My daughter was frustrated doing her 5th grade math homework. She had to add fractions with different denominators. She was confused for several days on this topic so I asked how her teacher showed her to solve them. She couldn’t say. I asked if she used Least Common Denominators and she looked confused.
My teaching methods are dated, true. I am 45, have an undergraduate degree in mathematics, and taught Developmental Math, aka remedial math, for college students. We consulted her student math handbook Investigations which has ten pages on fractions, two on addition. This 114 page book is a supplement to her text, kept at home for reference. Below you can see the five examples for adding fractions under the content title, Math Words and Ideas.
Samantha used a shaded strip method, Renaldo used percent equivalents, and Tamira used a number line. These are valid concepts to understanding, so we read on to the next page.
Deon used the clock model and Yumiko used the ever popular shaded strip method again. My daughter said they did examples like this in school. I turned the page, checked the Table of Contents. That’s it. Nothing on Least Common Denominators (LCD).
No wonder there’s confusion.
In addition to the variety of “ideas” suggested, there are a variety of students, both genders and multiple races: Asian, Hispanic, African American and white.
We have politically correct fractions.
Inclusion is good, but what’s the lesson about? Perhaps the focus should be how to add unlike fractions.
Here’s the rub. The method is not hard.
This is my daughter’s Saxon Math Homeschool text from 4th grade. It is not as colorful and there are no politically correct student names either, yet she learned to find a common denominator last year! Then summer came and went and public school math has replaced and erased any memory of it. She does have ideas about shaded strips, clocks, and number lines. These are a good place to develop conceptual understanding of fractions, but they are not a substitute for learning how to add unlike fractions.
These examples remind me of an Investigations style gym class where the teacher discusses the benefits of running, types of running, and the mechanics of the stride. At some point, the child must get out and RUN. Are we surprised to learn the childhood obesity rate is one out of three? Then again, the remedial mathematics rate in college is 40%. This means 4 out of 10 students attending college are not ready and need developmental math instruction. I know because I taught this class. The 40% is only for math remediation by the way, the general remediation rate for college is 60%! (National Center for Public Policy & Higher Education brief) If you have concerns about English today, check out my post on the amateur psychology which substitutes for literary analysis. (Literary Analysis or Amateur Psychology?)
I don’t mean to be cynical. I consider myself open-minded, willing to try new concepts. So when I attended the math parent meeting in October, I hoped to be enlightened. The district math expert gave us a slide presentation. She explained they were already doing everything for Common Core Standards and that memorizing math facts didn’t work for many kids, most kids I think she said. For example, she said 8 plus 5 is a tough fact to remember. So they had students add 8 to 2 to get 10 then add another 3 to get to 13. She dismissed concerns and criticism with the sweep of her hand and the announcement that she’d been doing this ump-teen years and knows.
Using Saxon math in homeschool, my children completed facts practice which they timed and graded every day, then recorded the results. They improved. At the beginning of the year, my daughter took 4 minutes and 30 seconds to complete 100 addition facts and got 99 correct. At year’s end she got 100/100 correct in 1:44. My son’s 64 multiplication facts took him 4:26 to complete 63 correctly. At year’s end he completed them in 2:40, 64/64 correct.
Practice. Time on task. There are studies and books on this topic now, the concept of deliberate practice and 10,000 hours. We’re not asking for 10,000 hours, but we should demand competence.
I disagree with the school math expert because I believe students can memorize as well as understand. When I taught remedial math, it was shocking to learn how many of my college students didn’t know their basic math facts: addition, subtraction, multiplication, and division. I had them tab their appendix for the fact charts and forbid calculator use. These were college students who needed basic math skills and pre-algebra, grade school and middle school math! I had 35 students per class and yes, there were some who should not have been in college. The rest, the majority, could do it. They had to practice.
During my daughter’s math open house, the young teacher was giddy over the curriculum. If and when students didn’t understand concepts she would spend time with them and she pointed to the back of the class. There were a bunch of pillows on the floor where she would sit to review mistakes and problem areas. In addition to a lot of bureaucratic stuff and Common Core slides, she pointed to the key areas of focus this year; on the wall in bright colors and bold letters were the words, READ, THINK, PERSEVERE.
I asked my daughter if she ever sat on the pillows with her teacher. She said other students often did, but she only had to once. When I asked what problems she had, she told me fractions.
Reading, thinking, and persevering. Nice “words and ideas” but if you’re teaching my daughter math, maybe you can start by teaching her how to add unlike fractions.
I leave you with the words of Richard Mitchell, the famous, the infamous, and sadly the now deceased classics professor and “Underground Grammarian” who understood the problem with education better than anyone else.
A colleague sent me a questionnaire. It was about my goals in teaching, and it asked me to assign values to a number of beautiful and inspiring goals. I was told that the goals were pretty widely shared by professors all around the country.
Many years earlier I had returned a similar questionnaire, because the man who sent it had promised, in writing, to “analize” my “input.” That seemed appropriate, so I put it in. But he didn’t do as he had promised, and I had lost all interest in questionnaires.
This one intrigued me, however, because it was lofty. It spoke of a basic appreciation of the liberal arts, a critical evaluation of society, emotional development, creative capacities, students’ self-understanding, moral character, interpersonal relations and group participation, and general insight into the knowledge of a discipline. Unexceptionable goals, every one. Yet it seemed to me, on reflection, that they were none of my damned business. It seemed possible, even likely, that some of those things might flow from the study of language and literature, which is my damned business, but they also might not. Some very well-read people lack moral character and show no creative capacities at all, to say nothing of self-understanding or a basic appreciation of the liberal arts. So, instead of answering the questionnaire, I paid attention to its language; and I began by asking myself how “interpersonal relations” were different from “relations.” Surely, I thought, our relations with domestic animals and edible plants were not at issue here; why specify them as “interpersonal”? And how else can we “participate” but in groups? I couldn’t answer. (Less Than Words Can Say, Richard Mitchell)
(Read the full essay here: Less Than Words Can Say)
A very good analysis showing experience and common sense. I also have degrees in math with years of continuing education through private employment. I believe that the fundamental childhood learning process for both reading and math begins with memorization then comprehension. I had to memorize the alphabet and words before reading and single digit facts before the next levels of math. I didn’t need to understand or think of 20 ways to describe the number 20.
Arthur, Thank you for reading the TreeHouseLetter. I’m glad to hear this from another math major and I agree that children must learn the basics before they progress. I subscribe to the classical model of education with a grammar or knowledge based stage, followed by logic, then rhetoric. It’s a tragedy that educationists skip the first stage entirely, no grammar or math foundation needed. We see the results.