Yes. I really think that the translation part is one of the hardest.
As a brief aside, I want to note that this conversation is happening in one of the languages we’re discussing and that could influence any conclusion we come to. I’m also going to suggest that we ignore Gödel for now
There are many people who are good at math. There are even a lot of people who are reasonably good at grinding through the mechanics of math. That doesn’t solve any of the problems you described above.
Statistics are a great example of this. Early statistics classics are mostly about the mechanics; here’s how you calculate the mean, standard deviation, confidence intervals, etc. 2 types of students generally come out of that class; math students who will forget all of that because they’re going to learn the “real” versions (eg they go through a huge number of proofs that involve calculus and linear algebra), and students who will forget all of that because the whole thing sounds like gibberish.
We teach natural languages the same way but we go much farther. Students learn vocabulary and grammar rules but they’re also expected to learn how to use them correctly. We had students current events articles and ask them to analyze them. We ask students to practice many writing methods including fiction and expository writing.
When I talk to my own kids about statistics I never write any formulas. I ask questions like, “What do you think ‘mean’ means?”, “If I have a bunch of <example item> does ‘mean’ describe it well?”, “What happens if I add an <example item> with <huge outlier>? Do you still think it’s a good description?” “How would you describe it better?”
If I ever had to design an introductory statistics course it would contain very little “math”. Classes, homework, projects, and tests would consist of questions like;
“Here’s some data and an interpretation, are they lying? Why or why not?”
“Here’s a (simple) data scenario. Tell me what’s going on.”
“Here’s some (simple) data. Produce a correct and faithful summary. Now produce a correct but misleading summary. Describe what you did and the effect.”
“Here’s a conclusion. Provide sample data that most likely fits the conclusion.”
“Change one word in the sentence, ‘Increase your chances by 80% means that there is now an 80% chance.’ to make it a true statement.”
Yes. I really think that the translation part is one of the hardest.
As a brief aside, I want to note that this conversation is happening in one of the languages we’re discussing and that could influence any conclusion we come to. I’m also going to suggest that we ignore Gödel for now
There are many people who are good at math. There are even a lot of people who are reasonably good at grinding through the mechanics of math. That doesn’t solve any of the problems you described above.
Statistics are a great example of this. Early statistics classics are mostly about the mechanics; here’s how you calculate the mean, standard deviation, confidence intervals, etc. 2 types of students generally come out of that class; math students who will forget all of that because they’re going to learn the “real” versions (eg they go through a huge number of proofs that involve calculus and linear algebra), and students who will forget all of that because the whole thing sounds like gibberish.
We teach natural languages the same way but we go much farther. Students learn vocabulary and grammar rules but they’re also expected to learn how to use them correctly. We had students current events articles and ask them to analyze them. We ask students to practice many writing methods including fiction and expository writing.
When I talk to my own kids about statistics I never write any formulas. I ask questions like, “What do you think ‘mean’ means?”, “If I have a bunch of <example item> does ‘mean’ describe it well?”, “What happens if I add an <example item> with <huge outlier>? Do you still think it’s a good description?” “How would you describe it better?”
If I ever had to design an introductory statistics course it would contain very little “math”. Classes, homework, projects, and tests would consist of questions like; “Here’s some data and an interpretation, are they lying? Why or why not?” “Here’s a (simple) data scenario. Tell me what’s going on.” “Here’s some (simple) data. Produce a correct and faithful summary. Now produce a correct but misleading summary. Describe what you did and the effect.” “Here’s a conclusion. Provide sample data that most likely fits the conclusion.” “Change one word in the sentence, ‘Increase your chances by 80% means that there is now an 80% chance.’ to make it a true statement.”