in which I rant about Common Core a bit

[cahn]

So Firefox/Pocket recommended me this article on the "New New Math" [Common Core] vs. the Old Math, and this reminded me I have been meaning to rant about this since the summer. Apparently ranting about pedagogy is a thing I do now? :P

Anyone in the US who has a child of school age is well aware that the US has recently revamped its elementary math pedagogy system and (more slowly) the curriculum to something called "Common Core." In principle I think is a great idea! Common Core emphasizes having a deeper understanding of the mathematical concepts instead of rote memorizing of equations and algorithms for doing specific calculations.

Until I had a child go through this, my chief issue with it was that teacher training and curriculum were not keeping up with the reforms, so that it was and is often implemented poorly. To be fair this is still often a major issue with Common Core. But I thought it was a great idea when implemented correctly! What's not to love about having a deeper understanding of mathematical concepts??

Now that I have had a child go through this system... In practice, it turns out that actually a lot of the time in arithmetic it does actually make your life easier if you just memorize the algorithm for doing a specific calculation, because that is what these algorithms are for. Here is what happened last summer that completely made me do a 180 on it: E took a programming class (Art of Problem Solving's intro Python class, which I was impressed by and which she enjoyed and which changed my attitude towards homework, but that's another story) and because it's a more mathematically intensive class than most beginning programming classes (the gestalt is that they teach programming at least partially as a tool to do mathematical computations, on an elementary level of course), there was a little test you could take beforehand to see whether you were ready to take the class. So I had E take the test. I didn't anticipate her having any difficulty.

Tears and meltdown and "I don't want to dooooo this!" Now, with another kid I might have understood this, but math is E's favorite subject! What?? I dug into it deeper and it turned out she didn't want to do the long division problems "because they take forever." (There were only three of them!)

I asked her to do a long division problem for me. She did one. (Easier when it's just one, and for an audience.)

And at the end of it I totally understood why she was melting down and refusing to do three of these. I would have too if someone had made me do long division the Common Core way! Because they wanted to make sure the kids understood what exactly was going on at each step in the process, they had to write down all the zeros at each step of the process and add everything up and... It's a lot. It's basically writing down the. entire. multidigit division process by hand. Whereas the whole point of doing long division the way you and I learned it in school is as a SHORTCUT so you don't HAVE to write all of these things down!

I taught her the "traditional" way of doing long division and told her that from now on she was to use solely that method (in order for her to practice it and get it solid in her brain) unless a teacher specifically told her she needed to use the Common Core method (which is unlikely with her current teacher, at least).

This fall I also had to repeat this process with multi-digit multiplication, which I also didn't realize was an issue until a very similar thing happened.

I'd also like to show you this graphic from the article that set off this whole rant today:



So...the graphic says it like it's a bad thing that kids learn the algorithm if a/b = c/d, ad = bc... I mean, I do agree with a lot of what the graphic is saying! I agree that (a) using that algorithm to solve that particular problem is way overkill, and (2) okay, tangentially, just teaching them the algorithm without teaching them why it's true is a terrible idea! Though it's tangential, I feel particularly strongly about (2) because my mom did exactly that and I was sooooo confused and didn't understand at all! (My mom, bless her, is amazing at math but completely and utterly terrible at pedagogy.) But... like... it takes two seconds for a math student to understand why this is true algebraically speaking. Multiply both sides by d, multiply both sides by b. Done.

And of course the Common Core way of doing it is great, it's awesome that kids are learning what proportions really mean, and the connection to linear graphs, that's awesome! And for the specific problem shown in the graphic the Common Core way is definitely better! I'm just saying that... at some point... by the time they get to higher mathematics... they also will find their lives easier if they've learned that if a/b = c/d, then ad = bc as a useful shortcut for algebraic manipulation :P

I fear that what will actually happen after Common Core is that we will turn out a generation of students who don't remember the deeper understanding of the mathematical concepts (because they don't need to know that in their everyday lives) AND don't know how to multiply two-digit numbers or do long division or find proportions that aren't 2/3, which... worries me.

Comments

[thistleingrey]

Hmm, interesting. Reason's school seems to have simplified Common Core silently--it's noticeably not what I learned (which is also probably what you learned!), but it's not nearly as much makework as what you show here.

One thing I do like about what Reason has been learning is that her current teacher bothers to name the algorithms for the kids and refer to back to them by name repeatedly. This means the kids can discuss things because there are labels! They don't have to write stuff out for each other and then be distracted! I never knew the names for any of the methods I learned (or deduced by myself), and it's so pleasing to hear young voices discussing methodology, even if in very basic ways, via Zoom breakout-room chat (small groups). (How am I an English major again? Well, anyway.)

[cahn]

Heh, well, good for her school! I actually think it's not a bad idea to go through the whole process once, but... you don't have to do it constantly, any more than you have to derive the quadratic formula every time you use it.

Oh, I like that her teacher names the algorithms! I don't think I know names for a lot of the elementary-school algorithms.

[hamsterwoman]

I actually think it's not a bad idea to go through the whole process once, but... you don't have to do it constantly, any more than you have to derive the quadratic formula every time you use it.

This!

I like the idea of Common Core where it's, like, explaining why you do things and introducing concepts early and spiraling, but the make-work aspect of it is ridiculous. Show the drawn-out way once, explaining why it works, back up to that if someone is not getting it, but exactly as you say with the quadratic formula -- you don't want to derive everything from scratch every time! It's not practical, and I'm sure a way to make even more people hate math...

(Fortunately, my kids only caught it at the pre-Algebra level, basically, so they still learned to do long division the normal way.)

[cahn]

Yes, a lot about it is great! But having to do all that every time you just want to multiply a couple of numbers!... ugh.

And yeah, I just don't know how the kids are dealing with it... E's best friend, according to his mom, was having a tough time with long division too. Now in his case he didn't have a very good math teacher last year, so there's that going on too and I'm not sure how to disentangle all the effects.

[thistleingrey]

Agreed re: not constantly. But also, because my mother has taught several levels within K-8, I'm afraid I give Reason 2-4 ways of doing a thing anyway, with the instruction to learn how to show work in the way the teacher wants and then decide for herself which of several approaches seems most useful for a given problem, instead of just learning it one way and thinking she's done.

I think I first noticed when the kids began discussing area model.

[cahn]

*nods* E is actually very good at coming up with creative ways to approach a math problem, and I hadn't made that connection to her being brought up with Common Core but it is clearly at least partially an outgrowth of that. (Partially it's just that she's been brought up with two math nerd/formerly-math-nerd parents, so we also do encourage different ways of thinking about a problem.) So yes, there are definitely good things about Common Core!

[thistleingrey]

I think that when kids can grasp enough of the parts of all that makework, it's helpful because it gives them little system components to swap in.

Also, I'm not really trying to think of all the ways in which Common Core can be better at times than how we learned :) but I think that in an era where people tend to memorize less and look things up more, Common Core is a reasonably good fit. For parts. If one does memorize some things anyway. But the way we did it with more front-loaded memory also benefited from having less awareness of the possibility of chatting with Siri, Google, Alexa, Bixby, etc. about one's homework.

[luzula]

Interesting! I do agree that only knowing the algorithm and not knowing why it works is not great, and one should absolutely understand why it works. But, yes, the purpose of an algorithm is to make it possible to calculate things efficiently. Somehow I don't think this Common Core approach would have been taken in the days before calculators/computers, because then humans (with the help of tables and slide rules) were the calculators, and it was obvious to everyone that you don't want to waste time when calculating.

I don't quite understand the graphic, because to me those are different problems (though of course they have the ratio in common). The first is about solving equations and about understanding what operations are allowable on an identity to get an equivalent identity (with the caveat that these aren't quite equivalent because what if b=0?). The second is about proportionality and straight lines. Like, you couldn't just replace problem 1 with problem 2 and have people learn the same thing!

[cahn]

Heh, there was the New Math in the 1960's? I don't know much about it, just that they were trying to teach set inequalities (I'm sure there must have been other things) and there was a huge backlash.

Yeah, the graphic isn't.. ideal for whatever it's trying to get across. In fact I had a more strongly-worded rant about solving equations until I decided that maybe what it was trying to say was that the first problem was trying to get people to understand how to do proportionality problems. But what that first problem actually makes me think of is how in, say, high school physics you have to do that kind of manipulation all the time, only with both numerators and denominators being variables. (And the ghost of my awesome, beloved high school physics teacher is saying, "No numbers first!")

[seekingferret]

I am putting a pin in this to come back when it's not Mystery Hunt weekend because I have FEELINGS about this from having to teach my little brother all the rote algorithms his school wasn't teaching him.

[cahn]

*elbow-bump of having to teach kids rote algorithms their school isn't teaching* I am looking forward to hearing from you! Because GAH.

[melita66]

I've got 2 kids in 3rd grade in California. I'm somewhat of a fan of some memorization because it can free your brain to concentrate on the concepts. In high school, my math 5 teacher (I guess it was pre-calc) died mid-year. The replacement...was not great and was teaching us about epsilon and limits which I just didn't get and she couldn't explain it well enough to me. After getting through calc 1 and 2 in college, I wish she had just taught us derivative algorithms so I wouldn't have had to memorize those while trying to understand calculus theory.

One kid is still shaky on some multi-digit addition. The outside-of-class app they used for memorization never clicked with him. He couldn't focus long enough to do a set. They're slowly memorizing multiplication now. I'm not doing much with them out-of-class because they both dislike school so I'm trying not to make it worse. *sigh*

I'm trying to introduce some tricks like even+even or evenXeven must give you an even number.

CC does show a different ways to solve a problem and each technique is named. I find trying to remember all of them a chore when 2 or more are introduced on the same day. The kids do too until they've had several days of practice.

[cahn]

I tell you what. Before I had kids in school I wasn't so much a fan of memorization, but NOW I am TOTALLY a fan. (I mean... obviously within reason, and my older one in particular memorizes very easily, which was why I wasn't a fan before because in lower grades she didn't have any trouble memorizing without doing any additional work.) As you say and as sub_divided said below, that sort of automaticity frees you up to be able to do more challenging things.

Ugh, I am sorry your kids are having trouble with school :(

I used Multiplication Rock when I was a kid to help me learn multiplication tables, and used them with my kids as well, don't know if that would be at all helpful. (Let me know if you need audio files.)

(We're in CA too, though private school right now. Older one is 5th grade, younger in K.)

[sub_divided]

I'm a high school math teacher... the trouble with common core isn't necessarily a problem with teaching concept-first, bc we also learned a lot of things concept-first, but the insistence that students need to justify themselves at every single step and show their work at every single step to PROVE that they have learned the concepts. It's a problem with teachers needing to collect evidence for everything, or make-work as another commenter said.

There's also the issue that students are just expected to learn MORE, including learning more about graphing and interpreting graphs and creating equations and so on... and to also learn these skills from an earlier age, bc there's some pedagogical theory that holds that even kindergarteners can solve equations and do other abstract operations if you call them "number sentences" or something like that. The higher level, more abstract concepts are pushed down to a younger age, but simplified to be age-appropriate, supposedly.

The problems are wordy and they're supposed to understand where the algorithms come from. You can't only add without taking away, so all the time you spend on vocabulary and applications and discovering the formulas and 'speaking math language' is time you can't spend on fluency with algorithms. In reality the students learn a lot of different methods, so they get confused between them, and they are very very very slow because they don't get enough practice with any one method.

The new standardized tests (like PARCC) also test for 'conceptual understanding' by only having a few, very high level problems on them because this is what engineering firms, etc, indicated they want the students to be able to do. We also have a lot more visualization problems now (reflection, rotation, dilation, nets, 3D solids of rotation) for the same reason, it predicts success in engineering.

These skills are also important, but the focus on interpreting graphs and solving multi-step problems is a tradeoff with speed on simpler problems. For example, on a military test like the ASVAB students are expected to solve a lot of straightforward problems quickly. They can't do it! All the training has gone in another direction.

P.S. In the school where I teach, 100% of students are minority and maybe half are first-generation immigrants. The students who learned via strict traditional methods tend to also be better on the conceptual problems because they have more automatic skills, so they aren't using brainpower for things that shouldn't need brainpower. But this could also be selection bias bc those students tend to come from better schools in general.

[thistleingrey]

This is really helpful as context--thanks for sharing your perspective!

[sub_divided]

No problem! It's really a pendulum, you need the conceptual understanding and the fluency. I think common core goes way too abstract way too early and requires young kids to do too much bothersome documentation of their thinking process. If you asked the questions verbally and had students explaining their thinking verbally, you could also teach concept-first, but you wouldn't have evidence that you'd done it.

[cahn]

Oh yaaaaaay I didn't know you were a high school math teacher and I'm very very happy to get your perspective as you're the actual one on the ground! (I know I only see a very little bit of it.) And yeah, I'm not at all dissing the concept-first approach, which I like! I like a lot of things about common core! It's just the stuff it doesn't do, or the stuff it does awkwardly, that I don't like :)

but the insistence that students need to justify themselves at every single step and show their work at every single step to PROVE that they have learned the concepts.

YES. Why does E still need to do this every time she multiplies out multi-digit numbers, even though she's long since progressed to other topics??

In reality the students learn a lot of different methods, so they get confused between them, and they are very very very slow because they don't get enough practice with any one method.

Yes, this is part of what worries me although I see i didn't articulate this well. I could still as an adult do long division without using any of my brain, just because it's so drilled into me. I'm not sure that's true of my child, even though I think she's better at math than I was at her age -- she still has to think about it, and then the way that CC asks her to do it is sooooo slooooow! (She's actually very fast at calculation in general.)

bc there's some pedagogical theory that holds that even kindergarteners can solve equations and do other abstract operations if you call them "number sentences" or something like that. The higher level, more abstract concepts are pushed down to a younger age, but simplified to be age-appropriate, supposedly.

Huhhhh, that's very interesting. I don't know how I feel about this at ALL -- again in principle maybe? But in practice I feel like you might encourage kids not to adopt an algebraic language because they're so used to abstract concepts only using words and visualization and such? (And then my child, who always does things by opposites, was actually so relieved when they finally got to variables and algebraic notation, and suddenly I went from her math teachers telling me how terrible her communication was to her math teacher telling me that she was a great communicator, which was a bit of whiplash I still think is hilarious.) But maybe you have more perspective on how this works in practice?

[sub_divided]

Yeah... even if they aren't doing the full song and dance to multiply multi-digit numbers, there's an emphasis on showing your work and explaining your reasoning because of all the online solvers.

And there's so much emphasis on showing work, I have students who think showing your work is a PART of the problem, instead of just a step to find the actual solution. They lose track of what's "process" and what's "answer" but I think this comes from bad prior level instruction, from teachers who weren't really sure themselves.

And like you said, at a certain point you just have to consider that they know how to multiply because there are bigger fish to fry...

***

I think we can waste some time re-teaching the vocabulary, sure. They had to learn what a number sentence is, then at high school you have to re-teach what an equation is! They end up learning the vocabulary twice. But hey, at least your kid can appreciate the notation :) Kids are always stumbling over notation like it's the most difficult part of doing math, when actually we invented notation to make things easier...

[cahn]

That makes sense! I mean... with the showing work, I suppose it's a bit of a pendulum swinging back, because I was never taught to show work properly until I had Awesome Physics Teacher, and I have certainly tutored my share of kids who weren't ever taught that. But like you say there's the difference between "process" and "answer."

You know what the notation thing reminds me of though, it reminds me of reading music, which of course is notation to make things easier. But bringing up kids playing violin or what-have-you in the Suzuki method, which does a lot (a LOT) of ear training in the early years, often results in kids who freak out a bit when reading music is introduced and never really get at all good at it, which really starts holding them back after a while. (I managed to escape this, along with my kids, by all three of us also learning non-Suzuki(-ish, in my case) piano, which is much stricter about teaching reading music from the very beginning, but e.g. my sister, who never really got into piano as much, has a much harder time, and I've met others who have an even harder time than she does.)

[sub_divided]

As far as pushing abstract concepts to an earlier age... I don't know, for some kids it's not a problem at all. In the district where I teach algebra is a huge stumbling block though. For a lot of kids Algebra I is a wall, they're stuck on one side of the wall and they just can't get to the other side. I've had a lot of students who told me they really enjoyed math when it was just numbers and they started hating it as soon as the letters came in.

[mildred_of_midgard]

I was planning to continue chipping away at my salon backlog today, but one of my coworkers decided he wanted to talk pedagogy with me, and he's the one I've been discovering much to my delight and horror that he does the thing you and I do, which is wall-of-text back and forth with each other with excited engagement, and haaaaalp I'm never going to get anything done! Why do the people I get naturally sucked into endless convos with both suddenly want to talk about my hot topic, pedagogy? :P (He's studying for an AWS certification, which is why it's on his radar.)

In conclusion, I must ignore this post, sorry. :P

[cahn]

hahahaha wait you can't just say that and not tell me what you guys were *talking* about with respect to pedagogy!! (Unless you were talking about your method for learning languages, which I gueeeeess we've already talked about so I'll give you a pass :P :) )

[mildred_of_midgard]

Oh yes I can, watch me. Watch me be responsible and prioritize sleep. :P

I will tell you some other time, but, um, some other time.