Action Research Paper

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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3. However, there is a domino effect in teaching. One small adjustment can
have profound repercussions. I was curious to see how making communication a
priority would impact student learning. I proceeded to investigate for myself, and
fortunately I was open for adventure. Communication in math class, like a great
many things, needs to be explicitly taught. I hadn’t considered this going in, but
luckily my students readily revealed what we needed to learn.
Foundational Steps
The first thing I learned was that asking questions alone did not produce
immediate results. Students need to be taught how to respond. When Lynne was
asked how she got her answer 305‐169=136 (Figure 1), she wrote, “I just subtracted
it.” When Hayden was prompted to explain why he put those little ones above the 5
and the 6 when vertically solving 567+259=826 (Figure 2), he proudly responded, “I
don’t know, but it still works!” I just did it and I don’t know weren’t the persuasive
statements for which I hoped. “In some instances, children’s inability to give
convincing arguments may stem from the fact that they are unaccustomed to giving
explanations; in other instances, their inability may reflect a lack of understanding”
(Flores, 2002, p. 274).
I realized my students would need many opportunities to verbalize their
thinking before they could argue an answer or write about their process. In some
instances guiding questions were helpful. What did you do first? What do these
numbers represent? Why did you do that? Using these prompts supported those who
needed help communicating and revealed misconceptions.
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4. A second important component to communication is listening. To have
meaningful conversations, my students needed to learn how to listen to each other.
How little my students paid attention to one another surprised me. Initially, they
looked to me for all truth and definitive answers and routinely ignored the
questions and comments of their peers. How could we have meaningful discourse if
I was the only one listening? I shifted the paradigm by standing back and allowing
student after student to share his thinking on the interactive white board. As
facilitator, I named the technique after the instructing student (Photo 1) and asked
for a different way to solve the problem. This prompt required students to listen to
their peers, to make connections between strategies, and to think creatively. Those
who repeated previously shared strategies often realized this during their
explanations. Amelia wasn’t sure her decomposing strategy was the same as
Alejandro ’s until she shared it (Figure 3). The class decided they were the same.
Seeing themselves and their classmates as mathematical thinkers was an important
element in developing mathematical communication. My class was beginning to
hear each other and reason together.
Just as asking questions did not initially produce meaningful answers, likewise,
having students represent their thinking in writing didn’t yield clear explanations.
To develop this skill, much of their writing was done in collaborative poster‐making
(photo 2). At first, the students’ posters represented their joy of using magic
markers and communicated very little mathematical thinking (photo 2.1). I wanted
the students to appraise, internalize, and communicate their thoughts, so I asked
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5. them to evaluate the posters for a particular problem and decide which was the
clearest and why. This conversation helped the students not only see the
importance of the words, but also how their proximity to the example, the sequence
in which they are written, and preciseness of their vocabulary all mattered (photo
2.3). “Over time, students should become more aware of, and responsive to, their
audience as they explain their ideas in mathematics class. They should learn to be
aware of whether they are convincing and whether others can understand them”
(NCTM, p.61). The clarity of their work became more evident as the purpose for
written communication gained value.
As one change led to another, I could see the dominoes continue to fall. Just by
laying this foundation, we had accomplished a lot. I was asking questions that
exposed their depth of understanding, and they were supporting their answers even
without being prompted. They were listening to and questioning each other and
putting their thoughts in writing. Our communication was evident. But as a
somewhat argumentative person, I stood back and said, “So what?” I wondered what
my students were actually gaining by all this communication. How was sharing their
thinking, asking questions, and explaining their ideas impacting student learning?
Sharing Thinking
Asking a question can start a discussion, and when students share their
thinking, we all gain from the conversation. The speaker strengthens, solidifies, and
deepens her thinking, and the listeners discover a fresh way to look at things.
During a study of subtraction, I pressed my students to share their thinking.
Ali shared how making a number line shows the difference between two numbers
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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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7. After my students discovered all triangles have an interior angle sum of 180˙,
they questioned whether squares, rectangles, and other quadrilaterals would have
similar measurements. Squares and rectangles proved simple, 90˙ times 4; but the
trapezoid and rhombus, like the triangles, required careful angle measuring. Once
they determined the interior angles of all quadrilaterals did indeed have the sum of
360˙, someone asked about the pentagon! Hexagon? Heptagon? What about all
polygons? My students had moved beyond asking clarifying questions and were
posing their own mathematical investigations.
Pairs of students set off to work. Armed with protractors, they measured and
determined the sum of the interior angles of regular and irregular polygons. They
confirmed their data with an interactive web site: Math Open Reference, Polygon
Interior Angles (http://www.mathopenref.com/polygoninteriorangles.html),
shared their findings, and made a group chart (Figure 8).
In making sense of their data, the class had a lot to communicate. Cate was
the first to show smaller polygons within larger ones. She saw two trapezoids inside
the pentagon (figure 9), but adding the sum of their angles was 180˚ too much. Why
was this? Building on this idea, Devin recalled in pattern blocks six green triangles
make one yellow hexagon, but adding the 180˚ of six triangles gave 360˚ too much
for the sum of interior angles of a hexagon. He went on to show the interior circle
formed where the equilateral triangles met in the middle (figure 10) and promptly
added that he needed to subtract that extra 360˚. Explain your thinking? Tracing the
hexagon with his finger, Devin said, “They don’t touch the sides.” Observing this,
Mark said he could make two equal trapezoids in the hexagon, add the sum of their
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8. angles, and that would result in the sum of the angles in the hexagon. Show me.
Vanessa said two hexagons would equal a decagon. How do you know? Amelia
noticed that each time we added a side to a polygon the sum of the interior angles
increased by 180˙. She observed that a straight line also measures 180˙ and
connected the additional side to the additional straight line. Mark conjectured that
the sum of the interior angles had to have something to do with triangles because
the triangle is the smallest polygon (the fewest sides), 180˙ is part of the pattern,
and 180˙ is also the sum of the interior angles of a triangle.
Asking questions like, “How did you do that?” or “I don’t get it,” which is
fourth‐grade code for, “Will you please explain that to me again?” was another way I
saw children take ownership of and become invested in their learning. They
measured angles with protractors (photo 3), drew interior polygons, made charts,
added, subtracted, and multiplied. Some students determined a formula (the
number of sides the polygon has, minus two, times 180 degrees), others used the
pattern from making a chart (add 180˙), and some drew smaller polygons (often
triangles) in the larger ones to find the sum of the interior angles for any polygon.
Using geometry tools in context and encouraging communication around their
discoveries took my fourth graders to a level of mathematics I never would have
thought to take them. They were deeply entrenched doing the work of
mathematicians: listening to each other, noticing patterns, testing theories, proving
their conjectures, asking questions, and representing their thinking.
When the question, “How do you know the sum of the interior angles of all
triangles is 180˚?” was answered, a class that was used to answering and asking
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9. questions took me down a weeklong digression of the interior angles of all polygons.
At that point, I had to decide whether I wanted to follow their line of questioning or
return to my fourth grade curricular objectives. I am glad I followed their lead. We
explored more topics more deeply, and my students developed their reasoning skills
though sharing and building on each other’s thinking when their questions were
valued.
Explaining and Justifying Ideas
When I began this research, I wasn’t sure how often I should press my
students to explain their thinking. Would I question every single answer? Marilyn
Burns (2004) states, “Teachers are accustomed to asking students to explain their
thinking when their responses are incorrect. It's important, however, to ask children
to explain their reasoning at all times” (p.17). I gave this idea a try and took it to
heart when going over routine homework. In previous years, I called on students to
share their answers, and we only paused when an incorrect response was given.
With my new mantra, I asked them to explain their thinking for each question. I
became a believer in at all times with this question, “What unit of measure would
you use to weigh a pencil? A. pound B. gram C. kilogram D. inch” (Enright &
Spencer, 2005, p. 11). When the student answering justified B. gram, in part, by
saying that a pound was a little too much, the lesson evolved from establishing a
benchmark for pounds to a deep discussion of the properties of even and odd
numbers.
John explained, “B, because a paperclip weighs a gram.” Heads nodded. This
was a benchmark the class seemed to agree on. He continued, “And a pound is a
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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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11. numbers.” He showed this by counting by two’s and drawing dots on the board. One
set of two makes two. Two sets of two make four. Three sets of two make six.
William, still grappling with differences between adding and multiplying odd and
even numbers made this conjecture, “An odd plus an odd makes an even, so an odd
times an odd must make an even.” He was surprised when we tried it and all the
products were odd numbers (7x3=21, 5x9=45). Then he said, “Oh, I get it.” William
talked and drew out his process with 3 x 3 showing with dots on the board like
Eugene had done that 3 x 2 was even, but when he added on the next group of three,
this odd number made the total an odd number (figure 11) . He concluded, “That last
odd number makes the product odd.” Like John before him, William refined his idea
while justifying his thinking.
They were discovering differences between addition and multiplication that
many adults miss. I discovered problems do not have to be inherently rich
mathematical tasks to produce deep and meaningful conversations. In a short time,
we had come a long way from determining it takes 72 pencils to equal one pound.
Explaining and justifying their ideas allowed my students to struggle through
disequilibrium, test conjectures, and make sense of the math they use.
Conclusion
Asking one small question can have a huge impact on teaching and learning.
“Show me,” changed the structure and dynamic of my classroom. “What is your
answer and how did you get it?” helps clear up misconceptions. “Why does that
work?” can lead to mathematical debates. All of this communication deepened my
students’ understanding of mathematics.
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12. This transformation doesn’t happen overnight. Students need to be taught
how to speak, listen, question, and write (figure 12). All of this takes time. In my
experience, it is time well spent. I told my students at the start of the year that they
would learn more from each other than they would from me, and I don’t think that
was an exaggeration (photo 4). Changing one thing, watching and responding to the
ripples of that change, helped me see its impact. I found the value of putting
communication at the center of my math instruction to have an exponential effect.
Yes, they learned to listen, speak, and write mathematically, but they also made
deep and meaningful mathematical connections, developed their mathematical
dispositions, and became a community of problem solvers. When I allowed space for
my students’ thoughts and questions to guide my instruction, ownership and joy
permeated math class. I will argue that allowing children to process content through
communication should never become an educational fad.
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13. Figure 1: Lynne’s subtraction solution. “I just subtracted it.”
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14. Figure 2: Hayden’s addition strategy
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15. Figure 3: Amelia’s and Alejandro ’s addition strategies
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16.
Figure 4: Ali’s Number line
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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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19. Figure 8—Class Chart, Interior Angles of Polygons
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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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21. Figure 11—William uses 3 x 3 to illustrate two odd factors give an odd product. He
draws three groups of three in a pyramid form. Then he circles the pairs. He
explains that 2 x 3 is even because each dot has a partner, but when we add on the
next odd number (three) there will always be one without a partner. Odd times odd
makes odd, but odd plus odd makes even!
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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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23. Photos:
Photo 1—Sharing Strategies: Maria’s Rounding Strategy show how she rounds up to
friendly numbers, adds, and subtracts the extras.
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24.
Photo 2—Making Posters: How can you best communicate your group’s thinking?
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25.
Photo 2.1—This group glued the questions to the poster, drew some solutions, and
connected their solutions to the questions with arrows.
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26. Photo 2.2—This poster is a response to, “What is the commutative property of
multiplication?” It follows the guidelines: title, names, proof, examples, and words.
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27.
Photo 3—Measuring the interior angles of octagons
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28.
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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