I won't even touch the political reflexive negativity surrounding Common Core.
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.
> 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").
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.
As a student I loved word problems and hated the straight equation ones. It always pissed me off that there were 35 explicit 45 - 17 = ? questions and 4 word problems.
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).
I've never personally heard any reasonable criticism of Common Core. The criticism tends to either be "The Federal Government is overreaching!" combined with some States Rights dogwhistling or "This wasn't the way I was taught and since this confuses me, a grown adult, I have decided that it's bad for children! [who have a completely different set of requirements and experiences]
Are there any sane, evidence-based criticisms? Because Common Core seems like a decent improvement to me.
From what I have seen from from a 1st, 2nd, and 7th grader the issue real issue is implementation. They want the 1st grader to understand that when you carry a 1 you are really carrying 10, so they have this huge diagram they draw out so it can be visualized. Great makes sense.
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 haven't seen any real arguments to the substance. I keep seeing that one picture reposted with the "Old Fashioned" method and the "Common Core" method[1], and although I don't know anything about Common Core, I can see that it is a clearly contrived example that is certainly very misleading.
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.
Um, I think that diagram is wrong. The common core example I have seen would break the subtraction into "32 - 12" = "32 - 10 = 22 22 - 2 = 20"
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.
The catch is: we engineers do that because we grok basic math so well that we can mentally juggle numbers fast.
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.
> 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.
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.
Most from-fundamentals education involves learning small steps which are later discarded in favor of radically different processes not easily understood without having gone thru those prior steps.
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.
To the contrary, I've never personally heard any reasonable support for Common Core. Of the many examples I've seen, including what my daughter has been inflicted with, it is leaping past basic competency going straight to advanced techniques which only are simple because of thousands of hours of practice with straightforward fundamentals. When asking even Common Core teachers about the curriculum, they're quick to express reflexive embarrassment and change the subject. Even (IIRC) the federal head of the curriculum sends his own kids to a non-Common Core school.
> I've never personally heard any reasonable criticism of Common Core
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.
> A buddy of mine who is an EE with a heavy math background has had trouble helping his child get the correct answer.
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.
> 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.
> I'm not really sure what you're talking about here.
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.
> Basically they are trying to do abacus math without the abacus.
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?
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...
Well, there's the 10th Amendment: "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people." And there's a quite reasonable position that says that an unbiased reading of the constitution does not give power over education standards to the United States government.
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...)
I've recently looked at one of the current "Common Core" textbooks for freshmen math in my sibling's school district in California. It is very different from the style of learning I was accustomed to. The textbooks I had introduced a new topic in some introduction pages, gave practice examples, and then a few pages of exercises of increasing difficulty.
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.
The argument about that Common Core is bad because it comes from the Federal government is also based on fiction. Common Core was created by the states (specifically the National Governors Association) as an effort to have consistent standards across multiple states. The only Federal Government connection I'm aware of is that adoption of Common Core standards is one way that states can score extra points for Race to the Top grants.
The big problem with common core is the reform package into which it is embedded. The standards themselves are in most cases as well thought out as other alternatives. Sure you can find details to argue about, but that is universal. The textbooks and workbooks have some rough edges because of newness. The real problem is the rest of the package: high stakes tests with no direct educational purpose taking weeks of class time, not to mention valuable tine now going to test prep. Privitazation of the testing, and increasingly of the schools themselves. Dimunuation and disenfranchisement of the teaching profession. The big picture is industrialisation of public education. This entails a loss of quality in favor of uniformity and cheapness.
> he big problem with common core is the reform package into which it is embedded. [...] The real problem is the rest of the package: high stakes tests with no direct educational purpose taking weeks of class time, not to mention valuable tine now going to test prep. Privitazation of the testing, and increasingly of the schools themselves.
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.
That's because you are looking at the wrong problem.
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.
Can't say I've been following Common Core too closely (my kids are 3 and 1 respectively, so there'll be a bit before it affects me directly). But these example questions just seem completely to be arguing with a strawman:
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?
Regarding the elementary one, there's a good reason for the larger numbers though. With 4x3, a student can get the right answer through a counting process.
1-2-3-4 5-6-7-8 9-10-11-12
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.
I also don't get their critique that the problems can be solved easily using simple mathematics. That's kind of the point of mathematics. Making the problems slightly more difficult (if it can be called that) doesn't really remedy the original problem, which is that they don't understand why something works.
The problem with Common Core, in my experience, has not been the math itself. It's the textbooks. The workbooks. The stubborn teachers.
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.
I've looked at some recent high school algebra textbooks. What a disaster. These books are literally written by committee, the "author's" page in the front has a dozen headshots complete with bios. The content inside the book is a mishmash of color photographs, historical information, brain teasers and junk.
The texts are distracting and takes away from what SHOULD BE an uncluttered exposition of the basics of mathematical reasoning.
I suppose it's just a result of "maths is boring" complaints, hence the attempt to make things a bit more attention-grabbing, but I agree that it distracts from the actual content. Maths does need deep thought and reasoning, which more plain textbook design facilitates.
I remember using the books of Durell[1] when I was in school, and those were definitely quite plain.
Teachers will complain about absolutely every change that's imposed on them. Perhaps it's because there isn't really an undeniable way to know what's good and what's bad (unlike most jobs where profits make that clear) so every teacher assumes what they're doing it good and any change must therefore be bad. Or maybe it's just a job that attracts people with that kind of personality.
> Teachers will complain about absolutely every change that's imposed on them.
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.
For clarity, these are the actual Common Core math standards. If it's not in here, then it's a problem with the curriculum instead of the Common Core. http://www.corestandards.org/Math/
What I'm seeing with my Maryland 3rd and 5th graders is lots of focus on number patterns and word problems using the integers. I think they are also getting fewer, and sometimes harder, problems than they would have previously.
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.
a. sounds like a perfect example of the concepts of CC being mis-applied. It might make sense to explain how they arrive at the answers in general, but separately explaining each answer is obviously nuts (and would have driven me crazy when I was in school).
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?
Well, they cancelled trig when I was in school because I was the only one taking it. It happens and there is no real excuse. As to how we learn in college? I gutted it out and did every damn problem I could. I would imagine that STEM is beat out of a lot of kids in elementary so it might just be the survivors are already used to doing the extra work.
There are several level of classes at the middle and high school levels. Normally this would be taught in elementary school. But there are some students who do need to be taught this again in high school. There are a whole debate of possible reasons why, but providing the basic math avoids sticking them in classes which they are guaranteed to fail
Exactly - my "pre-algebra" class in 8th grade put me beyond the entire "Algebra" class when I started high school. And my calculus teacher in college wasn't very good so I coasted through 2 classes based on what I learned in high school and then flunked out of the 3rd because I finally ran into new ideas.
This would be covered in Algebra I and pre-Algebra (under the old scheme, not sure what it falls under now). Those courses are typically taught somewhere between grades 7-10 (7 as the earliest for pre-Algebra, 10 as the latest for Algebra I) for most students depending on the particular track they're on.
It has to do with the particular phrasing of the question. The correct phrasing for your example would be, "Each shirt costs $6. How much money do you need for 7 shirts?" It may seem like a meaningless distinction but it isn't. The first phrasing sets you up to count, the second phrasing sets you up to multiply and grapple with what multiplication actually is.
I don't see any real difference in phrasing. They both involve multiplication, and you can always use repeated counting to multiply. Could you explain more where the difference is...?
One requires math. The other requires English comprehension before you can work out what the math problem is.
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.
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."
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.
My mom is a college math teacher, previously a high school math teacher, and helped develop the current Texas standards. She knows a number of the folks who wrote the California Common Core and has high praise for it in principle.
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.
> 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).
"What are two different equations with the same solution as 3(y-1) = 8?"
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.
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.
The examples given seem to be a bit of a strawman argument. Sure, asking a single, focused question tests less for understanding than the multi-part Common Core examples. But at least from what I remember, you'd be drilled on all the different permutations of these kinds of word problems anyway, so the net effect is the same.
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.
> 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,
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.
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
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.
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).
Exactly that kind of single by single digit mental multiplication is useful how in computer science?
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.
That's basic numeracy, the kind people need every day in their lives. This widget costs $7.99, and I need 7 of them, how much is this going to cost me roughly when I get up to the checkout counter? Is there enough cash in my wallet? That's ~$56, and I've got $60, will that cover the 5% additional sales tax? 10% of $56 is $5.60 (by the shift the decimal point method), half of that is $2.80. Guess I'm good.
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.
> 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.
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.
To me, this is the real problem. I see little reason to believe that Common Core fixes it.
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.
I am one of little faith. At the high school level, there's really only so much that can be done without the instructor understanding the end-game. It's just fundamentally difficult for an algebra teacher to effectively teach algebra is they haven't really groked any group or ring theory or linear algegra. And it's really hard for a Calc I instructor to teach effectively without having really mastered some elementary analysis.
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.
> 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.
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.
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.