Foundation Concepts

Foundation Concepts

Numeracy- is what we now believe to be the core deficit in math difficulties. It is an awareness of quantity and quantity relationships.
Subitizing - There are two levels of quantity pattern recognition. Douglas Clements describes them in his well known article, Subitizing: What It Is. Why Teach It.' This is a crucial skill for developing math concepts though and it is crucial that students receive lots of practice at composing and decomposing quantities if they are to develop number sense.

Perceptual Subitizing - You recognize the quantity you see. It is immediate recognition
Conceptual Subitizing- You recognize two or more groups as being part of a larger set you quantify. Look at the domino above. If you knew it was nine without counting, you used perceptual subitizing.

According to research the human brain can recognize quantities up to four but over that, the quantity needs to be in a pattern. (See the research of Stanislas Dehaene and the work of Brian Butterworth.)
After mastering quantity patterns of all addends up to and including the ordered pairs of ten (1+9, 2+8, 3+7, 4+6 & 5+5) the student can apply those patterns and relationships across place value to reason mathematically.
When students are learning numeracy patterns, they should do so using multisensory methods and not just use numbers and flash cards. They should construct and deconstruct patterns with many different representations: Unifix or linking cubes, ten frames, dice & domino patterns, sets of objects. They should apply the facts in problem solving.
Adding and subtracting with visual patterns in memory can aid LD students to work with numbers between 10 and 20 by using construction & deconstruction of quantity with a "Go To The Ten" strategy.
After mastering quantity patterns, students can apply those patterns across place value to begin really reasoning in math.
Avoid accommodation strategies which can inhibit numeracy acquisition. These may include too early use of calculators or touching & counting strategies. Students can be wedded to these compensations and be discouraged from really every learning facts or developing fluency with them.
Every attempt should be made to develop a strong sense of quantity independent of language retrieval without meaning.
The Multisensory Math Approach is dedicated to developing real comprehension of number quantities and using mastery to apply quantity relationships across place value to reason mathematically.

For a hierarchy of essential skills by grade level, see the Achieve the Core Math Focus by Grade website:
https://achievethecore.org/category/774/mathematics-focus-by-grade-level
This website offers a simple, clear view of what is expected of students at each grade level.