This is in response to Common Core vs Non-Core, What’s the Balance?

Some observations

Let me first state that I neither teach elementary school nor high school. When I teach, I teach mathematics mostly at the college / university level. With that said, I’ll give what I think most college professors see in terms of basic competency in mathematics and what it does to the courses we teach. This section is anecdotal and I wish I actually kept proper statistics, so take my observations with a grain or two of salt as I may suffer from some confirmation bias.

1. With amazing regularity, the distribution of students’ basic math competency is bimodal; that is, I either get students who know their arithmetic and algebra or don’t. It is rare that I see students with “average” competency. Of those students who do not have the basic competency, it is, unfortunately, mostly hopeless for them to progress in STEM fields. It is simply not possible to teach, say, a course in probability while simultaneously having to review fractions.
2. As a result of (1) classes like statistics and calculus become really painful / prohibitive for some students. This pushes students out of STEM fields. Ultimately this gives us articles that have to discuss about our culture of “I’m not a math person” .
3. For the late-in-mathematical-development student, universities do offer the basic class sequence and often require that students have placed out of these courses via some passing score on a standardized exam. The price for having to take these courses is high since these courses (a) are not eligible for credit towards an undergraduate degree, (b) still have the same financial cost as any other course, and (c) prolong the student’s graduation time table (certain courses are only offered in Fall and this delays entry into a two part sequence by a year — that sort of thing). Some students do successfully go through this sequence and do end up learning a great deal. But the majority simply muddle through hoping to get a “C” so they can get on with their major.

But how does (1) happen? It happens because those “core” subjects that students spend almost 12 years (!) on don’t amount to actual knowledge. They amount to an amalgamation of loosely connected facts devoid of meaning and context.

How is this possible?!?!

How do we have people spending twelve years in school and by the time they get out this problem is incomprehensible?
$$\frac{2}{3} + \frac{5}{8} = ?$$

Standardized testing is an easy scapegoat here. But it can’t be THE problem. If the pressure of standardized tests is as large as it is claimed to be for parents, students, teachers, administration, etc. then why is it that basic arithmetic is such a colossal failure?

I don’t know for sure.

All I have is a myriad of stories from math-phobic friends, other mathematics professors (who are not math-phobic but have probably become phobic of math-phobic people), quantitative and non-quantitative clients, and motivated and defeated students. All of these stories have a common theme and all of these stories begin with “My math teacher in grade \(N\) sucked.”. Yuck. These stories end with “… and then I was so lost in the next course; there was so much to memorize!” Double yuck. I grant that there may have been a teacher along the way with whom the rapport wasn’t ideal or that the teacher was actually just incompetent (every profession has their cast of incompetent individuals, teaching is no exception). But this can’t be THE problem either.

Synchronized Swimming

If I had to venture a guess, I think a root cause of this cascading failure is forced learning — what I call, “synchronized swimming”. By that I mean the notion that by a certain [short amount of] time students should know a particular topic. I can’t speak for other subjects, but with mathematics, this is very dangerous.

Math has to marinate. And for some students it has to marinate a little longer than “normal”.

I can not begin to count the number of times, I’ve just had an “aha moment”, days, weeks, months after being exposed to some mathematical concept. Our experiences shape us in all types of ways.

I remember several years ago I was explaining to an 8th grader how “mixing” problems work (If you mix a gallon of one liquid with a quart of another liquid and then you pour out one quart of the mixed liquid how much of each liquid type do you have?). I tried numerous examples (physical and virtual), teaching techniques, etc. but it just didn’t make sense to her. Later that evening, she was helping her mother with the dishes (liquids being mixed!), and suddenly, lightning struck and her mind had unlocked the secret to mixing problems. The next time I met with her, she showed me how she solved it. Boom! Just like that. This is math marinating.

Please, for the love of the student, let math marinate for as long as it needs to. It’s not synchronized swimming!!

I can regale you, my dear reader, with many such stories.

Curriculum

The way mathematics curriculum is set up requires that students progress through a sequence of singular courses that all build on each other. This is unlike climbing a ladder, where if one of the rungs above were broken you could potentially reach to the next rung and lift yourself up. History, English, Art, Social Studies, and even some of the sciences can be like this. But, mathematics curriculum is more like scaling a high wall by rope. If the rope is cut from above, you fall and the wall remains impassable.

Sadly, if a student does not “pass” math, the student likely fails a grade (I don’t know if this has changed since I was a kid). Failing a grade outright is a tremendous blow to any student’s self-esteem and social development. From stories I’ve heard, it seems that rather than have a student repeat an entire grade, the student is just passed through to the next grade. And this begins the math death spiral.

What is school?

School is more than just classes. School, from grade school to graduate school, is also a social environment. It’s about making friends and losing them. It’s about dating and falling in “love” for the first time and getting your heart broken. It’s understanding what maturity is. It’s about finding interests and hobbies. School is also about the “non-core”.

What the heck is “non-core”, anyway?

I am not a fan of this type of segregation. By categorizing something as “non-core” one may as well say “non-essential” which is a stone’s throw away from “irrelevant”.

Why is music not “core”? It’s not important? Art isn’t relevant? Mularkey! They’re not core because testing in those fields in a mass standardized way as inarguably as you could, say, mathematics, is not really well-defined. And Mozart help us if that day ever comes.

What’s the balance?

Any educational system that seeks to marginalize the arts in favor of the “core” (or vice-versa) is not an educational system. It’s a sweat shop.

Could an art teacher discuss the four coloring theorem? Maybe. Should he? Eh, if he so desired. Rather than have the art teacher spend class time on this, have the art teacher teach art and give to the students supplemental material for them to read and explore on their own. Call it extra credit if you must, but don’t force this in an attempt to give “a diverse and interconnected educational experience”.

Does a math teacher have to make everything pretty and wonderful with “real world” problems? Eh. Real world problems can be very messy and over simplifying them to make the “math work” goes to undermine mathematics. I work on “real world” problems regularly as a professional mathematician — this is what my clients hire me for. I can tell you that being creative is just as essential as being technically strong. And the real world (because school is the fake world), has some pretty annoying constraints like deadlines, budgets, and “can’t you just make the numbers work?”. No, sadly, I can’t just make the numbers work. Does school need to be like this? Maybe a little, but mostly not.

Schools ought to try to horizontally integrate classes at the faculty level rather than at the curriculum level by having / letting faculty share with each other what they’ve shown their students that could relate to the other classes. This will allow for faculty to know what their students have been exposed to and when to “interconnect”. Also, let there be room and time for the student to explore on their own time. Sometimes just telling students the existence of something is enough for them to go and look it up. Today’s student has access to so much information, they just need to be taught what to search for and how to search for it.

I don’t know if schools allow for asynchronous “grades”. Rather than have students move from grade 4 to grade 5 en masse, I think one solution would be to have grades per subject — this can’t be a novel idea. For example, a student can be in grade 7 Math, grade 10 English, grade 4 Art, etc at the age of, say 10. There’s still play time and recess and sports and hobbies and trick-or-treating to build bonds and camaraderie with the student’s age group.

I don’t think a solution to improving “core” competency is to marginalize “non-core” subjects. This is the pendulum swinging too far in the other direction. As Mr. Horne argues, “give me more balance” to which I will add “… but not too much”. Too much balance makes everything bland and same-y. And too much balance is thinking that one must do everything, always. That is a sure fire way to do nothing. So, let students go wild in one direction but bring them back to the center and nudge them in another direction. That would be a balanced education I think.