisher = {Information Age},
title = {{Early algebra and algebraic reasoning}},
title = {{The effects of classroom mathematics teaching on students' learning}},
year = {2007}
year = {2007}
}
}
Line 74: Line 58:
[[Category:Book Chapters]]
[[Category:Book Chapters]]
[[Category:2007]]
[[Category:2007]]
[[Category:Early Algebra]]
[[Category:Teaching Quality]]
[[Category:Algebraic Reasoning]]

Latest revision as of 22:00, 27 May 2015

The Effects of Classroom Mathematics Teaching on Students' Learning

Outline of Headings

  • Teaching Matters...But How Is Not Easy to Document
    • A Claim That Appears Obvious is Strikingly Difficult to Specify
    • Useful Theories of Teaching Are in Short Supply
      • Challenge 1: Different Teaching Methods Might Be Effective for Different Learning Goals
      • Challenge 2: Teaching is a System of Interacting Features
      • Challenge 3: The Influence of Mediating Variables
    • Documenting the Effects of Teaching on Learning is Methodologically Difficult
      • Accounting for Relevant Factors
      • Creating Appropriate Measures
    • Teaching Often Gets Interpreted as Teachers
    • Despite the Difficulties, Research-Based Claims Are Emerging
  • Opportunity to Learn: Still the Key Enabling Condition
  • Two Patterns: Teaching for Skill Efficiency and Teaching for Conceptual Understanding
    • Becoming Efficient in Executing Skills
    • Developing Conceptual Understanding
    • Teaching That Promotes Conceptual Development: Two Key Features
      • Feature 1: Teachers and Students Attend Explicitly to Concepts
      • Feature 2: Students Struggle with Important Mathematics
      • Teaching Features that Promote Conceptual Understanding also Promote Skill Fluency
      • Summary
    • Absence of Features That Support Conceptual Development from U.S. Mathematics Teaching
  • Future Directions for Research That Connects Teaching and Learning
    • Be Explicit About Learning Goals
    • Build and Use Theories
    • Set Realistic Expectations for What a Knowledge Base Can Do
    • Account for the Costs and Benefits of Different Research Approaches
    • Comparing the Effects of Different Instructional Methods
    • Correlating Features of Teaching with Students' Learning
    • Balancing the Benefits of Small-Scale Qualitative Studies and Large-Scale Quantitative Studies
  • Final Thoughts

Corrolary

APA
Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Charlotte, NC: Information Age.
BibTeX
@incollection{Hiebert2007,
address = {Charlotte, NC},
author = {Hiebert, James and Grouws, Douglas A.},
booktitle = {Second handbook of research on mathematics teaching and learning},
chapter = {9},
editor = {Lester, Frank K.},
pages = {371--404},
publisher = {Information Age},
title = {{The effects of classroom mathematics teaching on students' learning}},
year = {2007}
}