## Wednesday, May 20, 2009

### Autism Diaries X: following the directions:

J is really good at math, but has trouble following directions--so much so that he received failing grades on his first few math quizzes. I've therefore asked for school accommodations to include verbal clarifications that ensure that he understands the directions.

Whether this accommodation is met, of course, depends on the ability of the supporting staff to deduce whether J understands the directions. Sometimes, this may be quite a challenge.

However, when a child is seen using a ruler to measure shapes that implicitly aren't drawn to scale (implicit in "Suppose each figure has a perimeter of 24 centimeters"), you'd think it would be pretty obvious. Consider this:

When J's failure to follow directions is due to the school's failure to follow its directions, who should receive a failing grade?

#### 7 comments:

Mrs. C said...

To use a line from the test, "suppose" that the staff is working hard to accomodate J. Well, you know what happens when you assume. LOL

I looked at that and thought, you know, his mom is the same way. When I read your posts, I feel almost like you're pulling out the ruler and measuring things for yourself.

May God bless you for it.

Anonymous said...

This is a classic example where the directions and setting for the test were not thought out in advance. It is possible that the teacher did not understand the directions and was also confused. Obviously, a ruler was not needed or was it?

1. Consider what your son would have needed to know in order to find the area of an equilateral triangle. Could he have found the height of the triangle without the ruler? Not likely, unless he knew how to correctly apply the Pythagorean theorem (6th grade?)

I'm guessing his measurements are in inches? But once again measuring 1.6 inches would border on the ridiculous. Also, what is interesting are that his measurements at least appear consistent with what the problem is attempting to show the child. Increasing the number of sides, while holding the perimeter constant, increases the area inside the polygon and decreases the length of each side.

The logical conclusion is a circle of the same perimeter maximizes the area. (what the child is discovering?) This is not a sixth grade standard by the way.

While the answers to the first question is trivial (24/3, 24/4, 24/6), the answer to question 2 would be impossible to arrive at logically even using induction as the problem is suggesting from question 1.

You often see these types of problems in Connected Math - where for lack of time or space, not enough information is given to solve the answer correctly (e.g. maximize the perimeter of a corral that has one side shared by a barn).

Returning back to question 2. One could find the diameter of the circle if they knew that d = c/pi.

Where was pi defined? Could a sixth grader deduce pi from question 1? The answer is no. However, continuing along this line of reasoning, 24/3.14 = 7.64 (using a calculator) AND 7.64/2 = 3.82.

Note here that if a person incorrectly divided 7.64 by 2, a common mistake would be forgetting to carry over the remainder, so one would get 3.32.

My question to his teachers would be how did your son get 3.3. Was he being helped by anyone? Because if they were, they didn't do the problem correctly. Remember in the previous Singapore problem, pi was defined explicitly as 3.14 for the students. In this example, pi was not defined, unless the teacher wrote it somewhere on the board.

Finding the area of a hexagon is not usually treated until geometry, second semester (10th grade). So the hexagon would not be helpful (what could you do? estimate?)

This problem is very deceptive and there is no simple answer, because the problem is full of holes.

Anonymous said...

Just to correct the previous post, I do see the units are in centimeters. My bad, long day. But you are correct the shapes are not drawn to scale and that is the point. The teacher hasn't read the problem carefully - it shouldn't have been used as an assessment.

bky said...

Lefty -- if this was a quiz, that should mean that the class is being tested on problems of the sort that they have been working on. Or was this new stuff? I would expect that kids would know that the perimeter of a REGULAR N-gon is N times side length (but not in those terms).

Is this the case? Or are these problems new?

Also, the problem about diameter of a circle surely presupposes that the class knows the formula C = 2*pi*radius. Otherwise it is impossible except for Archimedes. And do they have calculators available to them to divide 24 by pi?

What I am wondering is autism really an issue here or preparation?

Also, is this really a quiz or a set of new problems?

Anonymous -- I would guess that for the last problem about area a correct answer would be something like "the circle has the largest area because it looks like I could fit any of the other shapes into the circle." So they don't need to be able to calculate any area at all. Lefty, do you know if such an answer is what they are looking for?

lefty said...

Bky, The issue is not the concept of dividing by N for regular polygons--a concept which J has understood since I don't know when. The issue is that he didn't notice the sentence "Suppose each figure has a perimeter of 24 centimeters" and therefore went and measured everything by hand (and somehow no one noticed him doing this!) This is where following the verbal directions comes in. What particularly upsets me is that this quiz would have been a piece of cake for him if someone had simply ensured that he had paid attention to all the directions.

Anonymous said...

"I would guess that for the last problem about area a correct answer would be something like "the circle has the largest area because it looks like I could fit any of the other shapes into the circle." So they don't need to be able to calculate any area at all. "

Its logical but that's not considered a standard answer for a rubric and would only get minor partial credit. If I were to do this with sixth graders I would use grid paper and then have students estimate areas. You could even have them fill in a circle that enclosed hexagon to come up with an estimate of pi.

In this case, there is no written explanation, although it is also possible he was explaining his thinking to whoever might have been helping? In which case, it might have helped the reader to write something down.

Most of the autistic kids I've taught seldom wrote and only talked when they had time to process what they had to say. Their answers were always carefully expressed and I couldn't interrupt or they would have to start all over.

I remember teaching physics for a whole year to an autistic child and he ended up doing very well on the final (I always kept my fingers crossed, that he was paying attention to me), which ended up surprising everyone. But then I take two years to teach one year of hs. physics. Most physics teachers won't do that.

Without more input, on a rubric this is a 1, no matter how you cut it.

The two problems with this quiz are providing rulers to kids where none was needed and then finding the diameter of the circle. Folding the circle in half would give you the correct diameter to measure with the ruler in centimeters, but not for a circle with a 24 cm-perimeter.

If the circle were reduced 50% or cut to one fourth its size the perimeter would be cut by half - then you would get a diameter close to 3.8 cm. Your son's answer is 3.3 cm, so I can only speculate that he had help or perhaps the entire class was helped on the second question. Or he drew the diameter in the wrong place. If you look at the sketch, the vertical chord he drew is not a diameter, but I don't think its far off.

A better question for this age group is to ask how do you find the diameter of a circle? The best answer is fold it in half.

How do you find a perpendicular bisector of a line segment? Fold the two endpoints together.

How do you bisect an angle? Fold the two sides of the angle together or fold the angle in half.

This problem (reducing and enlarging shapes to find the change in area and perimeter) is an advanced seventh grade standard.

Sometimes (for enrichment) teachers have students transform the coordinates of cartoon characters (to show stretching, reflecting, inversions,...). Its a challenge to use with low performing eighth graders.

I've graded thousands of problems like this and built rubrics for all of them. I'm done with that :)

Anonymous said...

I see I made another mistake. I'm having trouble remembering the picture while I'm typing.

He's probably measuring the height of the hexagon to get the diameter of the circle?