Grades K–2, Operations and Algebraic Thinking
Very young children have a natural curiosity to count and keep track of objects in their world. As children mature, they deepen their understanding of number relationships and grow in sophistication of strategies used to solve addition and subtraction situations. Algebraic Thinking becomes most evident as students begin to apply the additive properties of arithmetic (commutative and associative) and explain how these properties support solution strategies for addition and subtraction.
What interactions need to happen in order to support student development of these critical ideas?
- What mathematical experiences do students need to have in order to surface relationships among numbers and connections among operations?
It is these questions that teachers need to consider as they intentionally plan lessons to develop algebraic thinking in students grades K–2.
This module explores two domains of the Common Core State Standards for Mathematics (CCSSM), “Counting and Cardinality” and “Operations and Algebraic Thinking” that work together to develop algebraic thinking in students K–2. As participants engage in this module, they: (1) study focus standards from each domain that are significant in the development of number and operation sense for young students; (2) deepen understanding about the 12 story problem situations that develop meaning for addition and subtraction; (3) make sense of developmental milestones in students’ learning of counting and apply that knowledge to Level 1 counting strategies and extend that understanding to Level 2 and Level 3 strategies.
Throughout the module participants are challenged to consider different modes of representations used to model situations as well as engage in discussions focusing on five Standards for Mathematical Practice: (MP1) Make sense of problems and persevere in solving them, (MP2) Reason abstractly and quantitatively, (MP4) Model with mathematics, (MP5) Use tools strategically, and (MP8) Look for and express regularity in repeated reasoning.
- Become familiar with all word problem types, and become fluent with identifying, posing, and knowing when to use all problem types.
- Become familiar with developmental levels (Level 1, Level 2, and Level 3) for solving single-digit addition and subtraction problems and how each level is critical as students move toward computational fluency.
- Understand the role of representation in student learning, and expand the repertoire of and fluency with using representations.
Grades K–2, Number and Operations in Base Ten with Connections to Operations and Algebraic Thinking
In this module, participants will study the development of place value as it is outlined in domain Number and Operations in Base Ten, K–2. The study will begin with the Kindergarten domains of Counting and Cardinality (Standards K.CC.4, K.CC.5), Operations and Algebraic Thinking (Standards K.OA.3, K.OA.4), and Number and Operations in Base Ten (Standard K.NBT.1). The module will provide participants with experiences based on focus standards at each grade level K–2 that allow children to develop a firm foundation in their understanding of the base ten system. Understanding place value and the base-ten system is a complex task. For very young children the ideas start with their understanding of counting by ones, then grows to include grouping or unitizing numbers to ten and then eventually students will be able to apply their number knowledge to decompose and recompose numbers in order to make sense of addition and subtraction strategies.
What are the experiences students need to have in order to develop a solid foundation in base ten concepts? What instructional practices must teachers understand in order to support the developmental milestones of this thinking and help children “make sense” and “make connections” among all of the mathematical experiences K–2? In order to provide participants with a full perspective of the Number and Operations in Base Ten domain, participants will follow the progression of the standards related addition and subtraction within both Operations and Algebraic Thinking and Number and Operations in Base Ten up to the fluency capstone (4.NBT.4).
Work in the Number Operations in Base Ten domain is closely knit with work in the Operations and Algebraic Thinking domain, specifically connections/work with the Addition and Subtraction Situations (K.OA.2, 1.OA.1 and 2.OA.1). Children’s understanding of place value is foundational for the computation they are expected to do for each grade level. Both the NBT domain and the OA domain focus on properties of the operations and the use of and importance of representations and models, so the cross-referencing between these domains will provide a comprehensive, coherent development of addition and subtraction and the relationship between the two operations. The goal is to develop understanding of specific and generalized strategies for addition and subtraction based on place value, properties of the operations, and the Addition and Subtraction Situations, including contexts as developed in the Measurement and Data domain (2.MD.5). Specific references to the MD domain will help teachers see both the focus and coherence across the grades regarding these related foundational topics. There will be an in-depth study of subtraction strategies with connections to problem situations. Explicit connections will be make the Standards for Mathematical Practice, in particular MP2 Reason abstractly and quantitatively, MP4 Model with mathematics, MP5 Use appropriate tools strategically, MP6 Attend to precision and MP7 Look for and make use of structure.
- Deepen understanding of the structure of the base-ten number system based on the powers of ten (groups of ten as single entities or units).
- Develop understanding that the position of digits within a number determines the specific group of tens it represents and that the groupings of ones, tens and hundreds can be decomposed and recomposed in different but equivalent ways
- Become familiar with base ten strategies for addition and subtraction and understand the properties of operations that support the development of generalized algorithms.
- Understand how to use place value ideas meaningfully when engaged in a variety of problem solving activities.