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Unformatted text preview: Looking at How Mathematics teachers gain a wealth
of information by delving into the
thinking behind students’ answers,
not just when answers are wrong but also when they are correct. Marilyn Burns irst, a confession: Only during the last 10 to 15 years of my teaching career have I thought deeply about assessing students‘ understanding and learning progress. As a beginning teacher, I focused on learning to manage my classroom, plan
lessons, and hold students’ attention. Later, my focus shifted
to improving my lessons and expanding my instructional
repertoire. During those years, my attention was always
firmly on my teaching. Assessment was not one of my
concerns. Yes, 1 gave assignments and quizzes and examined
the results, but 1 did so more to determine grades than to
figure out what students were thinking. Assessment plays a much different role in my teaching
today. Although I‘m no longer a fulltime Classroom teacher, I
still spend time teaching students in elementary classrooms as
I try out new instructional ideas. 1 now approach assessment
in an intentional way and incorporate it into every lesson. No
longer am I satisfied to simply record students' performance
on assignments and quizzes; now, my goal is to find out, as I
teach, what the students understand and how they think. I
am still interested in honing my lessons, but along with plan
ning the sequence of learning activities, 1 also prepare to
question students about their thinking during class discus
sions, in individual conversations, and on written assign
ments. In addition, linking assessment with instruction has
become a key issue in the professional development I provide
to other teachers. After teaching a lesson, we need to determine whether the
lesson was accessible to all students while still challenging to
the more capable; what the students learned and still need to 26 EDUCATIONAL LEADERSHIP/NOVEMBER 2005 know; how We can improve the lesson to make it more effec
tive; and, if necessary, what other lesson we might offer as a
better alternative. This continual evaluation of instructional
choices is at the heart of improving our teaching practice. Uncovering the Way Students Think In my early teaching years. I was a devotee of discovery
learning, sometimes called inquiry learning. This instructional
approach involves designing learning activities that help Students Reason students discover concepts and make sense of facts and prin
ciples for themselves, rather than relying on textbooks or
teacher explanations. l implemented this approach by asking
the class a carefully prepared sequence of questions, in the
style of Socrates. if a student‘s response was correct, 1
continued to the next question If a student's response was
incorrect, other students would typically raise their hands to
disagree, and I‘d let a class discussion unfold until someone
proposed the correct response. Then I'd continue with the to SIEVE BARRETT next question. If no students obiected to an incorrect
response, I’d ask a slightly different question to lead students
to the right answer. Years later, I thought about why the discovery method of
instruction seemed flawed. The problem was that when a
student gave a correct response, I assumed that both the
student who had answered correctly and his or her class—
mates understood the mathematics hehind the problem. I
never probed students‘ level of understanding behind their
responses; 1 just happily continued on my teaching trajectory Students need to be able to look
at mathematical situations from different perspectives. As a result, 1 never really knew what students were thinking
or whether their correct answers masked incorrect ideas. I
only knew that they had given the answer I sought. 1 no longer teach this way. Aithough 1 still believe in the
value and importance of using questions to present ideas for
students to consider. l've broadened my use of both oral and
written questions so that I now attempt to probe as well as
stimulate students’ thinking. For example, when teaching fractions to a class of 4th
graders, I wrote five fractions on the board—[[4, ”/16. 3/3 , V1“,
and 3/4 —and asked the students to write the fractions in
order from smallest to largest. I then added another step by
asking them to record their reasons for how they ordered the
fractions. After giving the students time to solve the problem,
1 initiated a wholeclass discussion. Ruben reported first. He
said with confidence, "The smallest fraction is 1/10.“ in my
early days of teaching, I was accustomed to questioning
students when their answers were incorrect, but not when
they were correct. Now, however, I asked Robert to explain
how he knew that 1/”, was the smallest fraction. Robert read
from his paper, again with confidence, "Because 1/", is the
lowest number in fractions." The students had previously cut
and labeled strips of construction paper to make fraction kits,
and 1/", happened to be the smallest piece in their kits. The {\EhOLlAlltJN l'UR SUPERVISION AND CL‘RRILUIUM DFVELOP‘iiFNT 27 fraction kit—a standard tool in my
instructional repertoire that I‘ve always
found effective for deve10ping students’
understanding of fractions—had led
Robert to an incorrect generalization. By questioning Robert’s correct
response. I was able not only to clear
up his misunderstanding but also to
improve on the fraction kit lesson to
avoid this problem in the future. When
I teach this lesson now, 1 always ask
students to consider how we would
name pieces that are smaller than 1/16; I talk with them about how we could
continue to cut smaller and smaller
pieces and find fraction names for even
the teeniest sliver. Incorporating students‘ reasoning
into both written assignments and
classroom discussions was a crucial
step toward making assessment an inte—
gral and ongoing aspect of my class
room instruction. Now it’s a staple of
my math teaching. Assessment Through
i Students' Written Work
One of the main strategies 1 use to
assess students’ learning is incorpo—
rating writing in math assignments.
There are many ways to present writing
assignments that yield as much infor—
mation as possible about what students
are thinking (see Burns, 2004). Ashfor more than one strategy Solving
math problems often requires making
false starts and searching for new
approaches Students need to develop
multiple strategies so that they become
ﬂexible in their mathematical thinking
and are able to look at mathematical
situations from different perspectives.
Even when students are performing
routine computations, asking them to
offer more than one way to arrive at an
answer provides insight into their thinking. 28 EDUCATIONAL LEADERsHJPINGVEMBER 2005 PHOTO COURTESY OF MAHJLVN BURNS EDUCAH ON ASSOCIATES Incorporating students’ reasoning into Classroom discussions makes assessment an integral aspect of classroom instruction. For example, I worked with one 2nd
grade class that had been focusing on
basic addition. When they encountered
more difficult problems—those involving
numbers above 5, such as 9 + 6 and
7 + 8——the students' fallback strategy
was always counting. Over time, I
helped them develop other strategies
for addition. One day, to assess their
progress, I asked them to add 6 + 7 and
to explain how they could figure out
the answer in more than one way. Their
work was revealing. Daniel described
five methods, including the following
method. that showed the progress he had made with the important skill of
decomposing numbers—taking numbers
apart and combining them in different
ways: You take 1 from the 6 and 2 from the T and then you add 5 + 5. Then you add
on the l and the 2 and you get 13. Ryan, in contrast, was able to offer
only two methods, even when I pushed
him for more. He wrote. (1) You start with the 6 and count on 7
more. (2) You start with the T and
count on 6. Although Ryan's work showed that
he understood that addition was
commutative, it also showed that his
addition strategies were limited to
counting. Let students set parameters. A good
technique for assessing students’ under
standing as Well as differentiating
instruction is to make an assignment
adjustable in some way, so that it is
accessible and appropriate for a wider
range of students. For example, I
worked with one 3rd grade class in
writing word problems. For several
days, the students discussed examples
as a group and completed individual
assignments. Sometimes I gave a
multiplication problem—3 x 4, for
example—and asked students to find
the answer and also write a word
problem around the problem. At other
times I gave them a word problem—
such as, “How many wheels are there
altogether on seven tricycles?”—and
asked them to write the related multi
plication problem and find the answer.
Finally, I gave students the assignment
of choosing any multiplication
problem. writing a word problem for it.
and finding the answer in at least two
ways. Having the students choose their
own problems allowed them to decide
on the parameters that were comfort
able for them. Carrie chose 5 X 2 and
wrote a problem about how many
mittens five children had. Thomas
chose 102 x 4 and wrote a problem
about the number of wheels on 102
cars. Each student’s choice gave me
information about their numerical
comfort as well as their skill with
multiplication. Assess the same concept or skill in
different ways. I’ve often found that a
student's beginning understanding,
although fragile, can provide a useful By building and using a wide repertoire of assessment strategies, we can get to know more about our students than we ever thought possible. building block or connection to more
robust learning. Sometimes a familiar
context can help a student think about
a numerically challenging problem.
Using ﬂexible assessment approaches
enables us to build on students'
strengths and interests and help them
move on from there. In a 4th grade class, I watched Josh,
who was fascinated by trucks, over
come his confusion about dividing 96
by 8 when I asked him to figure out
how many toy 8wheeler trucks he
could make if he had 96 toy wheels.
Although his numerical skills were
weak, he was able to make progress by
drawing trucks and examining the
pattern of how many wheels he needed
for two trucks, then three trucks, and
so on. Take occasional class inventories.
Compiling an inventory for a set of
papers can provide a sense of the class‘s
progress and thus inform decisions
about how to differentiate instruction.
For example, after asking a class of 27
5th graders to circle the larger frac
tion— 2/3 or 3/4 —and explain their
reasoning, I reviewed their papers and
listed the strategies they used. Their
strategies included drawing pictures
(either circles or rectangles); changing
to fractions with common denomina—
tors (3/12 and 9/12); seeing which fraction
was closer to 1 (2/3 is 1,13 away, but 3/4 is
only 1/4 away); and relating the fractions
to money (2/3 of $1.00 is about 66
cents, whereas 3/4 of $1.00 is 7’5 cents). Four of the students were unable to
compare the two fractions correctly. I
now had direction for future lessons
that would provide interventions for
the struggling students and give all the
students opportunities to learn different
strategies from one another. Assessment Through
Classroom Discussion
Incorporating assessment into class
room discussion serves two goals: It
provides insights into students'
thinking, and it ensures that no student
is invisible in the class. but that all are
participating and working to under—
stand and learn. Here are some strate—
gies to get the most out of class
discussions. Ask students to explain their answers,
whether or not the answers are correct.
When I follow up on both correct and
incorrect answers by asking students to
explain their reasoning, their responses
often surprise me. Some students arrive
at correct answers in unexpected ways.
For example, when comparing 4/5 and
3/4, Brandon changed the fractions so
that they had common numerators—
12/15 and 12/16. He knew that 16ths were
smaller than 15ths, so 12/15, or “/5, had
to be larger! Students may also surprise
us by using incorrect reasoning to
arrive at the correct answer. lindsay a
3rd grader, used 7 x 3 = 21 to conclude
that 8 x 4 = 32. “Each number is just
one bigger," she said, and went on to
explain that 1 more than T is 8, 1 more ASSOCIATION FOR SUPERVISION AND CURRICULUM DEVELOPMENT 29 —l than 3 is 4, and 1 more than each of
the digits in 21 makes the number 32.
Although Lindsay's method worked for
this problem. it doesn‘t work for all
problems! Ask students to share their solution
strategies with the group. After a student
responds to a question that I pose and
explains his or her reasoning, I ask the
group, “Who has a dillerent way to
solve the problem?“ or “Who has
another way to think about this?“ I
make sure to provide sufﬁcient wait
time to encourage students to share
ideas. In addition to providing insights
into students' thinking and under
standing, this method also reinforces
the idea that there are different ways to
think about problems and lets the
students know that I value their indi
vidual approaches. Call on students who don't volunteer.
For many years, I called only on
students who had the confidence to
offer their ideas. For students who were Solving math
problems often
requires making false
starts and searching
for new approaches. less confident, I relied on their written
work. I didn’t want to intrude on shy
students and put them under addi
tional stress. I‘ve since changed this
practice, partly because of the insights I
gained from the excellent professional
resource Classroom Discussions (Chapin,
O'Connor, & Anderson, 2003). I now
tell students that it‘s important for me
to learn about how each of them thinks
and, for that reason, I need to hear
from all of them. I reassure them,
however. that if 1 call on them and they Purim a! KEVIN Davis don’t know the answer, they should
just let me know. I tell them, “It's
important for me to know when a
student isn‘t able to explain so I can
think about what kind of support to
give." I‘m always careful to check in
with the student later to determine
what kind of intervention I need to
provide. Use smallgroup work. This technique
is especially useful for drawing out
students who are reticent about talking
in front of the whole class. After posing
a problem, I‘ll often say, “Turn and talk
with your partner" or “Talk with your
group about this." Then I eavesdrop,
paying especially close attention to the
students who don‘t typically talk in
class discussions. Ask students to restate others’ ideas.
This is another strategy I learned from
Classroom Discussions. After a student
offers an idea or answer. I call on
someone else with the prompt,
"Explain what Claudia said in your own
words." If the student can’t do this, I
prompt him or her to ask Claudia to
explain again. If the student still isn’t
able to restate Claudia's idea, I ask
another student to try, reminding the
ﬁrst student to listen carefully and see
whether this alternate explanation
helps. After a student shares. I ask
Claudia, “Does that describe your
idea?" Depending on my professional
judgment about the student and the
situation, I may also return to the ﬁrst
student and ask him or her to try again. Improving Mathematics Teaching
According to the National Council of
Teachers of Mathematics (2000), To ensure deep, highquality learning
for all students, assessment and instruc
tion must be integrated so that assess
ment becomes a routine part of the 30 EDUCATIONAL LEADERSHIPINOVEMBER 2005 ongoing classroom activity rather than
an interruption. Such assessment also
provides the inlormation teachers need
to make appropriate instructional
decisions. Making assessment an integral part of
daily mathematics instruction is a Chal
lenge. It requires planning specific ways
to use assignments and discussions to
discover what students do and do not
understand. It also requires teachers to
be prepared to deal with students’
responses. Merely spotting when
students are incorrect is relatively easy
compared with understanding the
reasons behind their errors. The latter
demands careful attention and a deep
knowledge of the mathematics concepts
and principles that students are
learning. But the benefits are worth the eilort.
By building and using a wide repertoire
of assessment strategies, we can get to
know more about our students than we
ever thought possible. The insights we
gain by making assessment a regular
part of instruction enable us to meet the
needs of the students who are eager for
more challenges and to provide inter
vention for those who are struggling. References Burns‘ M. (2004). Writing in math.
Educational Leadership, 62(2). 30—33. Chapin. S. H.. O’Connor, C.. in Anderson.
N. C. (2003). Classroom discussions: Using
math tail: to help students learn. Sausalito,
CA: Math Solutions Publications. National Council of Teachers of Mathematics.
(2000). Principles and standards for school
mathematics. Reston, VA: Author.
Available: http://standards.nctm.org Copyright © 2005 Marilyn Burns. Marilyn Burns is Founder of Math
Solutions Professional Development,
Sausalito, California; 800—8689092;
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