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With so many expressions of LDs, it is not surprising that educators sometimes disagree about their nature and about the kind of help students need as a consequence. Such controversy may be inevitable because LDs by definition are learning problems with no obvious origin. There is good news, however, from this state of affairs, in that it opens the way to try a variety of solutions for helping students with learning disabilities.

Assisting students with learning disabilities

There are various ways to assist students with learning disabilities, depending not only on the nature of the disability, of course, but also on the concepts or theory of learning guiding you. Take Irma, the girl mentioned above who adds two-digit numbers as if they were one digit numbers. Stated more formally, Irma adds two-digit numbers without carrying digits forward from the ones column to the tens column, or from the tens to the hundreds column. Exhibit 2 shows the effect that her strategy has on one of her homework papers. What is going on here and how could a teacher help Irma?

Directions: Add the following numbers.

Three out of the six problems are done correctly, even though Irma seems to use an incorrect strategy systematically on all six problems.
Irma’s math homework about two-digit addition
42 + 59 ̲ 911 23 + 54 ̲ 77 11 + 48 ̲ 59 47 + 23 ̲ 610 97 + 64 ̲ 1511 41 + 27 ̲ 68

Behaviorism: reinforcement for wrong strategies

One possible approach comes from the behaviorist theory discussed in [link] . Irma may persist with the single-digit strategy because it has been reinforced a lot in the past. Maybe she was rewarded so much for adding single-digit numbers ( 3+5, 7+8 etc.) correctly that she generalized this skill to two-digit problems—in fact over generalized it. This explanation is plausible because she would still get many two-digit problems right, as you can confirm by looking at it. In behaviorist terms, her incorrect strategy would still be reinforced, but now only on a “partial schedule of reinforcement”. As I pointed out in [link] , partial schedules are especially slow to extinguish, so Irma persists seemingly indefinitely with treating two-digit problems as if they were single-digit problems.

From the point of view of behaviorism, changing Irma’s behavior is tricky since the desired behavior (borrowing correctly) rarely happens and therefore cannot be reinforced very often. It might therefore help for the teacher to reward behaviors that compete directly with Irma’s inappropriate strategy. The teacher might reduce credit for simply finding the correct answer, for example, and increase credit for a student showing her work—including the work of carrying digits forward correctly. Or the teacher might make a point of discussing Irma’s math work with Irma frequently, so as to create more occasions when she can praise Irma for working problems correctly.

Metacognition and responding reflectively

Part of Irma’s problem may be that she is thoughtless about doing her math: the minute she sees numbers on a worksheet, she stuffs them into the first arithmetic procedure that comes to mind. Her learning style, that is, seems too impulsive and not reflective enough, as discussed in [link] . Her style also suggests a failure of metacognition (remember that idea from [link] ?), which is her self-monitoring of her own thinking and its effectiveness. As a solution, the teacher could encourage Irma to think out loud when she completes two-digit problems—literally get her to “talk her way through” each problem. If participating in these conversations was sometimes impractical, the teacher might also arrange for a skilled classmate to take her place some of the time. Cooperation between Irma and the classmate might help the classmate as well, or even improve overall social relationships in the classroom.

Constructivism, mentoring, and the zone of proximal development

Perhaps Irma has in fact learned how to carry digits forward, but not learned the procedure well enough to use it reliably on her own; so she constantly falls back on the earlier, better-learned strategy of single-digit addition. In that case her problem can be seen in the constructivist terms, like those that I discussed in [link] . In essence, Irma has lacked appropriate mentoring from someone more expert than herself, someone who can create a “zone of proximal development” in which she can display and consolidate her skills more successfully. She still needs mentoring or “assisted coaching” more than independent practice. The teacher can arrange some of this in much the way she encourages to be more reflective, either by working with Irma herself or by arranging for a classmate or even a parent volunteer to do so. In this case, however, whoever serves as mentor should not only listen, but also actively offer Irma help. The help has to be just enough to insure that Irma completes two-digit problems correctly—neither more nor less. Too much help may prevent Irma from taking responsibility for learning the new strategy, but too little may cause her to take the responsibility prematurely.

Questions & Answers

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Need help solving this problem (2/7)^-2
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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Source:  OpenStax, Educational psychology. OpenStax CNX. May 11, 2011 Download for free at http://cnx.org/content/col11302/1.2
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