Houston, we have a problem
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The program is run by a non profit community organization which gets support from and seems to have some institutional connections with, a major university.
I am not a professional pedagogue by any means, though I do have some experience as a college instructor (computer science) and as the family "go-to" guy when family or friends need a boost in math or physics.
I am a retired EE volunteering in the GED program.
All of the students are older than "normal" high school age; I would say a plurality are young adults in the 20 to 30 yr old bracket, but some are middle aged and beyond.
Some students suffer only from lack of exposure to formal education. There is a sizable contingent of foreign born students who made it only through the fourth grade in the old country. Language is sometimes a bit of a stumbling block but basically I think they'll be OK.
There is another order of student that I just plain don't know what to do for or how to help. My co-teacher, who in her other life is a Social Worker, and a perceptive person, thinks that much of their problems are emotional and rooted in bad experiences with the "normal" education system, which explains in large part why or how they've managed to pass through that system without scoring a High School Diploma to show for it.
In a better world, I would hand those cases off to a specialist who knows, or at least is trained to deal with , the sorts of problems they bring to the table.
There are no such specialists available to them (or to us) in the program.
So for those students, the choice comes down to either:
1. me (or someone with the same lack of specialist training) or
So one part of me says "The magic word is
'triage'. Give help to those you can help and let the rest go. At least do no harm"
another part says "They've been badly treated so you're gonna pile on too, eh? Nice!"
So you guys are pros, what do you say? Can I do something non hurtful, and if so what and how, or, cruel as it may seem, is 'triage' the way to go?
Thanks for your hard work with those students!
We're given a list of items such as you might see on a shopping list.
For each item except one, there's a dollar amount.
A total is given.
The student is asked to find the price of the one and only item whose dollar amount is NOT given.
I take the student through the solution (The whole is equal to the sum of its parts), we add up the known
parts, we subtract that from the known total to get the missing part.
We do it (together) again with another list.
Then I show the same problem the n+1 th time, maybe with a variation (its so many miles from the start to the first stop, so many from the second stop the the third stop, etc. The end to end distance is given. Find the distance between the two stops where the distance has NOT been given. Nobody home.
We have a recipe. It calls for so many cups of milk (A given constant, lets say 3 cups) and makes so many cookies, lets say a dozen. But I only want to make half the amount of cookies; how much milk do I need? Nobody home. Actually now I'm not sure that if I had said "half" it would have been so bad. If I say 30%, for sure its spiral eyes time. But if I say what is 30% of (pick a number), no problem. It's not a case of her not knowing what a percent is or how to multiply a number by a decimal. This particular student is a native English speaker, so its not a question of not understanding what's being asked.
A couple of things to remember:
1. Even though they may be native English speakers, that doesn't mean they understood what you said or what is being asked (in fact, that's probably exactly where the problem is).
2. Most people struggle with word problems because they don't know how to get started and they don't have enough confidence in their answers to know for sure if they got it right or not.
3. Just because you gave another problem which to you looks exactly the same, it doesn't look them same to your students because they don't understand HOW to get the answer. Most of the time students fixate on the final answer rather than the process of how to get an answer to those types of problems.
4. Both the situations you described require algebraic reasoning (find the missing number), and that is HARD. It requires complex thinking skills. You need to model how to reason/logic/think your way through a problem out loud. What you may understand implicitly needs to be made explicit for them.
Here's the process I use for solving ALL math word problems:
1. KNOW - a) Read the problem through twice at least. If there are any words they don't understand, clear up the meaning and read the whole thing again. b) What information was given in the problem? Do I need all of this information or is some of it extra? c) What is the question? What type of answer do I need to give - is it a word answer, a number answer, the name of a person? It is often helpful to have the students rewrite the question in their own words.
2. PLAN - Decide which steps to take, and how many steps will be needed, to solve the problem. Decide which operation to use - ask yourself two questions: a) are groups coming together (addition, multiplication) or separating (subtraction, division)? b) are those groups the same size (multiplication, division) or different (addition, subtraction)? By asking those two questions you'll be able to choose one of the four operations.
3. SOLVE - Implement your plan, follow all the steps you decided on during the PLAN stage.
4. RECHECK - Reread the question. Does your answer match the question that was asked? Check your work with a calculator to make sure you answered it correctly.
Again, you will need to think aloud during the whole process but make your language as simple as you can. The hardest two steps are KNOW and PLAN, and if they don't understand the problem or make a good plan, they won't get the correct answer.
I would recommend teaching the above process with easy, one-step word problems first (until they have the process) and then go back to the problems you mentioned. You will need to repeat the process (thinking aloud, going through the steps) about 10 times more than you would for a typical learner. People do not necessarily learn in a steady, straight line going forward and upward. They learn some skills but not others, or may know something one day but not the next.
Have patience, you are doing good work! Hope this helps, and let me know if you have any more questions!
I've probably been assuming that our student who are native English speakers , and especially if they are a bit older, understand the question and can extract the information that they are given from the written presentation. UGH! What was I thinking?
It goes a long way to explaining why, if I ask ..."Was that too fast, too slow, or close enough" she's the one who will say "Too fast, way too fast!"
I'm sure you're also right about "Looks the same to an experienced eye, looks way different to the naive". I was probably assuming way more cognitive "oomph" than I should be. This is my first exposure to students
with that sort of problem. In what little teaching experience I do have, I've either taught engineering students (even there, what some kids came in not knowing floored me totally) or tutored family members.
I also see that most of our students, even the better ones, have trouble articulating a "plan". They read the problem, the better students find the solution method present in their minds and crunch out a number; whereas the weaker ones just get stuck and frustrated. Its kind of appalling to see them in that state and not know how to help them unstick themselves. Trying to get them to separate the plan from the execution is going to be a challenge; but thats nothing at all compared to making sure they understand the problem in the first place.
Its going to take me some time and practice to teach myself how to operationalize your advice, but it sure has the ring of truth to it. I'm meeting with my little group next week. Let me see if I can present "Step one (KNOW) " well enough for them to feel a difference.
Anyway, win lose or draw, now I have something to try, at least. Before, I was pretty much in a "stuck state" myself, wondering what to do beyond pure brute repetition.
My mom is a kindergarten teacher, and when I started my credential program, I was absolutely astounded when she told me that kids reverse letters pretty commonly when they are little (and most grow out of it). I had no idea that a 'd' was anything but a 'd', and certainly not a 'b', 'q', or 'p'! Now I work with kids who struggle with that (well into middle school). It's always good to get a perspective check!
The rubber met the road today. I had about half dozen out of maybe twenty or thirty students, the more advanced ones who could (more or less) handle "number facts"; i.e. add subtract multiply and divide whole numbers, decimals, and fractions.
It took a few goes around the mulberry bush to find the right level....problems that nobody found so hard as to go spiral eyes, but not so easy that they could all do it first go with no coaching. We spent all the time on word problems, doing them in two steps, first translating the narrative to an equation, and then solving the equation. All the equations were always (or almost always) one equation in the first power of the single unknown.
I went in thinking that I'ld stop with just setting up the equations (the idea being that there would be that much more time so we could do more examples of translating words to equations), but the students were
really not comfy with the idea that the equation could be solved and that solving equations was more "mechanical", i.e. more "step-by-step" and almost "rote", whereas setting up the equations required a bit of thought and often what I called "The chef's secret sauce", outside information that "everybody" knows so isnt actually given in the narrative, or is given in a very terse and gnomic manner.
That sort of surprised me, but I figured "what could it hurt?" and I coached them through solving the problem down to a numerical answer. I also got a chance to point out how similar some of the equations turned out to be. Different numbers, maybe different symbols, but
basically the same "shape" (eg one equation was something like A + A -6 = 20 which we simplified to
2A -6 = 20, and another was T + T -90 = 10 which of course simplifies to 2T -90 = 10 ) Students who hadn't seen the word problems as related DID see that there was some similarity in the form of the equations.
All in all they were engaged and starting to "get" it.
Now, memory is plastic, and maybe my memory of what I thought and felt, in eons past when I was in the 9th grade (Where I first saw "algebra") is faulty; but I don't remember ever struggling the way even some of these students are. We had something like
"Sally cuts a string into to parts (the original length of the uncut string is given) and one part is twice as long as the other part. Find the length of both parts."
So we assigned the symbol "S" to the length of the smaller part and B to the length of the bigger part.
They "got" B=2S but not first crack out of the box. It was hard for them. When I asked "So what if the problem had said the bigger part was three times the length of the smaller string", they got there but not
"pop!" just like that. Furrowed brows, sound of wheels turning, tentative voices "uh...um.....3s? " YAY! Daylight!
So thanks again and if you have any more gold like that, drop it on me, do!