Tag Archives: common core

Gross misuse of + and – and x and the one that’s not on my keyboard

Arithmetic is the art of processing numbers.
In ordinary language these words are verbs which have a direct object and an indirect object.

“Add the OIL to the EGG YOLKS one drop at a time”.
“To find the net return subtract the COSTS from the GROSS INCOME”.

In math things have got confused.
We can say “add 3 to 4″or we can say “add 3 and 4”.
We can say “multiply 3 by 4” or we can say “multiply 3 and 4”.
At least we don’t have that choice with subtract or divide.

The direct + indirect form actually means something with the words used,
but when I see “add 3 and 4” my little brain says “add to what?”.

There are perfectly good ways of saying “add, or multiply, 3 and 4” which do not force meanings and usages onto words that never asked for them.
“Find the sum of 3 and 4” and “Find the product of 3 and 4” are using the correct mathematical words, which have moved on from “add” and “multiply”, and incorporate the two commutative laws.

If we were to view operations with numbers as actions, so that an operation such as “add” has a number attached to it, eg “add 7”, then meaningful arithmetical statements can be made, like

“start with 3 and then add 5 and then add 8 and then subtract 4 and then add 1”

which with the introduction of the symbols “+” and “-“, used as in the statement above allows the symbolic expression 3+5+8-4+1 to have a completely unambiguous meaning. It uses the “evaluate from left to right” convention of algebra, and does not rely on any notion of “binary operation” or “properties of operations”.

If we want to view “+” as a binary operation, with two inputs then, yes, we can ascribe meaning to “3+4”, but not in horrors such as the following (found in the CCSSM document):

To add 2 + 6 + 4, the second two numbers can be added to make a ten,
so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

If + is a binary operation, which are the two inputs for the first occurrence of + and which are the inputs for the second occurrence of + ?
The combination of symbols 2 + 6 + 4 has NO MEANING in the world of binary operations.

See A. N. Whitehead in “Introduction to Mathematics” 1911.
here are the relevant pages:
whitehead numbers 1
whitehead numbers 2a
whitehead numbers 2b
whitehead numbers 3a
whitehead numbers 3b

And here are two more delights from the CCSSM document
subtract 10 – 8
add 3/10 + 4/100 = 34/100

In addition I would happily replace the term “algebraic thinking” in grades 1-5 by”muddled thinking”.



Filed under arithmetic, big brother, Common Core, language in math, operations, subtraction, teaching

School——>PARCC tweet———>suspension———->and then?????

Thank you Audrey Watters for leading me to this exposure of the behaviour of testing corporations.

These two are MUST READs, and should be passed on to everybody:



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Filed under big brother, education, testing

Smarter Balanced: Lacking Smarts; Precariously Balanced

It’s a bit long, but it sure takes the lid off the CCSS. Read it now.


In this time of  “public-education-targeted boldness,” the Common Core State Standards (CCSS) has made the American public one whopper of a “bold” promise:

The standards were created to ensure that all students graduate from high school with the skills and knowledge necessary to succeed in college, career, and life, regardless of where they live. [Emphasis added.]

There is neither now nor never has been any empirical investigation to substantiate this “bold” claim.

Indeed, CCSS has not been around long enough to have been thoroughly tested. Instead, the above statement–which amounts to little more than oft-repeated advertising– serves as its own evidence.

However, if it’s on the *official* CCSS website, and if CCSS proponents repeat it constantly, that must make it true… right?

Keep clicking your heels, Dorothy.

Now, it is one issue to declare that CCSS works. It is quite another to attempt to anchor CCSS assessments to the above cotton…

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Another Common Core Math Horror

I thought I had found them all, but NO.

Subtraction. Read this
Operations and Algebraic Thinking
• Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
What has subtraction got to do with taking apart ???
(The examples are all of the form 9 = 3 + 6 and so on).

Also there is NO mention at all of subtraction as a way of finding the difference between two numbers, or of finding the larger of two numbers (anywhere).

While I am in critical mode I offer two more, less awful, horrors from Grade 1:

“To add 2 + 6 + 4,…”  and  “For example, subtract 10 – 8″.

The poor symbols are clearly in great pain at this point. Just read aloud exactly what is written…..


Filed under algebra, arithmetic, language in math, operations, teaching

Commutative, distributive, illustrative-ly

Here they all are, apart from associative, as it belongs to algebra.

gif commutative add

gif commutative mult

gif distributive law


Filed under abstract, arithmetic, language in math, operations, teaching

Language in math, again.

“Is” is a very overworked word, to the point of illogicality.

which of these both

Technically in both cases none of them.

In everyday language we can get away with the question and accept the answer “The first one” even though it is actually a picture of the head of a dog.

In math we MUST be more precise, and ask “Which of these graphs is the graph of a function?”, or “Which of these graphs could represent a function?”.

A graph is NEVER a function, and a function is not a graph. If we actually followed the Common Core on this it would be even more troublesome, as a graph is DEFINED as a set of ordered pairs as below —
Functions 8.F
Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly
one output. The graph of a function is the set of ordered pairs
consisting of an input and the corresponding output.
But at least WE all know what a graph is…..or do we?


Filed under language in math, teaching

CCSS and Standardized Testing – Who shall pass and who shall fail ?

This excerpt is from the following:


by Anthony Cody, and you should read it.

The Department of Education in New York convened a panel of educators to set cut scores on the new Pearson Common Core-aligned tests. This article  http://www.lohud.com/story/news/education/2014/07/26/common-core-cut-scores-examined/13219981/  spilled the beans about the process.

Tina Good, coordinator of the Writing Center at Suffolk County Community College, said her group produced the best possible cut scores for ELA tests in grades 3 to 6 — playing by the rules they were given.

“We worked within the paradigm Pearson gave us,” she said. “It’s not like we could go, ‘This is what we think third-graders should know,’ or, ‘This will completely stress out our third-graders.’ Many of us had concerns about the pedagogy behind all of this, but we did reach a consensus about the cut scores.”

The result was that this panel of professional educators provided the state of New York with the cut scores that meant only about 30% of the state’s students were ranked proficient.

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Congruence, proof, and rigid motions: The Common Core says WHAT, not HOW

With all the stuff in the high school geometry about proving congruence by rigid motions we get this sample geometry question from PARCC
PARCC geom test proofdef
(get the rest from numberwarrior here)

Numberwarrior’s concerns are about the language and the formal properties of congruence and I agree with him on this.
My concerns are about the stated claims of the CCSS to specify the “What do the need to know/understand/be able to do”, and the PARCC test which says “This is HOW you do a proof”.
In this particular example there are other ways of proving the assertion, not least those using the definition of congruence by rigid motions.

Let us do it this way:
1: vertical angles are equal, as there is a rotation of line AD to GC through the angle CBD, and then AD is on top of GC, so angle ABD ABF is also the angle of rotation, and is therefore congruent to angle CBD
2:There is a translation of line HE to line AD, as they are parallel. So the translation of H to H’ puts H’ on the line AD, and so angle H’BF is congruent to angle ABF.
3: But angles are preserved by rigid motions, so angle H’BF is congruent to HFG, and therefore angle ABF and HFG are congruent.

So, if I chose to teach about proof using this approach (my “HOW”) the students won’t even understand the question. Test items MUST be “Method Free”.

Also, the so called Reflexive, Symmetric and Transitive properties of congruence are no different from a=a, if a=b then b=a, and if a=b and b=c then a=c for numbers, and in both situations these are so STUNNINGLY obvious that it is cluttering up the minds of the learners to burden them with this sort of stuff. It is clear to me that this is a contribution to the CCSS from the sole pure mathematician on the committee.


Filed under education, geometry, teaching

Commutative, associative, distributive – These are THE LAWS

Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.

I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c

But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.

The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.

It’s not surprising that kids have trouble with negative numbers!

Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?

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Filed under abstract, algebra, arithmetic, education, language in math, teaching

“Why do Americans stink at math?” Your weekend MUST-READ

I reached this long but well written and quite compelling article by Elizabeth Green via Dan Meyer’s blog. Everyone should read it, especially education administrators and decision makers, all the way to politicians.


Here are two excerpts:

With the Common Core, teachers are once more being asked to unlearn an old approach and learn an entirely new one, essentially on their own. Training is still weak and infrequent, and principals — who are no more skilled at math than their teachers — remain unprepared to offer support. Textbooks, once again, have received only surface adjustments, despite the shiny Common Core labels that decorate their covers. “To have a vendor say their product is Common Core is close to meaningless,” says Phil Daro, an author of the math standards.

Most policies aimed at improving teaching conceive of the job not as a craft that needs to be taught but as a natural-born talent that teachers either decide to muster or don’t possess. Instead of acknowledging that changes like the new math are something teachers must learn over time, we mandate them as “standards” that teachers are expected to simply “adopt.” We shouldn’t be surprised, then, that their students don’t improve.

Here is Dan’s post


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Filed under abstract, arithmetic, language in math, Uncategorized