I'm a fairly new private math tutor, and I'm good at math (I have a BS from Caltech with lots of graduate level math), but becoming good at teaching math is something else, which I strive to improve at every day.
I've starting working with my youngest student so far, a 6th grader. I'm dismayed by the problems in the way her class is taught. I'd like to know if this really is a problem or if I'm missing something.
Although I'm new to tutoring, I've been teaching non-math topics to adults for years. From that experience, I believe people learn best by starting in a secure place and moving by small steps. Of course they need to enter into new and potentially confusing situations, but the teacher can help them stay oriented to something solid along the way.
In the first session with my 6th grader, she was confused by word problems that tested her ability to know whether to use LCM or GCD.
The word problems seemed designed to be confusing. They read as if some teacher somewhere knew that kids get these two concepts confused in real-life models or word problems.
What I would do about this is introduce them as separate topics, probably spaced far in time. Let them develop a secure understanding of one before introducing the other. Only then mix them up.
However, this textbook introduced them at the same time and then gave them challenging problems in which it's hard to tell them apart.
That seems like exactly the wrong approach. I was able to help my student, however, by teaching her to differentiate between problems that involve "cutting up things" (GCD) and "extending things" or "laying things end to end" (LCM). Fortunately this was enough to solve her problem, in part because the language used by the word problems was consistent, so she wasn't being tricked.
The next two weeks, she was taught about multiplying and dividing by fractions and given word problems on those. The word problems asked her questions like "if I divide 2/3 of an acre of land into plots each 1/6 of an acre, how many plots do I get?" So that's division by a fraction.
But the problems also asked her to multiply by fractions, such as starting by referring to "1/6 of a garden" (of unspecified size) and then later mentioning the garden was 90 square yards, and asking what "1/6" of that is. So she had to multiply 1/6 by 90.
Having just been introduced to dividing by fractions, she wanted to divide 90 by 1/6.
I decided to teach her the difference between counting the number of pieces or sections, and finding the size of each piece. I ran through a bunch of simple examples, then started to model why division or multiplication would be appropriate.
But the next problem referred to a class of students, and asked "what is the number of students in the class?" Since I had just told her about counting the "number of" pieces, she naturally thought this was asking about counting sections or pieces. But it was really about the size of the whole class!
I think this is bad question-writing. If the sadists who designed these word problems were determined to confuse students, the least they could do is be consistent in their language.
This session was 90 minutes, because she needed so much comprehension built up from scratch, that's as far as we got. I skimmed the rest and there was no consistency. Each problem was a variation, without simple patterns that she could "hook" into to decide whether to add, subtract, multiply or divide.
I can only guess that the textbook writers think it's a good idea to help students differentiate concepts by mixing them together and making them as hard to discern as possible, but as I previously stated, I think that's bad teaching.
Looking at the biggest picture, most students don't grow up into jobs that require much math, so teaching hard math (and abstract math like algebra) is a kind of "rite of passage" that in my opinion, despite being employed as a math tutor, serves little purpose.
But given that our schools teach math (for possibly justifiable reasons), shouldn't they at least refrain from making it more confusing than necessary?
I'm fairly new to this, so maybe there's something I'm missing.