
Old Standards vs. Common Core: A Side-By-Side Comparison of Math Expectations - tempestn
http://excelined.org/common-core-toolkit/old-standards-v-common-core-a-side-by-side-comparison-of-math-expectations/
======
bsder
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.

~~~
thraxil
Totally agree on the importance of reasoning.

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.

~~~
bsder
> Word problems though are often hated because they're so ambiguous in
> unintentional ways that don't really relate to the problem.

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.

------
douche
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.

~~~
DanBC
> 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.

~~~
userbinator
_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.

~~~
maus42
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.

~~~
douche
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.

------
minikites
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.

~~~
protomyth
> 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.

~~~
nmrm2
_> 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.

~~~
protomyth
> 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.

~~~
nmrm2
_> Anecdotes and narratives are valuable._

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.

~~~
protomyth
> 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.

~~~
nmrm2
_> Common Core is based on a bunch of studies_

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.

------
jkingsbery
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?

~~~
tempestn
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.

------
sp332
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/](http://www.corestandards.org/Math/)

------
debacle
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.

~~~
angdis
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.

~~~
userbinator
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.

[1]
[https://en.wikipedia.org/wiki/C._V._Durell](https://en.wikipedia.org/wiki/C._V._Durell)

------
catawbasam
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.

~~~
tempestn
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).

------
lordnacho
If 3(y-1) = 8, then what is y?

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?

~~~
saboot
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

~~~
sp332
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.

------
charlesdenault
The first example is pretty bad. It could just as easily be "Each shirt costs
$6. How much do 7 shirts cost?"

~~~
minikites
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.

~~~
userbinator
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...?

~~~
TheOtherHobbes
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.

~~~
userbinator
_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.

------
niels_olson
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.

~~~
bsder
> 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_ ).

------
sixbrx
"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.

------
swills
At the bottom of this page you see:

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.

------
cline6
I will let this man do my talking.
[https://www.maa.org/external_archive/devlin/LockhartsLament....](https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)

