Tag Archives: language

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

Arithmetic is the art of processing numbers.
We have ADD, SUBTRACT, MULTPLY and DIVIDE
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”.

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

The Future

“How’s your Mary doing?”.

“She’s doing well. She’s 8 now. She’s in Grade 3. She really enjoys the Pre-Algebra and the Pre-Textual Analysis.”.

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Filed under algebra, education, language in math, 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?

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

This excerpt is from the following:

http://www.livingindialogue.com/education-reform-lexicon-paradigm-busters/

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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Language and Math again

Here’s another one, from a grade 8 test, same source:

9. What is the decimal notation of 7×10-4?   ( they mean 10 to the power 4)

a. 70,000
b. 7,000
c. 0.00007
d. 0.0007

Answer and EXPLANATION!!!!!!!!!!!!!!!!!!!!

9. D: Because the exponent of 10 is -4, the decimal which is located behind the 7 will move 4 spaces to the left, and any of the empty spaces will fill with 0’s. so 7x 10? -4=0.0007

I hope that the computer system used to present these tests can cope better with “strange” symbols. I looked as well behind the 7 and could not see a decimal.

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