MATH
Thinking Mathematically
action but does not involve making the
Supporting sense
initial group of 6. Here the student thinks
6 and says 7, 8, 9, 10, 11, 12, 13 counting
on her ﬁ ngers so she can look at her
ﬁ ngers and see that she needs 7 more
making: Thinking
stickers to get to 13. In solving addition and
subtraction problems students typically
begin by direct modeling, move to a
counting strategy, then use derived facts (6
+ 6 = 12, so 6 + 7 must be 13) and recall (I
mathematically
know that 6 + 7 = 13).
What is important is that we can
document the development of children’s
mathematical thinking across a number of
Students often have unique ways of solving problems, ways
subtraction, multiplication/division, place
value, multi-digit calculation, and the
that can surprise adults. Megan Franke explains how teachers
development of algebraic thinking. CGI
can use this knowledge to support math development
shares this research-based knowledge with
teachers, who take this knowledge and
use it to structure learning opportunities.
EVERY DAY, students bring intuitive helps teachers to know what to listen for We do not provide a curriculum or a set
knowledge and mathematical insight as they engage their students in explaining of practices that a teacher must use. The
to school. By building on this informal their mathematical thinking. Listening teachers use the knowledge in ways that
knowledge, teachers can ensure that students to students’ mathematical thinking may make sense for them in their situations and
learn with understanding. Consider how one sound easy, but students often do not within whatever curricula they use.
second grader solved the following problem. solve problems in the same ways as adults. CGI leads teachers to consider how to:
Teachers therefore need opportunities to

Choose and pose problems that support
learn how students will solve problems, and student learning;
how then to support them in developing

Allow students to solve problems in ways
more efﬁ cient and mathematically that make sense to them;
sophisticated strategies.

Structure classrooms to allow students to
Teachers often pose word problems that share their thinking; and

Probe student thinking.
“Lily has 6 stickers.
How many more stickers does Lily need
Understanding the
to earn so that she will have 13 stickers
development of students’
altogether?”
Harry solved the division problem 45
÷
9 Adults typically expect students to solve
mathematical thinking within
= ____ by decomposing 45 into 4 tens and a this problem by subtracting 6 from 13.
ﬁ ve (10 + 10 + 10 + 10 + 5). He then changed However, research shows that children
the 10's to 9 + 1. He took out the 4 nines naturally follow the action in the problem,
subtraction allows teachers
for 4 days of dog biscuits. He had 4 ones and start by counting out 6 objects, and to choose problems that
left and then along with the ﬁ ve, he made then add objects to those 6 until they have
students can begin to solve,
another group of 9. His answer was 5 days. 13. They then count the objects they added
Harry’s solution often surprises adults. to the 6 and ﬁ nd they need 7 more. Rather
and ones that challenge
What is apparent is that he chose to use a than seeing this as a subtraction situation, their thinking
strategy that made sense to him and used a the students see it as an adding or joining
basic property of number. situation, 6 + ___ = 13. If we teach students Choosing and posing problems to
to subtract to solve this problem and the meet students’ mathematical needs
Cognitively Guided Instruction students think of it as a joining problem, Understanding the development of students’
For the last 25 years, a professional we create a disconnect between students’ mathematical thinking within domains like
development approach called Cognitively existing ideas and the new mathematical addition and subtraction allows teachers to
Guided Instruction (CGI) has supported idea. We create confusion rather than sense choose problems that challenge students'
teachers to create classrooms in making. thinking. Teachers can also then choose
which students build on their existing Research also shows that the pattern problems that allow for multiple strategies
mathematical knowledge so that they of development in student strategies is and ﬁ t with where students are in the
connect their new ideas to existing ones.
1
robust and the strategies naturally build trajectory. They choose numbers that
CGI focuses teachers on what students on one another. In the previous problem, support student understanding. Engaging
know, not what they do not know. students start with the strategy that follows in sense-making around the problem opens
CGI provides teachers with knowledge the action in the problem, a strategy we up opportunities for students to participate
about how students solve problems and call direct modeling. They then move to because they understand the problem and
how strategies develop with experience. It a counting strategy that also follows the what is being asked.
16 Better: Evidence-based Education fall 2009
Better(US) Fall09.indb 16 14/10/09 13:05:18
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