In the Martha’s Carpeting Task, the cognitive demand is low because the
dimensions are given. Students should
know “long” is the length, “wide” is the width, and Area = Length X Width. I assume the teacher has introduced the area
formula to students; therefore, I would classify this problem as
Recall/Reproduction in Webb’s Depth of Knowledge because students only need to recognize
that “square feet” refers to area in the question and calculate (i.e. multiple
the dimensions) to find the answer.
In the Fencing Task, the cognitive demand is higher because only the
fencing (i.e. perimeter) is given.
Students need to know how perimeter and area are interconnected. Also, students must determine the maximum
area for the rabbits in both situations (i.e. 24 and 16 feet of fencing). I believe students would draw a picture, list
the factors of 24 as well as 16, and use trial and error to find the
answer. Based on their answers in questions 1 and 2,
they must organize their thinking and explain how to determine the most room or
area for any amount of fencing. Thus,
students would describe a strategy on how to derive an answer. Strategies may vary. I
believe this problem would be at the Skill/Concept level in Webb’s Depth of
Knowledge. I also considered the
Strategic level because this problem requires reasoning, drawing conclusions
about how to figure out area for any amount of fencing. This question goes beyond just memorizing a
formula.
Furthermore, I think the cognitive demand of any math problem would
also depend on the instruction. For
example, if the teacher reviewed several types of fencing tasks similar to this
one, then students are just regurgitating what the teacher has already shown
them. Therefore, deep thinking is
removed.
Shannon, I agree with your classification of these problems. Martha's carpeting task requires students to recall a formula and plug in values for length and width. Since these values/dimensions are already given in the problem, there is no preliminary work that needs to be done before students can plug in these values. The dimensions are in feet so there are not any necessary conversions to be done. Students would need to know that the unit of measure is square feet, something that is often forgotten, but the cognitive demand for this problem would be low.
ReplyDeleteThe cognitive demand for the fencing task is high. I worked out this problem myself and used a quadratic equation. Using the dimensions, I set up a quadratic equation and used the formula x=-b/2a to find the maximum. Then after I did a few examples, I noticed a pattern. The perimeter divided by four gives the solution making the figure a square. Once I saw the pattern, the problem seemed very easy, however, the pattern was not obvious to me at first.
This problem requires students to think about how perimeter and area are related. It also requires them to think about the effect that one dimension has on the other. For example, if I label the length x, then one side of the width would have to be one-half of 24-2x. Also, the problem having students come up with a process that works for all numbers requires them to consider even, odds, perfect squares, etc. This problem is high in cognitive demand not only because of the math involved, but because students must know about the mathematical relationships that exist.
Okay I was going to address this on my blog but since you started the dialogue I will post my thoughts here. I agree with you on the cognitive demands of both the problems. With Martha's carpet, parameters are already established and there is no connection to be made. Students should know the routine of multiplying LxW. The problem doesn't require a connection to a related concept.
ReplyDeleteThe Fencing Task does require a procedure and an understanding of doing the math. I actually drew pictures for each problem so that I could see how each pen would look. For me there was more thinking involve therefore more engagement. There were key vocabulary words and terms (such as organize) that I had to understand in order to complete the problem.
I do agree with you in terms of DOK levels-the first is definitely recall and the second is skill/concept. That allowed me as a reading teacher to understand the math complexity.
Group Members-
ReplyDeleteI agree that the cognitive domain of a problem depends on the teacher and how she "promotes" the learning process. Martha's Carpeting task is a low domain but can certainly be used as a basis toward higher domain problems. This is simply recall of the formula.
The fencing task depending on the student's ability requires a deeper understanding of mathematics and relationships. The students may need visuals while others could just figure it out. Perimeter and area should be taught as relationships and connections. Students will need to understand the concept of area as well as multiplications and factors.