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Feb 17, 2017

How Do Students Acquire Mathematical Understandings?

How Do Students Acquire Mathematical Understandings?

The Common Core Standards Writing Team produced a helpful guide to explain to educators how students acquire the ability to use numbers. This document, Progressions for the Common Core State Standards in Mathematics, explains that there are three models that teachers employ to teach students to solve single-digit addition and subtraction problems:

Level 1 – Counting All or Taking Away
Represent situations with groups of objects, a drawing or fingers.

Level 2 – Counting On
Methods of keeping track: fingers, objects, mentally imagined objects, body motions, other count words are used to monitor the count.

Level 3 – Convert to an Easier Problem
Students decompose and compose a part with another addend.

These three models are at the heart of how K-2 students learn to add and subtract.


In Kindergarten (or before), students master a concept called “subitizing.” Perceptual subitizing is the ability to recognize the number of briefly shown items without actually counting them. Perceptual subitizing leads to conceptual subitizing, which involves recognizing a number as a composite of parts and as a whole. For instance, a student will recognize an eight-dot domino without counting because he or she sees each side of the domino as four individual dots and as a four. They see the domino as composed of two groups of four and as one eight.

Students in Kindergarten work with three kinds of problem situations: add to, take from, and put together/take apart. The numbers involve addition and subtraction within 10. Students represent these problems with concrete objects and drawings, and they find the answers by counting.

The process begins with students learning counting words. Next, students pair each word with one object. They learn that the last number name of a group of objects represents the number of objects counted (K.CC.4a), and regardless of how the objects are displayed, this does not change the number of objects. (K.CC.4b). Later students can count out a given number of objects. Eventually, students will be able to count forward, beginning from a given number (K.CC.2). And finally, students will understand that each successive number name refers to a quantity that is one larger (K.CC.4c).

First Grade

Students in first grade work with all of the problem situations. The numbers in these problems involve additions involving single-digit addends and the related subtractions. Students represent these problems with math drawings and with equations. Students convert to an easier problem by changing the problem to an easier equivalent. This is important because this method involves making a 10.

In first grade, students compare numbers by adding and subtracting to find out “how many more” or “how many less” (1.OA.1). Eventually students write problems with equations. Counting on enables students to add and subtract easily within 20 because they do not have to use fingers to show totals of more than 10. Counting on should be seen as a thinking strategy, not a rote method. It involves seeing the first addend as embedded in the total, and it involves a conceptual interplay between counting and the cardinality in the first addend. Counting on for subtraction is easier than counting down.

Second Grade

Grade two students use addition and subtraction within 100 rather than within 20. They represent these problems with diagrams and/or equations. Most students work with two– step problems that involve single-digit addends. Students in second grade become fluent in single-digit additions and the related subtraction using level 2 and 3 strategies.

Summary of K-2 Development

Fluency in adding and subtracting single-digit numbers has progressed from numbers within 5 in Kindergarten, to within 10 in first grade, to within 20 in second grade. By the end of the K-2 grade span, students have sufficient experience with addition and subtraction to know single-digit sums from memory.

"Progressions for the Common Core State Standards in Mathematics.” The Common Core Standards Writing Team. 29 May 2011. Web. Accessed 17 Feb. 2017. https://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf

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