It is probably necessary to define truth first, which will then automatically define the real world, and show the problems of modeling. The truth can be defined using the following three sentences. (1) The laws of nature are the only truths. (2) These laws are created by the objects of nature and their characteristics. (3) Nature always demonstrates all its laws.

Clearly real numbers are not objects of nature, and therefore they must be false. Real numbers are points on a straight line. But straight line does not exist in nature, because all objects in the universe are continuously moving. Thus the foundation of mathematics is false, and therefore mathematics cannot describe truths.

Suppose I have a flower in my hand, why would I go to math modeling to understand it? I can feel, see, smell, touch, and even learn what the flower is telling me. My experience cannot be described by any language, not even by a camera photo. How can then mathematics, a symbolic language, describe the flower and my experiences? Why would I even use models, when I can directly experience all the truths?

Destiny is a law of nature, there are many examples to prove that. The following free book has chapters on Truth and Destiny, with many examples. Thus humans cannot have experiences. No experience can be used for future plans. Everything is controlled by destiny. We do not have freewill and choices. Once we know the laws of nature carefully, we will know that we are robots. https://theoryofsouls.wordpress.com/

To help students learn anything, teachers need to initiate the modeling process, eliciting early ideas, provoking students to determine their limits, and helping students develop their ideas further.

All learning is modeling. But not all teaching initiates the modeling process.”

In a nutshell. Thank you. To have this as the benchmark will strengthen my lesson choices and prompts. Yes, your ideas are dangerous. But in a good way!

I think Dan’s statement, “All learning is modeling” does float. All contexts are not created equal, perhaps.

We could choose to use the world “modeling” for “the process whereby a learner tests out her early ideas, determines their limits, and develops those ideas further” but we could just as well call it something else.

I don’t see “all learning” as synonymous with “the process whereby a learner tests out her early ideas, determines their limits, and develops those ideas further” but learning (a tough term to define in and of itself) does involve “testing out early ideas, determining their limits, and developing those ideas further”

The teaching practice of “eliciting early ideas, provoking students to determine their limits, and helping students develop their ideas further” does deserves the most attention here. Developing such practices is perhaps THE most challenging mountain for teachers to climb for many reasons, but it is a worthy mountain.

To “apply modeling pedagogy everywhere” (modeling as defined in something like the GAIMME report) is a very worthy goal, one that many have put effort into over preceding decades, under different labels, as noted by other commenters.

But that worthy mountain:

#lifegoals

Deserves much more attention. Continuing to discuss teaching so as to elevate the recognition of the complexity of the work is a Very Good Thing.

From Dan’s January 14 post:

When teachers express curiosity about diverse student thinking, students feel that and feel license to express even more diverse kinds of thinking.

The more perspectives on an idea a teacher can help students connect, the more students learn about that idea.

That all feels great so the teacher becomes more curious about student thinking and consequently re-evaluates her curriculum and instruction to emphasize tasks and pedagogy that are more likely to elicit diverse thinking.

The teacher becomes interested in learning more mathematics because the more math you know, the more you’re able to identify and connect diverse student thinking when you see it.

My only concern is school math retains the models used in the learning process as the meaning of the mathematics. It does not go back to show it is the limit of the modeling, not the modeling itself, that is mathematics. My point is not leaning how to make mathematics, it is the power I found in using it when I understood problem solving as the the deployment of abstractions. Before that I thought problem solving was retrieval of a model i encountered in school, and putting numbers into it.

I don’t think the following is true: All school mathematics can be learned as modeling.

Let me float this:

All learning is modeling, even learning in school mathematics. Any time a student has undergone conceptual change they have experienced the stages of modeling.

But not all teaching initiates that modeling. Many teachers fail to elicit early student ideas, to provoke them in ways that lead to growth, to help them resolve those provocations and develop new ideas.

All Learning is Modeling.

I’m slow but I finally am coming around to see its wisdom. It seems likely reasonable definitions of “learning” and “modeling” exist to make it true.

So what is the burr under my saddle? I don’t think the following is true:

All school mathematics can be learned as modeling.

I take school mathematics to be the mathematics students need to succeed academically and vocationally following k-12.

I find getting my students to do this work– mathematizing horizontally and vertically– is helpful.

In terms of Dan’s problem of the 1,2, 4, 7, 11… sequence I would suggest that this is vertical mathematizing since it is strictly inside the world of symbols.

To paraphrase a Japanese proverb, sometimes the shortest path between two points is around.

Beautiful.

We can probably agree that most of us use modeling of some sort when we are stating to learn a new concept. The question is, do we stop there?

Earlier in this thread Dan Peter provides a succinct description of the central role modeling plays in school math. This is how I understand him. Start at the lowest level.
Model it, do exercises based on the model until you are comfortable with the model, move to the next level. Iterate.

It appears we do stop there, we never reach conceptual understanding. At any level our only understanding is based on modeling in previous levels. We never reach the point of doing math on its own terms. The math provided by modeling is never self contained, never abstract, always understood as relations to models of itself. It gets bigger, not stronger. It does not prepare students for the time when they need the strength of conceptual understanding.

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