I’ve always thought of math as a language and I talk to my kids about it that way too.
Math is an other way to describe the world.
It’s very different from spoken languages and translating between the two needs to be learned and practiced.
Our math education doesn’t include enough word problems and it should be bi-directional. In addition to teaching students how to write equations based of sentences we should teach them how to describe what’s going on in an equation.
Yeah, it is kinda both in general. Though in this case, the math about this is well-defined: it’s possible to increase a percentage either with addition or multiplication and both of those can make sense, just the words we would use to describe them are the same so it ends up ambiguous when you try going from math to English or vice versa.
But the fact that switching between communication language and a formal language/system like math isn’t clear cut does throw a bit of a wrench in the “math doesn’t lie”. It’s pretty well-established that statistics can be made to imply many different things, even contradictory things, depending on how they are measured and communicated.
This can apply to science more generally, too, because the scientific process depends on hypotheses expressed in communication language, experiments that rely on interpretation of the hypothesis, and conclusions that add another layer of interpretation on the whole thing. Science doesn’t lie but humans can make mistakes when trying to do science. And it’s also pretty well established that science media can often claim things that even the scientists it’s trying to report on will disagree strongly with.
Though I will clarify that the “both” part is just on the translation. Formal systems like math are intended to be explicit about what they say. If you prove something in math, it’s as true as anything else is in that system, assuming you didn’t make a mistake in the proof.
Though even in a formal system, not everything that is true is provable, and it is still possible to express paradoxes (though I’d be surprised if it was possible to prove a paradox… And it would break the system if you could).
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.”
Why not both?
I’ve always thought of math as a language and I talk to my kids about it that way too. Math is an other way to describe the world.
It’s very different from spoken languages and translating between the two needs to be learned and practiced.
Our math education doesn’t include enough word problems and it should be bi-directional. In addition to teaching students how to write equations based of sentences we should teach them how to describe what’s going on in an equation.
Yeah, it is kinda both in general. Though in this case, the math about this is well-defined: it’s possible to increase a percentage either with addition or multiplication and both of those can make sense, just the words we would use to describe them are the same so it ends up ambiguous when you try going from math to English or vice versa.
But the fact that switching between communication language and a formal language/system like math isn’t clear cut does throw a bit of a wrench in the “math doesn’t lie”. It’s pretty well-established that statistics can be made to imply many different things, even contradictory things, depending on how they are measured and communicated.
This can apply to science more generally, too, because the scientific process depends on hypotheses expressed in communication language, experiments that rely on interpretation of the hypothesis, and conclusions that add another layer of interpretation on the whole thing. Science doesn’t lie but humans can make mistakes when trying to do science. And it’s also pretty well established that science media can often claim things that even the scientists it’s trying to report on will disagree strongly with.
Though I will clarify that the “both” part is just on the translation. Formal systems like math are intended to be explicit about what they say. If you prove something in math, it’s as true as anything else is in that system, assuming you didn’t make a mistake in the proof.
Though even in a formal system, not everything that is true is provable, and it is still possible to express paradoxes (though I’d be surprised if it was possible to prove a paradox… And it would break the system if you could).
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.”