More just in, Dept.

Looks like Education Dive isn’t the only one to write about how middle school math assignments lack “high cognitive demand”; Education Week is reporting on the “study” as well.

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.

6 thoughts on “More just in, Dept.

  1. Mr. Garelick,

    How did you get the school district to let you use Dolciani? Our district has terrible middle school textbooks and they move so slow. I have a Dolciani from the 1980s that I adore and am using with my daughter currently.

    You wouldn’t believe the pushback we are getting from my second grader’s teacher too. We picked up Beast Academy 2A/2B/3A-3D books and the old Singapore Math series. They want him to work in class on stuff he mastered in kindergarten. She sent his report home with a note essentially he needed to work less on “Home mathematics” and could we please reinforce the Bridges curriculum instead. Keep in mind they send home no homework at all.

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    • I didn’t tell them that’s what I’m using. I managed to procure about 15 of her 1962 edition of Algebra 1 at a penny per book; paid only for shipping. Good move, since now the same book is selling for $60. I like the book for its sequencing and problems; the explanations are too formal and confusing for some topics. But the scaffolding of problems is superb. I also draw from the official textbook (Big Ideas) for topics not covered in Dolciani (like fractional exponents, arithmetic series and some other topics, but not too many). Big Ideas has a dearth of good word problems and when they do have a standard work, or distance problem it’s a one-off type of thing–as if they are supposed to get the working of such problems from that one problem.

      Sorry to hear about your school. I used Singapore with my daughter and her friend; it works well. I was always curious about Bridges so thank you for the info.

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    • “… they send home no homework at all.”

      I distinctly remember one day in fourth grade when I told my son to do his Everyday Math homework and not leave it to the last minute. Ten minutes later I saw him playing and told him again to do his homework. He already did it – all four problems. Another parent with multiple kids complained that they were all working on the same material even though they were in different grades. I decided that EM wasn’t spiraling or scaffolding, but repeated partial learning (circling) where the onus was placed on the kids and parents. They used to tell us that “kids will learn when they are ready”, but it really meant that proper expectations and tracking were now hidden at home.

      Our schools finally got rid of CMP in 7th and 8th grades and replaced it with proper Glencoe textbooks because the parents complained that there was no curriculum path to honors geometry in 9th grade. The schools couldn’t argue against that – there it was in black and white. This drove the fuzzy math line back to the end of 6th grade and the end of strict full inclusion and Everyday Math. However, many students and parents now feel like they got whacked in the head with a brick when their CCSS exceeding expectations is not good enough for the top math track in 7th grade – the only one that leads to a possible STEM career.

      Why is it that K-6 is considered to be an all-natural education zone that completely changes in high school, college, and career?

      I tell other parents that they have to do it at home in K-6 – ensure proper mastery of basic skills. These skills include so much understanding that gets trashed by educators. Ask your schools exactly how they select students to get into their top math track in 7th grade. Get a copy of their test or their specific grades, rubrics, or decision process. My son had to show success on an Everyday Math test at the end of 6th grade. Ugh. I got a sample set of questions and worked on them with my son. You have to play their game, but do your own thing at home. After that, and into the high school AP Calculus track, math was traditional and proper.

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  2. This is the comment I posted twice on the above Education Week article site. It was deleted twice with no explanation.
    —–

    First, by definition, CCSS is not rigorous. It is defined from Kindergarten as a single slope curriculum and expectation path to no remediation for College Algebra. The split in math slope happens in 7th grade for most schools, so you can’t talk about whether middle school is rigorous or not – one track is a rigorous path to AP Calculus and the others do not offer any STEM possibility. My son’s old 8th grade Glencoe Algebra 1 textbook includes “Open Ended” problems such as creating a polynomial that requires two different factoring techniques. One could argue that teachers don’t assign those problems, but the implication here seems to be that understanding “Assignment B” problems should replace rote skill “Assignment A” problems. No. Separate mastery of skills problems are required and they use much more understanding than educational pedagogues like to admit or see. There are many levels of understanding. While open ended questions may be nice, their effects are hit-or-miss, and in many cases, neither necessary or sufficient. The key is that if you don’t end up on the upper math track in 7th grade, your STEM career chances are virtually over. The K-6 CCSS slope is now so low that this is impossible without outside help. As a STEM parent, I find educator talk of having parents be on board their math pedagogy team to help at home very disturbing.

    Second, there is a big change since I was young that few talk about – full inclusion. I got to calculus with absolutely no help from my parents, but I had to help my “math brain” son survive K-6 (enforcing mastery of skills at home) to get him to the higher slope AP Calculus track starting with a proper Pre-Algebra class in 7th grade. In high school, I didn’t have to help him one tiny bit, and now one of the degrees he will get in May will be in abstract math, where everything is about proofs. This K-8 outside basic skill help was common for all of my son’s AP Calculus track friends in K-8. I NEVER see any study of what students and we parents NOW have to do at home or with tutors. Go ahead. Ask us. We’re not looking for more open ended “thinking” questions in math. We are not so stupid to want just what we had when we were young. I’ve had a career of solving complex engineering problems with computer programming. However, I was lectured to by my son’s Kindergarten(!) teacher on understanding in math – one of many preemptive parental attacks I received.

    Now, to cover for the low CCSS math slope in K-8, the College Board is trying to push Pre-AP Math (algebra) in the 9th grade as a way to cross over, but they say little about how the extra missing math class is supposed to fit in. There is no magic engagement fairy dust that will make that work. You can’t let K-8 off the hook, but then put the onus on kids to catch up. It’s too late. Educators seem to see education now in statistical terms and not individual educational opportunities. They are happy if little Urban Suzie is the first in her family to get to the community college even if she could have gotten into Harvard. Is education now only a 2+ generation statistical solution?

    There is no math understanding issue for the AP Calculus track in math in high school – at least in the naive sense of educational pedagodues. There is no issue in college or lack of proper connection with careers. Colleges and careers are very clear about what they want, and no amount of vague business anecdotes will change that. The only issue is math in K-8, where educational pedagogues got it completely backwards. The “math wars” issue appears to be an educator academic turf battle to claim some sort of higher ground even though they’ve lowered the slope with full inclusion. The losers, however, are all of the kids in K-8 without parents who know STEM reality and how to fix things at home or with tutors.

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