As we expect every child to read fluently with comprehension by the end of third grade,  we should expect and work diligently toward, every child's mastery of numeracy with understanding by the end of fourth grade. With basic numeracy, all children can learn mathematics easily, effectively, and efficiently,  and can appreciate math's reach, power, and beauty. 

Math education in the school curriculum is about number concepts, number sense, and building the brain for numeracy and beyond. We must make it easier for students to understand and develop mathematical ways of thinking and communicating. 

Children can access basic and high-level mathematics once they achieve numeracy through systematic instruction and practice, and strategic training in mathematical reasoning and communication. 

Our brain is biologically wired for oral language and estimating the size of a collection of few objects. The brain is not wired for reading, writing, number processing, and math procedures. 

In order to successfully read, write, and calculate procedurally, the human brain must re-purpose regions biologically designed for other purposes. The brain must develop specialized neurons, new circuits and connections, specifically attuned to oral and written language, and mathematical thinking. Processing efficiencies eventually develop within these newly formed neural connections; the result is what neuroscientists call, functional connectivity.  It can be done.

Topics 

1. QUANTITY --> Number Sense (digit, quantity, word, image, 45-sight-facts for digits 1-10, making 10, teens numbers)

2. SPACE  --> Spatial Sense (left, right, directionality, words, arrangement, organization, position, relationships)

3. PATTERNS  --> Recognition, Extension, Relationships

  A. Math is the study of patterns.

  B. Arithemetic is the study of patterns in numbers, 4 operations, add, subtract, multiply, divide).

  C. Geometry is the study of shapes and relationships.

  D. Probability and statistics is the study of prediction.

  E. Algebra is the study of variability. 

4th Grade Focus on Division

Don't start division until students understand the 4 models of division. Introduce one model per day.

Understand why we start with ones on the right when doing multi-digit addition, subtraction, multiplication, and division, but start from the left when doing long division. 

Long division is a very important invention.

Didactic: What the student does not know.

Questioning: Questions instigate language. Language instigates models. Models instigate thinking. Thinking instigates meaningful understanding. Understanding produces competent performance. Competent performance produces long-lasting self-esteem. Self-esteem produces life-long learning. 

Teacher is Coach: What is the standard of performance? Help practice to the standard.

Outcomes: What is the desired outcome? What is the proof of mastery? What behavior is expected? What does competence look like?

Japan has a National Treasure award. The NT teacher travels anywhere with government support. A good question produces good answers.

Variety of Methods

A teacher said, "I only want to teach one method of division because  more than that will confuse my special ed students." 

Professor Sharma replied, "Do you only teach one way to spell /ay/ and forget about the rest because it may be confusing?" The teacher said, "No! We teach all the ways." Professor Sharma said, "Then, of course, you must teach all of the different ways to solve problems in mathematics! You cannot create terminal students incapable of functioning and learning outside of your classroom!"