Disclaimer: I hate high-stakes testing. Hate it. This post is not about high-stakes testing; it is about content that happens to be used in high-stakes testing. Imagine I’m writing about Holden Caulfield, and try to banish all memories of Catcher in the Rye plot pop quizzes from your mind.
My cousin asked me about my thoughts on common core. Here’s an actual paragraph from the common core website:
These standards define what students should understand and be able to do in their study of mathematics. But asking a student to understand something also means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.
This sounds awesome! I often think I understand something well, and then try to write it down only to realize that there are more subtleties than I thought, or that I had a fundamental misunderstanding, and in general that I am wrong.
On March 16th of this year I thought I had an answer to something I’m working on. Aaaand… today is August 27th and I’m still working on it. But I didn’t realize my answer wasn’t complete until I started writing, and I wrote “, because” and had nothing to follow it. This is frustrating and ridiculous, and I don’t expect every kid in America to become a mathematician and work on a problem for six months. But I do believe that trying to do something, thinking you’re right, and then trying to explain it and realize you’re wrong after six minutes is a good experience and makes you a better critical thinker. Question your beliefs, analyze your reasoning, explain yourself- all good things.
The above is all theoretical (I was a math and philosophy major in undergrad). So I looked up some common core examples.
Here’s a comment from this article, as reported by a Washington Post blog:
Try doing multiplication in long drawn out word form like this one: 3, 6, 9 what is the 12th number in this sequence? My son can’t just read that and think the 12th number automatically. He has to write them out. He also cries over this type of math.
The problem itself is great (I’m also not opposed to crying). Most parents can help their kids with 12 times 3 equals 36 “automatically,” but figuring out why 12 times 3 is 36 is hard. This question offers justification first, and encourages the student to explore and discover multiplication for himself. I was quite bad at my times tables as a kid (I still remember nonsense like “Six and eight went out on a date. When they came back, they were 48!”) because it didn’t make any sense. But writing out a sequence 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 is just like practicing spelling or lay-ups, and it makes sense.
I started finding more examples of common core type problems, but I think talking about one of my own past experiences is more helpful.
During my first semester of graduate school, I sat down to my first analysis exam. I’d never been a big analysis fan (though now that I’ve taken it umpteen times I actually enjoy it!), and I felt a bit shaky about the material before I walked in the room. The professor handed out the exams and said “you have one hour.” If this isn’t clear to you by now, I’m good at math. But as I leafed through the problems I started silently FREAKING OUT. I felt something like vertigo, and also felt very very cold as I thought about trying to finish this exam in the allotted time. My hands were shaking and I was a little sweaty. This was my first experience at math anxiety and it was TERRIFYING. My skin is prickling as I type this, remembering the experience five years later.
I can only imagine what it must be like to feel that with math for years in elementary-high school. Math anxiety is real, and unfortunately contagious. There’s been some amount of backlash to common core/shaking up math education, and I’m certain that some of it is rooted in past traumatic memories of math anxiety. All I can say is, I support students, I believe in them, and that “the most important thing is to stay calm.”
If you google “common core math” you’ll get lots and lots of mild vitriol, but you also get some good stuff. Here’s a side-by-side comparison of some math problems and explanations of why they switched. Here’s a cool explanation of the new subtraction.
Here’s an article called “The Ten Dumbest Common Core Problems” that I’ve run into a lot [I dislike it because it doesn’t give credit to where any of the pictures/examples came from], and my thoughts on 1-10. Again, not an elementary-high school educator, just a math person:
- 7 + 7 = 10 + 4 = 14. Maybe adding a word like “number bonds” is strange and new, but memorizing that 7+7 = 14 can be hard (I’m a terrible memorizer, but a good reasoner). Breaking 7 into 3 and 4, and then pairing up to get a 10? That’s how I add large numbers quickly, and I had to figure that out myself. Helping kids learn the trick is great.
- Awesome, visualizing subtraction. See above explanation on the new subtraction.
- This is a worksheet with a typo. So… unclear why it’s included. If it didn’t have typo, it’d be a good sheet.
- This worksheet’s picture doesn’t make sense. But the math does. (Figure out what’s unknown if you have parts of a whole)
- More visualization of addition and subtraction. I literally did this with 12 year olds when I was 15, using quarters on a table.
- I like this one too.
- I always hated “carrying the one” so I’m all for the new addition.
- I don’t understand this. This doesn’t mean I hate it, it means I want to know what it says.
- This isn’t new; I did this worksheet as a child.
- Another typo
Final anecdote in this jumble of a blog post: how I learned to divide fractions. They tell you to just multiply by the reciprocal. Like I said, I’m bad at memorization, so I’d often do random nonsense that seemed sensible (divide the tops, multiply the bottoms). No matter what I tried on my own, I felt like I just couldn’t get this dividing fractions thing. One day, my dad drove me to Baker’s Square to buy a pie. He asked how school was going and I told him that I would never get how to divide fractions and I’d always get it wrong and math is dumb and I was just a sad little 8 year old sitting in that passenger seat. He said, okay, well, if we split this pie among our family, how much pie does each of us get? And I said “one fifth.” What about if just you and I take the pie? “One half.” What if I give you the pie? “One.” What if half a person gets a pie, how much pie does one person get? “Tw-OHHHHHHHHHH.” Crystal clear, this concept that had been frustrating me for weeks. Why do we use the reciprocal? Because if half a person eats a pie, that whole person eats TWO PIES.
That’s my happy story of understanding the madness behind the method. I’ve seen dozens of students who know to multiply by the reciprocal, but who don’t understand why, which makes word problems quite difficult. Conclusion: so far, I’m all for common core math. The important thing is to stay calm. And ask questions. And maybe not help your kids with homework?
Oh one more thing: I’m interested in giving back to the community/doing some volunteer work and putting my skills to use. I told my advisor yesterday that I don’t have any skills, and he responded that I have math skills and I should go find something like the free clinic for math. Do you have any recommendations for me to help people with my math skills?
Baking and Math 
But what kind of pie is the person eating???
Child-yen:banana cream, definitely. Illustration: Jeune genevois plum pie, from Wikipedia. It’s traditionally eaten in Geneva on that holiday
Thanks for this great post. As for suggestions on how to help: Fawn Nguyen’s recent twitter post has some clues.
If there were regular sessions where parents could see this approach modeled and practice it themselves, that would go a long way towards reducing the contagious transmission of math anxiety. Maybe you could approach your local school, PTA, or community center about organizing some sessions of this type? I also think there is a lot of potential in math circles
http://www.mathcircles.org/
and community math centers
https://riverbendmath.org/
So you might want to investigate those.
The examples of “bad Common Core math” in the National Review article are really examples of individual teachers’ or textbook authors’ attempts to implement *their* interpretation of the standards — not examples of the standards themselves. In item #7, for instance, the vocabulary given there may or may not help to understand the math concepts but it does not come directly from the Common Core standards.
Click to access Math_Standards.pdf
The word “increase”, for example, only occurs 3 times in the entire 93 page document while “add” or “addition” appear over 130 times. So…
I have mixed feelings about the CC math standards, but it’s really discouraging to note that almost all criticism I see of the standards is based on specific local attempts to implement the standards, rather than on the standards themselves — and there doesn’t seem to be much recognition of the difference.
It’s great to see a professional mathematician discuss these issues thoughtfully. Come to think of it, that could be another way to help — encourage thoughtful and informed dialogue about math education amongst your colleagues.
This is great! I’d love to help parents help their kids, because of my fear of children. That South Bend math center looks incredible; I’ll investigate if there’s anything similar in Austin (there’s something at UT for military families…)
I agree, most of the craziness about Common Core out there is about local implementations rather than global. But that’s sort of the difference between mathematicians/scientists and others- non-professionals who are in the middle of the action might not have the luxury of debating more abstract principles; they have kids NOW who are in schools right now with these teachers and curricula. My baby goes to a national daycare chain, but I don’t really care about the company’s national policies vs. how they’re carried out in his specific classroom. So if I were to complain about something (we miss bear blocks!) I’d probably blame Bright Horizons rather than his specific branch of Bright Horizons. Not saying this is a good thing, just trying to say I get why it (conflating common core standards and implementation) is a thing.
Thank you for your thoughtful comment! Sorry for the delay in it showing up… I’m not particularly technically proficient.
Good point. It’s definitely the responsibility of CC advocates to make sure their standards are widely understood and to correct misunderstandings. Something they have not been very effective at.
I cringe when I see parents (and, even more sadly, teachers) offering explanations to kids, rather than asking the kids questions and helping them to investigate. Students, especially younger ones, generally find this kind of cooperative exploration to be fun. “What do you think… ?” and, “What would happen if… ?” are a lot more engaging and fun than, “Let me explain… ” (So that’s my protip, should you find yourself confronted by a crowd of surly 2nd-graders — or even apathetic undergrads!)
When I lived in Chicago, there was a community organization that supported a local elementary school, and I volunteered as a math tutor there once a week for a while. There might be similar groups for individual schools or school districts where you are.
I should’ve added that I’m not great with kids (or really, less-motivated people). Kids who want to be there = great (CTY was a blast!). But I tried doing 826 for awhile and I am just not great at being “fun” or making something “fun” that they don’t already think is maybe a little bit fun.
I love when mathematicians actually *read* the Common Core, because they invariably like (most of) what it says, especially the 8 “Mathematical Practices.” You might be interested in seeing what James Tanton has to say about Common Core:
And just a note that he approaches fractions the same way that made sense to you. Look at page 10 here: http://www.jamestanton.com/wp-content/uploads/2009/07/fractions-guide.pdf
Oh my gosh it’s beautiful! I love the stuff on page 54 onward; thanks for pointing me to him. I find K-12 education fascinating, partially because it’s where so many college students clearly didn’t get prepared for what they’re facing now. But children also intimidate and scare me. Hopefully as my baby grows I’ll become more comfortable around kids. I guess the way to make the biggest impact as a volunteer is by helping children, so I could try diving in. That homeless kids program sounds so good (in a virtuous way)!
Oh, I also am not so into kids. I actually work with teachers in outreach activities rather than kids. My rationale: every teacher works with hundreds of kids every day. Changing their approach makes a bigger difference in more kids’ day-to-day lives than a one-off (or even once-a-month) program for kids ever could. But the truth is that I just like grown-ups better.
Also, what kind of stuff are you interested in doing as a volunteer? Is there a math circle (for kids) or a math teachers’ circle in your area? (You can search for them online.) Some of our grad students also work with the local institute for human services providing after school math enrichment for homeless kids.
why would you “hate” carrying the one? Isn’t moving it to the next column showing how ten in one place value position becomes one in the next ( or visa versa for borrowing)? We used to have bundles of sticks in pictures which we would break up or bundle as necessary to solve the problems. By the time (grade) we got to carrying and borrowing we understood why we were doing it. It was a short cut. I fully appreciate showing why the math works the way it does as understanding, but when I actually have to add or multiply, understanding place values and carrying or with multiplication, adding an extra zero to each row, makes for a much quicker solution.
Two thoughts on this: First, bundles of sticks is great, and understanding is great, and i love that you remember this and have internalized the algorithm as a shortcut! I’ve definitely internalized “carrying the one” as an algorithm, and since I’m not a computer/super detail-oriented I mess it up ALL the time. That’s why I hate it, because I am prone to error and regularly mix up which locker in the gym is mine (was it 178 or 187?).
For an extreme example, if I wanted to add 3.78+4.94 I’d have to carry a one from the ones to the tens, and then remember to hang on to that one when I was adding the tens and carrying another one over, while holding in my head the answer to the ones place. Versus saying it’s 4+5 – (0.06 + (how I get from .78 to 1)), which I find a lot easier even though it’s more steps. They’re both shortcut approaches which don’t inherently preclude understanding, as you pointed out.
Thanks for commenting!