ERIC ED603932: Making Mathematical Thinking Visible

2020·1.6 MB·English

by  ERIC

Making Mathematical Thinking Visible
Johannah Nikula
Jill Neumayer DePiper
Mark Driscoll
Educational Leadership
December, 2019/January, 2020
M A K I N G

Mathematical
Thinking
V I S I B L E
For English language learners, diagrams
can be a powerful tool to develop and
communicate mathematical understanding.
Johannah Nikula, Jill Neumayer DePiper,
and Mark Driscoll
Imagine being a 6th grade student who is still learning English, sitting
in a mathematics classroom and trying to navigate the lesson. You
might wonder: What is the teacher saying I should do? Did my class-
mates solve it the way I did? Will the other students laugh at me
when I try to explain how I solved the problem?
Now imagine being a teacher in that same classroom. You might be
asking yourself: What can I say so that all my students will understand
what to work on? How do I know that all my students understand how
to solve this problem? How can I help students share their mathematical OCK
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ASCD / www.ASCD.org 77
For these students, diagrams served diagrams are a way that teachers and classmates
can see and hear what ELLs are thinking and
as a thinking tool, helping them to doing in their math work, thus positioning
ELLs as active participants in mathematics
navigate the quantities given in their
lessons (Driscoll, Nikula, & Neumayer DePiper,
2016).
word problems and find mathematical
To illustrate how diagrams can facilitate and
relationships to help solve the problem. make visible ELLs’ quantitative reasoning and
mathematical communication, we offer three
examples drawn from work by students we’ve
observed.
English. Unfortunately, students who are Jan, a 6th grade ELL, worked on the fol-
English language learners1 too often become lowing task: “Maureen bought 4 bags of soil
invisible in math classrooms because they lack for her gardens. Each garden needs 3/4 of a
proficiency in English and don’t have oppor- bag of soil. How many flower gardens can she
tunities to communicate their mathematical fill?”2 She drew a diagram (Figure 1)3, and then
thinking. It doesn’t have to be this way. explained it: “First, I drew the 4 bags then I cut
For the past decade, we have studied the role them into fourths. I shaded in three for each
that mathematical diagrams can play in giving one and circled it. That’s one garden. That’s
ELLs more productive mathematics experi- two gardens, that’s three gardens, that’s four
ences. We have collaborated with hundreds gardens, and that’s five gardens. So, she can fill
of middle-grades teachers across the country completely five gardens and she will have 1/4 of
to investigate how visual representations such a bag left over.”
as area models, number lines, and geometric
figures are used by students and how they
FIGURE 1. Jan’s Diagram of the
can facilitate ELLs’ mathematical reasoning
Flower Garden Task
and communication. By making mathematical
thinking visible, we’ve found, diagrams can
play a central role in enriching mathematics
education for ELLs.
Illustrating the Problem
Research has documented that mathematical
diagrams can be helpful for all students and
for ELLs in particular. Mathematics instruction Note that mathematical relationships are
that uses multiple types of mathematical repre- prominent in Jan’s diagram and explanation.
sentations reinforces the learning of concepts, Her work presented a repeated subtraction
processes, language, and norms of math- approach to division, subtracting 3/4 repeatedly
ematical communication (Chval & Chávez, from the four bags, and counting the total
2011; Woodward et al., 2012). Diagrams are number of times she can count 3/4 from the
of particular value because they can illustrate a four whole bags. The diagram helped her to
task’s mathematical structure and help connect keep track of the quantities and relationships
the text of a word problem to abstract concepts between quantities. In this sense, we can say
or the symbolic notation for solving it (Ng & that the practice of diagraming supported Jan
Lee, 2009). Diagrams are also useful tools for in becoming more proficient in the mathe-
supporting mathematical reasoning, problem matical practice of reasoning quantitatively and
solving, and communication. Furthermore, abstractly—a key mathematical practice under
78 EDuCAtionAl lEADErShip / DECEmbEr 2019 / JAnuAry 2020
the Common Core standards (NGA & CCSSO, counted in Spanish, “Uno, dos, tres. . . .”
2010). Carmine and Luis also had a set of printed
Sam, another 6th grade EL, took a dif- cards with sentence starters such as “This
ferent approach with his diagram on the same diagram shows . . .” and “I like this diagram
problem: because . . .” to help them structure explana-
tions during their pairs work. Using these
sentence starter cards, the students explained,
FIGURE 2. Sam’s Diagram of the
“This diagram shows 12 jumps,” and “I like this
Flower Garden Task
diagram because I see all the one-thirds.”
FIGURE 3. Number line for the task,
4 ÷ 1/3
He explained: “I made four flower gardens, In this case, the diagram, in combination
and I divided by fourths. Then I shaded in 3/4 with their use of their first language to support
of a garden because [. . .] the four boxes rep- thinking through a problem and sentence
resent the fourths and the part that’s shaded starters to frame their mathematical explana-
in represents the 3/4.” When the teacher tions, facilitated the students’ conceptualization
asked where the fifth garden came from, Sam of the task and provided the teacher a clear
responded: “When I shaded in with 3/4 in each representation of their thinking.
group, I found there was 1 left over in each
group because I used all the 3/4 I could find. I Supporting Sensemaking in Mathematics
took the 3/4 from the three other groups and For Jan, Sam, Carmine, and Luis, diagrams
made them into one group and I know there served as a thinking tool, helping them to
was one left over.” navigate the quantities given in their word
While Sam mistakenly said “I made four problems and find mathematical relationships
flower gardens,” he correctly labeled his four to help solve the problem. Diagrams can be
rectangles as “bags of soil,” and was able to use instrumental in facilitating this kind of sense-
the diagram to link quantities in the task to making and the selection and modification of a
mathematical operations needed for a solution solution strategy (Ng & Lee, 2009).
by attending to the mathematical structure of Of course, we also want students to acquire
the problem, another key mathematical practice and use the various forms of algebraic repre-
under the Common Core standards (NGA & sentations, but it is possible to build toward
CCSSO, 2010). such representations using diagrams. One of
In another classroom, two 6th grade ELLs, the research-based recommendations for math
Carmine and Luis, worked on a word problem instruction is to teach students how to use dia-
for which they needed to figure out what 4 ÷ grams, which can make them more effective
1/3 equals. The students used a number line problem solvers, rather than just focusing
(Figure 3), and as they proceeded, Carmine on algebraic representations (Woodward
pointed at the line, using her finger to show et al., 2012). By focusing on diagrams before,
jumps for each 1/3, starting at 0 and moving her or in addition to, algebraic representations,
finger closer to 4 with each jump, while Luis students have more opportunities to engage in
ASCD / www.ASCD.org 79
mathematical thinking. relationships they see, describe their problem-
Diagrams represent the critical mathematical solving processes, and deepen their under-
content—for example, fraction division in standing of embedded mathematical concepts.
the examples above—and they call students’ Through our work with ELLs, we’ve seen
attention to the quantities given in a problem, how students’ diagrams help communicate the
as well as the relationships between quan- richness of their mathematical thinking. Dia-
tities. Diagrams can also provide a structure grams help them showcase their mathematical
for understanding a term, phrase, concept, thinking to the teacher and other students,
or problem context because they can link the without having to rely only on words or having
words that set out a problem with the calcula- to know the “correct” words. In this manner,
tions that move toward a solution. In this sense, diagrams promote student agency and math-
they can provide an important bridge when ematical authority. For Jan, Sam, Carmine,
students struggle with the words-to-symbols-to- and Luis, all of whom have low-intermediate
numbers transitions. or beginner English-proficiency levels, their
As a tip for teachers, we have also found in diagrams were resources for communicating
our work that diagrams don’t have to be created their thinking in addition to serving as tools to
by the students to serve as powerful referents support thinking.
As shown in the example of Carmine and
By focusing on diagrams before, or in Luis above, ELLs can benefit from specific lan-
guage supports integrated with diagram use.
addition to, algebraic representations, Sentence starters, in particular, can help elicit
student explanations about their diagrams. In
students have more opportunities to
providing structures for students’ writing and
speaking, sentence starters can focus students’
engage in mathematical thinking.
attention on specific mathematical relationships
in a diagram and encourage students to com-
for mathematical thinking. Intentionally chosen municate their thinking and sometimes to use
or designed diagrams, or fictitious student work academic language they may not otherwise use
that presents one way of solving a task without (Driscoll, Nikula, & Neumayer DePiper, 2016).
suggesting it is the only method, can also
encourage student reasoning and expression Facilitating Engagement
(Booth et al., 2013). and Growth
To summarize, based on our research, our key
Supporting Student Communication recommendations to educators for supporting
In addition to supporting students’ sense- ELLs’ engagement in mathematical reasoning
making, diagrams can facilitate mathematical and communication are:
communication and language development. n Encourage diagrams as mathematical
To meet ELLs’ needs, math teachers must use problem-solving tools, to engage ELLs in pro-
practices that “specifically address the language duction of mathematical explanations, and to
demands of students who are developing skill support your analysis of their mathematical
in listening, speaking, reading, and writing in a thinking.
second language while learning mathematics” n Facilitate analysis and discussion of diagrams
(Celedón-Pattichis & Ramirez, 2012, p. 1). by students to support ELLs’ mathematical
We have found that integrating mathematical communication and their understanding of
diagrams into instruction—by having students problem-solving approaches.
both create them and analyze ones that others n Integrate language supports such as sentence
have created—provides students artifacts starters into mathematical tasks to increase
they can use to talk about the mathematical ELLs’ access to tasks and their communication
80 EDuCAtionAl lEADErShip / DECEmbEr 2019 / JAnuAry 2020
Through our work with
ELLs, we’ve seen how
students’ diagrams
help communicate
the richness of their
mathematical thinking.
about their mathematical reasoning.
Our work with mathematics OCK
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this way builds ELLs’ mathematical- ANT
reasoning, communication, and
strengths students bring to the classroom Chief State School Officers (NGA &
problem-solving skills, and increases context. CCSSO). (2010). Common Core State
overall task engagement. Further, 2This math task is adapted from Standards: Mathematics.
diagrams can provide teachers Hughes, G., Brendefur, J., & Carney, Ng, S. F., & Lee, K. (2009). The model
M. (2015). Reshaping teachers’ math- method: Singapore children’s tool for
with critical information about stu-
ematical perceptions: Analysis of profes- representing and solving algebraic
dents’ mathematical understanding
sional development task. Mathematics word problems. Journal for Research
(Driscoll, Nikula, & Neumayer Teacher Educator, 3(2), 116–129. in Mathematics Education, 40(3),
DePiper, 2016). 3The diagrams presented in this article 282–313.
Perhaps most important, diagrams are recreations of students’ work. Woodward, J., Beckmann, S., Driscoll,
M., Franke, M., Herzig, P., Jitendra, A.,
provide a means for ELLs to be full References
et al. (2012). Improving mathematical
and engaged participants in classes Booth, J. L., Lange, K. E., Koedinger,
problem solving in grades 4 through 8:
K. R., & Newton, K. J. (2013).
where they might otherwise be A practice guide (NCEE 2012-4055).
Example problems that improve
overlooked or neglected. EL student learning in algebra: Differen- Washington, DC: National Center for
Education Evaluation and Regional
Authors’ note: The research reported in tiating between correct and incorrect
Assistance, Institute of Education Sci-
this article was supported by the Institute examples. Learning and Instruction, 25,
ences, U.S. Department of Education.
of Education Sciences, U.S. Department of 24–34.
Education, through Grant R305A170297 Celedón-Pattichis, S., & Ramirez, N. G. Copyright © 2019 Johannah Nikula, Jill
and by the National Science Foundation (Eds.). (2012). Beyond good teaching:
Neumayer DePiper, and Mark Driscoll
under Grant DRL-1503057, both under Advancing mathematics education for
Education Development Center. Any ELLs. Reston, VA: National Council of
opinions, findings, and conclusions or Teachers of Mathematics.
recommendations expressed are those Chval, K. B., & Chávez, O. (2011). Johannah Nikula (JNikula@edc.
of the authors and do not necessarily Designing math lessons for English org) and Jill Neumayer DePiper
represent views of the Institute, the U.S. language learners. Mathematics ([email protected]) are mathematics
Department of Education, or the National Teaching in the Middle School, 17(5), education researchers at Education
Science Foundation. 261–265. Development Center, where they focus
Driscoll, M., Nikula, J., & Neumayer on equity and broadening participation.
1Note that in this article we refer DePiper, J. (2016). Mathematical Mark Driscoll is a nationally recognized
to students who are learning English
thinking and communication: Access leader in mathematics education who
as English language learners for clarity for English learners. Portsmouth, NH:
has overseen a wide range of math-
because that is how many districts Heinemann.
ematics teacher education, leadership,
identify them. However, we believe that National Governors Association Center
and materials development projects at
other terms, such as emergent multi- for Best Practices, & Council of
lingual learner, can better highlight the Education Development Center.
ASCD / www.ASCD.org 81