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 believed to be true for all cases,
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. Communicate their reasoning about their claims to prove, disprove, and revise through cooperaation. 
Zero Properties of zero Properties of zero for addition

Teacher has the following outcomes for learners: Construct a mathematical proof by creating ideas, articulating what they believe to be possible (conjecture), 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 the accuracy of mathematical conjectures (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. 

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Decides to assess or review what the learners know about zero and adding zero to another number. Don't press them to make a generalization. Give them additional number sentences for the same principle, but in this case the number sentence is false. 
Diagnositic assessment Content: Property of zero Property of zero for addition 
Mr. C.  Writes [58 + 0 = 58] on the board How about this number sentence? Is it true or false? Children  True. True. 
Pressing further to see how em 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 Properties of zero Property of zero for addition 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 others might still be in the formative and some may be having the light come on and then it could be summative. 
Mr. C.  How about this one? [Writes 78 + 49 = 78] Children  False! No, no false! No way! Mr. C.  Why is it false? Jenny  Because its the same number as in the beginning, and you already put more with it, so it would have to be higher than the number you started with. Mike  Unless it was 78 plus zero equals 78. That would be right. Mr. C.  Why is that true? We added something. Sieve  But that something is like nothing. Zero is nothing. Mr. C.  Is that always going to work'? Lynn  If you want to start with a number and end with a number, and you do a number sentence, you should always put a zero. Since you wrote 78 plus 49 equals 78, you have to change a 49 to zero to equal 78, because if you want the same answer as the first number and the last number, you have to make a zero in between. Mr. C.  So do you think that will always work with zero? How about this one? [Writes 789,564 + 0 = 789,564] Children  That's true. 

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 student's understanding is returns to helping to facilitate learning about numbers (content of mathematics). Asks students to revoice their ideas to probe their understanding. 
Diagnositic : Proces Proof  Proving an argument involves a conjecture about what is believed to be true for all cases, articulating that idea, refining it, and editing it until it is acceptable or proved false. 
Mr.C  How do you know that is true? Have you ever done that? Ann  I will tell you. All those numbers plus zero, you won't add anything, so it would be the same number. 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. Mike  No, because it was 100 plus 100 and that’s 200. Jenny  That's not what we are talking about. It doesn’t have just plain zero. Ann  I said!, umm, if you have a zero in it, it can't be like 100, because you want just plain zero, like 0 plus 7 equals 7. 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 numerals are different. (this can be problematic for learns at this age, since full understanding of 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 Properties of zero 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. Steve  Not true. 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? Ellen  No, added with another number, it equals that number. Mr.C.  Okay, So what do we want to say here? ... Ann? 

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: for the process concept of Proof 
Ann  Zero added with another number equals that other number. Mr.C.  Is that always going to be true for any number, even really big numbers that we haven’t tried? Children  Yes! Yes! Mr. C.  How do you know that? Ellen. Ellen  Because zero is nothing. You're not putting anything with the number, so it is still the same. Mr. C.  Do you all agree? Children  Yes 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? Lynn  I can. You can take away zero from a number, and you still get that number back. Mr. C.  Is that always going to be true for every number? Children  Yes. Mr. C.  How do you know? Steve. Steve  Because, it is just like the last one only it's take away. You are not taking anything away. Mr. C.  We should write these conjectures down so we can remember them and so we can share them with the rest of the class. 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 informati displayed so all can see (put up on the wall). He will introduce the term conjecture by using it in context. 
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? Sally  Yeah Whenever you lake a number away from itself, you get zero. Ann  [Excitedly] Hey, we have another conjecture here. A number minus the same number is zero. Mr. C.  Are you sure that's always true? How do you know? If you have some things and you take them all away, there's none left. It's zero. 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. 