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It's quite a shame that the first draft of the proposal basically makes small the place of Calculus in the high school curriculum, suggesting that it leaves stu

It's quite a shame that the first draft of the proposal basically makes small the place of Calculus in the high school curriculum, suggesting that it leaves students less prepared than had they taken a data science course instead. It is agreeable that data science is interesting, but one must ask what is data science without Linear Algebra or Calculus / Analysis.

eesmith 5y ago
This is certainly part of the debate over at least the last 10 years that I've heard of, whether "calculus is the capstone” - a view that ”originates with a committee of 10 men in 1892” - or if other forms of math should be considered as important alternatives. In general, many believe that statistics education suffers because of the focus on calculus. See for example this StackOverflow post from 7 years ago, https://matheducators.stackexchange.com/questions/2099/emphasizing-statistics-instead-of-calculus https://matheducators.stackexchange.com/questions/2099/empha... . It's not hard to find other links, like https://www.edweek.org/teaching-learning/calculus-is-the-peak-of-high-school-math-maybe-its-time-to-change-that/2018/05 https://www.edweek.org/teaching-learning/calculus-is-the-pea... , with more information on the problems with emphasizing calculus: > For a picture of how severe that inequity can get, one only has to look at calculus. ... Until about 1980, calculus was seen as a higher education course, primarily for those interested in mathematics, physics, or other hard sciences, and only about 30,000 high school students took the course. That began to change when school reformers glommed onto calculus as an early example of a rigorous, college-preparatory course, ... despite the rapid growth of calculus as a gold standard, university calculus experts argue it is a much weaker sign that a student is actually prepared for postsecondary math in the science fields than it appears Indeed, the L.A. Times piece describe it as "in high school, when they can choose advanced subjects, including calculus, statistics and other forms of data science" (emphasis mine). Note too how it classified statistics as part of data science. IMO, data science without statistics is even worse than data science without calculus.
- threatofrain 5y ago
  But analysis and linear algebra are at the bedrock of statistics. Otherwise, how do you even do a line of best fit?
  - eesmith 5y ago
    I don't know why you think analysis is the bedrock of statistics. I'll explain my understanding; perhaps that can help you clear up my confusion? "Analysis" has different meanings in mathematics than in general use. When you wrote "or Calculus / Analysis", I assumed you meant "the branch of mathematics dealing with limits and related theories" ( https://en.wikipedia.org/wiki/Mathematical_analysis https://en.wikipedia.org/wiki/Mathematical_analysis ). The real analysis course I took as an upper-level undergrad focused on the fundamentals of calculus. I did a lot of ε/δ proofs that semester. Statistics does not require that sort of analysis, so it cannot the "analysis" you refer to. "Analysis", less mathematically speaking, means "the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it" (quoting https://en.wikipedia.org/wiki/Analysis https://en.wikipedia.org/wiki/Analysis ). That goes back to at least Euclid ("Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth") (Ibid.) Historically, mathematics education taught Euclid's Elements - geometry - as a way to learn analysis. Lincoln, for example, "kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight"." (quoting https://en.wikipedia.org/wiki/Euclid%27s_Elements https://en.wikipedia.org/wiki/Euclid%27s_Elements ). Clearly, "analysis" doesn't require calculus, and surely can be taught independent of calculus. Indeed, Fermat and Pascal laid the foundation for statistics before Newton was a teenager. "Linear algebra" is also a very broad topic. When I learned to solve a problems like {given x + 2y = 5; 3x - y = 12; solve for x and y} in 9th grade, I was doing linear algebra, though it wasn't specifically told it was linear algebra. When I learned about eigenvalues and eigenvectors in college course actually named "linear algebra", it of course covered things in much more depth. But why does an introduction to statistics course requires knowing about eigenvectors, null spaces, and orthonormal basis? I mean, I vaguely remember my college statistic course, and it didn't require most of the mathematics I learned from linear algebra, real- and complex- analysis, or group theory? What I remember most strongly about that course was Bayes' theorem. That's such an important concept yet it wasn't taught in my math courses until taking a required class mostly for stats and math majors. And yet Bayes' theorem doesn't require analysis or linear algebra, and neither does it entail eye-balling lines SPSS memorization. At https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/Mathematics-and-statistics/Achievement-objectives#collapsible8 https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/M... you see some of the goals for teaching statistics to secondary school students in New Zealand: > Carry out investigations of phenomena, using the statistical enquiry cycle: > determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem using methods such as resampling or randomisation to assess the strength of evidence. > critiquing causal-relationship claims / interpreting margins of error. > calculating probabilities of independent, combined, and conditional events > applying distributions such as the Poisson, binomial, and normal. Clearly it is possible to have meaningful statistics education without calculus and linear algebra as prerequisites, so I don't understand your objections at all.

eesmith 5y ago

This is certainly part of the debate over at least the last 10 years that I've heard of, whether "calculus is the capstone” - a view that ”originates with a committee of 10 men in 1892” - or if other forms of math should be considered as important alternatives. In general, many believe that statistics education suffers because of the focus on calculus. See for example this StackOverflow post from 7 years ago, https://matheducators.stackexchange.com/questions/2099/emphasizing-statistics-instead-of-calculus https://matheducators.stackexchange.com/questions/2099/empha... . It's not hard to find other links, like https://www.edweek.org/teaching-learning/calculus-is-the-peak-of-high-school-math-maybe-its-time-to-change-that/2018/05 https://www.edweek.org/teaching-learning/calculus-is-the-pea... , with more information on the problems with emphasizing calculus: > For a picture of how severe that inequity can get, one only has to look at calculus. ... Until about 1980, calculus was seen as a higher education course, primarily for those interested in mathematics, physics, or other hard sciences, and only about 30,000 high school students took the course. That began to change when school reformers glommed onto calculus as an early example of a rigorous, college-preparatory course, ... despite the rapid growth of calculus as a gold standard, university calculus experts argue it is a much weaker sign that a student is actually prepared for postsecondary math in the science fields than it appears Indeed, the L.A. Times piece describe it as "in high school, when they can choose advanced subjects, including calculus, statistics and other forms of data science" (emphasis mine). Note too how it classified statistics as part of data science. IMO, data science without statistics is even worse than data science without calculus.

threatofrain 5y ago
But analysis and linear algebra are at the bedrock of statistics. Otherwise, how do you even do a line of best fit?
- eesmith 5y ago
  I don't know why you think analysis is the bedrock of statistics. I'll explain my understanding; perhaps that can help you clear up my confusion? "Analysis" has different meanings in mathematics than in general use. When you wrote "or Calculus / Analysis", I assumed you meant "the branch of mathematics dealing with limits and related theories" ( https://en.wikipedia.org/wiki/Mathematical_analysis https://en.wikipedia.org/wiki/Mathematical_analysis ). The real analysis course I took as an upper-level undergrad focused on the fundamentals of calculus. I did a lot of ε/δ proofs that semester. Statistics does not require that sort of analysis, so it cannot the "analysis" you refer to. "Analysis", less mathematically speaking, means "the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it" (quoting https://en.wikipedia.org/wiki/Analysis https://en.wikipedia.org/wiki/Analysis ). That goes back to at least Euclid ("Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth") (Ibid.) Historically, mathematics education taught Euclid's Elements - geometry - as a way to learn analysis. Lincoln, for example, "kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight"." (quoting https://en.wikipedia.org/wiki/Euclid%27s_Elements https://en.wikipedia.org/wiki/Euclid%27s_Elements ). Clearly, "analysis" doesn't require calculus, and surely can be taught independent of calculus. Indeed, Fermat and Pascal laid the foundation for statistics before Newton was a teenager. "Linear algebra" is also a very broad topic. When I learned to solve a problems like {given x + 2y = 5; 3x - y = 12; solve for x and y} in 9th grade, I was doing linear algebra, though it wasn't specifically told it was linear algebra. When I learned about eigenvalues and eigenvectors in college course actually named "linear algebra", it of course covered things in much more depth. But why does an introduction to statistics course requires knowing about eigenvectors, null spaces, and orthonormal basis? I mean, I vaguely remember my college statistic course, and it didn't require most of the mathematics I learned from linear algebra, real- and complex- analysis, or group theory? What I remember most strongly about that course was Bayes' theorem. That's such an important concept yet it wasn't taught in my math courses until taking a required class mostly for stats and math majors. And yet Bayes' theorem doesn't require analysis or linear algebra, and neither does it entail eye-balling lines SPSS memorization. At https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/Mathematics-and-statistics/Achievement-objectives#collapsible8 https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/M... you see some of the goals for teaching statistics to secondary school students in New Zealand: > Carry out investigations of phenomena, using the statistical enquiry cycle: > determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem using methods such as resampling or randomisation to assess the strength of evidence. > critiquing causal-relationship claims / interpreting margins of error. > calculating probabilities of independent, combined, and conditional events > applying distributions such as the Poisson, binomial, and normal. Clearly it is possible to have meaningful statistics education without calculus and linear algebra as prerequisites, so I don't understand your objections at all.