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.

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    It seems that the textbook you are using is targeted at stronger learners. Is there an option to use a textbook targeted at learners that need a more gentler approach? – Joel Reyes Noche Oct 16 at 10:17
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    @JoelReyesNoche I wish there was, but as a private tutor I have no control over the textbook used. The teacher gives additional homework which is even more confusing. My student reports the teacher offered sympathy and said that it's okay to get most everything wrong.. okay... but why not teach them in the way they can succeed? – composerMike Oct 16 at 10:35
  • I believe you that the text is nonoptimal and that a more progressive (meaning one by one, not meaning politically liberal) pedagogy would be better. Your question about "how can this be" shows naivete. There are a lot of poor pedagogy approaches. Look at whole language reading instruction. Net/net: just do your best and teach the kid. Break it down yourself. Find another problem source. Etc. – guest Oct 17 at 14:05
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    cM: One of the reasons you've been brought in is because the approach is nonoptimal. Like you said, you can't change the text, so stop worrying about it. Just do your best with the student you have. Nothing says you can't do some simple problems in your own sessions. Mix some in. Find another source. Adapt and overcome. I wouldn't complain about the text to the student because it will just confuse them. And I would spend part of the lesson on assigned problems. But mix in easier ones for 50% of the time. – guest Oct 17 at 14:08
  • @composerMike Would you mind naming the textbook the student uses at school? – Rusty Core Nov 27 at 18:11

I think there's a countervailing issue that the book you're describing is trying to deal with. I teach college students, so I don't know what the particular approach it's taking is age appropriate for a 6th grader, but it is trying to solve a real problem.

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.

The problem with this approach - and what I assume is exactly what the book is trying to avoid - is that you can't develop an understanding of (for instance) LCM in isolation. You can do a bunch of calculations, but if you try to ask problems which get at understanding LCM, students will just autopilot through them, taking the LCM of whichever two numbers they find.

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.

This is pretty clearly a pre-algebra problem. (Which I'm pretty sure is age appropriate; I think it's pretty common to start algebra in 7th grade, which makes 6th grade a good time to lay groundwork like this.)

Your student wants to just take numbers in the problem and combine them while ignoring the words, which is a pretty normal thing for students to want to do, but not the point of a word problem.

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.

If they used "consistent" language, students could shortcut through the problems by using those linguistic clues, rather than by understanding the situation. It sounds like that's exactly what your student is trying to do, so in that sense word problems of some kind are probably exactly on target. (Obviously I can't assess whether these specific ones are appropriate, and I imagine that's a hard thing to do, since it depends on how good the students' language skills are as well as their calculation skills.)

One thing I'd suggest is that it sounds like you're thinking of, for instance, LCM and GCD as the main topic here. I'm not sure the textbook agrees - it sounds to me like her textbook thinks the topic is word problems (that is, the process of interpreting situations mathematically), and things like LCM/GCD are incidental bits of content that provide topics to practice word problems on.

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    +1 excellent answer. OP must consider their goal as a tutor: "Learn LCM" and "Learn GCD" are likely not the best possible learning outcomes. The large-scale learning outcome we want is "Critically analyze a situation and choose a plan of attack." And the smaller-scale one for this situation is "Identify whether LCM or GCD are appropriate in a given context." – Chris Cunningham Oct 16 at 14:44
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    @ChrisCunningham, reflecting on it, I do think that I was trying to teach her to critically analyze a situation, but the textbook and teacher have already thrown her into the deep end of the pool. It's like saying, "Okay, we just threw this scared, struggling student into a raging river. Now help her not drown." There are so many better ways this could be done that would be truly successful. Perhaps schools need to evaluate what is realistic for most students and change the curriculum to something they can succeed at. – composerMike Oct 16 at 20:00
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    @composerMike: I'm skeptical that using language cues is actually scaffolding; I think most students learn an entirely different non-skill, memorizing some specific stock phrases, from such problems. I strongly disagree with the claim that it never makes sense to be confusing. Learning new things is confusing, and being confused is an ordinary stop on the road to understanding. (Of course, it may still be that your student's book is too confusing, or not age appropriate, or that your student has been dropped in too deep relative to her preparation.) – Henry Towsner Oct 16 at 20:44
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    @composerMike Many people who enjoy math assume that when someone finds "a mindless rule helpful" that they will later form a deeper understanding. This is only true for the tiny share of the population that likes math. A far more common response to "helpful mindless rules" is reinforcement of the notion that math is nothing more than mindless rules. Rote procedural testing further reinforces this. Hence, even in many post-secondary institutions, people still confuse $3\div 12$ vs $12\div 3$ (and equivalent mistakes) or, worse, they don't even know when or why they should divide at all. – WeCanLearnAnything Oct 18 at 15:09
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    @composerMike One more downside of teaching a mindless rule: Once students have memorized them, many will want to use that mindless rule exclusively. Many will tune out - if not angrily resist - any attempt to help them understand the rule. In the students' minds, they have a reliably mindless answer-getting heuristic, that's all they're tested and graded on, that's all math is, so your offer to help them understand is pointless. If you insist on understanding, they will find it infuriating. This may not have been our experience with math, but is the experience of most students. – WeCanLearnAnything Oct 18 at 16:31

I feel like I always post the same thing in these threads, but this again sounds like an issue of blocking vs interleaving. In this case, the textbook may have started interleaving different problems a little bit too early, but generally it should be introduced earlier than feels intuitively right.

That's because the algorithms for $\frac{1}{6}\times 90$ and $90\div\frac{1}{6}$ are not that tough to execute, but knowing what they mean and when to do them is the hard part. [If you don't believe me, just go to a community college and ask a bunch of students if the prior 2 expressions are greater than 90 or less than 90 or if it's not possible to tell and how confident they are.]

Here's a partial solution to your current textbook problem. Tell her:

  1. To practice algorithms on her own or with another book that has an answer key.
  2. Use your time with her to practice sketching and/or parsing the word problems. "Instead of calculating answers, today we're going to just read the word problems and you're going to tell me if LCM is relevant to each word problem or not and how you know."

This way, she does the easy stuff on her own and the much harder, more abstract stuff with you.

I would say that this book is trying to get students to think about what the calculations mean, rather than simply execute an algorithm, and I strongly believe that is something that should be done right from the beginning, not saved for later or for more advanced students.

I cannot tell you how many students I have worked with as a private tutor who, if you ask the question "How many pencils do you need if you want to put 7 pencils on every table and there are 12 tables?" will assiduously count by sevens until they reach 84... but if you ask those same students just five minutes later "What is $7\times12$?" they will immediately write down the standard algorithm for multiplication and find that the answer is 84. And then they look at those two questions, and their identical answers, and it never occurs to them that they are connected.

For too many students -- I am even tempted to say most! -- operations like multiplication, division, finding the LCM or GCD, etc., are "things you do with numbers" that are entirely disconnected from any meaning or sense. Everything we do as teachers or as tutors should be about making math reasonable, not just easy to learn.

  • I guess this answer is a bit ranty, but I'm in a ranty mood. – mweiss Nov 27 at 22:58
  • The word "rant" has been used very many times in math discussion groups, and I would think the less common version "ranty" would still show up a lot in a search, even just restricted to sci.math (I get 22 hits for all groups, using just one of my email identities), but according to this Math Forum search, "ranty" shows up only twice in sci.math (this being since 1996, when Math Forum's archive for sci.math began) and none for any of their other discussion groups. – Dave L Renfro Nov 28 at 10:31
  • "will assiduously count by sevens until they reach 84" - if you are talking about 6-yr olds, then it's ok. – Rusty Core Dec 1 at 0:23
  • @rustycore I'm talking about middle school students, as high as 8th grade – mweiss Dec 1 at 23:49

Even though your question seems like a rant about textbooks, I'm assuming that you are looking for advice on how to deal with this situation. So let me give my opinion on the topic:

From what you are telling, it sounds like this textbook wants to be very clever and do two things at once. When teaching these things like, for example, GCD and LCM, there are two things to do:

  1. Understand how to compute them.
  2. Understand when to use which one.

If you make such challenging tasks, you can do both with the same exercise. The students first have to figure out which one to use, and then they have to do a (likely difficult) computation to get the result.
I think your approach to split both points up to teach them to a struggling student is good. I would suggest to take it even further and, after the student understood the algorithms and concepts, start really slow with the second point, so instead of taking $1/6$ of a $90$ square yards, give her half a cake (real cake or figurative cake, you decide^^) and then let her eat half of it, so that she is left with $1/2*1/2 = 1/4$ of a cake. Make her understand the differences and develop a feeling for the concepts with such trivial examples, and teach her how to retract to such cases, e.g. "replace every integer by $1$ and every fraction by $1/2$ to figure out if you have to multiply or divide. Once you know that, do it with the actual numbers given in the problem."
Furthermore, it can also help to teach students how to check their solutions and develop an intuition for such things. Many students, especially struggling ones, are relieved when they finally have a solution to write down, so they don't even realize that "$1/6$ of the garden of size $90$ is $720$" sounds a little strange.
There are many different ways to develop that intuition, you might have her draw images, use the number line, etc. Just make sure that it is not you telling her "this is wrong, because..." but, whenever possible, ask her to verify her result. Of course also ask about verification of true results, don't let her associate verification with wrong work.

TL;DR version: Teach your student to verify her own results and help her to develop an intuition. Whenever possible, only give hints and let her do all the work. Do not teach algorithms blindly, always aim for an understanding of the underlaying ideas and concepts.

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    I like your answer, especially the part about replacing numbers in the problem with simpler numbers to check the logic and algorithm. That's an example of the "solve a simpler problem" strategy in "How to Solve It" by Poyla. – composerMike Oct 16 at 20:02

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.

LCM - the smallest number that can be divided by other numbers. GCD - the largest number onto which other numbers divide.

In school math GCD is used for simplifying/canceling fractions, which can be done AFTER the main portion of the exercise is done. This means one can simply forget about GCD when doing exercises until very end (or until the expression gets too bulky). LCM is used to find least common denominator when ADDING fractions. Again, can be completely avoided simply by multiplying the denominators. Ultimately, both can be put aside until/unless some simplifying/canceling is required.

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).

LCM and GCD are tools like hammer and saw. Your student should decide wheter to nail or to cut before using tools.

"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. Having just been introduced to dividing by fractions, she wanted to divide 90 by 1/6.

She does not understand the gist of the exercise, and neither her school teacher nor you are helping her. Her school teacher simply throws more exercises at her not even caring whether she solved them. Why? To get 50% or even 70% for "participation" and to raise overall class performance, from which teacher's own performance depends. Ugh.

$\frac{big\ chunk}{number\ of\ portions}=portion\ size$ is obvious even for a layman. Re-arranging the expression, we also get ${number\ of\ portions}\times portion\ size=big\ chunk$ and $\frac{big\ chunk}{portion\ size}=number\ of\ portions$ the less obvious variants of this formula. Here big_chunk can be a portion itself, dimensionless, or a whole, with a dimension.

Number of portions is usually positive integer, and has no dimension. The whole can be a portion of some other whole, dimensionless, then the result also will be a portion of that whole, no dimension. Or the whole can be measured in particular units, then the portion will also be in these units. This is a good time to explain dimensional analysis, like mulitplying yards by yards gives square yards, or dividing square yards by dimensionless number gives square yards.

The 90 square yards is the whole, with dimension. 1/6 is a portion. We need to find size of a portion, it will have a dimension. Pure mechanical solution after figuring out what to do.

The (2/3) / (1/6) exersize is that we are distributing 2/3 of the WHOLE parcel into 1/6 pieces of the SAME WHOLE parcel. Or you can do it with pizza: if we share 2/3 of a pizza, slicing it into 1/6 subslices of the WHOLE pizza (no one will do it in real life), how many subslices we get.

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