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An Appeal for Conceptual Progression of Teaching and Learning

Why should students be taught how skills and concepts develop?


A 5-year-old may know how to count but struggle with addition. To overcome this challenge, the child might be taught to memorize a procedure for adding digits that are lined up on a piece of paper. They may know how to "carry the one" when two digits added result in a number more than 10. This child might eventually be able to solve multi-digit addition problems quickly and correctly using this algorithmic process in their head.


When that child becomes a 7-year-old, they may know how to add but then struggle with multiplication. To overcome this challenge, they might be taught math facts via rote memorization, with flash cards, with games, mnemonics and visual patterns. They may learn that math facts with 6s are similar to math facts with 12s, and that facts with 8s are similar to facts with 4s. They might even be told to rehearse their addition patterns to memorize math facts that way, eventually being able to recall math facts up to a certain point quickly and correctly using a variety of memory and recall processes.


When that child becomes a 9-year-old, they may know how to multiply but then struggle with exponents. To overcome this challenge, they might be taught to multiply the 'base' however many times the 'exponent' says to multiply, and then simply use a calculator to complete the calculation. In fact, a calculator may be the solution for both the 5- and the 7-year old when learning about addition and multiplication. Knowing how to address exponents and their relative square roots can be addressed this way by teaching the student how to manipulate a calculator to simplify mathematical expressions.


In each of these examples, a student is taught to overcome mathematical obstacles relative to their appropriate developmental level. However, because the student is never taught with an expectation for cognitive demand beyond simply executing an algorithm or manipulating an electronic tool, the student's cognitive capacity for conceptual understanding of these mathematical obstacles is never fully developed. Instead, each concept is addressed individually, as isolated skills, with strategies that apply only to each skill independent of the conceptual progression which connects them.


When that child becomes an 11-year-old, they will be introduced to multi-step algebraic expressions with variables, involving addition, multiplication, division, and exponents. The number of isolated tricks and processes that will need to be automatic in order to simplify increasingly complex expressions will soon overload the student's cognitive capacity and the conceptual understanding behind the contexts in which those expressions are appearing will be lost to the student. At this point, math as an academic subject will become an exercise in task completion and point collection rather than an exploration and mastery of developmentally-appropriate mathematical concepts.


So what's different about teaching how skills and concepts develop?


A 5-year-old who knows how to count but struggles with addition wouldn't be told how to multiply or use exponents. Instead, the focus would be on helping them learn the words related to the ideas behind place value. Addition as a concept would become an ongoing conversation about counting different kinds of number groups, where any combination of number of groups that results in "more than ten" would be rearranged to visually reflect the concept of 'base ten' counting. Similar to word study or phonics in literacy development, celebrating the child's developing capacity to describe why they are rearranging different number groups into groups of ten (and a few ones left over) makes it easier to count, or add, quickly and correctly.


When that child becomes a 7-year-old, this conceptual understanding of addition likely won't cause the child to suddenly know all their math facts. Instead, the cognitive capacity the child developed for not just adding but communicating how and why they are adding will allow them to approach multiplication with that same capacity. Introducing the idea that multiplication is a shortcut for adding the same number groups quickly and correctly (e.g. six group of ten, or ten groups of six) will decrease the need for so many mnemonics and memorization tricks. As the child is celebrated in their ability to communicate the counting of same-number groups (e.g. 6 groups of 7), their ability to multiply quickly and correctly with larger numbers will improve over time.


When that child becomes a 9-year-old, the concept behind counting same-number groups as multiplication will feed into the concept of knowing how many of those same-number groups can result in a larger collection of same-number groups, organized with just a bit more detail. The child who is comfortable describing sixteen as "four groups of four" is likely more open to considering how many "four groups" of "four groups of four" would be. It is in celebrating the child's development of language to communicate mathematical concepts that redefines how we approach teaching how skills and concepts develop, instead of isolating each skill and concept and teaching isolated strategies for each skill or concept.


The challenge with conceptual understanding is that it requires time for the child to consider and visualize these ideas in order to assign the words to those ideas in ways that they can communicate them with confidence. This doesn't always happen according to a teacher's scheduled lessons which is why a self-paced development, as is the case in a more self-directed learning environment, allows for the child to explore suggested activities and be open about what they are seeing and hearing in a more conversational approach to challenging thoughts and ideas that meets a child where they are. Humans love to succeed, children most of all, and it's in connecting that feeling of success to the feeling of frustration that can develop a child's intrinsic desire to explore, ask questions, and focus on overcoming a particular challenge beyond simple memorization and celebrate their capacity for more cognitively rigorous presentation of skill and knowledge.


The image below refers to the four levels of Webb's Depth of Knowledge by which most math skills are assessed as proficient at a level of Basic Recall and Reproduction. If we are to develop our students' cognitive capacity for conceptual understanding, our rubrics and guides must be revised to addressed deeper depths of proficiency than that of Basic Recall.


Doing it isn't the same conceptually understanding it.

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