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Math Means You Know That You Know,
And If You Know,
You Know Making Math Real is the
Gold Standard of Math Education
Math means knowing that you know – not memorizing what procedure to use. There is a profound distinction between the two: the former refers to a deep integration of knowledge and comprehension in the brain and in the body, and the latter refers to applying a memorized series of procedural commands that do not require comprehension of the application, rather, sufficient access to memory in remembering what to do.
THERE’S NO MATH IN IT!
Since the teaching of math so frequently presents instruction as a series of commands to be memorized, instructional practices also frequently attempt to rely on procedural catch phrases such as “carry the 1′, “copy (keep), dot, flip”, “do the opposite” or “slash and dash”, “north and down”, “move the decimal to the right (or left)”, or “daddy-mommy-sister-brother” to help students to try to remember what to do. All these representative examples of procedural commands are dissociated from the comprehension of the mathematics that underlie the creation of them; and this is important to understand: the procedural commands replace the actual mathematical functions and applications – just do __. Furthermore, throughout my time in school and what I continue to observe occurring in classrooms, is these catch phrases are delivered as a hope and a prayer that the students will “just remember”.
- “Carry the 1” refers to renaming in addition or multiplication when a sum or product is between 10 and 19 inclusive in a place, and the new value, the “1”, is carried to the next place. The digit that is carried is expressed as a “1”, but it is referring to 1 new ten, or hundred, or thousand, etc., not a “one”.
- “Copy, dot, flip” is referring to the transformation of a divisor to a multiplier when dividing fractions. Since the catch phrase has no connection to the reasons why the divisor is transformed to a multiplier or to why it is exclusively the operator (divisor) that is “flipped”, students frequently forget which operand gets flipped, so they may flip the first fraction, or to be safe, flip them both, so the catch phrase is attempting to help them remember what to do, but without any understanding of why: copy: keep the first fraction the same; dot: multiply (the dot expresses multiplication); flip: make the reciprocal of the second fraction.
- “Do the opposite” and “slash and dash” refer to subtraction with integers. However, the failure rate with these catch phrases is exceedingly high as, unlike “copy, dot, flip” it does not attempt to account for which integer is to be transformed to its opposite signage. Consequently, since this catch phrase is dissociated from all conceptual connections to transforming signage as well as why subtraction with integers requires this transformation, students are frequently ungrounded and disconnected as they cannot remember which of the integers needs the opposite signage (the operator, not the operand). Frequent error response includes making the first and/or both the first and second integers with opposite signage; and since this procedural command has no grounding with the function and purpose of operations, comprehension of signage, and the understanding of why this is necessary, students feel free to do whatever – there’s no meaning to it anyway… just like “copy, dot, flip”…
- “North and down” refers to where the numerator goes: “n” for north goes with “n” for numerator, and “down” refers to where the denominator goes: “d” for down goes with “d” with denominator. This catch phrase is only targeting where each part of a fraction “goes” rather than the comprehension that “numerator” literally means “the one who numerates, which means the one who counts; and we count to find out how many; and denominator literally means the “one who names the group”. In addition, the fraction bar refers to “of the”, so the comprehension of all fractions is “how many of the group” which directly indicates “how many” is up, “of the” is the fraction bar in the middle, and “group” is below to express the totality: how many of the group.
- “Move the decimal to the…” refers to increasing or decreasing decimals by powers of 10. Moving the decimal is mathematically impossible because the decimal (point) is a symbolic wall that separates whole units from parts of a unit, and that “wall” can never be moved. Furthermore, this catch phrase requires “moving the decimal” in the opposite direction indicated: if increasing, the decimal is moved in the direction of decreasing; and if decreasing, the decimal is moved in the direction of increasing.
- “Daddy-Mommy-Sister-Brother” refers to an incomplete mnemonic device to help students remember to Divide-Multiply-Subtract-Bring down for long division applications. First, this mnemonic omits the crucial application of checking the values of the differences made from the “Subtract” step. However, most importantly, this catch phrase, similar to “north and down” for fractions, only targets remembering what to do: Divide-Multiply-Subtract-Bring down, but it does not provide any comprehension for why do we divide – what does divide tell us? Why do we multiply – what does multiply tells us? Why do we subtract – what does subtract tell us? Why do we check and for what are we checking? Why do we bring down – what does bring down tell us?
Since these catch phrases are all dissociated from the comprehension of why we do these things, I refer to each and all of these instructional practices as “There’s no math in it!”. I believe the reason math is so frequently taught as a series of procedural commands is due to an ongoing long-term cycle of educators passing on the same procedural commands they learned in school to their students without questioning and investigating what the actual comprehension and meaning of the mathematics really are.
WOULDN’T YOU LIKE TO LEARN THE
“WHY” OF THE MATH, ALL THE MATH?
Math means you know that you know, and wouldn’t you like to learn the “why” of all the math mentioned above as well as ALL the math from pre-k through calculus? Then, I look forward to seeing you in an upcoming Making Math Real: Overview K-12.
IF YOU KNOW, YOU KNOW MAKING MATH REAL IS THE GOLD STANDARD
Be well and stay healthy.
David Berg, E.T.
Founder & Director of the Making Math Real Institute
Creator of the Making Math Real Simultaneous Multisensory Structured Methods