# New math: Concepts precede skills

Under new math standards, students will be asked to explain why procedures work before they’ve mastered the  procedures, writes Barry Garelick in The Atlantic.

Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them “understand” the conceptual underpinnings.

Students are expected to:

Make sense of problem solving and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

These are “habits of mind that ought to develop naturally as a student learns to do actual math,” Garelick writes.

True habits of mind develop with time and maturity. An algebra student, for instance, can take a theoretical scenario such as “John is 2 times as old as Jill will be in 3 years” and express it in mathematical symbols. In lower grades, this kind of connection between numbers and ideas is very hard to make. The Common Core standards seem to presume that even very young students can, and should, learn to make sophisticated leaps in reasoning, like little children dressing in their parents’ clothes.

Teachers will need to adjust Common Core guidelines, Garelick writes. But will teachers have the freedom to do so?

Some people are misreading the standards, responds William McCallum, a University of Arizona math professor and leader of the math standards team, in a comment.  “The standards do not settle the debate on how fluency and understanding should interact in the curriculum,” he writes.  The phrases “critical thinking” and “collaborative learning” do not occur anywhere in the standards.

1. Bill says:

Another moronic idea from the educrats and course development peons. You cannot teach concepts until basic skills are in place (though you can explain why certain things work in math while teaching said skills).

UGH!

2. lee says:

I bet these same geniuses would insist that kids need to learn Newtonian physics before they are taught how to ride a bike, too.

I wonder if their probable ignorance of the internal combustion engine stops them from getting behind the wheel and turning the key in the ignition.

Fools, every last one of them.

3. gahrie says:

What is wrong with drill and kill at least through the primary grades? It worked for hundreds of years.

4. That’s the kind of nonsense you get when people who aren’t educators write the curriculum

5. Roger Sweeny says:

“William McCallum, a University of Arizona math professor and leader of the math standards team”

I’m sure he considers himself an educator, and he probably has as much experience teaching elementary school kids as any ed school professor.

Real teachers almost never write curricula.

• Why would you assume he has experience teaching elementary age students? He has a PhD in Mathematics.

He may understand it, but that doesn’t mean he can teach it, or decide what should be taught to elementary age schoolchildren.

• Roger Sweeny says:

I assume he has absolutely no experience teaching elementary school, just like almost all ed school professors.

6. “Under new math standards, students will be asked to explain why procedures work before they’ve mastered the procedures, writes Barry Garelick in The Atlantic.”

7. SteveH says:

“Real teachers almost never write curricula.”

Our teachers and administrators may not have “written” Everyday Math, but they picked it out. They could have picked Saxon or Singapore Math. Also, they love to claim that Everyday Math is not the curriculum. It’s just a guide for “their” curriculum.

Don’t tell us that real teachers don’t talk on and on about critical thinking and problem solving while dismissing any link between mastery of basic skills and understanding. I heard it over and over and over from my son’s “real” teachers. We parents went to an open house when our kids were in first grade to listen to a “real” teacher go on about how important it was for little Suzie to explain why 2+2 = 4 with MathLand.

While there may be differences of opinion between teachers, and things may change for some in high school, your meme of “don’t blame the teachers” won’t work.

• SteveH,

In 20 years of teaching I’ve never been allowed to give input on curriculum. I’ve been asked but the we were only allowed recommendations.

I have experience with Everyday Math and I can assure you the teachers did not select it. The district ignored the teachers’ recommendation, declared it the end all be all of Math instruction, and told us to follow it religiously. The result was as bad as you can imagine.
It is the publishers who write the curriculum, not educators.

• allen says:

Haven’t we been here before?

You lament how teachers have essentially no influence over policy, I point out that the structure of public education ensures that situation and you call me a name.

If you really are an adult aren’t you getting a little tired of having to be forced into a childish response as a necessary consequence of defending the system which renders you professionally impotent?

• I have never called you a name

• allen says:

Oh sure you have but leave that aside. Your lack of a response to the question of why you defend a system that renders you, by its nature, professionally impotent is conspicuous by its absence.

Do you really give so little thought to the “why” of the unimportance of teachers to the public education system or are you satisfied with your empty complaining about the fact?

I would have to think it’s the latter and that your fantasy is of a tax-supported, mandatory-attendance public education system with teachers in charge. But that’s a child’s fantasy as I’m pretty sure you know but refuse to acknowledge. As if that’s somehow better then accepting the bitter truth.

• I ignored your question b/c its one of your typically all or nothing questions, and stupid at that.

The decisions mentioned by me and other teachers don’t have to be that way. They are caused by the greedy companies people like you want to put in control of the schools. In states that have strong teachers’ unions I suspect things are different.

• Roger Sweeny says:

I teach in Massachusetts. Massachusetts has pretty strong teachers unions. My impression is that teachers don’t have a lot of input into curriculum decisions. (The grass is greener here but that’s because we get more rain than you do.)

I don’t see why greedy companies should be systematically biased in favor of bad curricula. It seems to me they make money no matter what the district buys.

• allen says:

You ignored the question, Mike, because you mustn’t even admit such a question can be asked.

The same reason you dismissed the question – you can’t even deal with the asking. There isn’t any worthwhile reply to make because teachers *are* not only at the bottom of the public education hierarchy but the political nature of public education makes that a proper state of affairs and one that can’t change.

So the decisions mentioned by you and others do have to be that way. The structure of the current public education model ensures it and no amount of complaining by the bottom-rung professionals in the public education hierarchy is going to change that at all. As you well know.

Those “greedy companies” are the tail being wagged by the dog of public education and if school districts and states demanded content-rich, affordable textbooks that’s what the greedy companies would offer.

But that’s not what the decision-making officials of public education want. They want books that insulate them from all criticism while displaying their eager pursuit of some mythical, educational cutting edge. That, along with your professional impotence, is also an inevitable result of the political nature of public education.

8. SteveH says:

“I have experience with Everyday Math and I can assure you the teachers did not select it.”

But you can’t speak in generalities. In our parts (New England), so many “real” teachers love EM and spout off about how good the spiral is and how kids will learn when they are ready. Ed school professors may never have taught Kindergarten, but they are pushing their ideas on future teachers, many of whom never seem to lose the influence.

All of our administrators are from the teaching ranks. Teachers have input on curriculum. They didn’t have to pick Everyday Math.

What did your teachers recommend for math in K-6?

• Crimson Wife says:

The district in which I lived from 2006-2009 picked EM over the parent-pushed Singapore based on a committee that had both teachers and administrators. The given reason was the Singapore allegedly “wasn’t appropriate for English Language Learners”. Somehow I doubt that it was the Russian and Asian immigrant kids that the committee was worried about not being able to handle the rigors of Singapore Math…

• I don’t remember to be honest, I was not on the committee. I do know the teachers felt they had been deceived throughout the entire selection process b/c a district official brought in EM at the last meeting and told them it was a done deal, even though they recommended something different.

EM may work fine with easy to educate kids, it doesn’t work well with others. My beef with it is that it was presented to the Board as the teachers’ choice and it was the end all be all of Math curriculum. 5 years later we are still trying to dig kids out of the whole they are in b/c they can’t add or subtract thanks to EM.

9. Bill says:

Crimson Wife,

The given reason was the Singapore allegedly “wasn’t appropriate for English Language Learners”. Somehow I doubt that it was the Russian and Asian immigrant kids that the committee was worried about not being able to handle the rigors of Singapore Math…

Ummm, it looks like it doesn’t address the latino/hispanic grouping of students, but if the Russian and Asian students can get ‘Singapore Math’, what is so hard for everyone else in the freaking world to not handle the concepts presented in ‘Singapore Math’?

ARRRGH!

In my district, teachers have no say whatever on the selection of textbooks, whether for math or anything else. These decisions are made by upper-level bureaucrats, who are often *not* from the ranks of teachers. Some in administration have never had any classroom teaching experience and have come from other fields — the military, nursing, business, whatever.

However, building principals have some discretion. At one school I was at for some time, there was no math text in use at all. The previous text, something similar to Everyday Math but less expensive, had ended up on storeroom shelves because the teachers hated it and parents complained about it. The principal decided not to throw good money after bad. So we ended up cobbling together math materials from various sources that matched the curriculum for each grade and this actually ended up working out better than using the texts, as we could provide more or less practice on specific topics, depending on the needs of the class or of particular students, and could more readily “differentiate” for students at lower levels of achievement, mainly those with sped classification or ones who would qualify for such but whose parents refused it. It also allowed for much more direct teaching and skills practice than the texts had done.

When I taught in a middle school in the late 90’s the text series we used was pretty good: clear explanations and examples, uncluttered pages, extra practice sections in the back, logical and cumulative sequence, but even then it was out of fashion. Teachers hoarded their copies because they were no longer available for purchase.

We seem to have a limited number of decision makers in high places who have any real grip on cognitive science and instructional issues. The relationship between conceptual understanding and procedural fluency is clearly a synergistic one.

11. SteveH says:

“5 years later we are still trying to dig kids out of the hole they are in b/c they can’t add or subtract thanks to EM.”

You’re not going to get any argument from me about this. I saw EM in action over many years. It didn’t even work for my easy-to-educate son.

I can see why many teachers are defensive about blanket criticism, but I’ve never seen teachers as a homogeneous group. While there are obviously some places where teachers have little input on curriculum, this is not the common situation, and all of the talk of critical thinking and understanding come straight from ed schools, not non-educators. I’ve never met a K-6 teacher who praised mastery of basic skills, but I’ve been told that some exist. They didn’t get that way from ed schools.

Our schools start requiring subject certification starting in 7th grade, and I’ve seen a noticeable difference between those teachers and K-6 teachers. I found the 7th and 8th grade teachers interesting in that they valued content and mastery of skills, but they live among the lower grade teachers who don’t. Instead of clashing with the lower grade teachers, their view seemed to focus on getting kids to take control over their own learning and getting them ready for high school. Along with dealing with the social pressures of middle school, students face the need to make a nonlinear change in their effort to get ready for high school. I used to teach an after-school SSAT class in 7th grade and saw lots of kids who really thought of themselves as stupid. The lower grades pump them along because they don’t want to act as a filter. The filter doesn’t go away. It just gets gets pushed back to a point where it so easy to blame the kids. Even the kids see themselves as stupid.

As I’ve said before, the big philosophical teacher and curriculum divide I’ve seen is between the grades where teacher subject certification is and and is not required. After this changed a few years ago to start in 7th grade, our school was able to get rid of CMP and replace it with hugely better Glencoe textbooks. The publishers don’t care.They don’t force curricula on schools. Publishers cover all of their options. They have multiple levels of fuzzy and non-fuzzy options. You can always tell the strongest middle school math series because it’s simply called “Pre-Algebra” and “Algebra”. The next lower level of rigor includes secondary titles, like “Tools for a Changing World”. These more hands-on and descriptive books cover less material and are less rigorous. The worst ones don’t even include the word “algebra”. They are the most hands-on and group learning oriented curricula.

In an odd sort of reversal of the saying, it could be that those who can, teach, and those who have never lost their love of what they learned in ed school, rise into controlling curriculum positions. I see that with our superintendent. But as a parent, why would I like to hear teachers complain about how it’s not them? Not all of us sit around blaming all teachers, but why would parents like to hear that there are big sytemic problems and then be told that they should not have school choice? There is also the big IQ excuse that was covered in depth on another thread, but some seemed to deny any real influence of systemic flaws.

• Roger Sweeny says:

“I used to teach an after-school SSAT class in 7th grade and saw lots of kids who really thought of themselves as stupid. The lower grades pump them along because they don’t want to act as a filter. The filter doesn’t go away. It just gets gets pushed back to a point where it so easy to blame the kids. Even the kids see themselves as stupid.”

Amen. We do kids no academic favor when we push them along when they’re not ready. They then take classes they are destined to do poorly in. They are frustrated and bored, and maybe act out. And many will think of themselves as stupid and failures.

• gahrie says:

I am a public school teacher and a member of the union. I have taught 7th through 12th grades. I believe the biggest problem in public education today is not the teachers. The biggest problem is unmotivated students and the culture of failure. (The whole “acting white” thing.) The second biggest problem is poor parenting.

I also believe in giving as much power as possible to those most responsible for learning, the parents and students.

I support school choice. I support charter schools. I support tracking. I support expulsion and alternative environments for those willfully disruptive and unwilling to learn. I support vocational education. I support a general education track to run along with the college prep track.

If I was suddenly made the philospher-king of the United States tomorrow, the first thing I would do is return educational practices and philosphy to the status quo in 1960.

Pushing kids along doesn’t work, but neither does retaining them in grade. If you “hold back” all the kids who fail to meet grade-level expectations, you end up with 15-year-olds in fourth grade: I know, because I taught in a district like that, and I had a fourth grade with over a dozen teen-agers.. Nobody “passed” to the next grade unless they scored well on the end of grade tests. The middle and high school teachers may have loved it (I can’t recall), because their classes were extremely small: most of the students dropped out before eighth grade.

So we need an alternative to marching everyone along in an age cohort vs. holding students back in an earlier grade. Neither of those two approaches is effective. Nor is assigning students to contained classrooms based on perceived “ability” or achievement in the early years: in several large meta-analyses, students of all ability levels did more poorly in such placements than did students grouped randomly and heterogeneously.

What might work — we haven’t really tried implementing it on a thorough enough basis to know for certain — is instructional level grouping in skill-based subjects (math, reading and written language) in the elementary grades. One variant of this is known as the “Joplin Plan” and involves grouping students by achievement level rather than grade placement, so students from second, third and fourth grade might be in the same math group based on their achievement. It is most often used for reading instruction, where the same organizational principle applies. This often requires harnessing additional staff, as groups vary in size; it can be a timetabling challenge as well, but less so if all grades teach math (or reading, or whatever) at the same time. This enables all students to be taught in their “zone” so to speak, and permits fast learners to accelerate and slower learners to spend more time consolidating needed skills.

I was surprised, when I did some research, to find that it did not have a lot of evidence of significant effectiveness. The other method with some data supporting it is within-class ability grouping: something I have seen especially in math, in many schools. All students would be working on the same topic (say, volume of solids) but different groups would we working on more or less challenging tasks at potentially different grade levels (an advanced group could be doing volume of cones and cylinders, a low group volume of cubes and rectangular prisms).

Some school administrators prohibit this kind of grouping, although “Guided Reading” a la Fountas and Pinnell — virtually universal in K-4 — is clearly a variant on this, so there’s some cognitive dissonance here. The research is limited but does show some positive effects, in both examples (Joplin Plan and within-class grouping), with an effect size of less than .5, rather underwhelming.

Another system that works well when it can be arranged is having teachers at the same grade level group students for math according to instructional level across classrooms (assuming there are 2-4 teachers per grade) and each teacher focusing on one or two groups based on their instructional level. My colleagues do this quite often when they can work it out in the timetable, which is not always possible. Unlike the experience of others here, I meet *many* K-6 teachers (including most of my colleagues) who are firm believers in the importance of skill mastery and allocate as much time as they can to developing it. Of course this will never be enough for all students: Engelmann and others did some studies back in the ’70’s that demonstrated that in the worst-case scenarios, a student might require 11, 000 repetitions to master a specific skill. We need some “system” plan to address those students’ requirements, too, as well as the needs of those who would benefit from acceleration (“enrichment,” for the most part, is rubbish).

Some overviews of research re Joplin plan et alia:
http://rer.sagepub.com/content/57/3/293.abstract

http://www.jstor.org/discover/10.2307/1000128?uid=3737720&uid=2129&uid=2134&uid=4576163247&uid=2&uid=70&uid=3&uid=4576163237&uid=60&purchase-type=article&accessType=none&sid=21101480076797&showMyJstorPss=false&seq=3&showAccess=false

13. Cal says:

Nor is assigning students to contained classrooms based on perceived “ability” or achievement in the early years: in several large meta-analyses, students of all ability levels did more poorly in such placements than did students grouped randomly and heterogeneously.

While I agree with your earlier statement that holding kids back is a terrible idea, I’m pretty sure that this statement here isn’t true, or at least isn’t complete.

Research consistently shows that ability grouping is the most effective way to reach all ability levels. That is, while heterogeneous grouping sometimes helps low ability kids, it does so at the significant expense of high and mid-level ability kids.

Grouping is really the only answer that makes sense, even if it won’t be optimal. The solution to an 8 year old who can’t read at third grade level is to move him along to “4th grade” with other kids who can’t read well, and continue their education at a skill level they can manage.

14. Bill says:

Cal,

Unfortunately, grouping students by ability (what was known as tracking in my public school days) is generally verboten by most school districts and politically correct morons.

Grouping students by ability worked just fine when I attended junior and high school, and if it worked 30 years ago, why wouldn’t it work today (two words…self esteem).

UGH!

• Crimson Wife says:

30 years ago, racial/ethnic groups being overrepresented in one track and underrepresented in another wasn’t an issue in most districts because there were hardly any non-white students (urban districts and rural ones in the South being the exception). Today, districts are way more diverse, and there is political pressure to get rid of any tracking if the result is anything other than a perfect representation of each demographic group in each track.

• allen says:

Oh, it’s not just “politically correct morons” that make tracking problematical. Parents are, quite naturally, disinclined to accept that their bundle of joy isn’t on the fast track to becoming a Nobel Prize winner/President of the United States/Olympic gold medalist so there’s another source of problems.

Then there’s the issue of how to assign students to tracks.

School districts, not having the “sorting” hat from Harry Potter, have to use tests. But as anyone interested in public education knows full well, tests are problematical as well. Tests that are used to differentiate between kids of different ability are also useful for differentiating between teachers of different ability which would open a whole new can of worms.

Parents would, naturally, want their kids to have the best teachers in the school and, unlike Lake Woebegone High School, not all teachers are above average. Now what? Fire the lousy teachers that preside over empty classrooms? Forget that!

It’s just much simpler, for the decision-makers, to make no distinctions between students using the tattered cloak of phony egalitarianism.

• SteveH says:

Our high school tracks by subject. We have four levels, Basic (remedial), College Prep, Honors, and AP classes. Teachers make recommendations, but students and parents can override their decisions. (Students can also flunk.) Nobody is yelling and screaming. Everybody mixes together in some classes. Some kids are better in languages than in math. No big deal.

What’s so special about K-6? Kids know who is smart (or able) and who is not. Parents figure it out pretty quickly too. In the “old days”, schools managed to keep kids together until Junior High. But with full inclusion, they can’t do that now. So they use a philosophy of education that diminishes the importance of content and skills and they use a mystery method called differentiated instruction to make it believable. Our schools managed to change that to differentiated learning to further put the onus on students. Then they needed to talk about things like critical thinking and understanding to hide the fact that their curriculum sets lower expectations. My son’s first grade teacher talked on and on about “voice”, but not how to form sentences and paragraphs that made sense. They track kids by age and pump the problems along. When kids hit the big high school filter, it’s too late. It looks like the kids are just stupid, and the kids will believe it.

All of this just makes it harder to separate and calibrate the variables in K-6. It also provides excellent cover by putting all of the onus on the students. By definition and philosophy, K-6 schools cannot identify academic skills well enough to separate students. My son’s Kindergarten teacher went out of her way to tell my wife and I that lots of kids can read in Kindergarten, but they don’t know what they read. His first grade teacher said that he had a lot of “superficial” knowledge. I wouldn’t trust them one tiny little bit to put kids into proper tracks.

15. SteveH says:

I’ve mentioned (at other times) a K-5 charter school in our area that claims to use what they call a full inclusion environment. The core academic courses are grouped by ability (results), and the rest of the classes are mixed. They felt that if differentiated instruction ended up separating students in the same classroom often enough, and the kids knew who the smart or the better students were, then why not separate them with walls. However, I think that they still suffer from a weak curriculum – my opinion.

In our local public schools, full inclusion is king in K-6. They know that the ability range of students is larger than long ago, but they hope that their full inclusion techniques (with a large dose of mixed ability groups) can make it work. Of course, they didn’t have any proof that these techniques would work when they decided to implement full inclusion, and I was told by the principal that some of the lower grade teachers resist differentiating at all. When our son passed through, the only differentiation he got was with homework.

I have also mentioned before that I have seen the benefits of full inclusion, but they don’t come without a cost. Unfortunately, that cost is passed on to the parents who have to keep their kids up to speed. More parents now send their kids out to the increasing number of K-6 private schools that never existed when I was growing up. Even in the recent elections, some school committee members were still claiming that these parents were elitist and are just giving their kids what they had when they were growing up. They don’t (want to) look very closely at this.

The fundamental flaw is that schools can’t keep all of the kids they used to send out-of-town, and then assume that with differentiated instruction, they can track kids by age. They then choose curricula like Everyday Math that tells them to “trust the spiral” and everything will be OK. It will all work by definition. It doesn’t. The schools could send home surveys to see how much parents help their kids with the basics. They don’t. They bought into the idea of full inclusion with the assumption that it can work. They tell parents they understand the problems. (My son had one member of his group in sixth grade who kept cutting up their projects. The teacher told them that they had to work it out. They were still going to get group grades.) In spite of things like this, our schools are bound and determined to not change their basic assumptions. These assumptions cannot be discussed. Even during their last 5-year strategic planning session, these assumptions were off the table. The best students will probably get by with help at home, but it’s my view that the middle kids are getting killed – especially when K-6 math curricula don’t ensure mastery of the basics. Unfortunately, CCSS won’t make full inclusion or mastery problems go away in K-6.

16. North of 49th says:

I don’t think grouping by instructional level in K-6 (or K-8) is such a pipe dream. One school I worked at did this in math and language. The tests used for placement in different instructional groups were criterion based, rather than mulltiple choice tests. If you met a certain criterion, you could be in group a, if you met a different criterion, you were in group b, and so on. This allowed students to work at their instructional level in different units of study or strands in math. Those who wanted to accelerate (quite a few kids did want to do this) could opt to take the summative assessment of whatever grade group they placed in, and if they were successful, they could move up to the next grade or level. It was initially more work for the teachers to organize things but it ended up making better use of learning time and allowing able students to move forward faster while average or slower students still moved faster than previously because their instruction was at the level appropriate for them. We even worked out acceleration plans for kids who could do the work in their assigned instructional level in half the time and then move forward. This was a few years ago and I don’t recall all the details now but it worked pretty well. It meant students were regrouped every six weeks or so, when units of study and/or strands changed.

Another thing that helped was that all the teachers involved were committed to this way of doing things and were stromng teachers. I was relatively new so I learned from my colleagues.

The hard part, like I said, was organizing it in the first place. But it is very doable. Anorther benefit, at allows for students who are working at different levels in different subjects, or even in the same subject. We had kids who were good at math but much better at geometry than algebra, or the other way around. This way they could zoom ahead in their strong areas and still make steady progress in their more average or less strong areas.

We found it more difficult to do in science and social studies because the curriculum was more grade-specific, but we still grouped according to achievement level so that every student could be challenged appropriately.

• SteveH says:

It is a pipe dream with full inclusion. It’s main goal is to not group by instructional level. I’ve had teachers give me examples of how differentiated instruction can work within mixed ability groupings. It’s hard not to laugh. The other limitation is that it only offers enrichment, not acceleration.