Procedural and computational fluency is achieved by building on a foundation of conceptual understanding. Fluency is not attained by simply memorizing facts, or following mechanical step by step procedures. The first step towards fluency is building number sense through problem solving. In the process of solving a problem, students engage in exploration and discussion, while discoveries of the properties of operations emerge into strategies and tools they apply and put to use. These informal approaches are refined and built upon, and eventually become tested and efficient methods that students can rely on. Students remain flexible in their choice of strategies, basing their decisions on understanding, efficiency, and accuracy.
Kindergartners Use Dramatic Play to Model the Structure of Addition and Subtraction Problems.
Kindergartners related math stories to birds, their current social studies and science topic, by using dramatic play to act out addition and subtraction problems. Through these stories, students explored the action in different structures of problems (see chart below). Some examples of addition structure bird stories that students acted out were result unknown, change unknown, and both addends unknown. “Students who are taught to approach problems by looking at their structures through the use of a visual model are more likely to perform better than students who do not” (NCTM, 2014, pg. 50). This approach helps students make better sense of the action that is happening in problems and rely less on using keywords that can often be interpreted in many different ways (Karp, 2014, pg. 21).
An Example of Student Work Showing Change Unknown Structure Along With a Structure Diagram:
“Six pigeons were pecking on the sidewalk on Bleecker Street. Some more pigeons came to see what they were pecking at. Then there were ten pigeons.” (CHANGE UNKNOWN/ADD TO).
Addition/Subtraction Problem Structures:
2014 Karp, Karen S., Teaching Children Mathematics. National Council of Teachers of Mathematics
2014 NCTM. Putting Essential Understanding of Addition and Subtraction into Practice
Laying the Foundation for Addition by Exploring Parts of Six.
Kindergarten students decomposed the number six by placing square inch tiles into different arrangements and then recorded how they saw the arrangements using numbers. One of the goals of this problem was to help students understand that numbers can be decomposed (broken apart) in many ways. Developing an understanding of part-part-whole relationships is the first critical step to grasping the concepts of addition and subtraction, properties, and algorithms (NCTM, 2014, pg. 11). For example, in later years, decomposition (part-part-total understanding) of multi-digit numbers are used in place value concepts and the operations of addition/subtraction, multiplication/division by applying the various properties of commutative, associative, and distributive.
Differentiating Through an Open Problem:
How Many Ways Can Six Tiles Be Arranged?
Each child arranged tiles in as many ways as they could. This open problem was accessible to every student in the class, and at the same time provided differentiation because of the challenging open-ended nature of the problem (see below, the same problem given at the college level). The class came together and examined all of the possible ways of arranging six tiles “side-to-side, and corner-to-corner”. Through this exploration, the geometric concepts of congruence and transformations (flips, slides, turns), were introduced. The students found 20 ways to arrange six tiles.
Six Tiles Problem by Professor David Meredith:
Below is the same six tiles problem given to mathematics students at San Francisco State University by Professor Meredith, of the mathematics department. Prior to San Francisco State Univ., David was a professor at MIT, and a Woodrow Wilson Fellow. Meredith’s students found 35 distinct non-congruent ways to arrange six tiles.
2014 National Council of Teachers of Mathematics, Putting Essential Understanding of Addition and Subtraction into Practice.
Students and Parents look forward to Family Math Night every year.
Students in grades one through four celebrate mathematics, as well as continue to hone their fluency in combination facts by playing fun games. Fourth grade students create their own math games as a capstone experience, and then teach them to family and friends during Family Math Night. Continue reading
Students in first grade have been working with the equal symbol, and the greater than/less than symbol. They’ve created number stories and equations using the data they collected from counting the number of security cameras the stores in the neighborhood have. Some of these equations are simply true statements, and some have missing addends, or missing sums, depending on the story they created. Continue reading
Students in first grade are learning that the equal symbol doesn’t necessarily mean to “do” something. It can just mean that a mathematical statement is “true”. Continue reading
What does: 1 + 7 = ___ + 6
have to do with: 3x + 9 = 5x + 5
…and why are first graders arguing with each other over the meaning of the equal symbol?