**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.

**Kindergarten**

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.

Reference:

"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