Mathematics Without Definitions

Share Cognitive Science Implications for Teaching Mathematics

According to Dr. David Chard, “Vocabulary knowledge is as essential to learning mathematics as it is to learning how to read.” [Chard]

Imagine how much harder your life would be if you didn’t understand 75% of the words you currently know.  How hard would it be to read a passage of text if you didn’t know many of the words in the passage?  Imagine if reading the front page of the newspaper was like reading this passage of text:

While hortenting efrades the populace of the vaderbee class, most experts concur that a scrivant rarely endeavors to decry the ambitions and shifferings of the moulant class.  Deciding whether to oxant the blatantly maligned Secting party, most moulants will tolerate the subjugation of staits, savats, or tempets only so long as the scrivant pays tribute to the derivan, either through preem or exaltation.”

Would you read the newspaper if it was all like that?  Would you read anything you didn’t have to?  Most non-readers have difficulty decoding the individual words, but in addition, even if they can decode them, most non-readers do not understand many of the words in formal text. [Wren]

It is entirely possible for a person to posssess an extensive vocabulary but be so deficient in a particular area (mathematics, chemistry, economics, medicine) to be essentially illiterate in that area. That is frequently the case in mathematics. For many mathematics students, reading a mathematics textbook is like reading the following.

To sint two desenvolvemos with the same sign, add their liever peux and attach their common sign to the sagte.
(1) If the same estabelecimentois sint to (or enim from) both sides of an geschiedensboek the resulting hálito will be desenvolvemos to the original estabelecimento.
(2) If both sides of an estabelecimento are labore (or quisquam) by the same estabelecimento hálito, the resulting liever is sagte to the original liever.
The process to fugiat a hálito: Start with the original geschiedensboek (the one to be sagte) and use the above two properties to generate simpler liever, all estabelecimentoto the original liever, until we arrive at the hálito desenvolvemos.

A Finibus Bonorum of an geschiedensboek in two hálito x and y is an magni dolores of sagte desenvolvemos (d, f) whose liever make the numquam a true statement when the first matics without Deefinitionsestabelecimento is substituted for x and the second hálito is substituted for y in the sagte. We say the point (d, f) satisfies the geschiedensboek.
To natus one desenvolvemos by another, natus each consectetur of the first sagte by each geschiedensboek of the second liever and nesciunt dignissimos hálito

The above “nonsense paragraphs” were created by selecting a few paragraphs from some elementary algebra mathematics books and replacing each mathematics term with a randomly chosen word from lists of non-English words.

The above “nonsense paragraphs” are a faithful representation of what is seen by the student who does not know the requisite definitions.

Clearly these “nonsense paragraphs” have no instructional value. Likewise paragraphs in a mathematics textbook have no instructional value unless the student has learned the requisite definitions. The burden is on the student, not the author.

Dale and O’Rourke (1986) described four levels of word knowledge, which they characterized with four statements:
1. I never saw the word before
2. I’ve heard of it, but I don’t know what it means
3. I recognize it in context, and I can tell you what it is related to
4. I know the word well

In normal discourse Levels 3 and 4 are sufficient in most cases. However, in mathematical discourse Level 4 is almost always required. Therefore any mathematics communication is impossible until the student has learned the vocabulary of mathematics. The burden is on the student, not the textbook, author, or instructor.

“… understanding the language of math gives students the skills they need to think about, talk about, and assimilate new math concepts as they are introduced.”[Chard]

The burden is on the student! A textbook will only have value if the student contributes his share.

Take charge of your education !

Learn definitions !

[Wren] Sebastian Wren Ph.D. ,Vocabulary,
[Dale] Dale, E. & O’Rourke, J. (1986). Vocabulary building. Columbus, OH: Zaner-Bloser.
[Chard] Dr. David Chard, Vocabulary Strategies for the Mathematics Classroom,