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 features the Common Core Grade 4 overview in blue, followed by its ambient counterpart as practiced by Waldorf Education and Math By Hand.
Operations and Algebraic Thinking
Use the four operations with whole numbers to solve problems.
Gain familiarity with factors and multiples.
Generate and analyze patterns.
The 4 processes are taught and learned side by side from the very beginning, so a relationship has been established, between and among all 4. Equations are written horizontally at first in Grade 1, as number sentences. Single digits totaling no more than 20 are worked with in addition, subtraction, multiplication, and division. After briefly reviewing the Grade 1 content at the beginning of Grade 2, the horizontal format is changed to vertical, using double digits with no regrouping, in all 4 processes up to 100. Since multiplication and division have been worked with so extensively, their long forms are taught mid to end of Grade 3. Factor patterns are explored as “trees” in Grades 2 and 3, then gone into at greater depth at the beginning of Grade 4, as a segue into fractions. A great emphasis is placed on math patterns and corresponding number “tricks” in both the Waldorf and Math By Hand curriculums.
Number and Operations in Base Ten
Generalize place value understanding for multi-digit whole numbers.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Place value is taught in Grade 2 using hands-on materials and manipulatives, and used primarily in regrouping with addition and subtraction. Regrouping is also used with short multiplication and division at first, then in mid-Grade 3, the long versions of both are taught with hands-on materials and manipulatives. Multi-digit arithmetic is performed with regrouping in all 4 processes from mid-Grade 2 on.
Number and Operations – Fractions
Extend understanding of fraction equivalence and ordering.
Build fractions from unit fractions by applying and extending previous understanding of operations on whole numbers.
Understand decimal notation for fractions, and compare decimal fractions.
There’s a readiness for the concept of breaking apart the whole in Grade 4. Fractiousness, along with the chaotic storms of growing up in a complex world are more real and present now, echoed in the Norse Myths or the trickster tales from other cultures. At the same time, caution must be taken with presentation. When new concepts are presented abstractly without concrete references, much harm results. Opportunities for teaching fractions concretely abound! Math By Hand begins with cutting up color coded squares (1 whole = red, 2 halves = green, 4 fourths = blue, etc., so equivalence can be readily grasped from the start. A set of playing cards is used with familiar, classic games that are adapted to learning fraction equivalence, and the rules for using the four operations with fractions are clearly laid out in a simple booklet that can be used as a learning aid as often as needed. Using coin rubbings to introduce decimal-fraction equivalence (see below) is both fun and effective.
Measurement and Data
Solve problems involving measurement/conversion of measurements from a larger unit to a smaller unit.
Represent and interpret data.
Geometric measurement: understand concepts of angle and measure angles.
Both time and measurement are introduced slowly, carefully, and creatively through anecdotal historical stories and lots of hands-on, real, and experiential application in Grade 3. This is expanded upon in Grade 4, with increasingly complex measurement, along with conversion to metric measurement. Data can remain informal, through collections and notebook recording (no Excel spreadsheets just yet). Angles and polygons wait until formal geometry in Grade 6, when instruments are used, featuring accuracy and aesthetics. One of Common Core’s mistaken principles is that if advanced concepts are pushed down to lower grades, math proficiency will more likely be achieved. False! Slowing down with generous slices of play and artistic expression all through the grades, while solidifying the basics, is so key!
Draw and identify line and angles and identify shapes by properties of their lines and angles.
As has been previously shown, form drawing is more than adequate to address all aspects of this standard. Here are two examples from Math By Hand Grade 4 form drawings. The compass rose is used to orient, as surroundings and local environment are mapped, beginning with home and radiating out from there. The second form is an example of the complex weaving and braiding found in many cultures’ symbolic figures, most notably the Celtic.
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 to continue with Common Core Math Standards and their ambient counterparts.