from Myrtle:
I've only recently discovered this feature on Google. For example, I've been using this book for supplemental word problems in Algebra:
New Elementary Algebra (1879)
Word problems start on page 139 and go on and on and on.
and:
Fifty Famous Stories Retold for free in Google Books
I've spent my share of time pouring over Google Books these past few weeks. Sometimes they decide I've abused my privileges & they cut me off.
a farmer and his sheep
A farmer has in his pasture 63 head of cattle, consisting of cows, calves and sheep. There are twice as many calves as cows, and twice as many sheep as calves; how many has he of each sort?
Ans. 9 cows; 18 calves; 36 sheep.
New Elementary Algebra
p. 139
Reading this "ancient" algebra text corrected a misconception I had.
ReplyDeleteI had always assumed that the LEAST in least common denominator was part of new-fangled language reform the way that carrying became regrouping. I see now that LEAST was used then. I prefer "lowest". I am not sure why.
I was also struck by the sense of "reducing" as used in the ancient text. The way I understand reducing a fraction, the numbers in the numerator and denominator get smaller. When you change a fraction to conform to the LCD, the numbers get bigger but are still being "reduced". I am at a loss here. What phrase is used nowadays to refer to this operation (making an equivalent fraction with the LCD)?
I am not familiar with this property of "least common denominator." What page is it on? Is it the smallest prime that can divide both denominators?
ReplyDeleteI think we've only needed the concept of "least common multiple" and "greatest common factor." I used standard algorithms to teach how to come up with these (Singapore algorithms from NEM 1)but explained "why" they worked by using the definition of multiplicative inverses, and prime factorization.
[I am not familiar with this property of "least common denominator."]
ReplyDeleteMy comments were probably too convoluted. I am just quibbling over terminology.
The ancient text says "reducing" when it refers to rewriting one or both fractions so they have the same denominator for addition or subtraction purposes. I think of reducing in terms of simplifying. However, we do say reducing to the lowest common denominator in figurative speech.
If you know German, I am trying to render the German phrase "auf den gleichen Nenner bringen" in concise English.
But lawyering math terms is great entertainment!
ReplyDeleteLatest quote of my conversational partner from my house , "I can't believe we are discussing 19th century math pedagogy terms."
But the topic did happily occupy several minutes of time before that realization.
I find that my charges have the most trouble grasping adding and subtracting fractions with unlike denominators and finding the LCD using prime factors. Using LCM is easy in comparison.
ReplyDeleteResearch shows (there is that phrase again) that five out of four people have trouble with fractions.
Research shows that African-Americans have different learning preferences and therefore need fuzzy math. One reason is harmony with their communities.
ReplyDeleteThe excerpt is from the Journal for Research in Mathematics Education as cited by David Klein:
Studies of learning preferences suggest that the African American students' approaches to learning may be characterized by factors of social and affective emphasis, harmony with their communities, holistic perspectives, field dependence, expressive creativity, and nonverbal communication...Research indicates that African American students are flexible and open-minded rather than structured in their perceptions of ideas...The underlying assumption is that the influence of African heritage and culture results in preferences for student interaction with the environment and that this influence affects cognition and attitude...
Even though math education in the Chicago public elementary schools is pretty dismal (a preponderance of constructivist junk, especially for the disadvantaged), there are isolated (very isolated) pockets where real math is happening and variables are being isolated among other algebra doings.
ReplyDeleteI am tutoring a seventh grader in a school set in a high-SES area populated by ambitious yuppies. There, the kid is doing algebra word problems from McDougal Littell's Pre-Algebra book. KTM readers might enjoy seeing algebra word problems. Here is one from the McDougal Littell resource book the kid is doing (in summarized form):
A fellow buys roses for $30 a dozen. They cost more when bought individually. With the money he has, he can buy 7 dozen and 4 single roses, or 64 single roses. Question: How much is one rose? How much money does he have?
I get $3.5/rose. He has $224.00. I hope I didn't mess up.