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Welcome to the Standards Solution blog! Here we’ll share our experiences, challenges, and insights in the age of the Common Core.

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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.

Nov 14, 2016
BY Standards Solution

The Value of Relationships

You were one of my favorite teachers! You made an impact on my life. I remember you treating me and some of the girls like real people and talking to us and respecting us. It's something I always try to do now that I'm a teacher and part of that is because of the memories I have of my year with you. 
So thank you!

Wow! That is what being a teacher is all about.

I received a Facebook friend request from a former student of mine, and upon minutes of accepting the request, she posted the above quote on my timeline. This student, who I had as an 8th grader and who is now a teacher herself, took the time, some 15 years later, to let me know the impact I had on her. I remember this student and her friends fondly, and it warmed my heart to hear from her. Since then, I’ve been able to “see” her and her family (including her brother, who was also my student) on Facebook and see how they’ve grown up.

What struck me as especially notable in her post is what she remembered about her year in my classroom. She didn’t mention the novels we read, the papers she wrote, or the projects she completed. It wasn’t the homework or the tests I assigned that made an impact on her life. It was how I treated her and her friends that made a difference to her. It was how I respected them that she remembers and emulates today with her own students.  

Noted educator Rita Pierson, in her TedTalk “Every kid needs a champion,” discussed the power of relationships, referring to James Comer, who said that “no significant learning can occur without a significant relationship” and George Washington Carver, who said that “all learning is understanding relationships.” She goes on to talk about how she worked to develop connections with all of her students, to see them succeed against all odds.

Many of the teachers I remember most from my own experiences as a student were those who were passionate about their subjects, who went the extra mile to show they cared about us as real people, and who saw our successes as their successes. It wasn’t what they taught us as much as how they taught us that made the difference. 

In recent years, educators have felt the weight of high-stakes assessments, legislature, evaluations, negative public opinion, heightened expectations, waning resources, and much more. It is easy in all of this to lose sight of what drew many of us to education in the first place: a desire to work with children and make a positive impact in their lives. We must not forget this as we enter the classroom each day. We must do whatever it takes to make the connection, develop the relationship, and be the champion every child deserves.

Jennifer Caldwell
Executive Director
Standards Solution Holding, LLC