2 Equality

  • See Table 2.1 p. 9 – 5 and 6 grade students 76% choose 12 and 98% give the wrong answer – WOW! 
  • Many if not most K-5 students have serious misconceptions about the meaning of the equal sign
  • Most students do not realize the equal sign denotes a relation between two equal quantities
  • Many interpret equal sign as a command to carry out a calculation similar to what a calculator does
  • This notion limits ability to understand
  • Arithmetic ideas
  • Flexibility of representation
  • Creates serious problem for algebraic understanding
  • Chapter discusses student conceptions of equal sign and sources of misconceptions
  • Objective to think about how to engage children in examining and revising their conception of the equal sign
  • Examine what children’s errors tell us about their understanding
  • We can make a significant difference in how children interpret equal sign
  • Discussions provide teacher with opportunities for modeling how mathematical ideas merge and how mathematical disputes are resolved

Children’s Conceptions of the Meaning of the Equal Sign

  • Five typical responses to 8 + 4 =  + 5
  • Lucy: The answer comes next: 8 + 4 = 12 + 5
  • The answer was the answer to the operation that came right before the equal sign
  • The equal sign was interpreted to be a command to carry out the calculation
  • The equal sign did not represent a relation between both sides of the equation
  • Randy: Use all of the numbers: 8 + 4 = 17 + 5
  • Took into account all of the numbers present
  • Did not recognize the order the symbols in the sentence came in made a difference
  • Only way to use all of the numbers was to add them all up
  • Barb: Extend the problem: 8 + 4 = 12 + 5 = 17
  • What Barb did made PERFECT sense to her
  • Used equal signs to show a sequence of calculations
  • Adults do this frequently too
  • THIS IS AT BEST BAD NOTATION and IS ALWAYS WRONG even when ADULTS do it!!! 

Making Sense

  • Methods were not unreasonable attempts to deal with unfamiliar problem
  • Generalized from their previous experiences that one of the rules for writing number sentences required the answer to follow immediately after the equal sign
  • Randy felt the equal sign was irrelevant
  • Errors were errors of syntax
  • For them the equal sign meant they were to do something rather than to read the entire equation and see the relation that existed there

Teacher Commentary 2.1

  • Struggling with the equal sign is a good struggle
  • Using trial and error is OK as long as you get there eventually
  • Important for students to understand the equal sign
  • Right from the first grade start writing relations like
  • 12 = 6 + 6
  • 3 = 3
  • and so on
  • Misconceptions are not trivial nor easily overcome
  • Some children hang on fiercely to the concepts they have formed – right or wrong
  • It is not sufficient to just “tell” the correct way to use equal sign
  • Ricardo: A relational view: 8 + 4 = 7 + 5
  • 8 + 4 is 12
  • Need to figure out what goes with 5 to make 12
  • Figured out it would be 7
  • Knows 12 cannot go in the box because 12 does not equal 17
  • Gina: A relational view: 8 + 4 = 7 + 5
  • 5 is one more than 4
  • Box has to be one less than 8
  • So the box must be 7
  • Is able to apply same reasoning to a two digit problem
  • Both viewed equal sign as expressing a relation between numbers
  • Both knew the expressions on both sides of the equal sign had to represent the same number
  • Neither imposed any arbitrary rules about number sentences having to be in a particular form
  • They were comfortable with operations occurring on either or both sides of the equal sign
  • Many students believe ALL number sentences must be in a particular form
  • When students become too accustomed to seeing number sentences in only one form they make errors when they try to adapt new forms to their previous models
  • In general, students think calculation NOT relation for the equal sign

CD 2.1

  • Both demonstrated knowledge of appropriate use of equal sign, but Gina showed deeper understanding of relational concept
  • Gina did not have to calculate the answer to left side first, she used the relations between the numbers present to determine the missing value
  • Critical skill to be developed by students for greater understanding of arithmetic and extension to algebra

CD 2.2

Developing Children’s Conceptions of Equality

  • Have challenge students’ existing conceptions of what the equal sign means
  • Engage students in a discussion where different concepts of the equal sign emerge and then help students resolve them
  • Can discuss as whole class or small groups, but students MUST be encouraged to articulate their ideas clearly and explicitly with appropriate vocabulary and mathematical language
  • Following clear and explicit expressions of their conceptions of the equal sign, then the teacher can attempt to help the students to resolve any differences voiced
  • Discussions address meaning, ways of thinking, and conversing associated with the principles of algebraic reasoning
  • Contradictions in concepts of equal sign provide context for examination and resolving inconsistencies
  • Alternative concept discussion forces students to be explicit about ideas that are often left implicit
  • Students must justify the principles they propose helping them to recognize and resolve conflicting assumptions and conclusions

A Context for Discussing the Equal Sign

  • Engage students in discussion of equal sign in the context of a specific task
  • Provide focus for students to articulate ideas
  • Challenge students conceptions by providing different contexts
  • Provide window into student thinking
  • True/false or open number sentences work well
  • Number sentences can be manipulated in a variety of ways to stimulate conversation
  • It is not easy to resolve the different perspectives that may be offered
  • Children cannot determine the meaning of the equal sign entirely by using logic
  • Variety of contexts open students to thinking more openly about their concepts regarding the equal sign

True/False Number Sentences

  • Even if students have not experienced these before, they are relatively easy to introduce
  • Not necessary to engage students in general discussion prior to using format
  • Works best to provide an example and then ask if true or false
  • Start simple and in expected format
  • Children quickly pick up critical ideas in this context
  • Try several, some true, some false in the expected format
  • After comfort obtained with these, switch to posing problems that challenge students to examine their concepts related to the equal sign
  • Place known true statements in familiar form next to forms that are different, but also true
  • 3 + 5 = 8
  • 8 = 3 + 5
  • 8 = 8
  • 3 + 5 = 3 + 5
  • 3 + 5 = 5 + 3
  • 3 + 5 = 4 + 4
  • Use several different ideas of equality to try to get at students understanding of this concept
  • This idea should be repeated using other operations as well depending on the curriculum the students are already familiar with
  • Important to discuss the need for applying rules consistently in mathematics
  • Open number sentences can be used to illustrate these same ideas
  • What number would be placed into the box to make it true?
  • Many numbers might be substituted, but only one number makes the sentence true
  • This links true/false work with open number sentences
  • Sometimes students respond differently to true/false and open number sentences
  • When the context changes, some students do not recognize the structure as being the same

Wording and Notation

  • Want to help students understand equal sign represents a relation between two numbers
  • Useful to use language that is more explicit in expressing the relation idea behind the equal sign
  • 8 is the same amount as 3 plus 5
  • Sometimes it is helpful to show the amounts on both sides of the equal sign have same numerical value

Teacher Commentary 2.2

  • Changing conceptions is not easy
  • Bidirectional logic is a problem – seeing 3 + 4 = 7 sometimes confused with 7 = 3 + 4
  • Posing these types of problems generates discussions
  • Children do not always possess the vocabulary they need to have a mathematically rich discussion
  • Even after all we do, the loudest response may still be the wrong answer

Classroom Interactions

  • Nature of discussion of mathematical ideas is critical
  • Goal is not just appropriate conceptions of the equal sign
  • Goal also to engage them in productive mathematical arguments

CD 2.3

  • Discussions of alternative conceptions of equal sign helps students to examine and justify their own conceptions
  • First step in helping students to make and justify generalizations about mathematics
  • Sets the stage for later ideas involving mathematical proof
  • Want children to be understand how fundamental differences are negotiated and resolved

Benchmarks

  • Everyone is different in how they attain knowledge, but the following are some benchmarks to work toward
  • Get children to be specific about what they think the equal sign means – right or wrong
  • Attained when children accept sentence as true not in the form a + b = c
  • Attained when children think/realize equal sign represents a relation between two equal numbers – Ricardo’s response is an example of this benchmark
  • Attained when children compare mathematical expressions without doing the calculations – Gina’s response represents an example of this benchmark
  • These are only a guide
  • Children will be heterogeneously distributed in class with respect to their achievement of benchmarks

What to Avoid

  • Do not use the equal sign in ways that do not represent a relation between numbers
  • Do NOT use the equal sign for the following
  • Listing ages or other numerical characteristic of people or things: John = 8, Marcy = 9, etc
  • Designating the number of objects in a collection:  = 4
  • Using the equality symbol to represent a string of calculations: 20 + 30 = 50 + 7 = 57 + 8 = 65
  • Using equality between two pictures:  = 
  • Once concept of equal sign means a relation is established important to not just give problems in a + b = c form – should alternate representations of problems

The Use of the Equal Sign as a Convention

  • Use of equal sign is a matter of agreement or convention
  • We cannot “prove” equal signs are used to represent relations like we can prove other things in mathematics
  • Students and teachers are required to back up their claims with justifications
  • Next have to discuss and decide what can be accepted as evidence for a claim
  • How many examples are needed for acceptance of an idea?
  • Important to adopt a consistent interpretation

Smoothing the Transition to Algebra

  • Limited conception of meaning of equal sign major stumbling block for learning algebra
  • Algebraic manipulations depend upon concept of equal sign forming a relation
  • Lack of understanding of equal sign limits students to the arithmetic calculations
  • Knowing concept of relation opens doors to better arithmetic and algebra

Where Do Children’s Misconceptions about the Equal Sign Come From?

  • Difficult to pin down where misconceptions come from or why they develop
  • Over use of the standard form, a + b = c, is a major contributor
  • Over generalization of a + b = c pattern by students
  • Even after developing appropriate meaning for equal sign students may revert back to previous misconceptions
  • Calculators can reinforce equal sign means carry out the operation instead of relation
  • Children may be predisposed to think of equality in terms of calculating answers instead of relation
  • Not simply a matter of maturation either
  • Situation must be directly addressed to help promote concept of relation
  • Computation ability not linked to ability to understand relation
  • Relation concept should be worked on throughout ALL elementary grades

Children’s Conceptions

  • Inappropriate generalizations symptomatic of fundamental limits in how mathematical ideas are generated and justified
  • Children believe they can prove their case by example
  • Challenging to get children to examine things they believe are true
  • Young children have difficulty in dealing with hypothetical situations
  • Children need to learn to answer the question of why things work out the way they do
  • Need to appreciate the limits of children’s thinking without imposing our own set of limitations on what they can accomplish