kindergarten - Third Grade Mathematic's Focus


This presentation is an attempt to encourage instruction and mathematical activities that facilitates students' construction of mathematical understanding through continual attainment of: 1) conceptual understanding and 2) thinking and understanding as mathematicians.

A brief look at Misconceptions in teaching and learning mathematics that reject continual attainment of: 1) conceptual understanding and 2) thinking and understanding as mathematicians.

Principled Procedures to guide teaching and learning mathematics through continual attainment of: 1) conceptual understanding and 2) thinking and understanding as mathematicians.

Conceptual and Procedural Definitions:

Conceptual understanding is the knowing and reasoning that leads to a belief in confidence in a person's solutions and answers.

Procedural understanding is knowing the steps and strategies used to solve various kinds of number tasks/ problems.

In Kindergarten - Third Grade the emphasis must be conceptual.

Teacher directed or indirect instruction must be 100% Conceptual 0% Procedural with the ONLY procedural knowledge being introduced by the students.

Conceptual Knowledge of NUMBER SENSE (link)

I have selected five strands of knowledge (Classification, Measuring, Counting, Number Sense, and Operational SENSE) that are essential for all students to master if they are to achieve understanding of basic mathematical number and operational sense. They are classified in these categories to assist teachers in the decision making process of bringing a comprehensive understanding of mathematics, children's intellectual and emotional development, and each student's disposition and present understandings of mathematics together to select appropriate and significant mathematical activities and tasks for students and to facilitate their conceptual understanding and ability think about and value math as mathematicians.

Developmentally Appropriate

While there are many ways to determine developmentally appropriate. The main standard chosen is borrowed from Piaget, with developmentally appropriate decisions being made with respect to the child's logical/ mathematical development as it was first described by him with subsequent changes since. The most significant factor is the Conservation Skills, with the operations or sub skills: Transformations and Reversibility essential for conservation along with the student avoiding (illogical thinking) Centering and Transductive reasoning. (learning theory web and characteristics of a child outline)

Activities (link

Dots for number sense (This program works in somme browsers (Firefox) and in some others only displays one dot at a time (bummer).
Dot for reversibility, transformation, hierarchical inclusion, combinations of addends
Add on Count Back


One example of how a pattern can be used to assist students' conceptualization throughout school and life.

Flash dots or squares (1-5), (1-10),(10-?) for students to recognize
Flash and ask for one more, one less, two more, and two less..
Have students make patterns with all kinds of objects, draw them, look for patterns inside other patterns..10 strips, ten frame, rolls..
Flash dots in square patterns and repeat tasks like above.. 100 chart, square..
Flash squares and rectangles and repeat tasks like above..
Use dots and squares to help students construct the concept of the inclusiveness of combinations of addends..

Multiplication grid
Multiplication grid double
Multiplication fractions
Algebra - see if you can do (x + y) * (x + y)

Suggestions, comments, questions welcomed! email

Dr. Robert Sweetland's Notes ©