10 Conclusion

  • Arithmetic can represent a collection of unrelated and arbitrary manipulations of numbers and symbols
  • Algebra is often perceived as a separate collection of meaningless procedures that are only remotely linked to arithmetic
  • We have learned looking at BIG IDEAS means viewing both arithmetic and algebra through a different lens
  • A small list of fundamental numeric properties accounts for all symbol manipulation in arithmetic and algebra
  • Properties of Addition
  • Identity – For every real number a, a + 0 = a
  • Inverse – For every real number a, there is a real number –a such that a + (-a) = 0
  • Commutative – For all real numbers a and b, a + b = b + a
  • Associative – For all real numbers a, b, and c, (a + b) + c = a + (b + c)
  • Properties of Multiplication
  • Identity – For every real number a, a x 1 = a
  • Inverse – For every real number a, a ≠ 0, there is a real number such that
  • Commutative – For all real numbers a and b, a x b = b x a
  • Associative – For all real numbers a, b, and c, (a x b) x c = a x (b x c)
  • Distributive Property of Multiplication over Addition
  • For all real numbers a, b, and c, a x (b + c) = (a x b) + (a x c)
  • Developing an intuitive sense of these ideas is a natural consequence of learning arithmetic with understanding
  • Do not expect all children to understand implicitly all properties demonstrated in book
  • Important that ALL students learn the big ideas linking arithmetic and algebra together
  • Big ideas alone NOT sufficient, must know applications
  • Need to learn
  • The importance of generalization
  • How to represent mathematical ideas accurately
  • Using natural language
  • Using mathematical symbols
  • To appreciate the value of justifying mathematical statements
  • To gain insight into mathematical justification
  • Need to develop disposition that
  • Mathematics makes sense
  • They can make sense of mathematics

Developing mathematical Thinking

  • Previously identified big ideas and how to engage children in learning these ideas
  • Influences on how children learn
  • The tasks they engage in
  • Interactions they have about those tasks

Tasks for Learning and Expressing Mathematical Ideas

  • Used two main ways to generate discussions
  • True/false statements
  • Open number sentences
  • Could also use basic word problems
  • Want children to be able to use their knowledge more flexibly and extend arithmetic notions to algebraic ones
  • Use true/false open sentences to focus on one mathematical idea at a time
  • Number sentences can be used to
  • Engage students in discussions about the appropriate use of the equal sign
  • Encourage students to use relational thinking
  • Foster students reliance on fundamental mathematical properties when learning
  • Number facts
  • Place value
  • Other basic arithmetic concepts
  • Help students generate conjectures
  • Can be used to focus on misconceptions to remediate them
  • Not just tricks to help recall number facts, but exploration of basic mathematical properties
  • Number sentences provide window into students’ mathematical thinking
  • Student generated number sentences
  • Insight into student mathematical thinking
  • Provide artifacts for class discussions
  • Tool for representing mathematical ideas
  • Need to understand what a variable is and how to use variables
  • Broad conception of variables good foundation for algebra

CD 3.2

Teacher Commentary C.1

  • First time using true/false number sentences made her nervous
  • Just started and the students responded
  • Tried to get each student to take a stand – true, false, not sure
  • Students from each category explained their thinking
  • Teacher posed another problem
  • They did it all again
  • Teacher learned a lot about
  • The number sentences
  • Students’ mathematical thinking
  • About the development of conversation around the ideas
  • Surprised by what some of the students said
  • Some students were really adamant
  • Couldn’t wait to try it again 
  • Still working on improving best number sentences to ask
  • Teacher improves skill at posing number sentences every time she does it
  • Worries about reaching all students – feels this is an ongoing reflective process
  • Number sentences are tools teachers need to take advantage of using in their classrooms
  • Have to develop classroom norms that allow students to question their ideas
  • Takes commitment by teacher to make this work
  • Skill on teachers part
  • Listening to students critical key

Interactions about Mathematical Ideas

  • Engagement significantly impacts what a student learns
  • Interactions with peers just as important
  • Teacher questions and responses shape student learning and perception
  • If want to teach for understanding, teacher must frequently ask
  • Is that true for all numbers?
  • How do you know that is always true?
  • Ultimately we want students to ask themselves these questions
  • Danger of our ideas replacing student ideas if we teacher direct too much
  • Important for teacher to value ALL student responses
  • Important that students feel that ideas and explanations matter

Mathematical Thinking for All Students

  • Learning with understanding means knowledge is
  • Connected
  • Integrated
  • Students make sense of new ideas by connecting them to previous ideas
  • Important to understand and build on what students already know
  • Elementary students can learn to engage in algebraic reasoning
  • Learning big ideas is for ALL students, not just gifted few
  • Need to be open to trying this and will be surprised at how MUCH students know 