A Year in the Life: Ambient Math Wins the Race to the Top!
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. Today’s post will focus on the fourth and last pair of Mathematical Practice Standards, listed in blue and followed by their ambient counterparts. Because there are so many parallels in standard #7 to the ambient Grade 1 math practices, this standard will be taken apart and looked at in detail. As for #8, most of the possible applications are too advanced for Grade 1 so it will be just briefly reviewed.
The Standards for Mathematical Practice, Grade 1
#7 and #8: Seeing structure and generalizing.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure.
The Waldorf Grade 1 curriculum begins with a steeping in structure and pattern. Form drawing provides a first look at geometry as it relates to each number from 1 to 12. This awareness of pattern is essential since it not only translates later to an understanding of the principles of geometry, but more importantly it provides a bright and engaging glimpse into the truly beautiful face of mathematics. And this above all else can generate the spark that motivates learning (and teaching)!
Young students for example might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.
The first, noticing equivalency as a pattern happens easily when using the color calculation strips (mentioned yesterday) side by side to show relationship within each or between two or more processes. Secondly, sorting shapes may not happen by exploring shapes using a tub of plastic, foam, or wooden manipulatives, but will register on a deeper level through exposure to the properties of geometry through form drawing. For example, an instinctual recognition of the different properties of the square and triangle results when they are both presented with a story and/or picture for support.
Later, students will see 7 x 8 equals the well-remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property.
Having seen examples of Common Core practices and worksheets where a simple 2 or 3-step process becomes an elaborate, overly complicated, multi-step procedure, I have little faith in the effectiveness of exposing K-3 students to this level of abstract thinking. It’s confusing, so rote memorization along with looking for patterns in the times tables as prompts are much more effective at this age. The distributive property is taught with extensive story and pictorial support, but not until Grade 2.
They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.
Form drawing builds an effective foundation for this advanced function as the significance of every line is recognized as essential to the structure as a whole. An early exposure to the compelling beauty of geometry is more advantageous than anything else to grasping its abstract principles later on.
8. Look for and express regularity in repeated reasoning.
This points to a recognition of repetition as the foundation for using methods and shortcuts in calculation. If the right solid, foundational elements are provided now it paves the way for this sort of recognition later. One element, that of maintaining oversight of the process, while attending to the details, may be applicable at this level. Awareness of the “big picture” along with the building blocks that make it up is a consistent focus in all Waldorf grade levels. The macro is not sacrificed to the micro or vice versa but instead, a healthy balance between the two is always maintained. This may very well result in a responsible, morally integrated individual in right relationship to the world, the community, and the environment.
Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of that knowledge as a worthy goal. Tomorrow we begin to look at the Common Core Grade 1 Operations and Algebraic Thinking Standards along with their ambient counterparts.