AUDIENCE THEORY -CULTIVATION THEORY - GERBNER.pptx
Communication speaks2 -kinman
1. Abstract
Recently I turned my attention to the NCTM Principles and Standards and
was surprised to see “communication” as a key factor. Metacognition? Math
journaling? Are we still doing this? I wondered what would happen if I put
communication at the center of my math instruction? I was surprised by the results
of my action research. Communication is not a passing fad! Sharing thinking, asking
questions, and explaining and justifying ideas belong in the very heart of every math
class.
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2. Communication Speaks
“In a grades 3–5 classroom, communication should include sharing thinking,
asking questions, and explaining and justifying ideas. It should be well integrated in
the classroom environment. Students should be encouraged to express and write
about their mathematical conjectures, questions, and solutions” (National Council of
Teachers of Mathematics, 2000, p. 193). What would my fourth grade classroom
look like, how would I teach, and what would my students need if I actually strove to
meet this standard in a real and substantive way?
Where to Begin?
A somewhat argumentative person myself, I have always enjoyed teaching
essay writing as a convincing and organized argument. I was compelled by the idea
of having my students argue and justify their mathematical thinking in much the
same way they support their theses in my writing classes. But I wasn’t sure where to
begin. I decided the only thing I would do differently in my math instruction was to
ask my students to explain their thinking and see where it would take us. This one
question took many forms: “What is your answer and how did you get it?” “Explain
your thinking.” “Why does that work?” “How do you know?” And because we are in
Missouri, simply, “Show me.” My students became accustomed to my mantra and
explained their thinking prior to being asked. They learned that giving an answer
wasn’t enough and began to explain their processes. As we became immersed in
these questions, it was easily sustained. Conversations were expected and math
class was interesting and engaging for both my students and me.
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6. (figure 4). This gave a logical and visual reason for why the answer to a subtraction
problem is called the difference. Sharing their thinking provided a context for the
correct use of math terminology. To be understood, they needed to use the right
vocabulary.
Subtracting with regrouping in the hundreds place in order to rename a zero
in the tens place was a confusing procedure for many of my fourth graders. Some,
like Lynne, remembered the algorithm and just subtracted without questioning how
it works (figure 1). Cate’s approach got around this complication and revealed her
understanding of place value and what subtraction means. She decomposed the
subtrahend and started “taking away” with the hundreds (figure 5). Eugene
presented a similar technique, but he counted backward using an open number line
(figure 6). Wyatt’s subtraction strategy (figure 7) revealed what he knows about
place value and negative integers. His method and the others shared allowed
students to see the process of subtraction beyond the cross‐out and regroup
algorithm that was difficult to remember. Learning multiple strategies, seeing a
problem from different perspectives, using correct terminology, and solidifying
their understanding were all results of students sharing their thinking.
Asking Questions
As my questioning became part of the fabric of the class, I noticed my
students were not only eagerly answering them, but they also started asking
questions themselves. “The most productive discussions around mathematical ideas
seem to happen in classrooms where questioning is an almost spontaneous part of
the way children talk to one another about their work” (Kline, 2008, p. 146).
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10. little too much, a kilogram is a lot more, and inches are silly because they are for
distance.”
I wasn’t sure the class had a solid benchmark for pounds so I pressed, “Is a
pound just a little more? How many pencils would equal a pound?” I dropped a
pencil on the kitchen scale. It barely registered. I added a few more until we dumped
in three boxes of pencils (12 per box) and still hadn’t reached half a pound. Six
boxes later, one pound was showing. I asked, “How many pencils is that? How many
make a pound?” John offered, “93,” but Christopher interjected, “No, it can’t be an
odd number!” John explained that he knew 2 boxes was 24 pencils, so he mentally
multiplied 24 three times. When he did this on the board he wrote 24+24+24 and
corrected his answer to 72. Christopher spoke out, “I knew it couldn’t be odd!” I
asked him to explain his thinking. He said, “An even plus an even is an even, so an
even times an even is even. Twelve (pencils in a box) times six (boxes of pencils) is
an even (number) times an even (number) so the answer’s got to be even too.” But
the way John solved it—24 times 3—was an even number times an odd number.
The class knew an even plus an odd gave an odd answer. How could an even number
times an odd number produce an even answer? They were connecting what they
knew about how even and odd numbers behave in addition and applying it to
multiplication. Disequilibrium set in. We did some additional multiplication facts
with mixed even and odd factors and consistently found even products. I asked,
“Why does this work?”
Eugene offered, “It doesn’t matter how many times if you have an even
number, it (the product) will always be even because you are counting by even
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17. Figure 5—Cate’s decomposing the subtrahend strategy 504‐169=335
Figure 6—Eugene’s Open Number line 504‐169=335
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18.
Figure 7: Wyatt’s subtraction strategy 12/05/08
Wyatt decomposed the numbers and started with the ones. He figured 5‐9= ‐4. He
moved to the tens and subtracts 0‐60=‐60. Then he worked out the hundreds, 300‐
100= 200. Last, he combined his differences: 200‐60‐4. He did this in two steps. 200‐
60=140 and then 140‐4=136. Wyatt shows us that 305‐169=136.
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20. Figure 9: Cate sees two trapezoids in this pentagon. The red arc shows where the extra
180˙ is found.
Figure 10: Devin sees 6 triangles in the hexagon and subtracts out the extra 360˙
produced by the circle of extra angles in the center.
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22. Figure 12
Steps I used to establish and promote communication:
1. Provide a safe environment that promotes risk‐taking
Set behavioral norms with the class.
Prompt: What will you need to do your best learning?
What are your hopes and fears about math class?
2. Develop discourse in math class
Ask questions and wait for answers. Hear all voices.
Strategies: turn and talk; think, pair, share; call on everyone
3. Expect listening to the ideas of peers and allow grappling to understand them
Ask students to paraphrase, compare ideas, question, and add on to each
other.
Prompt: Who can explain how she figured it out? How are these strategies
alike? What questions do you have? Can anyone add on to that idea?
4. Allow processing of content through writing
Use poster‐making, journaling, and exit tickets with clear guidelines (title,
names, proof, examples, and words).
Prompts: How did you solve this problem? Pretend your friend is sick: write
a letter explaining what we learned today. Describe what you learned today.
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Photo 4: Sharing strategies: The Open Number Line
References
Burns, M. (April 2004). 10 big math ideas. Instructor, 113(7), 16-19, 60.
Enright, B. & Spencer, D. (2005). Test Ready Plus Mathematics. North Billerica, MA:
Curriculum Associates.
Flores, A. (January 2002). How do children know that what they learn in
mathematics is true? Teaching Children Mathematics, 8(5), 269-274.
Kline, K. (October 2008). Learning to think & thinking to learn. Teaching
Children Mathematics, 15(2), 144-151.
National Council of Teachers of Mathematics. (2000). Principles and standards
for school mathematics. Reston, VA: Author.
Page, J. (2008). Polygon interior angles—Math open reference.
http://www.mathopenref.com/polygoninteriorangles.html.
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