Virginia DOE proposes math change up, stops advanced math in lower grades.

[Benjamin Frank]

I disapprove.  For many different practical fields such as probability and statistics or analysis/modeling, there's a basic understanding you can gain, and then a more advanced understanding that only comes from having a solid foundation in calculus and other intermediate math concepts.  If you're really good at math, you should be focused on building that solid foundation as quickly as possible.  Learning the practical stuff early is kind of pointless because you're not going to use it yet, so you're just going to have to learn it again later.

That said I don't think an extra course on "practical math" in the middle of high school would hurt.  Especially teaching people how to think about probability/statistics, that's a fundamental skill that most Americans are missing.  But it shouldn't come at the expense of advancing in algebra/trig/calculus as fast as possible.

Totally disagree.  Calculus is very useful if you complete at least first level differential calculus and can pass ( ) first level integral calculus.  However, most high school math ends with pre calculus (AKA algebra II) which by itself is entirely abstract and meaningless.  Multiple studies show that algebra II turns many high school students off of mathematics to the point where they fail to graduate and can't attend any post secondary institution as a result.

Also, for those students who do pass Algebra II in high school (which is the majority to be sure) most of them, if they go on to college are not going to take calculus in college.  So, again, teaching algebra II by itself is still a waste of time in high school.  

So, teaching algebra II by itself is a bad thing.

At the end of the day, all the curriculum makers have in favor of teaching algebra II in high school are snobbish arguments about the supposed wonders of teaching students meaningless abstractions.

As to teaching 'practical math', probability and statistics.  It is, I gather, correct, that a full understanding of probability and statistics does require calculus, however, it is completly false to argue that there is no benefit to critical thinking skills from teaching one or two probability and statistics courses that do not require any knowledge of calculus.

So, ultimately there is something of a choice: either teach probability and statistics in high school or teach advanced mathematics to the level that first level differential and integral calculus are taught as well (students in high school physics classes would be very happy with that, I'm sure.)  

I don't necessarily have a preference one way or the other, but as somebody else here mentioned, the quality of the mathematics teaching in high school tends not to be great and calculus teaching tends to require better teaching than teaching probability and statistics.  Contrary to some of the arguments, since it's actually impossible to magically conjure up thousands of better high school math teachers, I argue teach probability and statistics in high school, and leave algebra II for those who take calculus in college.

Of course, if high school students want to take algebra II as an elective, that's wonderful.

[Benjamin Frank]

Improve equity in mathematics learning opportunities
Empower students to be active participants in a quantitative world
Encourage students to see themselves as knowers and doers of mathematics
Identify K-12 mathematics pathways that support future success
Collaborate with multiple stakeholder to advance mathematics education

As I posted above, I largely agree with these goals (I don't agree with doing away with advanced class math electives) but the number of buzzwords/phrases used here is cringeworthy.

1.equity
2.learning opportunities
3.empower
4.active participants
5.quantitative world
6.knowers and doers
7.pathways
8.collaborate
9.stakeholder

Leaving out articles and prepositions, I counted 33 words in that explanation, meaning around 1/3 of the total words used are obnoxious buzzwords/phrases.

Well, I'm off right now to facilitate participation with a stakeholder I need to touch base with.

[Benjamin Frank]

How does it benefit anyone to withhold advanced mathematics from students capable of learning it successfully and forcing them to share in the same lessons as children who struggle at math? Terrible idea.

I was in advanced classes, then they said there wasn’t enough space available in 4th and 5th grade for me; I was part of the unlucky who lost that lottery at my elementary school. I became bored of school, my grades declined, and they never truly recovered until my senior year. I’d hate to see other advanced students face that same fate. Fortunately for me, I was still invested in learning for learning’s sale and chose to educate myself on a variety of subjects that were never addressed in school; however, I do believe I would’ve benefitted more from advanced classroom education - and so would most students.

This is why there need to be charter schools that can set their own curriculum (with minimum requirements of course.)

[Benjamin Frank]

I think the unique challenge with maths curricula in secondary education (in general; I’m not that familiar with the US*, but this applies everywhere) is the wide variety of uses pupils go onto have for maths. For most, it will just be being functionally numerate. But for the more able, they will actually go onto use it in their university studies - and within this group, you further have a spectrum from those who need it for maths itself or physics, to somewhat less mathematical natural sciences like biology and chemistry, to the social sciences. For this reason, I do think it really makes sense to stream maths curricula from the age of 13 or 14.

*Could someone clarify for me what students in their final two years of high school typically learn in the US in maths? In the UK, for those who choose maths (students at this age in generally have a free choice of any three or four subjects - you can also choose further maths, in addition to normal maths (as I did), the only subject for which this is possible, so that maths takes up half your timetable), it’s a range of pure maths (functions, trig, calculus, logarithms, proof, vectors, series, binomial expansion) and statistics (hypothesis testing, conditional probability) and mechanics (constant and variable acceleration, forces on an object, moments, projectiles). Further maths extends these topics and introduces ones such as complex numbers, matrices and differential equations.

Generally the most base standard track would develop all the principles till 7th grade and starts Pre-Algebra in 8th. Algebra in 9. Geometry in 10 . Algebra 2 in 11 and Pre Calc in 12. The Algebra > Geometry> Algebra 2 gap is a pretty big deal IMO and really should not be done. Accelerated paths also exist and for example one would start Pre Algebra in 7th or 8th and then be doing Calc in 11th or 12th.

1.My understanding is that Algebra II and Pre Calculus are the same thing.  Pre Calculus is what Algebra II is called in college for those who want/need to take remedial math courses.

Edit to Add: the previous poster was correct.  Pre Calculus is in grade 12, but it is an extension of Algebra II.

2.Algebra II is the third math course in high school and will guide you through among other things linear equations, inequalities, graphs, matrices, polynomials and radical expressions, quadratic equations, functions, exponential and logarithmic expressions, sequences and series, probability and trigonometry.

Pre calculus in high school ends off with students able to answer the 'fundamental theorem  of calculus.' So, all of the hard work with meaningless abstract math, but none of the payoff.

3.What the previous commenter referred to as 'mechanics' is part of physics classes.

[Benjamin Frank]

This is why I support public and non profit charter schools rather than a 'one size fits all' model (at least for grades 8 and up), but most of my fellow Democrats (if I lived in the United States) don't agree with me.

[Benjamin Frank]

The geometry split of algebra is a tough nut to crack. On the one hand, geometry, if taught properly, is extremely useful for teaching rigorous proof and reasoning skills as opposed to simply rote computation; thus it or an adequate replacement should be part of any reasonable path.

On the other hand, splitting algebra in two like that is very awkward indeed and puts a lot of people in a strange place before heading into Alg II.

I know you mentioned geometry and not trigonometry, but trigonometry is also very useful to show ratios and the like if it's taught using fractions, but in most if not all cases, it's taught with formulas with students using calculators.  An entirely pointless rote exercise.

I appreciate that students will think 'why should I learn this when a calculator/computer can do it for me anyway.  But, to go against what I said about the 'once sized fits all' approach, if it were up to me, calculators would be banned in math class.

I realize that I shouldn't use my own personal experience to apply to everybody, but I 'failed' my grade 12 math exams (60% combined) before the finals, but had to still take the final.  While studying for the final, I noticed that trigonometry (at least in high school) was nothing but ratios, so I ditched the calculator and did all the math using fractions with pen and paper (or in my head) and I got an solid A on the final.

[Benjamin Frank]

Classes are split this way for the benefit of all of the students and it has nothing to do with leaving anyone behind or privileging certain students. In fact, placing struggling students in classes that require them to learn faster than they're capable of will probably hurt them even more than it hurts the advanced students to be placed in the more basic classes. It isn't even a question of intelligence so much as aptitude with a certain subject since I imagine it's not too uncommon for a student to be in a standard math class while being in an advanced English class or vice versa.

Isn't that what grades are for? Assuming the difficulty is tailored towards the median student you should have the gifted students getting A or A+ results while less gifted students will get just bare passes with a C or C-; so no need to split students based on ability.

Also, for the people who are completely inept at math, they should be able to drop math alltogether after like year 10 or so.

Plus, for what is worth, in my anecdotal experience, separating the good students from the bad students just creates more inequality. Sure, perhaps the good students might end up performing better, but the class with all the bad students will end up underperforming and those students learning even less than with a "mixed" class

How does it benefit anyone to withhold advanced mathematics from students capable of learning it successfully and forcing them to share in the same lessons as children who struggle at math? Terrible idea.

I was in advanced classes, then they said there wasn’t enough space available in 4th and 5th grade for me; I was part of the unlucky who lost that lottery at my elementary school. I became bored of school, my grades declined, and they never truly recovered until my senior year. I’d hate to see other advanced students face that same fate. Fortunately for me, I was still invested in learning for learning’s sale and chose to educate myself on a variety of subjects that were never addressed in school; however, I do believe I would’ve benefitted more from advanced classroom education - and so would most students.

    Advanced tracks for gifted students are critical, and we hurt them a lot by denying them those opportunities. Otherwise the concept of tracking has merit, but is at odds with the nature of university education which focuses in many departments on training students for academia and expecting them to manage the transition to industry if they choose to go that route.

Thing is, separating kids into "advanced classes" and "¿Standard? classes" has a very negative effect for those who get placed in the lower performance groups, both psychologically and in terms of those less gifted students learning even less than they would have if students from all performance levels were together in the same classroom.

In a way that most left of centre people will understand separating kids in such a way might make the gifted kids even more gifted, but it makes the less gifted kids perform even worse; raising inequalities. You can add to that for #woke points if you want that the people in less gifted groups will come from less advantaged socioeconomic groups and what not.


For people more interested on the broad concept of having students from all performances be together in the same classroom, that is what is called "Comprehensive schooling". I am not going to claim it is perfect (indeed for gifted students, which are probably overrepresented in Atlas, it is actually a downgrade). But it is a model that in theory does reduce inequalities and i am surprised at so many people left of centre being against it (indeed, repealing or watering down those models is a routine right wing education reform here!).

https://en.wikipedia.org/wiki/Comprehensive_school

1.Part of the issue with mathematics I think comes from the snobbery of those in charge of the curriculum, that 'academic math' (Algebra II/Precalculus) is 'better' than applied or practical math (probability and statistics.)  If they believed and explained to teachers and students that they are different (at the high school level) but that neither is 'better' I think that would go a long way to ending any stigma at that level.

I think practical math is very important for the development of critical thinking skills and I strongly disagree with the notion that outside of the handful of people who have a mental condition that makes it impossible for them to do math, that there are any people who can not do basic mathematics.

How many people accept it if somebody says "I no can do no sentences.'

In a broader context, separating kids need not be a problem.  Just as the 'snobby educational elites' look down upon practical math, so do they look down upon other practical skills, so cooking or other trades in schools are looked down on in schools compared to algebra or history.  It is the rare student who is better at every subject, so the students who do not excel in the primarily academic subjects might likely do better in cooking or 'shop' (if that's still available, and it should be.)  Learning we are all individuals with our own unique abilities that are neither better or worse than other people's abilities is an important life lesson but that what's important is applying your abilities.

Edit to add: I admit that 'shop' is looked down on by more than just the education elites (or at least it was in my day) but I've never understood why.  if 'shop' isn't applied mathematics (and physics) I don't know what is.  Of course this is the same thing with skills training and apprenticeships being looked down on compared to academia.  I've never understood that.  Carpenters, plumbers, ironworkers... are highly skilled people requiring a practical understanding of many academic subjects.


2.In the old British T.V show Yes, Minister though this was from the sequel Yes, Prime Minister, Prime Minister Hacker is arguing in favor of reinstating the draft with basic training and argues "we can give them a comprehensive education...to make up for their comprehensive education."


Those who are unable to do math at all have a condition called dyscalculia.

https://www.mentalfloss.com/article/62436/11-facts-about-math-disorder-dyscalculia

I looked it up after writing the above, and a quote from a professor there argued the same as I just did:

A big part of the general population's unfamiliarity with dyscalculia has to do with our culture’s general discomfort with numbers, and our ingrained belief that math—compared to reading—is just supposed to be hard. Dr. Gavin Price, an assistant professor at Vanderbilt University who has researched dyscalculia in several countries, says, "When I teach classes, I’ll ask at the beginning, 'How many people think they’re not good at math, they’re bad at math?' And half of them put their hands up. Then I ask, 'Are any of you bad at reading?' And nobody puts their hand up."

[Benjamin Frank]

I mean tack, the counterpoint is that you are essentially taking certain levels of mathematics (as well as access to rigorous math education overall) and just declaring them to be forbidden for no reason. Forbidding people from learning stuff like that is quite frankly ridiculous (and I especially take offense at it in my own field).

Admittedly, what I'm not so sure about is the whole idea of 'testing' or 'placement' into laning - arguably, in an ideal world it would be entirely up to choice as to what track or level of rigor you wanted. Sure, there would be choices that are inadvisable - but again, I don't think anyone should be deprived the choice of taking a more rigorous math course.

Also, a note on math specifically - building a very robust knowledge of a set of mathematical concepts is essential to being able to learn future concepts on top of it. For instance, it is entirely possible to muddle and/or bash your way through a standard Algebra (I or II) course. But doing this will leave you lost if you try to learn anything that builds on top of it. Thus, for anyone who seeks to do anything math-related in higher education etc., taking courses that ensure this level of knowledge (and cannot just be muddled through for a grade) are pretty much essential unless they are to waste substantial amounts of time later "relearning" the same thing with more rigor. And this is what - at least in principle - the advanced or honors lane provides which a unified lane could not.

It is not really "forbidding" things though? A model like the one I studied with definitely doesn't forbid kids from taking "pure math"; which included stuff like basic integrals, derivatives, geometry, matrices and a couple other stuff in year 12 (I can remember less well the contents of other years and I'd have to look them up).

For a student that is so brilliant and so excellent at math that they are so good, you can argue they are being held back by not being presented with even harder stuff and they are! (I guess this is what you mean by "forbidding") But it is not for "no reason". It is best for the whole of society is precisely that model, where  the most brilliant students get held back so that the less brilliant ones don't fall even further behind than they should.

This of course gets into a whole philosophical argument where whether individual liberty or the greater good of society should matter most; and an argument where there is no right or wrong answer. I am not saying the "comprehensive education" model is superior; but is is a model that does reduce inequalities. (which is indeed why I was so surprised at so many left of centre people in Atlas being against it. Just watch the UK example, with the reform being implemented by the Labour governments of Wilson and Blair; and being watered down by a certain Margaret Thatcher, first as secretary of education and then as Prime Minister)

A student that is bad at math and would be placed in those "standard" classes; let's face it, is unlikely to later pursue a mathematics heavy path in college (like say, Engineering or indeed a mathematics degree). That student would be better served from taking an applied maths class centered around statistics and what not; or possibly dropping math alltogether and in its place pursuing other classes like world history or art, instead of being forced to learn math against their will all the way until they graduate HS. Math beyond a certain level which gets achieved approximately in year 9 or 10, is not really for everyone.

This basically means your 3rd paragraph, while true, isn't really all that relevant? Someone who wants to be say, a lawyer, or a historian, or a translator, or something like that; will not need many mathematics other than super basic stuff in college (or in their professional life for that matter)

I do agree with your 2nd paragraph that if laning happens, people should not be stopped based on academic results; and that any reports they get should be exclusively non-binding. I just don't think laning based purely on academic performance is a good thing

For what is worth you could argue a model like I grew up with is de facto laning and I would even agree with it; humanities students (the ones that don't get math; or physics and chemistry and other sciences for that reason); are generally stereotyped as "dumber". But that is a stereotype and the humanities path isn't intended to be easier in any way. I would have been just as miserable learning about "History of Art" (or even "Biology", which I dropped after year 9) as a humanities student would have learning calculus or physics

I was thinking this is also a philosophical argument on liberalism.  Younger liberals (progressive?) like you are clearly arguing here on the basis of 'self esteem.'  Older liberals like me argue that real self esteem is derived from real accomplishment and that accomplishment need not be academic but can be from sport, art or a trade skill.  And that, we need to emphasize that we are all unique individuals with individual skills that are neither better or worse than anybody else's (except for the 1% or so who are truly superior at everything) but that what matters is making use of your abilities, but also that we all (with noted exceptions) have enough minimum skills to be able to do most everything at a basic level (sports like dancing of course can require physical abilities that some people don't have.)  In my case, I have no sense of timing, so, as much as I'd like to, I can't even play a musical instrument at a basic level.

[Benjamin Frank]

I think the unique challenge with maths curricula in secondary education (in general; I’m not that familiar with the US*, but this applies everywhere) is the wide variety of uses pupils go onto have for maths. For most, it will just be being functionally numerate. But for the more able, they will actually go onto use it in their university studies - and within this group, you further have a spectrum from those who need it for maths itself or physics, to somewhat less mathematical natural sciences like biology and chemistry, to the social sciences. For this reason, I do think it really makes sense to stream maths curricula from the age of 13 or 14.

*Could someone clarify for me what students in their final two years of high school typically learn in the US in maths? In the UK, for those who choose maths (students at this age in generally have a free choice of any three or four subjects - you can also choose further maths, in addition to normal maths (as I did), the only subject for which this is possible, so that maths takes up half your timetable), it’s a range of pure maths (functions, trig, calculus, logarithms, proof, vectors, series, binomial expansion) and statistics (hypothesis testing, conditional probability) and mechanics (constant and variable acceleration, forces on an object, moments, projectiles). Further maths extends these topics and introduces ones such as complex numbers, matrices and differential equations.

Generally the most base standard track would develop all the principles till 7th grade and starts Pre-Algebra in 8th. Algebra in 9. Geometry in 10 . Algebra 2 in 11 and Pre Calc in 12. The Algebra > Geometry> Algebra 2 gap is a pretty big deal IMO and really should not be done. Accelerated paths also exist and for example one would start Pre Algebra in 7th or 8th and then be doing Calc in 11th or 12th.

What (briefly) is the contents of these courses? So do you spend the whole of 10th grade doing geometry and not touching other (more relevant, IMO, or maybe that’s just because I never particularly liked geometry compared to algebra ) areas of maths?

YMMV but what I remember it was:

Algebra I
Standard inequalities and linear equations, things like that.  I remember algebra being really difficult for me to grasp the first time I encountered it because I was like "I know that if x + 2 = 3 then x = 1, why do I need to do this (x+2)-2 = (3-2) crap?"

Geometry
This was all the basic details of circles, triangles, 3D objects, and other shapes.  Most of it was memorizing equations for area/perimeter and how to calculate things based on angles.

Algebra II
Exponents, logarithms, wave functions, polynomials (including the quadratic formula), various algorithms to solve equations, different types of numbers like rational/irrational, complex, imaginary numbers (kill me), etc.

Pre-calc/Trigonometry
In my experience these were the same thing.  There were a few new concepts introduced such as binomial equations, but most of the class involved doing algebra with the previously-learned concepts.  There were a bunch of rules to memorize and then apply to simplify.  Most tests were just "Simplify" at the top, and then a whole bunch of complicated-looking equations.

Calculus I
Derivatives and limits.  IMO there is a pretty large jump in difficulty from "pre-calc" to calc.

Calculus II
Integrals

Calculus III
Multiariable functions, partial derivatives, integrals of 3D objects

My high school offered all the way up to Calc III, but I only made it to Calc II and ended up taking III in college.

IMO everything up to Algebra II should be mandatory for all American high school graduates as Algebra II is basically a survey course of different advanced mathematical topics and important to understanding anything math-related you read for the rest of your life.  Even if you never use "imaginary numbers" ever again, you still need to understand logarithms and exponents for day-to-day life.  But if you want to be college-ready you absolutely have to take through Calculus II.  Integration is a fundamental concept to all the hard sciences and engineering.  You can take Calc I and Calc II in college, but then if you want to do hard science or engineering, you'll be a year behind everyone else and taking the weed-out courses for your major your sophomore year.  That's how you end up with kids changing majors halfway through college.

Calculus III was only really useful in graduate-level courses.  Unless you plan on reading a lot of research papers for math/engineering/hard science, I wouldn't take it.

Linear algebra, though, is extremely useful and applicable to all science/engineering disciplines as well as a lot of "soft science" disciplines, and IMO should be offered at high school level and treated with the same respect that probability and statistics courses get.  You can take linear algebra after Calculus I.  Linear algebra is basically anything involving vectors and matrices, basically reasoning about the relationships between collections of numbers.

Calculus I.  Derivatives, yes, but also differential!  (How could you mention the integral without mentioning the differential?)  Differential and integral calculus, the reverse processes.  

From my experience, differential calculus was no more difficult than Algebra II leading up to the Fundamental Theorem of Calculus, but integral calculus was far more difficult.  However, once understood, differential and integral calculus make calculations in physics much, much easier (and also highlight relationships between concepts in physics.)

There is a reason why the circumference of a circle is taught as 2*Pi*r and not as Pi*D (at least it was in my day) obviously because it's easier to show the relationship with the area calculated as Pi*r^2.

I agree that Algebra I should be mandatory (which is where imaginary numbers are taught) but Algebra II should be an elective for those planning to take STEM subjects in college.  I argue probability and statistics should be mandatory at the high school level as it is far easier to integrate into other classroom subjects.  

Edit to add: looking closer, a whole bunch of the things that you list as part of Algebra II, are part of Algebra I.  I learned quadratic equations, for instance, in grade 8.  Quadratic equations are nothing more than an application of basic arithmetic and mathematical concepts.  It shouldn't be hard to teach people to grasp a fuller meaning of 2 + 3 = 3 + 2 (The Commutative Property of Addition.)

As I argued previously, there is no point for a student to leave off high school having only taken Algebra II.  If Algebra II is going to be mandatory then so should at least differential calculus.  If the 'meaningless' abstractions of Algebra II are going to be taught, then the students should at least leave with some understanding of what all those abstractions mean.