4 Making Conjectures About Mathematics

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
How do students articulate, refine, and edit conjectures?
How do we identify important conjectures from students to make conjectures about?

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

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

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:
Do we know these big ideas will always work?
How will this big idea help you?
Why is it important for us to think about this?
Why would knowing you can switch the order of numbers to add help you to do math?
Can you think of when that might help you?
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
Only difference is the order of the numbers is reversed
Once one conjecture is established the other follows from it
Conjectures in table are related in interesting and important ways
Parallels between addition and subtraction of zero and multiplication and division by one
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

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

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
Adding two numbers always gives a bigger number
only true for positive numbers
not true if one number is zero
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
Describe basic properties of numbers and operations on them
Most important conjectures for students’ learning of arithmetic and algebra
Addition conjectures of this type chapters 8 and 9
Conjectures about classes of numbers
Include conjectures about even and odd numbers
Conjectures about factors and divisibility rules
Descriptions of procedures
Rules for carrying out specific computational procedures
Descriptions of procedures involve outcomes of calculations
Not usually amenable to being expressed in terms of open sentences
General descriptions of outcomes of calculations
Notions like
Addition and multiplication result in larger numbers
Subtraction and division result in lower numbers
Quite global conjectures
If a is grater than b and c is greater than d, then a + c is greater than b + d
Definitions
Should not be considered as conjectures
Definitions cannot be justified
They are true by definition

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