4 Making Conjectures about Mathematics
 Children have a great deal of implicit knowledge
 Usually not a part of mathematics class to make student knowledge explicit
 Children do not usually get an opportunity to explore why what they know works
 Explicit knowledge really helps students to understand conceptually how the mathematics is working
Making Implicit Knowledge Explicit
 Want children to make conjectures about their implicit knowledge to make it into explicit knowledge
o How do students articulate, refine, and edit conjectures?
o How do we identify important conjectures from students to make conjectures about?
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Teacher Commentary 4.1
 Important to have conversations with students about mathematics
 Conjectures help to focus the conversation to a manageable set of ideas
 Students learn a lot from defending their ideas and questioning the ideas of other students
 Conjectures are a way to talk about big ideas that involves the whole class in the discussion
Articulating, Refining, and Editing Conjectures
 Goes beyond simply engaging students in communicating
 Students need to use precise language in stating mathematical ideas
 Students confront important mathematical ideas
 Students engage in basic forms of mathematical arguments
 True/false number sentences are a good way to begin these conversations
 Operations involving zero easiest for students to identify and talk about
CD 4.1
 Conjectures are made and posted
 Conjectures are added to throughout the year
 Generating conjectures should become a norm for the class
 By applying properties and justifying solutions by stating properties students transition from implicit to
explicit knowledge
 Let students correct each other and help each other to refine conjectures
 Encourage students to edit conjectures to contain precise language
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Teacher Commentary 4.2
 Language and mathematics do not have to be separate endeavors
 Kids who struggle with language should be encouraged to use language more often
 Some kids will need help with vocabulary when they start talking about mathematical ideas
 Have students restate a classmate’s ideas in their own words
 Mathematical discussions can nurture learning
Editing Conjectures
 Initial description of a conjecture will often include several examples
 Students who disagree should give specific examples to say why a conjecture needs to be stated more
clearly
 When conjectures seem ambiguous, editing can help to make them more precise
CD 4.2
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Teacher Commentary 4.3
 Writing down a conjecture is only the beginning of the discussion
 Still need to discuss the big ideas present
 Need to ask questions:
o Do we know these big ideas will always work?
o How will this big idea help you?
o Why is it important for us to think about this?
o Why would knowing you can switch the order of numbers to add help you to do math?
o Can you think of when that might help you?
o How is it going to help you with your arithmetic?
 Putting the conjecture on the wall is not the end of the discussion either
Some Conjectures about Basic Properties of Number Operations
 More than just making big ideas explicit
 Want students to make conjectures because they explore important mathematical ideas
 Conjecture discussions empower students to learn new mathematics, to solve problems, and to understand
the mathematics they are currently learning and doing
 See table 4.1 for a list of basic properties of number operations p. 54-55
 Conjectures initially written in natural language
 Large numbers tend to draw out conjectures better than smaller numbers
 Conjectures 1, 4, and 7 have two similar statements
o Only difference is the order of the numbers is reversed
o Once one conjecture is established the other follows from it
 Conjectures in table are related in interesting and important ways
o Parallels between addition and subtraction of zero and multiplication and division by one
o As students discuss these conjectures it is valuable for them to see these relations
 Extending conjectures like a + 0 + 0 + 0 + 0 = a does not really add to the basic conjecture even though it is
true
 If extended conjectures come up they can be interesting to discuss and can provide some good insight
 Be wary of special case conjectures – conjectures that are true only for a particular case or isolated set of
cases
 Want to be economical in writing conjectures
 Conjectures can be combined to form new conjectures
 See table 4.2 p. 56
Invalid Conjectures
 Sometimes conjectures may sound good initially, but are generally untrue
 Look a lot like valid conjectures
 Come from over-generalizations
 See table 4.3 p. 57
More Conjectures
 Will be discussed in later chapters
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Teacher Commentary 4.4
 Students thought there were no numbers smaller than zero
 Lager number minus smaller number gave them zero
 Money to the rescue!
 Writing and talking about conjectures solidifies and clarifies knowledge
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Generating conjectures is an ongoing process
A wide variety of conjectures will be made
Some will be quite different from ones explored here
Some conjectures which seem true at first blush may prove false under other circumstances
o Adding two numbers always gives a bigger number
 only true for positive numbers
 not true if one number is zero
o Decisions have to be made of how to proceed
 Can let the conjecture stand – not generally a good idea as it causes problems for the
students later on
 Teacher can edit the conjecture without going into a detailed explanation
 Teacher can take the time to go into a brief or detailed explanation – requires a major
time commitment
 Need to deal positively with conjectures that are not true
 Want the children to take responsibility for deciding the truthfulness of conjectures
 Teacher is generally obligated to guide students into correcting untruthful conjectures
 Ok to post false conjecture as long as the truth of the conjecture is in doubt and is eventually determined to
be false
 Important to edit conjectures – keep edits as evidence of students progress in understanding a concept
Definitions
 Frequently proposed by students
 Distinction between definition and conjecture is there is no way to justify a definition
 Definitions are somewhat arbitrary
 Making this distinction may be very informative
 Students struggle with defining mathematical terms they are familiar with
 Students need to articulate and edit definitions too
Rules for Carrying Out Procedures
 Students often propose algorithm as conjecture
 Even though we use these rules as a convenience, they are for generating answers not for increasing
understanding
 Probably best to steer students away from these ideas
Conjectures about Even and Odd Numbers
 When you add two odd numbers, you et an even number
 Conjectures about even and odd numbers offer a good opportunity to examine what it means to justify a
conjecture
Summary: Types of Conjectures that Students Make
 Conjectures about fundamental properties of number operations
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o Describe basic properties of numbers and operations on them
o Most important conjectures for students’ learning of arithmetic and algebra
o Addition conjectures of this type chapters 8 and 9
Conjectures about classes of numbers
o Include conjectures about even and odd numbers
o Conjectures about factors and divisibility rules
Descriptions of procedures
o Rules for carrying out specific computational procedures
o Descriptions of procedures involve outcomes of calculations
o Not usually amenable to being expressed in terms of open sentences
General descriptions of outcomes of calculations
o Notions like
 Addition and multiplication result in larger numbers
 Subtraction and division result in lower numbers
o Quite global conjectures
o If a is grater than b and c is greater than d, then a + c is greater than b + d
Definitions
o Should not be considered as conjectures
o Definitions cannot be justified
o They are true by definition
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Teacher Commentary 4.5
 Conjectures can be challenging for special education students
 Can do workbook pages well, but do not understand concepts behind rote operations
 May not get right the first time, but after editing they do get it
 For first and 2nd graders, this is amazing