Mixing
Don't these word problems seem designed to be confusing?
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
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.
fractions word-problems
asked 6 hours ago
composerMike
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
1
down vote
favorite
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.
fractions word-problems
asked 6 hours ago
composerMike
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
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
4 hours ago
1
@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
4 hours ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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.
fractions word-problems
asked 6 hours ago
composerMike
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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.
fractions word-problems
fractions word-problems
asked 6 hours ago
composerMike
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 6 hours ago
composerMike
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 6 hours ago
composerMike
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 6 hours ago
composerMike
1212
asked 6 hours ago
composerMike
1212
1212
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
composerMike is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
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
4 hours ago
1
@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
4 hours ago
add a comment |Â
1
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
4 hours ago
1
@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
4 hours ago
1
1
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
4 hours ago
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
4 hours ago
1
1
@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
4 hours ago
@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
4 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
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.
answered 1 hour ago
Henry Towsner
6,1332144
1
+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
19 mins ago
add a comment |Â
up vote
1
down vote
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:
- Understand how to compute them.
- 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.
answered 3 hours ago
Dirk Liebhold
78927
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
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.
answered 1 hour ago
Henry Towsner
6,1332144
1
+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
19 mins ago
add a comment |Â
up vote
3
down vote
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.
answered 1 hour ago
Henry Towsner
6,1332144
1
+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
19 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
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.
answered 1 hour ago
Henry Towsner
6,1332144
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.
answered 1 hour ago
Henry Towsner
6,1332144
answered 1 hour ago
Henry Towsner
6,1332144
answered 1 hour ago
Henry Towsner
6,1332144
answered 1 hour ago
Henry Towsner
6,1332144
6,1332144
1
+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
19 mins ago
add a comment |Â
1
+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
19 mins ago
1
1
+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
19 mins ago
+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
19 mins ago
add a comment |Â
up vote
1
down vote
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:
- Understand how to compute them.
- 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.
answered 3 hours ago
Dirk Liebhold
78927
add a comment |Â
up vote
1
down vote
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:
- Understand how to compute them.
- 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.
answered 3 hours ago
Dirk Liebhold
78927
add a comment |Â
up vote
1
down vote
up vote
1
down vote
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:
- Understand how to compute them.
- 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.
answered 3 hours ago
Dirk Liebhold
78927
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:
- Understand how to compute them.
- 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.
answered 3 hours ago
Dirk Liebhold
78927
answered 3 hours ago
Dirk Liebhold
78927
answered 3 hours ago
Dirk Liebhold
78927
answered 3 hours ago
Dirk Liebhold
78927
78927
add a comment |Â
add a comment |Â
composerMike is a new contributor. Be nice, and check out our Code of Conduct.
composerMike is a new contributor. Be nice, and check out our Code of Conduct.
composerMike is a new contributor. Be nice, and check out our Code of Conduct.
composerMike is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmatheducators.stackexchange.com%2fquestions%2f14672%2fdont-these-word-problems-seem-designed-to-be-confusing%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
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
4 hours ago
1
@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
4 hours ago