CU Concept Paper. The purpose of this paper is to define KIPP’s point of view on conceptual understanding (CU) in mathematics. For guidance on
planning and executing lessons which support CU, consult KIPP’s Conceptual Understanding Toolkit.
“All young Americans must learn to think mathematically and they must think mathematically to learn.”
- Adding it Up: Helping Children Learn Mathematics
The Data That Matters
On the 2011 Trends in International Mathematics and Science Study (TIMSS), the United States scored behind the
following regions:1
Singapore | Korea | Hong Kong-China | Chinese Taipei | Japan | Northern Ireland | North Carolina2 | Flemish Belgium
There were an additional six countries that scored in the same range as the United States, effectively ranking the United
States 15th in terms of mathematical performance on the TIMSS. In order to understand why the United States was so
outperformed, it’s important to the approaches the other countries take to math instruction with that of the United
States.
Singapore vs. United States Approaches to Math
Singapore, which has consistently been a top performer on the TIMSS assessments, teaches an average of fifteen topics
per grade. The United States, on the other hand, has traditionally taught anywhere from 20% to 160% more topics per
grade.3 Because teachers in the United States are expected to cover more topics, they are not able to go into as much
depth as teachers from Singapore. Although the adoption of the Common Core State Standards is shifting the United
States to more depth over breadth, the United States still covers more topics per grade than Singapore.
However, it is not just that Singapore covers more topics per grade that has led to their top position in the international
comparisons. Textbooks in Singapore are designed to build deep understanding of mathematical concepts. According to
“What the United States Can Learning from Singapore’s World-Class Mathematics System,” “lessons in the Singapore
textbooks are structured to provide depth and conceptual understanding. The textbooks use an approach that begins
with concrete pictorial representations of content and later moves to abstract approaches to teach mathematics.”4 In
comparison, textbooks in the United States have shorter lessons and focus on applying definitions and formulas to
problems, which does not support students in understanding how to use the math in real-world scenarios.
The Shift We Need to Make5
For too long in the United States, and at KIPP, the role of the teacher has been to provide students with all of the
information necessary to learn the basic procedures that enable them to be successful on a standardized assessment -generally the state exam or a norm-referenced test like MAP. We tell students exactly what definitions, formulas, and
rules they should know, and we demonstrate how they should use this information to solve mathematics problems.
1
NCES Statement on PIRLS 2011 and TIMSS 2011, National Center for Education Statistics. Web. 11 December 2012
Florida and North Carolina participated in TIMSS at grade 4 in order to receive their own state results. Florida's average score (545)
in mathematics at grade 4 was higher than the average in 43 education systems and lower than in 6. It was not measurably different
from the U.S. average. North Carolina's average score (554) was higher than the average in 47 education systems (including the
United States) and lower than in 5.
3
American Institutes for Research (2005), What the United States Can Learn From Singapore’s World-Class Mathematics System
(and what Singapore can learn from the United States), Washington, DC, 32
4
Ibid., 45
5
For an illustration of a classroom that has made the shifts described, please see “Making Mathematics Reasonable in School” by
Deborah Loewenberg Ball and Hyman Bass (2003) in A Research Companion to Principles and Standards for Mathematics (2003),pg.
27 – 44. Reston, VA: National Council of Teachers of Mathematics.
2
Students are challenged to apply the mathematics only after they have mastered the basic skills. This has led to a culture
of math that values the end product over the process and puts the heavy cognitive lifting on the teacher rather than on
the students. By not asking students to engage in the intellectual exercise of exploring, discussing, and applying
mathematics to novel situations, we are stifling their ability to problem solve and reason. In order to ensure that our
students are college, career, and STEM ready, we need to shift our math instruction and culture.
NCTM has identified a number of principles of learning that provide the foundation for effective mathematics
teaching. Specifically, learners should have experiences that enable them to:6


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

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Engage with challenging tasks that involve active meaning making and support meaningful learning.
Connect new learning with prior knowledge and informal reasoning and, in the process, address preconceptions
and misconceptions.
Acquire conceptual knowledge as well as procedural knowledge, so that they can meaningfully organize their
knowledge, acquire new knowledge, and transfer and apply knowledge to new situations.
Construct knowledge socially, through discourse, activity, and interaction related to meaningful problems.
Receive descriptive and timely feedback so that they can reflect on and revise their work, thinking, and
understandings.
Develop metacognitive awareness of themselves as learners, thinkers, and problem solvers, and learn to monitor
their learning and performance.
What This Means for KIPP
At KIPP, we believe that a deep understanding of mathematical concepts is the gateway to success in higher level
mathematics in high school, college, and beyond. In its publication Principles and Standards for School Mathematics, the
National Council of Teachers of Mathematics states that “those who can understand and do mathematics will have a
significantly enhanced opportunity and options for shaping their futures.”7 In order to make the necessary shifts at KIPP,
our math instruction must have a focus on conceptual understanding and mathematical reasoning. We know that
conceptual understanding is a critical component in developing our students’ ability to know and do mathematics.
When students have conceptual understanding, their knowledge goes beyond facts and procedures. They are able to
understand the importance of the mathematical concept and the contexts in which it would be useful.8 Conceptual
understanding allows students to build on their existing knowledge and connect new ideas to “existing conceptual
webs”.9 These webs lead to increased retention and recall of concepts.10
We also know that mathematical reasoning is the backbone of conceptual understanding and algorithmic proficiency.
Mathematical reasoning can be defined in two ways. First, it is the ability to construct knowledge through inquiry
(reasoning of inquiry). Second, it is the ability to justify and prove mathematical claims (reasoning of justification).11 It
must be modeled, taught, and practiced, and students need consistent opportunities to make meaning of the math they
are learning. This is done through the regular use of rigorous mathematical tasks that allow students time to grapple
with ideas and refine them through writing, speaking, and listening to the ideas of others. The Standards for
Mathematical Practice from the Common Core State Standards play a crucial role in fostering mathematical reasoning.
Where content standards tell us what students are required to understand, know, and be able to do, the practice
6
National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics (Executive Summary)
Washington, DC.
7
National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics (Executive Summary)
Washington, DC., 1
8
Ibid., 118
9
Van de Walle, J., Karp, K., Bay-Williams, J. (2013) Elementary and Middle School Mathematics: Teaching Developmentally. Boston:
Pearson, 29.
10
Ibid.
11
Ball, D.L. & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W.G. Martin, and D. Schifter (Eds.) A
Research Companion to Principles and Standards for Mathematics (pg. 27 – 44). Reston, VA: National Council of Teachers of
Mathematics.
standards inform us of the student actions that we need to foster. Through the practices, students develop the ability to
reason, understand why what they are doing works, and apply their thinking to real-world situations.
To support conceptual understanding and mathematical reasoning in our classrooms, instruction needs to shift from the
traditional “I/We/You” to a “You/We/You.” This model for instruction will put the heavy lifting on students and help
them construct their own knowledge, leading to deeper understanding of concepts. In short, our students need to be
able to explain “why” and not just “how.”
How Can Conceptual Understanding and Mathematical Reasoning Be Taught Effectively?
Planning. A classroom that builds students’ conceptual understanding and ability to reason mathematically starts with
planning. It requires a deep understanding of the mathematical content. Teachers must anticipate student mistakes or
misconceptions, plan questions that will assess or advance student thinking, be prepared with multiple ways that
students can show understanding of a concept, and plan tasks that are worthy of instructional time.
Worthy Tasks. A worthy task is the foundation for building conceptual understanding and mathematical reasoning. A
worthy task must be cognitively demanding for students and allow them to make connections to known concepts and
knowledge. The task should have a variety of ways in which students can approach it and demonstrate their
understanding.12 It is also important to note that a task must have a relevant context for students. This engages students
in the learning process.13
Productive Struggle. Students are engaged in productive struggle when they have opportunities to grapple with the
content. Productive struggle involves students diving “deeply into understanding the mathematical ideas, instead of
simply seeking correct solutions”.14 Teachers must scaffold the content so that students make progress but must be sure
not to override student thinking or step in to do the work for them.
Questioning. It is necessary for questions to advance or assess student thinking. Students should expect to be asked to
elaborate, explain, or clarify their thinking. Effective questioning promotes mathematical reasoning and builds towards
conceptual understanding by helping students understand the “why” behind a solution or approach and make
connections between concepts.15
Discourse. Discourse builds towards a shared understanding of mathematical ideas and a focus on reasoning and
problem solving is essential for developing conceptual understanding.16 Discourse in the classroom should focus on the
“why” behind the concept, not the “how” of the procedure, in order to build towards conceptual understanding.
Students should make conjectures or predictions and build on the reasoning of others.
Making Thinking Visible. Evidence of student thinking is necessary to assess progress towards understanding and adjust
instruction as necessary.17 Teachers must be prepared with questions that can be answered independently in order to
gather data about student progress. It is also necessary for teachers to have a full understanding of what to look for and
count as evidence of mathematical thinking from students.18
12
Ibid.
Ibid., 38
14
Leinwand, Brahier, & Huinker, 48
15
Van de Walle, Karp, Bay-Williams, 45
16
Leinwand, Brahier, & Huinker, 30
17
Leinwand, Brahier, & Huinker, 53
18
Ibid.
13