Old Standards vs. Common Core: A Side-By-Side Comparison of Math Expectations
excelined.org
excelined.org
However, the only non-political (I'm being nice ...) criticism I continue to see is in the mathematics. And it's always because of "word problems". Common Core likes "word problems" and reasoning a LOT.
I have taught math. With very few exceptions, everybody hates word problems. Most students just want "Give me the question and give me the procedure so I can regurgitate the procedure. Don't make me think." They don't want to reason.
You can see this even in senior year of high school in the US in science. Just try to get the students to explain why they are doing an experiment (In short: "state the hypothesis"). It's almost always a disaster. After a year, you might get a majority up to the point where they can actually state it coherently.
I have this problem even with some really senior engineers when debugging things. "We did X, Y, and Z. They didn't work." "Um, okay, why did you do X, Y, or Z?" "Huh?!?!?!" "Why should X, Y or Z have worked? Is there a relation between X, Y, or Z and the problem? What is it?" <puzzled stares>
So, I see the objections as positive evidence that the Common Core standards are doing their job. The fact that adults have trouble with some of these problems is no surprise. Most of them skated through reasoning and word problems themselves.
This starts very young. Most "science for kids" on the Internet is little more than "here's a demonstration of something, and here's a sciency explanation of what's going on". Most sites don't suggest making any changes to the setup, or ask any questions about what happens if you do stuff differently.
(I'm keen to find good science content for children that does teach scientific method, so please let me know of any good sites).
Word problems though are often hated because they're so ambiguous in unintentional ways that don't really relate to the problem. Eg, in the middle school example on the page, when it says "She makes $4.40" do they mean net or gross? In this case, it turns out that only one of those gives you an integer answer, so you can work out what they meant, but it's not uncommon to see problems where that's not the case and you're stuck making a 50/50 guess. With the current emphasis on test scores, that adds a lot of stress.
Statistical analysis helps this dramatically.
I remember all of the SAT prep books I used to have. I could never score much better than just very good. Never understood what the problem was.
Then I got actual SAT's from previous years. Perfect scores, no problem.
The difference was that the SAT questions had been vetted by statistical analysis. Any question that was ambiguous was removed or reworded.
I wonder what, if anything, that says about people. Should I start asking people which problems types they preferred during interviews? (that's mostly rhetorical...i think).
Are there any sane, evidence-based criticisms? Because Common Core seems like a decent improvement to me.
These are more implementation problems, but as a parent they've been the biggest issue.
In contrast "Integrated Learning 1" by CPM takes a very different approach. It is very much continuous where the "text" of a topic is interwoven with a couple problems to do. Class room exercises and experiments constitute a lot of the main lessons. This is in contrast to Khan Academy where the progression of math knowledge is easily understandable and the step-by-step approach for learning by example and doing is emphasized.
I was a bit taken aback by this approach. I can see it possibly working better in a class room setting. However I did most of my learning by reading. I'm also concerned that this approach will make it more difficult for students to catch up if their previous math education was not great as this continuous flow style doesn't lay out which topics to study.
I'm not sure, but I'm not currently convinced. I'll be seeing how my younger sibling fares next semester, especially as our parents are not great at math and his education has some large holes in it (which we are working on). Thankfully he really enjoys Khan Academy so I anticipate him using that more if Common Core falls short.
Every other criticism is consistent with your experience, essentially calling it overreaching or teachers complaining that they shouldn't have to learn it to teach their kids.
[1] http://www.thefederalistpapers.org/wp-content/uploads/2014/0... (this was the first Google hit for the image I was looking for)
Which is how almost every engineer I know would do the math in their head. If they didn't just go "30 - 10 or roughly 20" (obviously most engineers can do that simple a subtraction completely--but the number of times I estimate something by lopping off everything except the leading digit is quite high).
Note that this is teaching "estimation" almost by osmosis. The "standard" procedure of starting from the least significant digit and "borrowing" obscures estimation.
Yes, seems Common Core is trying to teach those methods - but to children who barely grasp the subject, and thus must memorize rote processes to scribble down all the obligatory steps without understanding why they're doing it.
Children need learn math by concrete manipulatives, handling the physical-world form until they (individually!) internalize the process, then growing comfortable enough with those internalized processes to develop more sophisticated AND casual ways to mentally tear numbers apart & rearrange them to perform calculations fast.
The rote processes Common Core teaches are closer to what you and I use. Why teach subtraction from least-significant to most significant manipulation and then make students throw that away and relearn the other direction? That is especially meaningless when tabular clerical calculation skills have no existence in the modern world.
We don't teach children to read from right to left and then force them to relearn reading left to right to read fast.
> Children need learn math by concrete manipulatives, handling the physical-world form until they (individually!) internalize the process, then growing comfortable enough with those internalized processes to develop more sophisticated AND casual ways to mentally tear numbers apart & rearrange them to perform calculations fast.
And, yet, most significant to least significant is almost always how we manipulate things in the real world.
I cut material down to rough size, and then I cut it to detailed length. Imperial recipes measure multiple cups and then fractional cups. A bolt looks like a 1/2 inch head, so I try the sockets around my estimate. etc.
Better reading analogy [I read very fast]: we don't teach children to glance at a page, absorb random words, predict likely filler content, identify interesting sections, and repeat process on those sections - instead, we teach children to read left-to-right, word-for-word, until they can do so well enough & fast enough that they internally develop content prediction techniques leading to "page at a glance" reading.
No such thing. Common Core doesn't specify methods, it specifies that students must attain specific things by specific times. Methods are left up to individual districts, as they have always been.
The biggest problem with common core is that it codifies social promotion under the guise of "schools know better" while linking promotion to money received by the schools.
None of that has anything to do with Common Core, and, AFAICT, politically, the groups that most oppose Common Core are the groups that have most supported the policies that have done the most to produce the effects you complain about.
Problem is they make the kids draw out this giant ridiculous diagram every time, and they get graded on whether they draw the picture error free. So instead of teaching the concept and then reinforcing the application of the concept they are reinforcing how well you can draw the needless picture you were taught. Parents then get mad because they can't help their kids because they didn't learn the rules of the diagram drawing.
I'm a away from my desk, but the examples that are causing the most problems are not the ones shown in the referenced article (those seem quite reasonable). The problem is how they teach addition and multiplication. Basically they are trying to do abacus math without the abacus. They also accept estimates instead of actual answers and a total removal of the memorizing of tables. A buddy of mine who is an EE with a heavy math background has had trouble helping his child get the correct answer. The worst part is he taught his daughter math early and she struggles because she knows the answer but doesn't know how to express it the way her teacher wants. This is a true pain because he thought he was raising a kid excited about STEM and now its just a wall.
As to the state rights and such. Yes, that is a valid issue, but is much broader than Common Core and a continuation of lunches and NCLB. Common Core is just more evidence to an old argument.
My kids both had abacuses at their desks in Elementary school.
> They also accept estimates instead of actual answers
I haven't seen this at all. Like any math class, partial credit is given if the method was sound but there were arithmetic errors.
> a total removal of the memorizing of tables.
This has nothing to do with the Common Core, and was a decision made by the school. Both of my kids have numerous tools at their disposal, from the school, and mastery of basic math facts up to 12 is required as part of my daughter's second grade competency.
You're conflating a lot of different things under the umbrella of "Common Core." When was the last time you went to a school board meeting?
Yeah, and I'm not sure that is a good thing http://www.ee.ryerson.ca/~elf/abacus/feynman.html
> I haven't seen this at all.
I have and the estimate is what was supposed to be the answer given, not the actual answer. This causes stress in parents and children who know math.
> You're conflating a lot of different things under the umbrella of "Common Core." When was the last time you went to a school board meeting?
Never went, also a school board meeting is not the be all and end all of the standard. I have a nephew and niece coming up and have seen the homework of my peer's children. Also, I've read the standard http://www.corestandards.org/wp-content/uploads/Math_Standar...
Yeah, these anecdotes of supposedly brilliant people incapable of learning a new way to add numbers are always about engineers and accountants and stuff. Never an actual mathematician. It's almost like they went through an entire college degree pushing symbols and were never forced to stop and come up with a real proof on their own...
It's possible to have a heavy math background and still be shit at math. Especially if that background stopped short of anything proof-based (but even then, plenty of people memorize proof techniques instead of actually understanding the arguments and the subject).
Calculus sequences and ODEs courses that EEs take are particularly bad. They are just symbol pushing like in middle and high school. More symbols and more complicated pushing, but nothing essentially deeper.
If I had a dime for every vector calculus student who didn't understand basic facts about the Reals (or even the Integers)... and most of them get decent marks, too.
So, new rule: any rant about common core mathematics can only contain as many words as the number of words in the longest proof the speaker has written outside the context of a homework assignment or exam.
No, he is "not shit at math", he loves it, tried to pass that love on, got that dream promptly resisted, and you don't make the rules. Anecdotes and narratives are valuable. Looking at a lot of these studies, they are just gussied up anecdotes with questionable method and results.
But when those anecdotes and narratives are ultimately nothing more or less than an appeal to authority ("I'm an ENGINEER/accountant/etc. and can't do/didn't do/don't need to do this -- it must be crap!"), then it's absolutely reasonable to question that authority.
What, exactly, was the problem that your friend had trouble finding the correct answer to?
I'm extremely skeptical claims with the form "I'm an X and can't do CC problems" for a reason. There's no shortage of anti-common-core accountants/engineers/etc. who take the the blogosphere with complaints that "even they" don't know how to work a problem.
But when you look at the problem, it's just performing addition using a non-standard algorithm or setting up and solving for a linear relationship. Not exactly rocket science. And then you look at Calc III classrooms and see students who clearly haven't internalized division. Which leaves only one conclusion -- being an engineer or accountant who made it through a few calc courses doesn't exactly equate to "good problem solver" or "understands anything about mathematics".
> Looking at a lot of these studies, they are just gussied up anecdotes with questionable method and results.
I'm not really sure what you're talking about here.
I don't need empirical evidence that understanding multiple algorithms for a arithmetic procedures is a useful and crucial exercise. Just like I don't need empirical evidence do know that there's a lot less value in memorizing quick sort than there is in seeing multiple different sorting algorithms and comparing them.
> and you don't make the rules
Obviously :-) But it's a good sanity check on what it means to be well-trained in mathematics.
If you've never written a proof of substantial length, you really don't know what mathematics actually is. In particular, the mathematics courses US engineers and accountants take are mostly unsubstantiated symbol pushing (warrant: find me a calculus student in the US outside of Chicago or a few other places who can prove the fundamental theorem), which isn't mathematics.
Common Core is based on a bunch of studies, look up the list cited in the documents on it.
> If you've never written a proof of substantial length, you really don't know what mathematics actually is. In particular, the mathematics courses US engineers and accountants take are mostly unsubstantiated symbol pushing (warrant: find me a calculus student in the US outside of Chicago or a few other places who can prove the fundamental theorem), which isn't mathematics.
When you decide that only one place in the US has any idea what mathematics is, then this discussion is not going to go any further. I guess Harvard, MIT, etc. don't count.
I'm not an educational researcher and I don't have a thorough understanding of the research methodology or the issues involved in designing those studies. Every Education researcher I've talked to thinks anti-common core people are a bit nuts and/or fundamentally don't understand what common core even is (I think most of them would put you in this second camp, since you're complaining about specific assignments).
But as a mathematician, some problems are obviously the sort of problems that anyone with a passable mathematics education should have no problem solving. The common core problems people complain about are decidedly in this set.
So when people say "I can't solve this common core problem", I mostly take it as an indication that they've had a really shitty mathematics education rather than an indication that common core is flawed.
And yes, even someone who has passed through a calc course at Harvard can be bad at math.
> I guess Harvard, MIT, etc. don't count.
Harvard, MIT, etc. have excellent Mathematics departments and, following my criteria, any Math major for either of those institutions could have a lengthy conversation about common core. Indeed, among the many mathematicians I know with undergraduate degrees from Harvard College, I've never heard a single one complain that common core problems are obtuse or difficult.
The distinction I was drawing is that US-based Calculus for Engineers and ODEs for Engineers courses aren't proof-based except in a small handful of cases. And, those courses are often easy to skate through with little or no mathematical understanding. Yes, even at elite universities.
Which goes back to my original observation -- if you really can't add numbers in a novel way or setup and solve for a set of linear equations, then you're apparently not very good at math. Even if you are an intelligent pattern matcher who made it through a few calc courses by applying templates and performing rote calcuations.
Why is that a valid issue?
And the other side will state that it's covered by the commerce clause or general welfare clause or something. (But everything can be argued to be covered by those to clauses, and the 10th Amendment meant something to the authors...)
The only difference in the elementary school question was bigger numbers.
The first middle school example contains only one additional step (being able to do algebra instead of just arithmetic). The explanation for the "old way" contains the word "simple" gratuitously - you might as well say that the CC version of the problem only requires "simple algebra."
For, the second middle school example, I definitely had questions that were multi-part like that in middle school. Also, each part is just applying the same formula a few different times - but the explanation given makes it seem like it's a whole new thing!
The explanations for the high school example is really the worst one though. "This question is an example of solving equations as a series of mechanical steps" and "This question is an example of solving equations as a process of reasoning" mean basically the same thing, just one is put in a derogatory manner.
The CC version of the question itself is also vague: can I pick any two equations that have the same solution? Should I pick equations that are the intermediate steps toward the solution? Does "y=5/3" count as one of the equations?
For someone who hasn't either memorized multiplication tables or learned faster ways of working out the answer, that's likely the approach they will take. With 6x7 on the other hand, it's natural to look for a faster (better) way to solve the problem.
I agree that the last question was vague / poorly state.
Math is math. Knowing more than one way to solve 345 * 14 is a valuable skill. I would rather my children learn it on a whiteboard than from a workbook, and I think generally the quality of the modern texts is very poor for actually teaching math.
I also think the teachers are bemoaning the Common Core much more than they need to. Explicitly teaching the algorithms for mental math is important. Learning how to do the lattice method, even if it's not the preferred method for multiplication, is important. I've talked to far too many teachers who have complained unendlingly about the weakness of the standard, and all that the kids are being forced to learn. It feels very much like a "This is change and change is bad." mentality.
Disclaimer: I haven't encountered the Common Core as a parent at a high school level yet.
Often for good reason. Remember "new math"? Most of my math textbooks as a kid had things about ordinality and arithmetic in non-base 10 (generally base-12--why?--I have no idea) down as far as like Grades 4 or 5. It turns out that if you don't have a very clear grasp of your own arithmetic, you can't change bases very easily.
Teachers who have been around for 20+ years have seen bad education fads through several cycles.
The texts are distracting and takes away from what SHOULD BE an uncluttered exposition of the basics of mathematical reasoning.
I remember using the books of Durell[1] when I was in school, and those were definitely quite plain.
Some of that is good. At times they really are being required to think more deeply.
The problems I have seen are: a. Emphasis on explanation even when it doesn't make sense. In 2nd grade they brought home worksheets with 20-30 simple addition problems where they were asked to 'explain' each answer. b. Focus on a narrow range of topics, to the point where they are not always learning the basic vocabulary of mathematics. c. In particular, there has been very little attention given to geometry.
On what planet is this a high school math problem? How do people learn linear algebra and vector calculus in college if this is what they're at in high school?
And in the "Write two equations..." example the English is very vague. There's no suggestion that what it really means "Show the steps you use to rearrange the equation as you work out what y is."
As I read it, it's clearly not stating the real problem correctly. If you write two arbitrary rearrangements that are correct but aren't steps on the way to solving the equation, do you get marked down or not?
The first example is clearly specious. What is being taught? The concept of multiplication, the memorisation of times tables for multiplication, or the use of multiplication to solve simple problems?
I have no problem with a unified national curriculum, but I'd prefer it to be a unified national curriculum supported by hard evidence that it improves outcomes and understanding. This looks more like pointless tinkering around the edges - especially if the textbooks are really bad, as others have suggested.
I did not get that meaning at all from the wording. (I'm not in the US.) My best guess was that it was a really open-ended question and any two equations with a solution of 11/3 would do, which seems like a rather odd thing to ask.
In practice, she is quite frustrated when she tutors my daughter. She has worked with her Over FaceTime, using Pearson's own remote tutoring app, and over a two week "Grammy Camp" in the summers.
Thanks for posting the side-by-side. I wish they would tell the parents something like this, once per testing period or so. My undergrad is in physics and I too struggle with how to help when asked.
In what way? Is it that you're trying to replicate an unfamiliar procedure or an unfamiliar concept.
Maybe as an engineer I'm just unusual in that all of the Common Core procedures I have seen so far actually map to the way I calculate in my head. But, I've seen carpenters, woodworkers and machinists all calculate similarly when they do (normally they use advanced geometry to avoid calculating at all).
How is this scored if the student just adds 1 to both sides, then 2 to both sides, for the other two equations?
3(y-1)+ 1 = 8 + 1
3(y-1)+ 2 = 8 + 2
Is that a full-credit answer (it is technically correct)? Or do they want movement towards a solution? This seems very ambiguous as to what they want which I've noticed is common in CC questions.
All content Copyright © 2010 – 2015 Foundation for Excellence in Education
Wikipedia says the Foundation for Excellence in Education is a think tank established by Jeb Bush and has received funding from Bill Gates, who I understand has also funded development of Common Core.
Also, I really don't see how their reasoning follows on their first example. 3 shirts, $4/per shirt = $12. 7 shirt, 6 buttons/per shirt = 42. All that multiplication is, on the integers, anyway, is repeated addition. The only real difference I can see is that you can do the first example on your fingers and toes, whereas you'd have to scratch some tally-marks in the dirt or on paper to do the second if you don't know multiplication tables.
What I wonder about with these Common Core curricula, is whether students will ever get enough practice working through mathematics operations to really be comfortable with it. When I was taught math (and it really wasn't that long ago), we were never allowed to use calculators, and we did sheet after sheet after sheet of additions, subtractions, multiplications, divisions, polynomial expansions, equation simplifications, etc. Maybe five or ten minutes, every day; they were probably graded enough to make us do them, but not enough to really matter. Most of the Common Core style math worksheets I've seen require far too much rewriting the algorithm from mathematical notation into grade-school English to ever allow that kind of repetition.
A child can do sheet after sheet of calculations and co-incidentally get the right answer, without having any understanding of what they're doing.
They can manipulate these symbols because they've memorised a routine. But because they lack understanding they stumble when they move onto something more advanced - they need to forget what they thought they knew, re-learn it properly, and learn the new thing.
I agree that the first example is a confusing example to use, and that there doesn't seem to be much difference.
* identify a rote procedure to apply when a problem of a particular form is seen
* execute the rote procedure
* come up with "<equation here> / 0"
* ... and then turn the thing in anyways.
But then, I have a bit of an unusual perspective on the problem. My position isn't that the old curriculum is good and this new one is bad. They're all pretty darned bad. Common Core has some neat bits in it, but it's still polishing a turd; Lockhart's Lament still applies to it in full force.
Especially the second one. I'm not sure what a calc I teacher is supposed to say if a student has any foundational questions, which any bright student probably will.
No, I wouldn't!
Take much easier examples: Percentages.
Ask someone what 10% of 230 is, and most people can give you the right answer. They might not understand what's going on, but they can type the right buttons on a calculator.
Now ask them to find what 37 is as a percentage of 390 and an alarming number of people will struggle. Or ask them "This TV now costs $240. It's had a %10 discount applied. How much did it cost before the discount?"
It's alarming because so much information is presented as percentages.
I think the intent of those is to let the implication of what multiplication really is sink in as they do those calculations repeatedly - as they start to notice patterns in the numbers.
But because they lack understanding they stumble when they move onto something more advanced
On the contrary, if they memorise the tables they can multiply quickly without much thought, which becomes useful later on when they move onto more advanced concepts. The alternative is to "understand" multiplication; but not having done it much, fumble around and be error-prone when solving a more advanced problem actually requires the use of multiplication as part of its solution. It's distracting to the thought process and inefficient to have to exert effort to recall "how to multiply" (and possibly "how to add", but hopefully not "how to count" too...) when one's focus should actually be on a higher-level concept.
I worked with some first-year CS students and the number of them who had difficulty with a single digit by single digit multiplication (e.g. 7 x 8) makes me think mental arithmetic is one of those skills that really needs to be drilled into heads. Understanding the concept is good, but being able to effortlessly put that into practice is even better.
Except we have tons of evidence that multiplication drills do not get lots of people to that effortless level.
I was a product of the old multiplication system and I think it's humorous that you chose 7x8 as an example, as its one I can never remember. I always have to calculate it using the actual underlying math. And guess what, that's always served me perfectly well even in a highly technical profession.
Here's a sheet of subtraction.
What can we tell of the student's understanding from this sheet? Take something simple like number placement - can we tell if the student understands there's a unit column and a tens column?
Here's a video of a girl with a pile of counters. She's asked to separate them into piles of tens, and asked to count the left overs. Then she's asked how many she has, and how she'd find out. She counts each counter individually. She's asked to write the number, and she correctly writes the tens and units in the right place. So there's some understanding, but she's not able to transfer it from one area (writing the number) to another (counting the counters).
I do have some difficulty coming up the answer to 7 x 8: I have to work that out from 8x8. This takes a couple of seconds, tops. Assuming one can still understands how multiplication work and can actually find a correct answer in a matter of seconds, or maybe tens of seconds (and a pencil and paper), that's really quite enough. (edit. If you meant that students can't come up with the correct answer at all, okay, then you have problem, but that seems quite improbable so I assume you meant 'remember instantly'.)
As a CS major, things I work with include proofs, set theory, mathematical logic, or even abstract algebra (and here the important thing to realize about multiplication is how it relates to addition and taking powers and how that can be generalized or abstracted), and most importantly, everything resembles general abstract mathematical problem solving, not memorizing multiplication tables. Even within realm of basic arithmetic, the idea of logarithms (and some familiarity with the powers of 2) is far more useful, and that's more of the abstract concept side of it (edit. okay, some general feeling of logarithms helps, too).
I feel positive about any mathematical education that tries to train those important skills and at the same time portray mathematics as an art of exact reasoning, not just rote memorization.
Unfortunately, that is the case I've seen. They either must reach for a calculator(!), or get it completely wrong if forced to do it manually.
Being able to estimate and do mental math quickly is important for everybody, and the key to doing that is rules-of-thumb and having the basic facts memorized. Think of it as caching/memoization and heuristics, if that puts it in a more computer science frame.
Both occur. I was very averse to the calculator use in the classroom, but from what I have seen that part of the curriculum is more about learning how to use a calculator than doing math with the assistance of a calculator. Memorization of arithmetic is still very important and focused on.