Fleming County Schools
Kindergarten Mathematics
21st Century Learning: Six principles of learning
1. Being literate is at the heart of learning in all subject areas
2. Learning is a social act
3. Learning establishes a habit of inquiry
4. Assessing progress is part of learning
5. Learning includes turning information into knowledge using multi-media
6. Learning occurs in global context
4 C’s of 21st Century Learning
1. Creativity
2. Collaboration
3. Critical Thinking/Problem Solving
4. Communication
***Information in the explanations and examples found in this document came from Arizona Department of Education and Common Core. Information in resources and assessments came from the Georgia Dept. of Education
Key Instructional Shifts in Mathematics
1. Focus strongly where the Standards focus / Rather than racing to cover everything in today’s mile-wide, inch-deep curriculum, teachers use the power of the eraser and significantly narrow and deepen the way time and energy is spent in the math classroom. They focus deeply on only those concepts that are emphasized in the standards so that students can gain strong foundational conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom.2. Coherence: think across grades, and link to major topics within grades / Thinking across grades: Instead of treating math in each grade as a series of disconnected topics, principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication develop across grade levels and students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.
Linking to major topics: Instead of allowing less important topics to detract from the focus of the grade, these topics are taught in relation to the grade level focus. For example, data displays are not an end in themselves but are always presented along with grade-level word problems.
3. Rigor: require conceptual understanding, procedural skill and fluency, and application with intensity. / Conceptual understanding: Teachers teach more than “how to get the answer” and support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by solving short conceptual problems, applying math in new situations, and speaking about their understanding.
Procedural skill and fluency. Students are expected to have speed and accuracy in calculation. Teachers structure class time and/or homework time for students to practice core functions such as multiplication facts so that students are able to understand and manipulate more complex concepts.
Application: Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content.
PACING and SUGGESTED TIMELINES
Not all of the content in a given grade is emphasized equally in the standards. Some clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness. In addition, an intense focus on the most critical material at each grade allows depth in learning, which is carried out through the Standards for Mathematical Practice.
To say that some things have greater emphasis is not to say that anything in the standards can safely be neglected in instruction. Neglecting material will leave gaps in student skill and understanding and may leave students unprepared for the challenges of a later grade. The following table identifies the Major Clusters, Additional Clusters, and Supporting Clusters for this grade.
Achieve the Core.org
Major ClusterSupporting Cluster
Additional Cluster
Required Fluencies in the Common Core State Standards for Mathematics
When it comes to measuring the full range of the Standards, usually the first things that come to mind are the mathematical practices, or perhaps the content standards that call for conceptual understanding. However, the Standards also address another aspect of mathematical attainment that is seldom measured at scale either: namely, whether students can perform calculations and solve problems quickly and accurately. At each grade level in the Standards, one or two fluencies are expected.
K / Add/subtract within 51 / Add/subtract within 10
2 / Add/subtract within 20*
Add/subtract within 100 (pencil and paper)
3 / Multiply/divide within 100**
Add/subtract within 1000
4 / Add/subtract within 1,000,000
5.NBT.5 / Multi-digit multiplication
6 / Multi-digit division
Multi-digit decimal operations
7 / Solve px + q = r, p(x + q) = r
8 / Solve simple 2x2 systems by inspection
Fluent in the Standards means “fast and accurate.” It might also help to think of fluency as meaning the same thing as when we say that somebody is fluent in a foreign language: when you’re fluent, you flow. Fluent isn’t halting, stumbling, or reversing oneself. Assessing fluency requires attending to issues of time (and even perhaps rhythm, which could be achieved with technology).
The word fluency was used judiciously in the Standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity. In fact, the rarity of the word itself might easily lead to fluency becoming invisible in the Standards—one more among 25 things in a grade, easily overlooked. Assessing fluency could remedy this, and at the same time allow data collection that could eventually shed light on whether the progressions toward fluency in the Standards are realistic and appropriate.
*1 By end of year, know from memory all sums of two one‐digit numbers
**2 By end of year, know from memory all products of two one‐digit numbers
***This information is cited from: http://www.engageny.org/sites/default/files/resource/attachments/ccssfluencies.pdf
Kindergarten
Grade K Overview
Counting and Cardinality (CC)· Know number names and the count sequence.
· Count to tell the number of objects.
· Compare numbers.
Operations and Algebraic Thinking (OA)
· Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
Number and Operations in Base Ten (NBT)
· Work with numbers 11–19 to gain foundations for place value.
Measurement and Data (MD)
· Describe and compare measurable attributes.
· Classify objects and count the number of objects in categories.
Geometry (G)
· Identify and describe shapes.
· Analyze, compare, create, and compose shapes. / Mathematical Practices (MP)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
In Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and operating on whole numbers, initially with sets of objects; (2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.
(1) Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set after some are taken away.
(2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes or orientations), as well as three-dimensional shapes such as cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.
Learning Targets:
Learning Targets are “I can” statements in student friendly language that guide students’ learning to reach mastery of the intended standard. While they may be formatively assessed, they are not intended to be summatively assessed separately. Teachers need to work with their students to be sure they understand the learning goal. See example below:
Display Intended Standard: K.CC.3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).
Ask students, “What is this asking you to do?”
Student response- “I should be able to count how many things there are and write it down.”
Teacher Response: “Do you think I expect you to be able to do this by the end of today?
Student Response: “No”
Teacher Response: “You are right. I know this is going to take a while. We are going to work on this for several weeks. Let’s set a learning target for this week.”
Learning Target: “I can count how many things I see and write it down. (1-10)
Daily Target: Today, “I can continue working on counting and writing how many things I see up to 5”.
Standards for Mathematical Practice /Standards / Explanations and Examples /
Students are expected to: / Mathematical Practicesare listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. /
K.MP.1. Make sense of problems and persevere in solving them. / In Kindergarten, students begin to build the understanding that doing mathematics involves solving problems and discussing how they solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Younger students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” or they may try another strategy.
K.MP.2. Reason abstractly and quantitatively. / Younger students begin to recognize that a number represents a specific quantity. Then, they connect the quantity to written symbols. Quantitative reasoning entails creating a representation of a problem while attending to the meanings of the quantities.
K.MP.3. Construct viable arguments and critique the reasoning of others. / Younger students construct arguments using concrete referents, such as objects, pictures, drawings, and actions. They also begin to develop their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking.
K.MP.4. Model with mathematics. / In early grades, students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed.
K.MP.5. Use appropriate tools strategically. / Younger students begin to consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, kindergarteners may decide that it might be advantageous to use linking cubes to represent two quantities and then compare the two representations side-by-side.
K.MP.6. Attend to precision. / As kindergarteners begin to develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning.
K.MP.7. Look for and make use of structure. / Younger students begin to discern a pattern or structure. For instance, students recognize the pattern that exists in the teen numbers; every teen number is written with a 1 (representing one ten) and ends with the digit that is first stated. They also recognize that 3 + 2 = 5 and 2 + 3 = 5.
K.MP.8. Look for and express regularity in repeated reasoning. / In the early grades, students notice repetitive actions in counting and computation, etc. For example, they may notice that the next number in a counting sequence is one more. When counting by tens, the next number in the sequence is “ten more” (or one more group of ten). In addition, students continually check their work by asking themselves, “Does this make sense?”
Table 1. Common addition and subtraction situations.6
Result Unknown / Change Unknown / Start UnknownAdd to / Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now?
2 + 3 = ? / Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How
many bunnies hopped over to the first two?
2 + ? = 5 / Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before?
? + 3 = 5
Take from / Five apples were on the table. I ate two apples. How many apples are on the table now?
5 – 2 = ? / Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat?
5 – ? = 3 / Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before?
? – 2 = 3
Total Unknown / Addend Unknown / Both Addends Unknown1
Put Together / Take Apart2 / Three red apples and two green apples are on the table. How many apples are on the table?
3 + 2 = ? / Five apples are on the table. Three are red and the rest are green. How many apples are green?
3 + ? = 5, 5 – 3 = ? / Grandma has five flowers. How many can she put in her red vase and how many in her blue vase?
5 = 0 + 5, 5 = 5 + 0
5 = 1 + 4, 5 = 4 + 1
5 = 2 + 3, 5 = 3 + 2
Difference Unknown / Bigger Unknown / Smaller Unknown
Compare3 / (“How many more?” version):
Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?
(“How many fewer?” version):
Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie?
2 + ? = 5, 5 – 2 = ? / (Version with “more”):
Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?
(Version with “fewer”):
Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?
2 + 3 = ?, 3 + 2 = ? / (Version with “more”):
Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?
(Version with “fewer”):
Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?
5 – 3 = ?, ? + 3 = 5
\6Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).