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Among those in the reform math area, there has been a push to interpret the SMPs (Standards of Mathematical Practice) along reform math ideologies that push certain mathematical “habits of mind” outside of the context in which such habits are learned, as well as a predominate use of collaborative group work and inquiry-based learning. This article provides the description of each SMP as written in the Common Core math standards. (http://www.corestandards.org/Math/Practice) It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.Here's some of his commentary on SMP 1: Make sense of problems and persevere in solving them:
The SMP writeup describes a problem solving mind-set as well as a variety of problem solving strategies that students should have. It is important to realize that the goal of this SMP comes about after years of experience and practice. The ability to solve problems and think mathematically develops over time. Problem solving cannot be taught directly; rather, it is based on mastery of many basic skills. (See (http://www.ams.org/notices/201010/rtx101001303p.pdf )It's a great series, I highly recommend it.
Requisite for learning how to solve problems is an explanation of how specific types of problems are solved using worked examples and practice with routine problems. A set of problems can then escalate in difficulty through careful scaffolding: i.e., by changing aspects of the problem so that students must apply their knowledge of the basic procedure to new forms of the problem. In this way homework is not just a set of repetitive “exercises”. Students progress from simple routine problems to those which increase in complexity and are non-routine. The non-routine problems can then be extended into even more challenging problems. Such challenging problems should definitely be given but students must be able to use prior knowledge of skills and procedures in solving them. The goal of math teaching is to provide sufficient opportunities to apply skills and knowledge so that students know how to turn “problems” into routine exercises.
While the approach described above is a sensible and effective interpretation of this SMP, the reform math ideology that is dominating Common Core implementation is likely to reject it. That philosophy is to regard math as some sort of magical thinking process. It holds that “understanding” the problem and seeing the big picture is math, while the mechanics of problem solving are just a rote afterthought. Worked examples and routine problems are generally disparaged as “non-thinking” and “routine achievement”. The reform approach usually manifests itself as giving students a steady diet of “challenging problems” in an effort to build up a problem solving habit of mind that is sometimes referred to as “sense-making”. Such approach does not accomplish this, however. Instead, the constant pursuit of “challenging problems” stands in the way of developing fluency with certain classes of problems and building on what one already knows.
4 comments:
I've made a cursory inspection of the Common Core standards and they seem more or less what I was teaching almost twenty years ago when I did Course I/II/III math at the high school level.
How do they really differ?
We route a huge number of students each year into remedial versions of the classic Algebra I and II at the community college where I now teach. After looking at the Common Core stuff, I find myself wondering to what extent we're doing them a disservice. Back in the day when our community college course outlines were put together, we were remediating what they theoretically got at the high school level. In other words, the high schools were teaching classic Algebra I and II, and we were reteaching the same stuff. But now the high schools are teaching something quite different, so for the students, the "remedial" Elementary Algebra and Intermediate Algebra taught at the community college level (and at the CSU) is actually not so remedial. I gather they haven't seen much of the stuff before, or if they have, it was handled so poorly by their materials (text, handouts) that they never really learned it.
What used to bug the sh!t out of me when I was teaching the Course I/II/III crap was that the texts were so poorly written. They would be full of PC photos showing multi-ethnic groups of kids grinning like idiots as they say around in a circle, graphing calculators in-hand, obviously delighted to be doing group work in an ideal math class, but were short of meaningful examples that could actually help the students to understand what it was they were supposed to do. Additionally, the problem sets were designed to avoid the straw-man "drill & kill" scenario, so students were expected to become experts with <10 problems for a given topic.
The district where I used to work could afford to buy new laptops for incoming freshmen to take for granted and abuse, but they only bought textbooks every ten years or so. Thus I was thinking, what's a teacher to do, if they are required to teach Common Core out of a book that was written before Common Core became the standard-du-jour?
At the school were you teach, what texts do you use for Algebra I/II? Do you still call the classes Algebra I/II, or something else? How much has the embrace of Common Core really changed what you do on a day-to-day basis?
CCSS leaves itself open to interpretation for both pedagogy and level of expectations. Barry Garelick shows that many educators interpret it as meaning only one thing - a validation of their own beliefs. CCSS, however, is specifically written to be pedagogy neutral.
One of the goals of CCSS was to strengthen the skills portion of the balance of skills and understanding that all educators want. However, the standard is so vague that one has to look at the emerging tests (like PARCC) to determine what level of skills are enforced. The signs are not good. PARCC defines it's top "distinguished" PLD level as making sure that all students could pass a college algebra course. There is no talk of STEM preparation and the needs of those students are ignored in K-6. There is no curriculum path for those students. Perhaps the most able kids will make the non-linear change to strong high school math by themselves or with the help of parents, but many kids in the middle never get the support they need to have a chance at a STEM career. It will be all over by 7th grade. Worse yet, they will believe that they are just not good in math. Educators will see some students who succeed in math, but never ask why.
K-8 educators will continue to hope that the Chesire Cat smile is all students need while ignoring all of the skills work done at home (or with tutors) by the parents of the best math students. Educators with the least knowledge of math cannot seem to understand what is really required for success in math. They continue to ignore the skills portion of the balance. They hope to approach the problem from the top (smile) down and assume that concepts will create skills.
Most all high schools include a traditional math sequence: algebra I, geometry, algebra II, pre-calc, and calc. Most middle schools offer algebra I in eighth grade so that some students can get to calculus as seniors. Generally, integrated math in high school is only taken by weak math students, if it's even offered. Some integrated math high school curricula claim that they prepare kids for the AP calc test, but more often than not, "integrated" means lower expectations and rigor, in spite of all of their talk of understanding.
Our middle school used to use the integrated CMP math curriculum, but it never covered algebra well. They tried to add in extra algebra concepts to prepare some of the kids for geometry in high school, but it didn't work. Parents demanded that the middle school offer (as an option) the exact same honors algebra course as the high school. This caused the school to dump CMP so that all students took a more traditional pre-algebra and algebra sequence. The more able kids got to the full algebra I class in eighth grade and the other students got the same material, but went at a slower pace. They are set on a pace to get to a strong algebra course as a freshman in high school.
So, instead of forcing kids to make a big curriculum leap in high school, the big leap now exists between 6th and 7th grades; between low expectation Everyday Math to a higher expectation and more rigorous traditional algebra sequence with proper textbooks.
A lot of this comes down to sorting those kids who have the ability and interest to make it to a STEM career and those who don't. Unfortunately, CCSS allows schools to start everyone on a path to no STEM career in K-6. It doesn't solve the K-6 problem. Educators may talk about understanding and critical thinking, but those are just cover for low expectations. What does understanding mean if students don't have the skills to do the problems? The parents of the best students know better. They are the ones enforcing mastery of skills at home.
So, in K-6, we have full inclusion and a lot of fuzzy talk of differentiated instruction and critical thinking. We have low expectations on the skills portion of the balance. Then, in 7th grade, when kids are sorted in math, problems look like they belong with the students, peers, parents, and society, not the fact that K-6 math did not offer the students a proper curriculum path to algebra in 8th grade.
The lack of skills mastery in K-6 will continue to haunt many students throughout high school. Even though they will get to algebra concepts in 8th grade and a full algebra course starting as a freshman, they will struggle to get to the point where they can successfully pass a college level course covering the same material - and this is what PARCC calls "distinguished". What on earth are these poor kids doing in
math for 4 years in high school?
I'm all for offering kids many math options in high school, but only if schools solve the K-6 math problem. CCSS does not do that. People thnk that CCSS sets a high standard, but it does not in K-6. All of the talk of critical thinking and understanding cannot cover over the fact that K-6 schools still cannot deal with the hard work necessary to achieve mastery of the skills. Go ahead and start with the smile, but show me that it works. It hasn't in the last 20 years. Traditional math has been gone from K-6 for at least that long in many school districts, but still it's used as the bogeyman to prop up ineffective K-6 math curricula which don't offer a STEM path and don't enforce mastery of the basics. CCSS does not fix the K-6 problem.
I appreciate the detailed responses.
I'm getting some data from our in-house researchers regarding the high schools that feed us, and over the next couple of years I expect to come in touch with people at our feeder schools to get a clearer picture of what they are doing with the students before they come to us.
About 90% of our students place into remedial math, and we have remedial courses that go down to four levels before transfer, i.e., basic arithmetic, pre-algebra, beginning algebra, and intermediate algebra (plus geometry, though not many take this). The proportion of students who pass these courses is typically around 50%, so the probability that a typical student can even make it to transfer level (100-level math courses or above) if they're starting at pre-algebra (where 25-30% of our incoming students place initially) is usually between 0.10-0.15.
There is a growing push in California higher education to "shorten the pipeline" and do something, anything, to get the success rates up. Even at four-year institutions. I heard from a friend who is a math proffie at a big CSU campus that administrators have been after the math & engineering folks because of low pass rates in their courses. At this campus, many service math courses, which used to be capped at 30-40 students, are now run in large lecture formats with 150-200 students/section. This has been mainly motivated by budget cuts, but these changes are also part of a big picture policy change, where the CSU and the UC are basically giving community colleges carte blanche to take care of pre-baccalaureate remedial math coursework as they see fit.
Many schools with sufficient computer infrastructure are doing non-traditional remedial math courses where students are put in front of a computer, using ALEKS or Pearson's MyLab & Mastering (or equivalents), and they are allowed to work at their own pace until the software says they have learned a certain percentage of the material. Others have discarded everything but the names of the remedial math courses (especially intermediate algebra, which is a required prerequisite for anything that transfers to CSU/UC), and they do their own thing. There is a course called Statway that is taught at a number of community colleges in CA, which is nominally equivalent to intermediate algebra ("there is no legal definition of intermediate algebra," the originators explain, "so we have redefined it to suit our purposes."), in which they teach a small amount of intermediate algebra and a large portion of descriptive statistics.
All of these approaches have pros and cons, but I admit that I don't agree with a lot of what is being done. Aside from what seems to be soft bigotry-- "gee, you students seem to have a hard time meeting these standards, so we'll lower them for you"-- bachelor's degrees are already considerably devalued, but few (if any) seem to care, and they are actively working to dumb things down even more. I don't want to be complicit, but at the same time I believe we (my employer, my colleagues) can better serve our students, and it's helpful to have a clearer picture of where our students are coming from.
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