Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

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Math Fragments Perpetuate Fragmented Learning

Posted by mark schwartz on June 8, 2016


Carla (not her real name) indicated that she had missed a lot of education because in the religious school she attended, time was spent on theological not academic issues. She has very fragmented math information and it seems that she tries to take any new material and fit it into the fragments she knows. She seems to have a need to preserve what she knows, not realizing that it interferes with learning. It is likely not a conscious act.

The Story.

An example of this is in her work on a quiz. I gave a bonus question. Typically, they aren’t too hard but do incorporate some material that has been learned but also include something that if you are aware, makes the problem somewhat easy.

The problem was

.25 ─ 1/2  + .4 ─ 1/4  + .5 ─  2/5.

Her work and her answer was

51/20 + 17/20 + 27/20 = 95/20 = 19/20

These numbers were puzzling until I realized that she had seen the problem as

25 1/2 + 4 1/4 + 5 2/5

The first thing to notice is the blurring of the “—“ sign. It disappears, thus giving her 3 mixed numbers to add.  She said that she saw the minus sign as dashes separating the numbers from the fraction.

Somehow one of the fragments she recalled was that mixed numbers can be converted into improper fractions, so the weight of that fragment also drove the disappearance of the “─”.  It is also possible, since it had been noted in class, that the notation of a mixed number “A b/c” really means “A + b/c”. This fragment seems to have stayed, but the “+” got replaced with “─”.

The next fragment that emerges is that in order to add fractions, you have to have a common denominator. I think this explains why there is a “20” in each denominator, rather than the 2, 4, and 5.  There was an awareness of this and she found the correct lowest common denominator, but the fractions weren’t converted to 20 correctly.

The decimal points seem to have been totally ignored; when we talked about this, she said she had never seen them before and when she saw them, she decided it was a printing error.

Finally, there is a fragment about reducing fractions. It was demonstrated that “canceling” really is looking for a common factor in numerator and denominator. Somehow, only the part about the numerator was operating.

Carla really wanted to learn math; she was motivated and trying very hard and I applauded her effort but talked to her about fragments and how the fragments were dominating her learning. When we talked about this she kept clinging to the fragments she knew because she had seen this material before and was sure that what was going on now had to fit into what she already knew. After presenting her with visual and kinesthetic procedures which didn’t offer her the chance to use her fragments, she slowly but surely started to restructure what she knew, rather than trying to restructure the new information to fit her fragments.

Recognizing a student’s learning problem sometimes has nothing to do with what we see, but rather is an invisible impediment to learning … and not easy to find, so keep snooping around.


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