Articulating, Editing, and Refining Conjectures in a Second or Third Grade Classroom
Processes 
Content 
Teacher planning and reflection  Assessment  Classroom view 
Mathematical Reasoning and proof Proving an argument involves a conjecture about what is believe to be true for all cases, articulating that idea, refining it, and editing it until it is acceptable or proving it is false. Communication  It's important to use precise language in stating mathematical ideas so people know exactly what is meant and will be able to understand and communicate with one another. 
Zero Properties of zero Property of zero for addition

Teacher has the following outcome  students will construct a mathematical proof by creating ideas, articulating what they believe to be possible conjectures, refine them through reasoning and comparison to information, and edit conjectures to make them more accurate and meaningful as new ideas are presented.  Activity  Discussion about the accuracy mathematical statements written in a TRUE and FALSE format. The following scenario is from Thinking Mathematically: Integrating arithmetic & Algebra in Elmentary School. by Carpenter, Franke, & Levi. 


Decides to assess or review what the students know about zero and adding zero to another number. He doesn't press the children to make a generalization. He gives them another number sentence for the same principle, but in this case the number sentence is false. 
Diagnositic: Content

Mr. C.  Writes [58 + 0 = 58] on the board 

Pressing further to see how strong or fragile the student's understanding of zero and the property of zero for addition. Uses tranformation one of the mental operations that concrete operational thinkers are begining to develop an understanding that can be used to transform things in different ways. This way is from the infintisimle small to the infinite large. 
Diagnositic, formative, summative, and generative: Content If all the students are understanding as well as those speaking, then it probably is generative or summative if large numbers and zero were discussed before. If not, then other students might still be in the formative and some may be having the "light come on" and it could be summative. 
Mr. C.  How about this one? [Writes 78 + 49 = 78] Mr. C.  Why is it false? Mr. C.  Why is that true? We added something. 

Satisfied with their understanding he presses on with what he has planned for reasoning and proof by diagnosing what students know. Of to a good start the discussion gets side tracked awhile with the idea of zero as part of a number rather than a value. Mr. C aware of how fragile students understanding is returns to helping to facilitate learning about numbers (content of mathematics).

Diagnositic: Proces Proof  Proving an argument involves a conjecture about what is believe to be true for all cases, articulating that idea, refining it, and editing it until it is acceptable or proving it is false. 
Mr.C  How do you know that is true? Have you ever done that? Mr. C.  So we kind of have a rule here. Don’t we? What’s the rule? Ann  Anything with a zero can be the right answer. After some additional discussion to clarify what the children are talking about the number zero not zero in numbers like 20 or 500, the children are challenged to state a rule that they could share with the rest of the class. 

Addition of zero and zeroes in a numeral are different. (this can be problematic for students at this age, since place value is probably still a year away)  Satisfied, again, with their understanding he presses on with what he has planned for reasoning and proof by refturning to formative assessment mixed with ideas being suggested by what students know or are discovering.  Diagnositic, formative, summative, and generative: Content Going through a quick cycle of formative assessment, summative, and genertative before returning to processes and proof, when satisfied that students understand. 
Ellen  When you put zero with one other number, just one zero with the other number, it equals the other number. Mr.C.  Wait let me make sure I got it. You said, "If' you have a plane zero with another number." With another number like  like just sitting next to the number? 

Uses vocabulary "conjecture" but does so nonchalantly. This is a sort of priming of the pump for another day.
Reviews today's / yesterdays activities and is probably pretty pleased. Decides that the conjectures should be shared with the rest of the class, and the sheets put up on the wall. 
Formative and moves to summative: Process 
Ann  Zero added with another number equals that other number. Children  Yes! Yes! Ellen  Because zero is nothing. You're not putting anything with the number, so it is still the same. Mr. C.  So we have a conjecture here about adding zero to any number. Can anybody think of another different conjecture about zero that is true for all numbers? Mr. C.  Is that always going to be true for every number? Mr. C.  How do you know? Steve. Mr. C.  Writes each conjecture on a separate sheet of paper. Zero added with another number equals that number. 

Day 2  
Property of zero for subtraction  Reviews today's / yesterdays activities and is probably pretty pleased. Decides that the conjectures should be shared with the rest of the class, and the sheets put up on the wall.

Diagnostic or review, move to formative for conjecture and proof Generative for Property of zero for subtraction Summative and or Generative for conjecture 
Mr. C.  How about this one, is this true or fake? [Writes 785 – 785 = 0] Mr. C.  Those are pretty big numbers. Are you sure? Mr. C.  Write his conjecture (A number minus the same number is zero.) on another sheet of paper and adds it to the two conjectures already posted on the wall. 
Dr. Robert Sweetland's Notes ©