# Trivium and Conceptual Approach to Math

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Hey everyone,

I have recently been working through choosing another math program for my struggling 2nd grade son. I've already posted elsewhere addressing programs that might be a good fit. And while helpful, I began to see that whereas before I thought one simply needed to determine which style (spiral or mastery) when choosing a curriculum, I now understand this other element of conceptual vs procedural comes into play as well.

How do the conceptual and procedural approaches mesh with the ideas set forth in the Trivium? As I see it, in the grammar years there is a greater emphasis on memorizing and filling the head with information and what-not, and that exploration of the "why" of things happens more in the dialectic stage. Wouldn't working in a heavily conceptual curriculum be more like "cutting against the grain" during the grammar years? if that makes sense.

Thoughts?

Leah

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Hey everyone,

I have recently been working through choosing another math program for my struggling 2nd grade son. I've already posted elsewhere addressing programs that might be a good fit. And while helpful, I began to see that whereas before I thought one simply needed to determine which style (spiral or mastery) when choosing a curriculum, I now understand this other element of conceptual vs procedural comes into play as well.

How do the conceptual and procedural approaches mesh with the ideas set forth in the Trivium? As I see it, in the grammar years there is a greater emphasis on memorizing and filling the head with information and what-not, and that exploration of the "why" of things happens more in the dialectic stage. Wouldn't working in a heavily conceptual curriculum be more like "cutting against the grain" during the grammar years? if that makes sense.

Thoughts?

Leah

Excellent question. Bump.

:lurk5::bigear:

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Why wouldn't a conceptual curriculum not be teaching the facts but more thoroughly?

Rosie-mad insomniac, so not good at thinking just now.

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Thoughts from a total and complete non-expert:

I don't think the two are necessarily in an either-or relationship. The emphasis in the grammar years according to the classical model is on memorization, but that doesn't mean conceptual understanding needs to be put on hold. In fact, I'm remembering the very brief couple of sentences in TWTM about bundling pencils into bundles of tens and leaving single ones separate to show a child the concept of "carrying." And that's just one example. Many of the math programs TWTM recommends, and other people on the boards use, employ manipulatives extensively to help kids grasp the underlying concepts before moving into the relatively abstraction of written numbers. A lot of math picture books likewise seek to show kids the larger picture, the "whys" as well as the "hows" of playing around with numbers and shapes.

That said, I think kids vary a whole lot in their intellectual leanings: some will do better with the "Yours is not to reason why; just invert and multiply" approach, while others seek understanding and finding answers to the whys from a young age. There have been some studies on middle-schoolers that show boys like to just plug in the formula, while girls like to know why it works -- a gender gap in logic stage approaches.

My own daughter did better with the emphasis slanted toward conceptual math when she was younger. Now that she's older and into the textbook phase of her education (algebra), she just wants to get through the book and doesn't care why something works. I think that's a loss, and it's partly related to the way the textbook divides up math into discrete "skills" and topics. So we're switching to CoMap mathematics next year, which teaches algebra, geometry, and trigonometry together, as they apply to a series of scenarios from the real world (data collection, satellite imaging, animation, robotics, topology, construction, etc.). In other words, her "why" has changed from why you do a problem a certain way to why anyone would bother thinking about these things and how they would be used -- a different stage of "why."

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I don't know of any math programs that take a formal theory-based approach (with proofs and the like) in the elementary grades. Even something like Singapore, which is considered a "conceptual" approach, does the whole concrete -> pictorial -> abstract thing.

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I wouldn't characterize it as a dichotomy: conceptual versus procedural. Some math programs, like MM or Singapore, teach mathematical procedures along with the underlying conceptual framework, and some programs just teach the procedures. And even then it's really a continuum: some take a basically procedural approach but throw in a bit of conceptual explanation, others focus very heavily on the conceptual side of things and recommend the student use flashcards or separate worksheets to master math facts as a separate skill, and others fall somewhere in the middle.

It's also important to remember that the concept of grammar/logic/rhetoric "stages" is an invented one; it's not something that is a universal component of child development. In classical education, grammar, logic, and rhetoric were three different subjects, they were not stages of development. That is a modern use of the terms, and not one that many child development specialists would agree with. The idea that children in elementary school can only learn to memorize facts, and that teaching them the conceptual basis for mathematics somehow goes against their natural development is contradicted by the fact that many Asian and European countries teach math in a very "conceptual" way starting in Kindergarten, and these countries tend to do much better in math than the US does. Liping Ma's Knowing and Teaching Elementary Mathematics has an excellent and very eye-opening discussion of the differences between the way China and the US teach math, and why the "Asian way" is generally more successful.

Jackie

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• 1 month later...
I don't know of any math programs that take a formal theory-based approach (with proofs and the like) in the elementary grades. Even something like Singapore, which is considered a "conceptual" approach, does the whole concrete -> pictorial -> abstract thing.

This is what Classical Liberal Arts Academy's Classical Arithmetic course is. Proofs and rules first. Application second. It is studied by students as young as 7 and as old as college/adult.

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