3.OA 2: Grouping to Divide & Subtract (#187)

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 187

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times. Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. CCSS math standards are listed here in blue, followed by ambient math suggestions. Note that this post is largely a repeat of yesterday’s because the two are so closely related.

Operations and Algebraic Thinking 3.OA
Represent and solve problems involving multiplication and division.
2. Interpret whole number quotients of whole numbers; eg., interpret 56 / 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 / 8.

Is this enough to drive any teacher crazy, or is it just me being dense? Common Core proponents would say the latter. But if looked at from a Waldorf perspective, this really is much ado about nothing. This sort of standard, even though Common Core philosophy repeatedly says that CCSS is just standards and not scripted curriculum, is the source of all those scribbled diagrams, showing the above-mentioned shares and groups. Not pretty to look at, and I am sure tedious and boring to create, these so-called arrays could be avoided by teaching all 4 processes together from the beginning.

With both the Math By Hand and Waldorf methods, this concept is begun in Grade 1, as division and subtraction are compared, side-by-side. Manipulatives help to make this clearer and more understandable. The glass gems pictured above are used to correlate division and subtraction as follows.

Color-coded strips differentiate each of the 4 processes:

addition/plus = green
multiplication/times = yellow
subtraction/minus = blue
division/divide = red

Each strip is folded into six 3″ square sections, with a white square that can be placed anywhere for the missing number. To compare division and subtraction, a red strip could be placed next to a blue strip, with corresponding equations. For the minus equivalent to the divide equation 10 / 5 = 2, 2 gems would be placed in each of 5 squares. Same principle as above, with smaller numbers.

If this is demonstrated repeatedly and consistently in Grade 1, with the 2, 5, and 10 times tables, it will easily translate to the higher tables in later grades. No need to have students draw endless arrays of dots or other objects to reinforce the concept. And, on the subject of scribbling (because that is what it looks like when you draw 7 groups of 8 circles each), the Waldorf philosophy holds that it’s essential for everything the young child creates in the learning process to be beautiful. (And beyond, as middle and high school Waldorf students’ work testifies.)

Math should be full of fun and games as well. Here is a star/folding activity for learning or reinforcing the 6 times table. Both rote memorization and a deeper understanding of math concepts should coexist. The times tables do need to be memorized, and this is accomplished with games, rhythmic movement, recitation, singing, and handwork. Patterning, on the other hand, takes it all to a deeper level, so that math is appreciated for its complexity and compelling beauty.

Interest and beauty, as prime motivating factors for grade school age children, will carry the day as the love of learning grows into a lifelong quest. The “reasoning” or “putting it into words” aspect of Common Core math, as groups of students are required to discuss how they arrived at their answers, may very well be a mistaken goal for math as a whole, as well as for the child until age 12-14 when abstract reasoning first appears. Until then, we need to have a little faith that the process of learning will naturally occur if conditions are favorable and the right ingredients are gathered and presented.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 3 math CCSS and their ambient counterparts.

August 11, 2014 Common Core Grade 3

About the author

Marin Lipowitz holds an MS in Waldorf Education, with years of classroom teaching experience. She is an advocate for age appropriate, arts based, Waldorf inspired learning for all children, and is dedicated to providing hands-on programs that meet these goals. Her gift lies in creating unique, effective, project-based solutions and applications to enliven basic, standards-aligned curriculum.