They give an example from the study of two assignments. Assignment A is considered purely procedural, while Assignment B is considered to require more cognitive demand:
A: Factor completely, and state for each stem what type of factoring you are using.
x4 + x3 – 6x2
B: Create expressions that can be factored according to the following criteria. Explain the process you used to create your expression.
A quadratic trinomial with a leading coefficient of 1 that can first be factored using greatest common factoring. The greatest common factor should be 2x.
I find the wording of Assignment B a bit confusing but that’s besides the point. It appears that the people who did this “study” (and yes I will keep using quotes around that word, however offensive that may be to some) are not happy with a focus on procedural type problems. We are not given a complete view of the homework problem sets, so we don’t know if the problems scaffolded in difficulty. The “study” also did not examine the textbooks/assignments in K-6, which from what I have seen take an “understanding first, standard algorithmic procedures later, and only when understanding is attained” attitude. (See here for a clarification of what I mean).
Since I teach in a middle school, I see directly the casualty cases from what passes as mathematical education in K-6. Middle school teachers in general realize they have to prepare students for high school math. Given that burden, and having to teach students who are still counting on their fingers to add and subtract, don’t know their times tables and are flummoxed by the simplest of problems, it doesn’t take a brain surgeon to figure out what’s going on. And what is going on is that middle school teachers are having to focus on the basics rather than the “critical thinking, depth of knowledge problems” held so dear by those who believe Common Core’s content standards are only there to support the platitudes known as the Standards of Mathematical Practice (SMPs).
I’ll also make a distinction here. We are trying to teach students to solve problems, not “problem-solve”. The latter is a term generally used to describe the process of solving one-off problems with little or no instruction in how to even approach them.
As for students taking algebra in 8th grade, I mentioned I use a 1962 textbook by Dolciani. Here are two problems taken from the book. The first is about factoring:
“In the following problem, the given binomial is a factor of the trinomial over the set of polynomials with integral coefficients. Determine “c” . 2𝑥−3; 10𝑥2−3𝑥+𝑐 “
And this is a word problem students are expected to solve:
“The plowed area of a field is a rectangle 80 feet by 120 feet. The owner plans to plow an extra strip of uniform width on each of the four sides of the field, in order to double the plowed area. How many feet should he add to each dimension of the field?”
My students are not concerned with the “relevance” of the problem or whether it meets “real world” criteria. They want to solve such problems and draw on solid, explicit instruction and mastery of procedures in order to do so.