Don't these word problems seem designed to be confusing?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

My mathematical teaching experience is limited to helping my kids with their homework, but have you considered getting her to draw the problems? Pizzas, chocolate bars, stickmen, whatever - depending on the context.

Answer 5

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.

Answer 6

Just read this post and see why students still fail at math. The first guy after reading your response confidently claimed ,’no, not a problem with the confusing language the problem is you have a slow learner’.

The mark of a great teacher is defined by their results not theories or assumptions of why a student can’t understand a particular subject. If your student fails it’s because you failed them.

It’s so easy to blame the student because it’s convenient but it takes true intelligence like you yourself must have to question the status quo and see if it can be improved upon.

One user said for example,’oh if you make it simple they’ll just use that same routine without ever thinking analytically (in a nutshell) .

It blows my mind. Grown adults who’ve went off, studied at prestigious schools, raising future generations , who in the face of the obvious say ‘no nothings wrong with the material, how it’s worded, it’s the student’

Lazy, ineffective, and worthless. Just waiting on their pension marking off students as smart, slow, or dumb because deep down they really can’t teach , they just have a degree.

Having a degree in education and actually being a teacher who can provide results are two different things.

You’re on the right track. Disregard all these others.

Find ways to show how all this complex is easy. Model/guide , quiz in the moment like a drill, allow student to think for themselves (a human will naturally be analytical when trying to solve a problem therefore it’s stupid to believe forcing a particular wording is the only way to bring this about and that’s the problem the wording)

Likewise make it fun. This whole learning isn’t supposed to be fun comes from people who have degrees but can’t teach that doesn’t sound like you it sounds like you have what I wish every student had someone who got it.

It just goes to show this is why the education system is the way it is. No one wants to say it but these teachers suck.

They’re boring. Can’t put together a creative, stimulating, and challenging exercise to produce a result to save their lives.

Don’t be like them, trust me years later you have no idea how grateful your students will be some you’ll mark forever because you did it differently.