To add 194 + 207, do you still need to write it down?
Let's help children understand math, not do math problems.

Building Math Vocabulary

Asking Questions that Support Conceptual Understanding 

Math Common Core Standards Categories:
  • Counting & Carnality: Know the number names and the counting sequence
  • Operations & Algebraic Thinking: Understand the principles of, and facts for, addition, subtraction, multiplication, and division
  • Number & Operations in Base Ten: Work with numbers in relation to their base ten values to solve problems
  • Number & Operations—Fractions: Work with fractions and mixed numbers to solve problems
  • Measurement & Data: Classify, describe, measure, and analyze different types of data
  • Geometry: Identifying, describing, analyzing, comparing, and measuring different shapes
  • Ratios & Proportional Relationships: Understand ratio concepts and use ratio reasoning to solve problems
  • The Number System: Under and use rational and non-rational numbers
  • Expressions & Equations: Solve real-life and mathematical problems using linear and non-linear equations with numerical and algebraic expressions.
  • Functions: Use a variety of functions to model relationships between and among quantities
  • Statistics & Probability: Use statistical data to interpret categorical and quantitative data, make inferences and justify conclusions, and apply probability rules to make decisions
Counting & Carnality: Know the number names and the counting sequence --can the child count forward, backward, starting from a random number?  As I think about some of our third graders, they literally are missing this most basic kindergarten skill.   They can't, without effort, determine where a number goes, what is before and after it, start at a given number and move forward and backwards fluently, effortlessly.  This basic understanding MUST be in place.

My ahha has been that adding, subtracting, multiplying and dividing are just fast ways to count....duh, Lisa.  ... just being honest....

Number & Operations in Base Ten: Work with numbers in relation to their base ten values to solve problems --can the child make sense and use of our number system, based on 10? Can they tell you that a teen number is not a 1 and 5, but that it is a group of 10 and 5 more? Can the child tell you, without using their fingers, that 17 + 5 = 22 because 5 can be a 3 and 2, so 17 + 3 = 20 + 2= 22. How about counting out a random pile, can they/do they organize using groups of 10 or do they just randomly count and then loose track, especially with larger quantities?

Composing: Composing is student's knowledge about how numbers can be put together to create different numbers--can the child quickly put 5 + 6 to make 11, or 7 + 4 to make 11? ...Without their fingers.... Not that using fingers is not important in early development, but by the time they are a bit older, they need to fluently and effortlessly compose (put together) quantities to make a larger quantity. How many "simple mistakes" are made when adding larger numbers because they simply couldn't compose (put together) the numbers?

Decomposing: Decomposing is student's ability to put together and take apart numbers by using "parts" and a "whole."--Does the child understand that 10 and 7 make 17, or 2 and 3 make 5? Can they decompose (take apart) because they understand that the whole is made up of parts? Part, Part, Whole...Whole, part, part.

Subitizing: Subitizing refers to the ability to instantly and accurately recognize groups of objects. --Does the child know it without having to count. They can "see" the group of 3 and group of 4 and know it is 7? They can "see" the group of 2, group of 2, group of 2 and know that it is 6?

Math Focus by Grade Level
Going deep, not wide
​Achieve The Core
(To enlarge image, right click and open in a new tab)

Standards of Mathematical Practice

How do they learn the language?

We teach it to them.  We teach them how numbers work, the vocabulary necessary to converse their thoughts.  We teach them to model with mathematics and to look for patterns.  We teach them using the Standards for Mathematical Practice.

Math Talks/Number Talks​

elementarynumbertalks.wordpress.com/Elementary Number Talks
Please feel free to use the Number Talk Resources in your Classroom!
schoolwires.henry.k12.ga.us/Page/44118​ROCKY CREEK ELEMENTARY
number_talks_spring_representations.pptx
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number_talks_summer_representations.pptx
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A kindergarten student demonstrating her Number Sense foundation.  Thank you for sharing with us!

Kathy Richardson

 The 9 concepts that we want children solid with. 
​ Solid means without extra effort in their thinking, using fingers, etc.  It is instant understanding.
​www.didax.com/media/pdf/AMC-Goals-and-Phases.pdf
Assessment Goals and Critical Learning Phases:

Counting Objects Topic:
  • Counting Goal: To determine if a child can count and keep track of an unorganized pile of up to 32 counters and can make a pile of up to 18 counters.
    • Critical Learning Phases:  Keeps track when counting objects • Remembers “how many” after counting • Reacts to estimate and makes a closer estimate • Knows “one more” without counting • Knows “one less” without counting • Interprets and writes numerals to label quantities. 
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Changing Numbers Topic:
  • Beginning Number Relationships Goal: To determine if a child knows if a number is larger or smaller than another number and can change one quantity to another.
    • Critical Learning Phases: When changing one quantity to another: • Tells whether to take some away or get some more • Figures out how many to add or take away by counting on or removing extras • Knows (tells) how many to add or take away



More / Less Trains Topic:
  • Comparing Numbers Goal: To determine if a child knows if a number is larger or smaller than another number and can change one quantity to another.
    • Critical Learning Phases: • Uses what is known about one amount to determine another • Adds or takes away from one group to make it the same as another group • Tells how many more when groups are lined up • Tells how many more when groups are not lined up • Tells how many less when groups are lined up 1

Number Arrangements Topic:
  • Identifying and Combining Parts Goal: To determine if a child can recognize parts of a number and combine these parts without having to count all.
    • Critical Learning Phases: • Recognizes groups of numbers to 5 in a variety of configurations • Recognizes and describes the smaller parts contained in the larger numbers • Identifies one or more parts and counts the rest (counting on) • Combines parts of arrangements by knowing




Combination Trains Topic:
  • Number Combinations Goal: To determine what number combination the child knows. To find out if the child can use related combinations as a strategy for getting answers.
    • Critical Learning Phases: • Combines parts by using relationships • Knows doubles • Uses doubles plus one • Uses doubles minus one • Combines parts by knowing

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Hiding Assessment Topic:
  • Decomposing Numbers Goal: To find out which number combinations a child knows by determining if he/she can tell the missing part of a number without having to figure it out.
    • Critical Learning Phases: • Figures out missing parts of numbers when unknown • Knows missing parts without needing to figure them out • Uses addition to solve subtraction 2


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​Ten Frames Topic:
  • Ten and Some More Goal: To combine single-digit numbers by reorganizing them into a 10 and leftovers. To use their knowledge of the parts of numbers to 10 to subtract from numbers up to 20.
    • Critical Learning Phases: • Describes a 10 as a single entity even though it is composed of 10 single objects • Organizes numbers into groups of one 10 and leftovers • Knows 10 plus any number from 1 to 10 • Tells how many needed to make 10 • Tells how many leftovers when removing 10 for numbers from 11 to 20 • Combines quantities by reorganizing into one 10 and leftovers • Subtracts quantities by breaking numbers apart and recombining whatever is left

Grouping Tens Topic:
  • Numbers as Tens and Ones Goal: To determine if the child can tell “how many” in a quantity if the number of tens and ones is known, and to determine if the child can add ten and take away ten without counting. To also determine if the child knows that the total number does not change when counted in a different way, what it means to count by 2’s or 5’s, and how well the child can do this.
    • Critical Learning Phases: • Organizes quantities into tens and ones • Instantly knows total quantity when the number of tens and ones is known • Knows 10 more for any two-digit number • Knows 10 less for any two-digit number

Two-Digit Addition & Subtraction Topic:
  • Combining and Separating Tens and Ones Goal: To determine if a child can add and subtract from two-digit numbers by mentally breaking numbers apart and reorganizing them into tens and leftovers, and to determine how the child solves problems presented symbolically.
    • Critical Learning Phases: • Tells how many needed to make the next 10 • Combines numbers by reorganizing them into tens and leftovers when necessary • Breaks apart tens when necessary and reorganizes what is left into tens and leftovers

Games and Activities

Addition and Subtraction Cards

Counting

Comparing Quanties

Ten and Some More

Place Value

Multiplication/Division

Choral Counting and Counting Collections

What is the purpose and what is the difference?

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Counting Collections focuses in on collections, or groups of objects, in which students are asked to draw on the counting principles in relation to one another as they figure out how many objects they have.

Choral Counting supports students to count together as the teacher records the number sequence in a specific manner.  As they do so, students examine number relationships that enable them to identify, to discuss, and to use patterns and the structure of the number system.

Choral Counting and Counting Collections--Frake, Kazemi, Turrou

How Do the Concepts Build?

Using the Coherence Map from Achieve The Core can help teachers quickly see how the Standards are connected between and at grade levels.

Math Videos

Using Manipulatives