There has been a growing trend to say that Americans need less algebra education. In The Math Myth, Andrew Hacker argues that algebra and other upper-level high school mathematics like geometry and trigonometry are largely unnecessary for students and can even get in the way of students preparing for the life ahead of them. Hacker is not alone in his assessment. The chancellor of the California community college system, Eloy Ortiz Oakley, is trying to remove algebra as a requirement for non-STEM (science, technology, engineering, and math) majors. This idea has been getting play in outlets from the Huffington Post to the New York Times. I don’t disagree that there are problems in mathematics and the way that it is taught. However, Hacker’s solution of simply getting rid of it (or transforming it into a practical math course) is largely throwing the baby out with the bathwater.

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Mathematics, at its core, is simply abstract reasoning and logic. It is what happens when you strip away all of the details, and focus on the reasoning and logic itself. Hacker suggests that there are other, more practical ways to hone reasoning skills. For instance, Hacker suggests that practicing being on a jury helps build reasoning skills that are even more practical than those of mathematics.

Now, I have no problem with training people how to use logic and reason when on a jury. However, the difference is that reasonable people can disagree over how to weigh evidence on a jury. In mathematics, we can actually show you how reasoning works concretely, definitively, and exactly, and in such a way that we can mostly evaluate your progress in this matter.

Being able to manipulate an algebraic equation is essentially the same as being able to abstractly manipulate thoughts. More.

Why are social justice warriors so focused on depriving the people they are supposedly trying to help of critical thinking skills? In an age when STEM-type skills are more important than ever.

*See also:* Algebra is not racist.

and

Nature: Stuck with a battle it dare not fight, even for the soul of science. Excuse me guys but, as in so many looming strategic disasters, the guns are facing the wrong way.

Mathematics is also one area where naturalism is unequivocally false.

Mathematics is the [study of the] logic of structure and quantity. To rob people of algebra and geometry is to not only rob them of gateway studies for sci-tech in a civilisation pivoting on such, it impoverishes our minds. Math also starts with those patently real and empirically astonishingly effective abstracta, numbers. I was recently amazed to see how a local skeptic reacted to outlining von Neumann’s construction of the naturals, by way of bringing this out and by way of highlighting just how grossly erroneous is the view that only empirical, physically observable entities hold reality. KF

PS: Actually, we should be using multimedia approaches to get first steps of calculus into the standard educational toolkit of the ordinary student. Somewhere, I understood that 5 years of calculus and 4 of physics were standard for Russian high school students. I could be wrong on specifics but they sure made an effort.

KF –

“I was recently amazed to see how a local skeptic reacted to outlining von Neumann’s construction of the naturals”

Can you expand this?

“Actually, we should be using multimedia approaches to get first steps of calculus into the standard educational toolkit of the ordinary student.”

I’m working on this myself. I am working on a high-school Calculus text that (a) doesn’t spend lots of time on irrelevant things, (b) tries to mold a mathematical intuition into students, (c) teaches students how to develop formulas, and (d) deals with some of the interactions between philosophy, theology, and mathematics.

KF @2,

Well stated. Agree.

johnnyb,

Spot-on. Thanks.

johnnyb,

here’s a problem maybe you can help me figure out:

in your insightful article you mentioned that somebody has proposed to “remove algebra as a requirement for non-STEM (science, technology, engineering, and math) majors.”

It seems like that refers to undergraduate university studies, doesn’t it?

If that’s the case, then perhaps many folks out there will agree.

However, the problem I see is quite the contrary. The so called ‘high school’ education should be complete in its coverage.

All high school students, regardless of whether they are STEM-oriented or not, should learn fundamental concepts of math logic (set theory, etc.), calculus (differential & integral), algebra, geometry, physics, chemistry (inorganic & organic).

Perhaps that’s not enough. I’m open to include more prerequisites.

Why?

We need to develop abstract thinking skills, but also respect for absolute standards, open-mindedness, thinking out of wrongly preconceived paradigms, attention to details (look at the fish!), discerning skills, testing everything and holding only what is good.

What for?

Well, for example, just look out there to what’s happening in leading-edge science in academic circles these days: multidisciplinary research teamwork. We see control systems engineers, physicists, computer scientists, biologists, biochemists, mathematicians, collaborating in biology-related research. They must share a basic communication protocol. Otherwise they can’t understand each other.

At the same time, basic teaching of language skills, world geography and history, classic world literature, classic music, classic world arts, basic philosophical concepts, should be required of all students, including those pursuing STEM-oriented careers too.

High school education should be complete. Many students are undecided when they start their undergrad studies.

Actually, early childhood learning of basic concepts and principles should help their mental development.

Once the students get well immersed into their undergrad education, they should have a better idea of what career to pursue. But maybe some will require longer time to decide.

#6 PS:

Young minds are generally more capable of learning.

Why then rushing them through incomplete exposure to fundamental principles and concepts?

Specially now that average lifespan seems extending its duration.

Actually, learning never ends. The more we know, more is there for us to learn. We see this specially in biology research papers these days.

As outstanding questions get answered, new interrogations are raised.

God has graciously blessed me with an 8-month grandson and a 3-month granddaughter. I’m fascinated observing their development. It’s far beyond my understanding. Simply unbelievable, but I’m seeing it right before me.

However, I’m concerned about the world they will grow up in and pray that God will guide and protect them.

All I can do now is help with their learning.

BTW, those completely dependent creatures have the power to wrap me all around their little fingers to the point of making me their submissive servant. Now, let the ‘string’ AI experts out there explain that. Can they build a robot that will have a similar experience? 🙂

Dionisio, could you please clarify for me what High School students in E Europe and esp Russia are/ were required to do with Math including Calculus and Physics? I know there was a major, decades long initiative to support sound educational approaches at secondary and tertiary levels in the USSR. I have several of the old MIR textbooks to prove that, including Yavorsky and Pinski Physics at HS level. This is comparable to an A Level book and is one of my treasures. KF

JB, could you elaborate on your textbook initiative? I am very interested. KF

JB

Let me try to sum up on the exchange in a small business office.

As you know, just about anywhere in the world now you have aggressive skeptics who trot out Dawkins-style New Atheist, long past stale date talking points and/or Dan Brown new age-ish dismissive myths as a proof of how clever they are and how dumb religious people — believing in invisible, intangible fairies/ friends — are. They are highly dubious about the history of say the NT, and are utterly unaware of implications of such undue hyperskepticism.

Most Christians in this region have only Sunday School level religious/ theological/ Bible Study/ Issues exposure and are utterly ill-equipped to deal with skeptic’s typical talking points. This BTW is a key point on the need for street level theology training for a critical mass, hence this in context on w/view frameworks, this too on history and this slideshow.

What happened in a nutshell is the skeptic started to dismiss belief in God, asserting that only what is evident to the five senses is real. He was dismissive of NT historicity etc also and evidently equated faith in God to believing in the tooth fairy or an invisible childhood friend.

After answering questions on what faith is (by definition) and the like and seeing resistance to doing a discussion of roots of arguments and faith-points as roots of worldviews, I decided to use a test case on abstracta only accessible through thought that are very powerful in the empirical world. This is a case of a man who made his fortune by working in a technological domain and who has to respect accounting to handle money. So, this is a direct challenge to the sort of hyperskeptical naive empiricism that goes with a somewhat older fashioned form of atheistical or fellow traveller skepticism. (Ironically, from separate encounters, there is a sort of Golden Age Egypt neopagan myth that is eagerly attended to.)

I pointed out the significance of numbers and asked, what is a number. Of course, there was no coherent knowledge.

There was another business man there (the owner of the office) who is Christian. His presence moderated the discussion.

I then outlined that von Neumann, a famous mathematician, had helped us understand numbers based on sets, which roughly are identifiable collections. So, we look at the set that collects nothing, to start:

{} –> 0

{0} –> 1

{0,1} –> 2

ETC, endlessly (thus, infinity)

I then summed up that from this, we can construct the rest of number systems etc.

He tried to dismiss the exercise as nonsensical, rubbish, trashing in effect the fool who could come up with such. I noted that JvN is the man responsible for the architecture of the computers sitting on the desks in front of us and in our smart phones, also that he was one of THE master blasters behind the atom bomb etc. That is, dismissal to the man will not work as JvN is not on trial, our responsiveness to substantial evidence on the reality and power of abstracta is.

A bit more bluster and the other businessman intervened, in effect that the skeptic was trying to dismissively fend off facts and reasoning that challenged his views.

I hope this helps?

KF

KF –

Thanks for sharing that story! If you would like to take a look at the current version of the textbook, please email me at jonathan@bartlettpublishing.com.

OK cf email, but I think you took a security risk. KF

KF,

Of course I am also of the opinion that numbers and other abstract entities are “real” in some way, so I disagree with your skeptic interlocutor.

However, does the von Neumann construction provide any evidence that numbers themselves are real?

I would suspect that one who denies the reality of numbers also would deny the reality of sets (especially the empty set) and the set operations required for the construction.

In effect, the von Neumann construction is simply a different numeral system, equivalent to the usual arabic system we use. One could accept that these concrete entities (marks on paper, e.g.) exist, yet deny that the underlying numbers they allegedly represent do not exist.

KF,

As far as I remember, by the end of the 1960s and beginning of the 1970s the high school system in the so called “socialist block” generally (with exceptions) included differential and integral calculus in the 11th and 12th grades. However, each country had its own variations. I think the DDR segregated students by their skills into different secondary school levels. Perhaps all of them followed similar approaches, but some were more strict in their implementation. It’s been a while and my limited memory has been filled up with biology-related information lately. 🙂

Here’s a description of the current education system here in Poland:

http://eurydice.org.pl/wp-cont.....14_www.pdf

Although the soviet education system emphasized strong math and physics curriculums, I would not take them as role models. They lacked important aspects that affected the general education of their students.

The current education system in their country doesn’t do well either, according to a 5-year old report by DW:

http://www.dw.com/en/russia-an.....a-15906118

KF,

When MIR published the two-volume physics textbook you have, I was past HS, halfway through engineering school. When the accompanying problem book was published I had graduated. But you’re right that such a textbook shows their strong HS physics level.

Some of my former classmates migrated to Israel, Western Europe or USA. As far as I know none came here to Poland. I keep contact with some of them. I’ll try and ask them about their HS math and physics curriculum.

Dionisio, thanks. KF

DS, I’d say the JvN construction is a construction of N, and that the sort of skeptic being discussed cannot otherwise come up with a reasonable conception of numbers and their effectiveness. KF

F/N: Cf Booher here on the ordinals construction of N: http://math.stanford.edu/%7Ejb.....promys.pdf

DS, the way that N’s members are real is as abstracta connected to the logic of structure and quantity. They are objects of thought that manifest constraints and possibilities that apply to the physical world as this is a domain that manifests structure and quantity, being logically ordered by way of distinct identity, coherence and linked issues, including that no distinct entity P can be such that two core characteristics X and Y stand in mutual contradiction. For instance, there can be no square circle for this reason. So, objects of thought directly relate to objects of physical manifestation. KF

KF,

I would probably take a tack similar to your #17 and #19.

I might use the game of Go as an example. Even a skeptic would have to admit they can distinguish legal Go positions from illegal ones, and would be able in principle to count (by pairing up only concrete entities) the legal positions on a 19 x 19 board.

Moreover, every being with the intelligence and resources would be able to carry out this counting process, and will always arrive at the same answer (after translating between notation systems, if necessary). It doesn’t matter whether you asked a human or a “Grey” from Zeta Reticuli—the answer will always be the same.

I suppose I have to take a leap here to conclude that this number is “real” as opposed to a mere useful fiction. But it certainly is very awkward (if not impossible) to characterize this statement about Go positions without referring to abstract numbers.

Go is a great game. And yes we should teach algebra and geometry to all high school students, as well as some probabilility, statistics, data analysis in general; and I think some informal calculus concepts could be taught with benefit to many.

However, I think the way we go about teaching these things could use quite a bit of improvement. One thing I did in my math teaching career was work on a curriculum of upper-level math skills for the non-college bound, and the approach turned out to be beneficial for all students. My geometry modules were adopted by a number of regular geometry teachers, and the technique of having real-world math lab projects to illustrate and unify concepts was successfully extended to pre-calculus and calculus courses.

NOTE – the discussion about the second part of the article is here.