After reading the NCTM article, “3 Strategies for PromotingMath Disagreements” written by Angela Barlow and Michael McCrory, I decided to watch
for opportunities to try these strategies in my third grade classroom. The article encourages teachers to watch for
differing opinions during classroom discussions and use this time as an
opportunity for students to debate, prove, and deepen their thinking about a
mathematical concept. This is the perfect atmosphere for

**MP3**Construct viable arguments and critique the reasoning of others.
A few days later,
my students were discussing why fractions have equal parts. I quickly realized
that several were confused about how to turn our “Crusty the Clown” pattern
block model into equal parts. The textbook and student workbook pages always neatly
did this for the children, so they never had to think about forming equal parts
of an irregular shape on their own.

As stated in the NCTM article, I asked students to choose a side
of the classroom that represented their viewpoint. Immediately, two of my
sharpest students went to the wrong side, declaring that Crusty had 5 equal
parts. Jeffrey and Jorge were correct that

*part*of Crusty was made up of 5 equal blocks, but they were not looking at the*whole*clown shape. Others who were unsure or undecided trusted the boys and stayed with them—even after much proof and evidence was offered by students on the other side of the room that Crusty had 14 equal parts. The children were delighted to debate and state their views with a microphone in hand. Emotions surged and intensified on both sides of the discussion; however, I stayed neutral and continued to encourage them to listen to each other and clearly state their opinions.After exchanging pattern blocks several times in an attempt to get equal parts, the students hit a deadlock. I told them to take a breather and read library books for ten minutes to clear their thoughts. When we came back together, some children were more open to actually hear what others were saying, and at last cleared up the misconception about “What does it mean to have equal parts?” And "Where is the

*whole*?"

Students realized that the entire shape had to be equally divided and that if you add area to the original shape you have just changed the whole. In the end, I told the both sides that I was extremely proud of them for standing their ground and methodically proving their ideas. The smaller group consisted primarily of soft-spoken Latinas. Although they were outnumbered by aggressive, dominant boys who

*knew*they were right, I was delighted to watch the girls become more confident and show their evidence of why the green pattern blocks had to be used to evenly divide up the entire shape without redefining the whole.

14 equal parts of Crusty the Clown |

The next day, we used snap cubes to show “5 of 6 equal parts of 18,” and I noticed that the children were immediately engaged and knew how to do it. Several enjoyed making the connection back to dividing our pattern block Crusty into equal parts as well. Success! I now have them beginning to listen to each other and starting to support their views with concrete evidence and simple proofs.

Source of the Math Disagreements article: Teaching Children Mathematics, v17 n9 p530-539 May 2011, NCTM

4 equal parts--but is the whole complete? |

6 equal parts--what happened to Crusty's hat? |

5 equal parts--but is it still Crusty the Clown? |

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