This is how I felt when I first tried to figure out what common core math was, after I kept hearing about it. It sounded like the people who made it were trying to create a process around what I would do in my head to try and solve a problem when I didn’t have paper or a calculator handy.
What common core, and this reading approach, seem to miss is that those are things that come after learning the foundations and rules.
I was taught with phonics, but 99% of words I read today are simply seen and recognized. Much like the 3 queuing method tries to create a process around. However, I think that is something that naturally develops through repetition. When someone reads the word “horse” and “house” enough times, the need to sound it out or read each letter goes away, but that’s not the starting point.
As much as I like to think I can determine a word’s meaning through context, when put to the test, I often miss the mark. Reading on an e-reader lowers the barrier to looking up unfamiliar words and I find myself doing it more often. I find my assumption based on context can often be less than correct. Maybe I’m close, but there is more nuance to the actual definition. Sometimes I’m completely wrong and the whole meaning of the paragraph changes. Not to mention, if I look a word up to learn the actual meaning, it might be something I can introduce into my own speech and writing without sounding foolish.
I relied on context a lot growing up, because I was too lazy to look things up. These days, with it being so easy, I don’t know why kids wouldn’t be encouraged to look up words they don’t know. I think my vocabulary would be much better had I grown up learning the words I didn’t know, instead of simply bypassing them.
Common core math has its issues, especially with teacher training, textbooks, and bad assessments. But I wouldn't lump it in with three-cueing, which sits squarely in opposition to decades of research.
For example, I don't think that kids who struggle with common core math and then switch to the "normal way" (the way their parents learned) will have a leap in ability the way that kids who switch from three-cueing to phonics do.
At least half of the problem with common core math is that parents get a pit in their stomach when they look at their kid's homework and discover that they aren't familiar with the methods. The angst will lessen over time as we start to have kids whose parents grew up with the new methods.
There's certainly a long road ahead in getting math education right, but I imagine the long-term solution will look more like common core than the 20th century curriculum.
> For example, I don't think that kids who struggle with common core math and then switch to the "normal way" (the way their parents learned) will have a leap in ability the way that kids who switch from three-cueing to phonics do.
I'm not quite sure that's true. In my city, there is a local charter school that teaches math with only Common Core methods. The local public school teaches primarily Common Core, but also supplements it with Houghton-Mifflin workbooks of the sort that we learned 30 years ago. By 4th grade, the public school's math proficiency rates are 20 points higher than the charter school's. They draw from similar socioeconomic demographics - if anything, the charter school parents tend to be a bit more affluent and involved in their kids' educations than the public school's.
As said elsewhere, there's not really such a thing as common core mathematical methods.
The main way they differ from what came before is that the CC standards expect students to be able to explain more of what they're doing and why, which is pushing a lot of people building curriculum to emphasize visual methods, bar models, etc. But that's not part of CC per se.
By "common core methods", I'm lumping together a bunch of approaches where, in the words of the immortal Tom Lehrer, "the important thing is to understand what you're doing rather than to get the right answer".
Understanding what you're doing is important. Understanding what you're doing on the way to getting the wrong answer is not really, unless your goal is to get the wrong answer. IMHO, getting the right answer should be the starting point for math education. It's not all of math education - there is a vital role for understanding what you're doing. And wrong answers have their place as well - it's actually very illuminating to work out what happens in number systems where 1/0=1 or 1/0=0 or is anything other than undefined, and then show how that means that every number is equal to every other number. But the big lesson from working these lessons is usually that mathematical conventions are what they are because it makes other branches of math simpler, and they could be otherwise (that's the whole point of having axioms), but then you'd have to deal with the consequences, which is often that certain results we take for granted don't hold.
In software engineering, we have a saying: "Make it work, make it right, make it fast." It means that you don't shoot for perfect code up-front. There's too much learning involved in just making useful code. But once you have something that works, you can anchor your explorations around something that already has a quantum of utility. At that point, you work on simplifying the approach, making it understandable, cutting out abstractions that aren't necessary and adding abstractions that make the problem shorter. And then when you're done with that, you add optimizations to make the program run faster, but only if necessary.
The point of starting with working code is to bound the search space, though. If you don't have that, you can find that you spin your wheels forever and never generate anything useful. Likewise, the point of phonics is to get you somewhere close to the right word. It bounds the search space, while whole-word or 3-cue approaches leave the beginning reader guessing indefinitely. And the point of drilling math problems until you're at least familiar with basic arithmetic is so that you can enter the conversation about "What does this all mean? Why do these approaches give the same answer? What happens if I take a slightly different set of axioms?" with knowledge of what that answer was in the first place.
> By "common core methods", I'm lumping together a bunch of approaches where, in the words of the immortal Tom Lehrer, "the important thing is to understand what you're doing rather than to get the right answer".
Well, maybe that's really misdirected then, isn't it? Because the actual Common Core standards aren't really behind your complaints.
I don't really agree with the rest of what you say, though. Rote and math drilling are very important, but understanding is pretty important so you know what method to select. IMO, a good math class is 1/3rd rote, 1/3rd intuition/pictures, and 1/3rd efforts to add rigor.
It's worth noting the whole lot of the world introduces pictures and geometric methods early, like some of these curriculums are trying to do, and gets better results in math education than we have. Though I don't really think most US teachers know how to do that well, yet.
> Understanding what you're doing is important. Understanding what you're doing on the way to getting the wrong answer is not really, unless your goal is to get the wrong answer.
Everyone wants the right answer. A huge proportion of our population basically gives up on math at some point from 4th to 7th grade; they decide they're bad at it, and this self-assessment is more predictive of their future performance than their actual performance. This is really a general pedagogical thing rather than being rooted in standards or even curriculum, but a lot of the reason some teachers and programs are trying to celebrate partial success is to reduce this problem. The background trend is particularly harmful to girls, who for a variety of reasons tend to self-assess more harshly than boys.
In any case, this all has very little to do with Common Core, other than it started happening in the same decade. (Buying new textbooks to align with a new standard is a good chance to change things.) And it's really confusing to those of us with some domain knowledge who can't figure out the specific, non-CC reason why a particular person is complaining.
My background: I advise on math curriculum selection and development at a small private school; I coach a competitive MS math team that routinely wins (crushes— we are 4% of the local MS pop but had 8/12 of the top students last year) regional competition despite us being tiny. Our elementary school that feeds us adopted visual and explanatory methods early, and devoted a whole lot of effort at getting good at teaching this way. (Note, we didn't abandon rote practice. Also note that we are not really required to align with state standards, but for math we could do so with very little change).
We get a huge proportion of our student body through AP Calc; my eldest did it in 8th grade.
> This is how I felt when I first tried to figure out what common core math was, after I kept hearing about it. It sounded like the people who made it were trying to create a process around what I would do in my head to try and solve a problem when I didn’t have paper or a calculator handy.
> What common core, and this reading approach, seem to miss is that those are things that come after learning the foundations and rules.
The CA Common Core for math¹ is a very tiny spec that can be easily read in a day for the entirety of K-12. It's a spec meant to coordinate textbook publishers, test makers, and other content creators.
It offers very little restriction or guidance, in other words wide latitude, on pedagogical vision.
And what's there isn't too different from what's before, though pedagogical ideas that are now more in vogue (use of tape models) are mentioned prominently, and there's a lot more emphasis on students being able to explain what they've done and why it works.
I agree there is a development path to learning to read. My son is learning and during the phase of learning the phonics he would use the pictures a lot to help get context. You could tell because he would pause on each new page as he observed the graphics.
But as he became more proficient he started using the graphics as a crutch to guess the words rather than sound them out and blend them. So I started covering the pictures with my hand as I turned the pages and his progress jumped ahead in the following weeks.
He is still learning and falls into guessing patterns too easily. However through accompanied parent reading every night he is progressing well (IMO).
I'm sorry, but as a mathematician and educator, the comparison to Common Core is unwarranted.
When you say it sounded like something to you, you communicate that you didn't understand what it actually meant and deduced the meaning through pattern-matching - which is exactly the problem here.
Common Core aims to rectify the problems introduced during the "New Math" era (and before it, too), which, like the "three cueing", took sense out of the art of reasoning and imagination we call mathematics.
To understand what I am talking about, I beg you (no exaggeration) to read Mathematician's Lament by Paul Lockhart[1].
It will give you an immediate understanding of how we teach mathematics wrong - like the article we are discussing exposes how we teach reading wrong.
This was an incredible interesting read, thanks for sharing. I actually fear that this is a generalized problem with subjects in school, it's not math alone that suffers from this. I remember being asked to learn by hearth poems that I didn't even like, making me hating poetry in general. Or even art, just a bunch of names and artworks to learn and memorize characteristics and whatnot. I enjoyed math and physics, but I realize now it was because I was good with algebraic manipulation rather than proper understanding. And I rather struggled later at University studying physics indeed.
Indeed it is, sadly. But I'm very glad you got to read this paper, and get thoughts from it! The more people see things this way, the closer we are to changing the education system for the better.
That was an interesting read. I wasn't expecting to actually read 25 pages as a result of a comment here, but the author wrote with enough passion to hold my interest and presented some interesting ideas that I hadn't heard before. Thanks for sharing it and begging me to read it.
I also went a watched a quick video on common core (which used a multiplication example) after reading Lockhart's paper, to try and better understand where I was going wrong, and where you were coming from. Prior to this, my last look at Common Core was many years ago, and fairly brief. It also lacked appropriate context, just the news, so I was entering with a bias (more on that bias later).
I do agree that providing a formula and telling kids to use it 100 times to drill it into their head is probably not a great way to teach. A person can't really extrapolate on ideas when they aren't understood, but rather memorized. It wasn't until after collage that I stumbled across some gifs showing how/why various formulas in geometry are what they are, and I was really upset that those types of things weren't shown in school. All I had was Donald Duck in Mathmagic Land, lol. I find it much easier to remember things if I can conceptualize them and understand why something is the way it is. If I can deduce it myself, to the paper's point, that is even better. That way, even if I forget the formula, I know how to get back to it.
My opinions on any change of the math curriculums have been colored by my own experience 25-ish years ago when my high school introduced Integrated Math, to replace the traditional track. My class was the first one without a choice. At the time we were told the idea behind Integrated Math was that no one needed high level math, so why bother preparing kids for it... which was an idea also touched on in the paper. I was planning on going into engineering and needing high level math, so that program kind of screwed me. To live in the author's perfect world, college would likely need to change as well to not assume knowledge of various things going into college level math programs. I liked math (or at least what I thought was math before reading that paper), and Integrated Math destroyed that for me. It was very unfortunate. A year after I graduated, I received a letter from my high school asking me about the program and what I thought of it after having been out for a year. I wrote a pretty scathing letter in response. The program was abandoned some years later. I tried to jump on the traditional track in college, but I did too well on the placement test, so they wouldn't let me take algebra, but didn't really get the foundation to effectively move forward into calc.
To this day, I still feel like I have a gap due to missing out on the traditional math track in high school, and the ramifications that had on my college and career (not that I'm doing bad, I'm probably better off for it). Hindsight being 20/20, I didn't actually need it, but things do come up from time to time that make me think I missed out. I've attempted, to go back a few times to learn on my own. Without being in school, and the boring nature of most mathematics instruction, I've never gone that far.
I like the ideas presented in the paper about viewing mathematics as art. At the risk of going against every point he was trying to make, are you aware of anything resources that teach mathematics through this lens that I could check out? Some sort of framework to give me problems to try to solve to lead me in the right direction, while also showing how to see math problems in the everyday and the solutions as art, instead of wandering around in the desert for the rest of my life hoping I stumble upon the problems/solutions that took centuries to uncover. Are some of the Common Core resources out there a good place, or is there something in the spirit of Lockhart's paper that is better?
Thank you so much for reading that paper and writing this comment.
There definitely are resources that I can recommend. Can't do this at the moment (work awaits!), but will come back to this in the evening.
One book that did a lot for me was What Is Mathematics?[1] by Courant and Robbins. Godel, Escher, Bach[2], while not strictly a math book (and not a book I actually finished reading), was another. These would be my go-to recommendations without thinking much.
There are also many more playful mathematics books that might be an even better fit for what you want, but I'd need to go through my bookshelf to pick some.
> It wasn't until after collage that I stumbled across some gifs showing how/why various formulas in geometry are what they are, and I was really upset that those types of things weren't shown in school.
What common core, and this reading approach, seem to miss is that those are things that come after learning the foundations and rules.
I was taught with phonics, but 99% of words I read today are simply seen and recognized. Much like the 3 queuing method tries to create a process around. However, I think that is something that naturally develops through repetition. When someone reads the word “horse” and “house” enough times, the need to sound it out or read each letter goes away, but that’s not the starting point.
As much as I like to think I can determine a word’s meaning through context, when put to the test, I often miss the mark. Reading on an e-reader lowers the barrier to looking up unfamiliar words and I find myself doing it more often. I find my assumption based on context can often be less than correct. Maybe I’m close, but there is more nuance to the actual definition. Sometimes I’m completely wrong and the whole meaning of the paragraph changes. Not to mention, if I look a word up to learn the actual meaning, it might be something I can introduce into my own speech and writing without sounding foolish.
I relied on context a lot growing up, because I was too lazy to look things up. These days, with it being so easy, I don’t know why kids wouldn’t be encouraged to look up words they don’t know. I think my vocabulary would be much better had I grown up learning the words I didn’t know, instead of simply bypassing them.