Teaching statistics before calculus?

[bhobba]

And in the physical sciences and engineering, they really need calc-based stats, which means they need calc first.

Well said.

I did my degree part time while I did a pretty dead end job in the public service. It didn't really worry me but some people I knew thought my talents were being wasted and every now and then would contact people in the PS hierarchy and I would pop up for them to look at my qualifications.

I well remember one guy looking at my subjects and he saw Mathematical Statistics 1A, 1B, 2A, 2B and said what's that? Do you do things like the t-test? Of course. Then what is this mathematical thing? Its code-word for uses calculus. His eyes glazed over and said I need to speak to someone. He went away then came back and said - not sure if that's of any value here. I internally shook my head but didn't say anything. As soon as I finished my degree left for elsewhere. Sad really.

Thanks
Bill

 

[Astronuc]

Here is a short (and old) TED talk where a mathematics professor suggests we teach stats and probability in depth before teaching Calculus because it's math that is more relevant to a wider range of people. Have we got our math curriculum wrong? Thoughts?

There are some applications of probability and statistics that can be addressed with algebra, e.g., counting discrete events of a finite population such as the number of coin tosses that are H or T, or roll of a die, or dice, and the probabilities of the number of times a number appears, or a combination of numbers appears.

https://www.mathsisfun.com/data/
https://www.mathsisfun.com/data/probability.html

At some point, calculus is necessary in order to deal with probability theory and continuous distribution, probability density functions and so on.

I remember learning a little of chance or probability in 4th grade science (genetics) and later in 10th grade biology, which was a year before I took a formal calculus class. I had actually bought a book on calculus in 10th grade and was studying it on my own, with my father's encouragement. It was clear to me that I needed calculus to study physics.

 

[Dale]

Several posts about the anti-vaccination crowd have been deleted. Please keep the discussion polite and do not reopen the anti-vax topic in this thread.

 

[Loowee]

Here in the Philippines, the K-12 curriculum was launched nationally only last month (June 2016). In the current curriculum, kindergarten students are introduced to statistics and probability concepts. The topics build up slowly from grade to grade culminating in combinatorics in grade 10. Calculus is taught beginning grade 11 culminating in integral calculus in grade 12. That looks good on paper. I sure hope the private schools that dared to open senior high (grades 11 and 12) are up to the challenge. Public schools have to offer senior high whether they like it or not. Before the advent of K-12, high school was only up to fourth year (grade 10). Grades 7 to 10 are now called junior high school.

We Filipinos usually think that (our) public schools are not good, with a few exceptions like the University of the Philippines. Private schools have a better reputation. I came from a rotten private school (with no reputation at all).

When I was in my beloved but rotten high school, we studied algebra in the second (grade 8) and fourth (grade 10) years; statistics, business math and taxation in the second year; geometry in the third year; and trigonometry in the fourth year. My alma mater was a Chinese school until 2000. In 2001, the Chinese curriculum was removed and the school became exclusively English. In the Chinese curriculum, calculus was taught along with trigonometry in the fourth year. I never learned calculus in high school because I did not take the Chinese classes.

Two of my nieces migrated to Canada when they were 5 and 6. My sister, their mother, was surprised that Canadian children could not read well at that age. I was surprised as well. I expected more from a developed country. Socialization is more important than learning to read at a young age in a developed country to produce well-rounded people.

Educated Filipinos (allow me to emphasize “educated”) learn to read and do arithmetic at an early age. That is why I was against the K-12 curriculum which seems to be the standard worldwide.

Some schools (in any part of the world) teach advanced subjects which the majority of schools do not include in their curriculum, e.g. graduate-level subject in an undergraduate degree. Are these early “inductees” better than the regulars?

I searched the Net for a federal K-12 standard but so far found only the standard course of study of mathematics for North Carolina public schools. North Carolina already has statistics and probability by the sixth grade. As proven by earlier poster/s to this thread, statistics is taught in a number of American high schools before calculus.

Look at us Filipinos. Our beloved country, the Philippines, apparently has had a relatively high standard of language and math education long before K-12 rocked our education budget, yet our country is poor. The Philippines remains one of the most corrupt countries in the world. Make no mistake: I may pass on malignant remarks about my country, but I will always love her more than any other in the world.

Is it really that important which math is taught first in your country? I expect there is corruption anywhere even among your ranks. However, I do love the creed you live by, “that all men” – and women – “are created equal…”

It is good that schools in countries like ours are given the freedom to choose which subject to teach first.

If I were to choose though, I would choose statistics first, not because local curriculum made me take it first anyhow, but because statistics is more practical for any citizen, with or without a college degree – this essentially quotes “it's math that is more relevant to a wider range of people.”

I do not mean to lecture. Just my two cents. Thank you for the indulgence.

 

[Dr. Courtney]

I do not mean to lecture. Just my two cents. Thank you for the indulgence.

I appreciate the level of detail and the personal insights. And I never mind a bit of preaching from the heart.

My view is somewhat skewed from having served in an administrative position at the Air Force Academy that had a close-up view of our nation's need to improve STEM education and especially STEM majors.

Lack of stats is not the bottleneck in producing the needed numbers of STEM majors. Lack of Calculus (and good enough pre-calc to succeed in Calc and Physics) is the bottleneck. Many more aspiring majors and careers in science, engineering, and medicine are stopped cold by the inability to succeed in Calculus and Calc-based Physics than weaknesses in stats.

 

[Diaz Lilahk]

Lack of stats is not the bottleneck in producing the needed numbers of STEM majors. Lack of Calculus (and good enough pre-calc to succeed in Calc and Physics) is the bottleneck. Many more aspiring majors and careers in science, engineering, and medicine are stopped cold by the inability to succeed in Calculus and Calc-based Physics than weaknesses in stats.

Yet again, I think we should be a little careful with this. In the high school where I teach there is a drive on the part of students to get to BC Calculus as quickly as possible. The ultimate goal is to complete it by the end of junior year so that they can have it on their transcripts and since no one gets below a B, well then they have among "the most rigorous courses" on their transcripts with a pretty decent grade. What I have observed as a teacher of algebra and calculus based physics of these students is that their calculus knowledge is extremely superficial and their knowledge of algebra, geometry and precalculus is the same. The problem solving skills are practically non-existent. What is worse is that they see themselves as being successful in mathematics. Very few of them are actually prepared for a rigorous STEM major unless they are willing to acknowledge and fill in a lot of gaps in their knowledge. I would argue that they are far better off getting a rigorous pre-calculus education involving proofs of elementary algebra, geometry and trigonometry accompanied by sophisticated problem solving, ala https://www.amazon.com/dp/0395524326/?tag=pfamazon01-20 than they are having a standard calculus course.

 

[Dr. Courtney]

Yet again, I think we should be a little careful with this. In the high school where I teach there is a drive on the part of students to get to BC Calculus as quickly as possible. The ultimate goal is to complete it by the end of junior year so that they can have it on their transcripts and since no one gets below a B, well then they have among "the most rigorous courses" on their transcripts with a pretty decent grade. What I have observed as a teacher of algebra and calculus based physics of these students is that their calculus knowledge is extremely superficial and their knowledge of algebra, geometry and precalculus is the same.

Yeah, I agree. When I was on the USAFA faculty, we saw lots of students with As and Bs in AP Calculus who could not even pass the pre-calc exam and get placed in Calculus.

Grade inflation is horrible, and every high school teacher who awards As and Bs in AP Calc who shouldn't even be in the course should be fired for fraud.

I'm talking about students needing real mastery, not a grade in a course.

The problem solving skills are practically non-existent. What is worse is that they see themselves as being successful in mathematics.

Yeah, I've had a lot of students get to my physics courses who could not even solve algebra based kinematic problems (projectile motion, etc.) yet they thought they were good at math because of their high school grades. But the fraud of the teachers who passed them does not remove the real need for competence in Algebra, Trig, and Calculus. I stopped confusing real competence with having passed courses a decade ago.

Very few of them are actually prepared for a rigorous STEM major unless they are willing to acknowledge and fill in a lot of gaps in their knowledge. I would argue that they are far better off getting a rigorous pre-calculus education involving proofs of elementary algebra, geometry and trigonometry accompanied by sophisticated problem solving, ala https://www.amazon.com/dp/0395524326/?tag=pfamazon01-20 than they are having a standard calculus course.

There is no need for Calc in high school. I believe a 30-35 on the ACT in math much more than a grade in any high school course, including AP Calc. But I've known high schoolers to score in this range on the ACT in math with nothing more than an honest and rigorous Algebra 2 course.

 

[Diaz Lilahk]

Grade inflation is horrible, and every high school teacher who awards As and Bs in AP Calc who shouldn't even be in the course should be fired for fraud.

Who will do the firing? The very people who encourage and endorse the behavior. I think you are largely blaming the wrong people. I know personally that the only grade which I hold a line on is an A+ and to a lesser extent an A, below is my grade distribution for my AP Physics courses this year - pay close attention to the number of drops. Every other letter grade should be a full grade to two grades lower, and everyone below a B should have failed the course.

Grade Total
A+ 3
A 9
A- 10
B+ 9
B 8
B- 9
C+ 3
C 1
C- 0
D+ 0
D 0
D- 0
F 0
Drops 30

Now you have to realize that my grade distribution was not politically acceptable particularly the number of drops and I was told point blank to decrease the rigor of my course and that my grades were too low. Had these students actually earned the grades they deserved, there is no doubt in my mind that I would have been fired this year, and I still may be fired next year. Now you can argue that I should just walk out, but I can tell you that it is the same if not worse elsewhere. In fact, in all the places this is the school where I had the most academic freedom. Furthermore, I am without question considered the toughest teacher in the department when it comes to content and grades and have had the reputation of being an extremely tough teacher in every place that I worked. The problem is at a higher level and I really think a lot of this could be solved if admission offices required submission of AP tests/scores. Another thing that needs to happen and if teachers were forced to submit their final grades to the College Board to audit and run a correlation between AP scores and student grades, schools AP programs would then be put on probation if there is a large disparity.

Anyhow I want to tie this back to the original thread, which is teaching statistics before calculus. I think it is really important for someone like the person who gave the Ted talk, Arthur Benjamin, to spend a few years as a high school math teacher so that he can get an idea of whether this type of proposal would have any impact at all.

 

[Dr. Courtney]

Who will do the firing? The very people who encourage and endorse the behavior. I think you are largely blaming the wrong people. I know personally that the only grade which I hold a line on is an A+ and to a lesser extent an A, below is my grade distribution for my AP Physics courses this year - pay close attention to the number of drops. Every other letter grade should be a full grade to two grades lower, and everyone below a B should have failed the course.

Grade Total
A+ 3
A 9
A- 10
B+ 9
B 8
B- 9
C+ 3
C 1
C- 0
D+ 0
D 0
D- 0
F 0
Drops 30

Now you have to realize that my grade distribution was not politically acceptable particularly the number of drops and I was told point blank to decrease the rigor of my course and that my grades were too low. Had these students actually earned the grades they deserved, there is no doubt in my mind that I would have been fired this year, and I still may be fired next year. Now you can argue that I should just walk out, but I can tell you that it is the same if not worse elsewhere. In fact, in all the places this is the school where I had the most academic freedom. Furthermore, I am without question considered the toughest teacher in the department when it comes to content and grades and have had the reputation of being an extremely tough teacher in every place that I worked. The problem is at a higher level and I really think a lot of this could be solved if admission offices required submission of AP tests/scores. Another thing that needs to happen and if teachers were forced to submit their final grades to the College Board to audit and run a correlation between AP scores and student grades, schools AP programs would then be put on probation if there is a large disparity.

Anyhow I want to tie this back to the original thread, which is teaching statistics before calculus. I think it is really important for someone like the person who gave the Ted talk, Arthur Benjamin, to spend a few years as a high school math teacher so that he can get an idea of whether this type of proposal would have any impact at all.

I feel your pain. I've suffered a couple non-renewals of my contract because I would not bow to pressure to pass students who did not deserve to pass and otherwise refused to gift grades. I've known other physics and math teachers who experienced the same thing. No doubt the admins who pressure faculty bear the greater guilt, but my conscience has never allowed me to trade unearned grades for a paycheck.

The phenomena is widespread, but it is not ubiquitous. I can say with great certainty that there was no gifting of unearned grades (or pressure to do so) when I taught at the United States Air Force Academy, and my wife has the same experience and testimony when she served on the Physics faculty of the United States Military Academy (West Point). A very dear longtime friend bounced around quite a bit due to his refusal to award unearned grades, but now he is happily teaching (Math and Physics) at a special school for math and science and reports that there is no pressure to gift grades at his current institution.

Who will do the firing?

A few years ago, my wife and I realized that as parents, we had to do it, and the only way to ensure a quality education in science and math for our own children was to pull them from the public schools and home school them.

It has taken some effort (and is still dicey), but we've also had some success finding good teachers (for our teens) in college dual enrollment programs who maintain a high level of academic rigor and do not seem to be gifting grades.

But if grades are routinely gifted, the discussion of Calculus vs. Stats is arranging the deck chairs on a sinking ship.

 

[houlahound]

I perceive a real pressure at the undergraduate level to get and keep butts on seats even at the expense of what is best for the student long term, the economy, the course quality...etc.

Faculties want to exist, to exist they need butts on seats.

There are accusations around that some universities let the standards slide downhill to capture full fee paying in advance foreign students, it is seen as a special market that raises revenue to subsidize the fees of domestic students and creates cash for infrastructure and research.

Of course I doubt any uni admins would agree.

 

[Dr. Courtney]

I perceive a real pressure at the undergraduate level to get and keep butts on seats even at the expense of what is best for the student long term, the economy, the course quality...etc.

We address this issue in some detail in this paper:
https://arxiv.org/ftp/physics/papers/0612/0612117.pdf

Abstract: As education systems move toward business models of operation, there is a strong tendency to misidentify the student as the customer. Misidentifying the student as the customer leads to interpretation of the course credit or degree as the product. The true product is the additional knowledge, skill, and ability that course credit and degree should represent. Consequences are potentially disastrous, because the notion that “the customer is always right” can lead to the perceived product (course credit or degree) meeting the desires of the misidentified “customer” (student) rather than the real product (value added to student) meeting the standards of the properly identified customers (future employers and taxpayers).

Faculties want to exist, to exist they need butts on seats.

Sure, but they do not need the absolute maximum numbers of butts in the seats.

My experience is that holding the line on academic rigor is a game of chicken. Once students realize they really need to work hard to reach their grade goals, they will work hard and earn the grades. But they will avoid working hard until they are really convinced they need to. Sometimes, a faculty member needs to have high rates of Ds, Fs, and Ws for a semester or two until word gets around and the students start working.

There are accusations around that some universities let the standards slide downhill to capture full fee paying in advance foreign students, it is seen as a special market that raises revenue to subsidize the fees of domestic students and creates cash for infrastructure and research.

This is part of it at some schools. But at other schools, it's a simple numbers game: the state money to the school depends on the number of students in the seats. If there are other nearby schools (driving distance) that will gift grades, a school that won't gift grades will lose enrollment and dollars to the ones that will.

 

[Dale]

Can we bring this back around to statistics vs calculus rather than a generic critique of the education system?

 

[houlahound]

Statistics is a huge field, can someone post specific topics in a logical order, would you include the null hypothesis, what year level.

Is mathematics modeling statistics? It forms the basis of almost all gov public policy that affects everyone.

 

[Andy Resnick]

Statistics is a huge field, can someone post specific topics in a logical order, would you include the null hypothesis, what year level.

Is mathematics modeling statistics? It forms the basis of almost all gov public policy that affects everyone.

I think the Common Core curriculum lays it out very well, the 'null hypothesis' is introduced in 6th grade and Baysean methods are discussed in High School:

Kindergarten: Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

1st grade: Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

6th grade: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Summarize and describe distributions.

7th grade: Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models.

8th grade:Investigate patterns of association in bivariate data.

High School: Summarize, represent, and interpret data on a single count or measurement variable. Summarize, represent, and interpret data on two categorical and quantitative variables Interpret linear models. Understand and evaluate random processes underlying statistical experiments. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Understand independence and conditional probability and use them to interpret data. Use the rules of probability to compute probabilities of compound events. Calculate expected values and use them to solve problems Use probability to evaluate outcomes of decisions.

 

[Dr. Courtney]

I think the Common Core curriculum lays it out very well, the 'null hypothesis' is introduced in 6th grade and Baysean methods are discussed in High School:

Kindergarten: Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

1st grade: Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

6th grade: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Summarize and describe distributions.

7th grade: Use random sampling to draw inferences about a population. Draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models.

8th grade:Investigate patterns of association in bivariate data.

High School: Summarize, represent, and interpret data on a single count or measurement variable. Summarize, represent, and interpret data on two categorical and quantitative variables Interpret linear models. Understand and evaluate random processes underlying statistical experiments. Make inferences and justify conclusions from sample surveys, experiments, and observational studies. Understand independence and conditional probability and use them to interpret data. Use the rules of probability to compute probabilities of compound events. Calculate expected values and use them to solve problems Use probability to evaluate outcomes of decisions.

What percentage of students who will pass the assessments for these learning objectives do you reckon will really be competent with them?

I see a lot of very good standards in science and math. But then I see most students who reach college with high grades in each subject nowhere near the level of competence their grades would indicate.

Coursera and ALEKS both have excellent statistics courses that I've overseen high school students taking.

I would trust success in those to have actually delivered mastery of the learning objectives (outlined above) much more than I would trust the competence of most students coming out of a common core assessment framework.

 

[Andy Resnick]

What percentage of students who will pass the assessments for these learning objectives do you reckon will really be competent with them?
<snip>

There is a plethora of other threads devoted to discussing this. I don't see the need to re-hash what is at best empty speculation and at worst, character assassination.

 

[Dr. Courtney]

There is a plethora of other threads devoted to discussing this. I don't see the need to re-hash what is at best empty speculation and at worst, character assassination.

Empty speculation and character assassination? Who is the target of this alleged character assassination?

Empty speculation? I've taught college level physics labs for many years, as well as both algebra-based and calc-based statistics.

My observations of the level of statistics preparation students have is based on first hand experience and working with thousands of students in intro courses.

My first hand experience gives me a high level of confidence in students who complete a stats course through ALEKS or Coursera, but I have less confidence in other pathways. There are some dual enrollment opportunities for high school students to take intro stats courses at local colleges and universities, but the quality of statistics learning here will come down to the local institution and individual instructor.