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I've tutored math SAT prep for some years now, and have developed a routine with certain texts and exercises that I have found to be fairly successful in improving scores overall. I recognize that many students face a challenge in translating word problems into algebraic expressions, and am usually able to make progress toward this problem with most of my students.

However I am currently tutoring a student for whom this issue presents a greater challenge than I've encountered. I say "greater challenge" because my metric is the disparity between his comprehension of and familiarity with the underlying algebra (which is fairly strong) and his ability to put word problems into algebraic terms (which is almost non-existent). I've looked online for instances of this disparity occurring at such a remarkable level, but have only found literature on the general difficulty faced by many children.

Does anyone have any experience with this sort of problem, and are there any techniques, exercises, methods of explanation, etc. that have exhibited notable success toward remedying such an issue?

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    $\begingroup$ I'm just curious: are the word problems in the same language as the first language of the student? $\endgroup$ – Joel Reyes Noche Oct 25 '17 at 13:43
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    $\begingroup$ I'm intrigued. This sounds like a case study waiting to be published (as done in medical circles). $\endgroup$ – Daniel R. Collins Oct 25 '17 at 15:11
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    $\begingroup$ Does the student exhibit a general difficulty in the area of written language, i.e. not limited to language about mathematics? $\endgroup$ – shoover Oct 25 '17 at 16:51
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    $\begingroup$ I asked about rudimentary levels because there must be gaps somewhere along the way and you need to find out what they are. He may be high school level calculus but some part of him is way below that. Therefore you have to find out where the disconnect occurs. Sometimes the way to do that is to find out what level he missed. In your example, does he know what a linear relationship is? Can he translate that to they are on the same line? etc. $\endgroup$ – Amy B Oct 25 '17 at 19:22
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    $\begingroup$ I've worked as a math teacher in a technical school (hotels and tourism), some students were not interested at all. At the age of 14, one of them was asking "But sir, why are you always working with letters instead of numbers?" as I worked with variables for generality reasons. I've explained it to her again, but she simply did not want to understand it: once the difficulty reached a certain level, the limit of her brain/ willingness to understand were reached, and I just bumped into a wall. Some students simply lack the ability to understand even simple things, whatever effort you spend. $\endgroup$ – Dominique Oct 26 '17 at 9:53
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Recently, I got to teach a new high school class called "Essentials of College Math." It sounds like hard content, but it was basically just a transition course from Algebra II to Precal. Anyways, this is the teacher's manual for the course (which also includes all the student handouts). The first unit was called "algebraic expressions" and dealt with this topic of translating words to equations quite a bit with some good activities, especially the card sorting activities. I found Task #5 ("the swimming pool") to be quite useful. In fact, I did a whole 2 or 3 days after that task just devoted to dot patterns (from visualpatterns.org) and having students create their own expressions for the patterns. We proved that two different expressions generated the same pattern. I would also give them a list of different solutions and ask them where they see the different aspects of the expression in the visual picture. (similar to this task, I found Levi's comment on the task particularly insightful).

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  • $\begingroup$ "Entry Event: Play initial clip of Bucky the Badger." - did you say it was a high school book? $\endgroup$ – Rusty Core Oct 2 '18 at 0:24
  • $\begingroup$ Yes, this is a high school book. What is your question about Bucky the Badger? Did you have a look at the task? $\endgroup$ – ruferd Oct 2 '18 at 12:06
  • $\begingroup$ I had not looked. I looked just now. This is a brutal progression. Just did not expect to watch videos in a high school math class, but if it works than hats off. $\endgroup$ – Rusty Core Oct 2 '18 at 18:58
  • $\begingroup$ I'm not sure what you mean by "a brutal progression," but the Bucky the Badger activity did work quite well when I used it my class. Anything that helps to get students arguing / debating / discussing mathematical ideas with each other is a success. $\endgroup$ – ruferd Oct 2 '18 at 19:21
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I suggest you break it down into smaller parts. In fact, don't start on equations. Start on translation of terms:

  • Double of a number, translation: $2x$
  • Sum of two numbers, translation: $x+y$
  • The triple of a number after adding five units, translation: $3(x+5)$

Next combine translations into equations:

  • The double of a number is equal to the triple of another, translation: $2x = 3y$
  • The sum of two numbers is equal to the triple of the first after adding five units, translation: $x+y = 3(x+5)$

Make sure you work through many standard terms before moving on.

A table may be helpful: one column for the word terms and another for the algebraic translation.

Treat each line of the table as one translation step as you read a word problem: start with "The double of a number...", and the second column reads $2x$; then continue with "...minus 30...", and the second column reads $2x-30$; complete with "...equals the number.", and the second column reads $2x-30=x$.

Sounds tiring, but worth the effort if your student understands and internalizes the process to the point of no longer needing such device.

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    $\begingroup$ I like this answer a lot; the goal would be a realization by the student that many statements in English can be translated into algebra. $\endgroup$ – Chris Cunningham Oct 26 '17 at 5:59
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    $\begingroup$ I don't know about more recent texts, but it was fairly standard in older beginning algebra texts (before late 1950s, when new math entered the scene) to have an introductory section or two, sometimes an entire chapter, devoted to translations from verbal to mathematical (both expressions and equations). For example, the first 23 pages of this book from my bookshelves deals with this, but unfortunately it doesn't seem to be freely available on the internet. But coming up with practice examples should be easy. $\endgroup$ – Dave L Renfro Oct 26 '17 at 16:55
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    $\begingroup$ @DaveLRenfro: This is still common in algebra texts today. Each of the half-dozen or so recent algebra texts on my shelf does this. For example: OpenStax Elementary Algebra, Section 1.2, "Use the Language of Algebra": cnx.org/contents/CImQfPDv@2.44:faHRo7tY@3/… $\endgroup$ – Daniel R. Collins Oct 27 '17 at 14:13
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    $\begingroup$ @Daniel R. Collins: The reason I used the cut-off of the late 1950s is that the very few more recent beginning algebra texts I have are from the 1960s to early 1970s, and translation exercises such as these are mostly absent in them. I don't have anything more recent at the beginning algebra level, so I was playing it safe by saying "I don't know about more recent texts". $\endgroup$ – Dave L Renfro Oct 27 '17 at 16:15
  • $\begingroup$ I wonder how many students might think that "the triple of the first after adding 5 units" would be 3x + 5. We are used to some very formalized ways of specifying things. But most people aren't at all, and don't see much difference without a lot of experience. $\endgroup$ – Carl Offner Sep 28 '18 at 23:27
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You need to make sure he really understands what kind of actions each of the basic operations serve as a model.

So before algebraization of problems, try to work on identify the actions involved. Next, Singapur bar models can be useful as an intermediate way of modelling what's happening, specially on the relevance they put un identifying the total involved.

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  • $\begingroup$ Mind elaborating on what Singapur bar models is? I'm not familiar with it and so a reference would be great. $\endgroup$ – Chris C Nov 2 '17 at 16:02
  • $\begingroup$ Here is a nice guide to it thedailyriff.com/WordProblems.pdf But they're a model halway between pictorical and abstract representation of a situation, where you can have one or more bars, usually one of them representing some kind of total, and it's split according to the parts involved forming this total or comparison. $\endgroup$ – leapedalea Nov 3 '17 at 22:14
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    $\begingroup$ Do you mean Singapore bar models? $\endgroup$ – Joel Reyes Noche Feb 1 '18 at 12:59

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