3 Developing and Using Relational Thinking

  • Use of the equal sign correctly, but still calculating important step
  • Use of the equal sign correctly and realizing calculation not required, only need to compare both sides critical part of learning
  • Goal of this chapter to consider how to engage students to reach this critical stage
  • 1st show 2nd grade student making transition from computational thinking to relational thinking
  • 2nd show how one teacher challenged her students
  • 3rd show students can write their own sentences to show their understanding of relations
  • Final section focus on using number sentences as tool to develop specific arithmetic concepts or skills

Emma Learns to Look for Relations

  • Emma is at benchmark 3, she uses equal sign correctly, but calculates both sides of the equation
  • Key transition in thinking occurs after asked to solve 15 + 16 = 15 + 
  • Emma realizes the connection between this problem and the other problems and advances to benchmark 4

CD 3.1

A Class Discussion of Number Sentences Involving Relational Thinking

  • Engages 4th grade class in discussion about true/false questions beginning with standard form addend + addend = sum or sum – addend = missing addend
  • Gives both true and false examples in standard form
  • Next starts giving problems with relational ideas in them
  • Start with problems students can compute and then extend to problems that require relational thinking to solve easily
  • Moves on to problems with 2 numbers on the left and three on the right, where one on the right is missing
  • Children determine various patterns for figuring out relations – some work all the time, some do not

Another Context for Engaging Students with Relational Thinking

  • Students can write true/false number sentences themselves
  • Students often write more interesting problems than teachers
  • Sharing student problems opens up rich discussions
  • While many problems may not pose much difficulty in terms of calculation, they still can provide interesting challenges to use fundamental ideas of arithmetic to avoid or simplify computation

CD 3.2

Using Relational Thinking to Support the Learning of Arithmetic

  • Many fundamental mathematical ideas involve relations between different representations of numbers and operations
  • Relations need to be given to students in a variety of forms to increase their flexibility
  • Once relations are learned they can be used to support the learning of many arithmetic concepts and skills

Teacher Commentary 3.1

  • In the beginning, like most things – we are unsure of how things will work and where exactly to begin
  • Teaching relations helps kids to think
  • Helps kids to know that math is more than just procedures
  • Helps students to understand
  • Don’t think of algebra as just one more piece to add – it really fits with what needs to be done in mathematics

Learning Multiplication Facts

  • Relations can make learning number facts easier
  • Learning commutative property reduces number of facts to be learned
  • Understanding the relation between addition and multiplication helps students relate learning multiplication facts to addition facts
  • Focusing on specific relationships can make it possible for students to build on facts they already know
  • Relation between addition and multiplication helpful when first learning multiplication facts
  • Ok to revert back to benchmark 3 in the beginning, want to progress to and use benchmark 4
  • It will take time for students to learn this, but will pay big dividends in the end by making learning meaningful

CD 3.3

Teacher Commentary 3.2

  • Many children miss big ideas in the curriculum when focusing on procedures
  • Breaking apart numbers is a big idea in algebra
  • Mathematics is about understanding
  • Algebraic thinking should be the foundation for building a child’s mathematical knowledge and for building mathematical curriculum
  • If you are not tying arithmetic to these “Big Ideas,” you are wasting precious time

Thinking Relationally

  • Important for 2 reasons
  • 1st facilitates students learning of arithmetic
  • algebraic thinking is NOT a separate topic
  • 2nd provides a foundation for smoothing the transition to algebra
  • Traditional learning makes operations just procedures to be carried out
  • Traditional learning gets in the way of understanding and being able to do algebra