Common standards: The newest new math

I employ a highly unscientific, but I think also highly defensible, approach to evaluating most proposals for changing how a  particular course is taught. I talk to an experienced colleague who actually teaches that subject, and teaches it well.

Blog readers will know by now that I don’t have a lot of time for much of what passes for pedagogical theory, especially as taught in education schools. But I have enormous respect for pedagogical experience, and for teacher reality checks.

So as I continue my posts on the common core standards, I reached out to Stephanie Sawyer, a fellow teacher at Juan Diego Catholic High School in Utah. She has been working very hard to figure out how to improve her students’ math understanding, and performance.  I asked Stephanie what she thought of the common core math standards.

Here’s the first part of her thoughtful response. I’ll post the second part soon.

Should the state of Utah adopt the Common Core Standards for Mathematics and English Language Arts? That’s a moot point now because they have. The Common Core Standards for Mathematics (CCSM) were adopted in 2010. The right or wrong of that is worthless to debate at this point, because it’s a done deal. Mathematics teachers throughout the state will have to deal with the fallout from that decision.

Utah has chosen to adopt, for mathematics, what the CCSM folks call the “international model” for meeting the standards through curriculum. What this means to most high school mathematics teachers is that the traditional Algebra I, Geometry, Algebra II sequence is gone. It has been replaced with the rather vague-sounding Secondary Mathematics I, Secondary Mathematics II, and Secondary Mathematics III sequence. Not to worry, though; you will still be teaching the same material, just in different courses. Sounds good on the surface, doesn’t it? But it doesn’t quite work out that way. More on that later.

As a parochial high-school teacher, I am often asked if we have adopted the Common Core Standards. The answer is yes and no. Yes, we think the standards are okay – more than a little ambitious, but they are good standards. But no, because we don’t think we need to adopt them as they are  presented sequentially, nor do we think we need to teach all the standards (the ones without a + next to them) as a minimum. I believe that these standards attempt to do too much in too little time.

The lesson from the last TIMMS (Trends in International Mathematics and Science Study, 2007) assessment was that the US mathematics curriculum is a mile wide and an inch deep. These standards seem to indicate that we are to expand our curriculum to two miles wide and six inches deep…in the same amount of time that we were unsuccessful in teaching the previous standards. We are supposed to provide more depth to the old topics, and add in a whole new strand of Probability and Statistics that used to comprise an entire high-school statistics course. While the new standards certainly spell out what our students need to know to be college-ready in the 21st century, the recommended curriculum for how to accomplish mastering them is decidedly UN-realistic. At least as how things now stand.

When Mary asked me to write this post, I wasn’t sure how to approach it. To me, the standards-assessment-college-ready discussion has become a hydra: every time you address (lop off) one head, two heads grow to replace it. I think that in lopping off the standards head, we must now confront pedagogy and grades.


The new standards are very focused on conceptual understanding. In the early elementary grades, I’m not sure one needs to understand the concept of addition to add, or the concept of multiplication to multiply. While understanding these concepts becomes essential when approaching word problems and applications utilizing these skills, I don’t think one needs to understand WHY something works in order to be able to understand HOW something works. In the elementary grades, could we please teach the students HOW to do things, like add, subtract, multiply, and divide integers, decimals and fractions? And in the middle school, could we please teach them how to comfortably move from decimal to fraction to percent? And how to apply decimal, fraction, and percent to the real world? I know this material is taught; my contention is that it is not mastered, precisely because there are just too many standards to hit in these grades.

You don’t have to go to college to get a certification to sell real estate, but you sure as heck have to be able to do math to pass the test that allows you to sell real estate. I understand that CCS have this built into the standards, but I’m not sure that education schools have trained teachers to go beyond the “fun of  learning”  into the necessary drill required to master these skills. The term “rote learning” has become synonymous with “not thinking.” But don’t we want our students to be able to handle basic operations without having to really think about it? Because drill is what it takes to actually master anything – mental, physical, emotional. No citation here, just life experience. And “drill” has become anathema in education schools.

I suppose I see mathematics education a lot like I see learning a sport. In baseball or basketball, there are “fundamentals” – things you have to have mastered in order to progress to the next step. If you want to play basketball, every practice, regardless of how long you have been playing the game, includes lay-up drills and wind sprints. Do the kids who have made the varsity basketball team already know how to do lay-ups? And how to run? Of course they do! But the point of drilling is to make it second nature – something they don’t have to think about so it is automatic.

Likewise, my 8-year-old son is a baseball player. He pitched for the first time this week. He’s never had a pitching coach, but he understood that his main goal was to get the ball in the strike zone. Which he did. Sure, his pitches were slow. But to my mind, this is fine. He can pitch. He can learn about the nuances – the WHY – of pitching now that he has his accuracy nailed. Now he can worry about things like how high up he needs to cock his front leg on his lead-in, or how much his back leg has to follow for velocity, and if making these adjustments will change the accuracy. He can now focus on these issues because he has mastered the fundamental getting-the-ball-where-you-want-it-to-go.

If I may, I’d like to give one more example of this mastery from drill idea, but back in the math realm. When I talk to parents and colleagues over, say, 35, and ask them how to add fractions, they can actually tell me. Most of them haven’t had to add fractions in 20 years or more, but they can still tell you things like “the bottoms need to be the same” and “you multiply the top and bottom by the same thing.” Why is it that these folks can still, years after they learned this skill, recall how to do it? Yet we have students in Honors Geometry who have been looking at and reviewing fraction addition for at least four years, and they will say “I don’t remember.”

If one wants to be able to do, much less understand, high-school mathematics, there are just some things that have to be automatic. And you get to automatic by drill.

Until we in mathematics education acknowledge that mathematics is a combination of skill and understanding, it won’t matter how great the standards are, or how cool our classes are. If our kids can’t do the most fundamental math, it’s pretty much a waste of everyone’s time.


  1. Howard Beale

    I really don’t think this core talk matters much. I think three simple things would make all instruction, including math instruction go better. The problem is these simple things are difficult to do and might require some money and certainly sacrifice by many.

    1) Put at least two qualified teachers in every classroom.
    2) Math takes practice, guided practice and independent practice. Too many high school students don’t do their homework. The Block schedule (where students meet every other day) is problematic and exacerbates this problem. Students don’t do homework, skip class which is really two classes of instruction. Let’s say a student misses a day, you might not see your math teacher for nearly a week. You can’t really do math well (well, most people) unless you practice, practice, practice. The way school, especially secondary schools are structured, this practice, or need for it, is short-circuited. Parents also need to work, really work, with their children at an early age to get math fundamentals taught and reinforced at home. Parents don’t work enough with their children and it’s showing,
    especially in math.
    3) Segregate the genders, especially at high school. Do this simple thing, all learning, especially math learning, will increase many-fold.

    Again, these simple things may seem radical. The first one would require money and a new commitment to education but it is done in Finland and really makes a difference. Two teachers can relate to most students, manage the class better and get around the class better to give individual attention. This can only help in math instruction. Having 40 kids in a typical Algebra class with an overworked, underpaid teacher isn’t going to get good results in most cases…

  2. Yak_Herder

    Howard? One room with two teachers and forty students? I vote for two rooms with one teaecher and twenty students in each.

    Of course, a respectable salary would be nice, too. Let’s not get silly.

  3. howard beale


    Dream big, how about one room, two teachers and 25 students!

    Of course, this would take a huge investment and paradigm shift. My solutions are easy but I’m not sure they are easy.

  4. Aaron

    “In the elementary grades, could we please teach the students HOW to do things, like add, subtract, multiply, and divide integers, decimals and fractions?”

    Thanks, Stephanie. I’ll second that!
    In our elementary grades students get the mile wide and inch deep array of concepts. It’s a flood of ideas that students struggle to keep organized in their minds. I am often told that the current program in elementary math is research-based, which somehow means it is the only reasonable approach. Then I see that the program is soon replaced by another “researched-based” program. Well, they can through all the research at me they want, but I still start my middle school math class with a majority of students who can’t add, subtract, multiply and divide! This always impedes the learning of other concepts.

  5. Mary McConnell

    You aren’t going to like this reply – but hiring more teachers isn’t a paradigm shift. It’s pretty much the strategy that schools followed over the past two decades. The number of teachers has risen much faster than enrollment.

    From the National Center for Education Statistics:
    “For public schools, the number of pupils per teacher—that is, the pupil/teacher ratio—declined from 22.3 in 1970 to 17.9 in 1985. After 1985, the public school pupil/teacher ratio continued to decline, reaching 17.2 in 1989. After a period of relative stability during the late 1980s through the mid-1990s, the ratio declined from 17.3 in 1995 to 16.0 in 2000. Decreases have continued since then, and the public school pupil/teacher ratio was 15.3 in 2008. By comparison, the pupil/teacher ratio for private schools was estimated at 13.1 in 2008. The average class size in 2007–08 was 20.0 pupils for public elementary schools and 23.4 pupils for public secondary schools.”

    “The number of public school teachers has increased by a larger percentage than the number of public school students over the past 10 years, resulting in declines in the pupil/teacher ratio. In the fall of 2010, there were a projected 15.6 public school pupils per teacher, compared with 16.0 public school pupils per teacher 10 years earlier.”

    It’s not at all clear to me that this increased hiring produced increased learning.

    Salaries, on the other hand, HAVE been largely stagnant in real terms. From the same source: “The average salary for public school teachers in 2009–10 was $55,350, about 3 percent higher than in 1990–91, after adjustment for inflation. The salaries of public school teachers have generally maintained pace with inflation since 1990–91.”


    These facts are surely not unrelated. As Rick Hess, among others, has pointed out repeatedly, school districts have put money into hiring more people, not raising their salaries. Given that states will face even tougher budget constraints for the next year (driven in many states, including my now home state of California, in part by the rapidly rising cost of government, including teacher, pensions), I don’t see much prospect for increasing the number AND salaries of teachers.

    So here’s a proposed change that really might be a paradigm shift. Why not hire bright college students to help in classrooms – in other words, tap in to some of that talent that’s applying to Teach for America in droves. Ask any math department chair – it’s tough to find qualified candidates to teach those upper level math classes. But it might be possible to offer part-time work to future engineers, financial analysts, etc. Ditto for writing. Here I can speak with more experience. Over the past few years I’ve worked with talented writing assistants who help me grade (and therefore allow me to assign) more essays. If I design the rubrics, train student reader/graders, and supervise the results, I can make use of what are (one hopes) my professional skills while increasing my productivity.

    Now, I know that Utah has large classes and low salaries. So I agree that your argument carries more weight here. But given the state’s large student population, the per taxpayer burden here is actually pretty comparable to other states. . . and I don’t see the legislature approving big expenditure increases, at least any time soon.

    So maybe we should be exploring ways to use online courses, teaching assistants, etc. to leverage the workforce we already have . . . and maybe, eventually, even increase teacher salaries.

    In the long run, I think the best way to convince legislatures and school districts to pay teachers more is to provide more convincing proof that we’re doing a better job of educating students. No, I don’t think teachers deserve all the blame for our educational woes. There’s plenty of blame to go around. But it will be interesting to observe whether those states and school districts that have been more willing to introduce value-added assessments and other measurements that tie student performance to teacher contributions ultimately end up paying (at least demonstrably successful) teachers more.

    And now I’ll duck and run for cover.

    • Yak_Herder


      I’d like to see comparable statistics for Utah’s teacher-student ratios. I’m willing to bet they aren’t comparable. A distinction between national and local trends needs to be made.

    • Aaron

      The problem with many student to teacher ratios is that the teacher side includes administrators who often are licensed teachers but do not have any classes of students. The ratio could be low yet a teacher has 35-40 students in the classroom. A much more realistic measure of average class size is…well, class size–go into the classroom and count the students! A student to teacher ratio that includes several individuals that don’t really “teach” in a classroom gives us an inaccurate perception.

      • Mary McConnell

        Good point, and one I’ve made before – but should have added the rising number of administrators to my comment.

        In fairness to administrators, the growing number of administrative mandates enacted by state and federal governments has simply required schools to employ more administrators. I still think this is an area where schools can find more efficiency . . . but maybe we should be scaling back the micromanagement as well.

        I didn’t mention this in my recent op-ed on the common core curriculum, but it’s another concern. How many more administrators will need to be hired to track all these complicated “curriculum mapping” requirements?

  6. howard beale

    At the high school in our neighborhood FTE’s have been decreased though the student body will be about the same or slightly higher. The Social Studies dept. at this school went from 9.5 teachers to 6.5 teachers. The school has lost teachers in all departments. Several programs are being downsized or eliminated altogether. Maybe you cite trends from whatever sources you would like, but I live in reality and the reality is that schools are reducing staffs (teachers and support staff), class sizes are getting larger and teacher morale and student success is suffering. Maybe this is a Utah thing???

  7. howard beale

    On the salary thing Mary, you have to be kidding right. Salaries have kept up with inflation for teachers. Go talk to any teacher in a public school that has taught say 15-20 years and find out what has happened to their salary the past five years, perhaps even 10 years. Many districts recently have furloughed days, not honored steps and even lanes. Most have raised insurance premiums. The Provo School District raised salaries 1 percent this year, the highest raise since 2008. Also, with this particular district, things went south for teachers and staff way before the “great recession” hit due to poor financial management. Teachers lost benefits and salaries for this district went stagnant before other districts hit the pinch.

    Many teachers at the local high school teach well over 200 students. Many staff members have had their hours cut or lost their jobs altogether, custodial staffs have been cut, and needed repairs to facilities have fell by the wayside. Again Mary, I’m not sure where you’re getting your stats but maybe you ought to get down in the trees a bit to see what is happening in the forest.

    However, I like some of your ideas of using college students better etc. Thankfully, the local schools around BYU actually tap into that a bit but things like you suggest aren’t bad thoughts. I still think Mary your paradigm on education wants to blame teachers for the ills you see in education, this is too bad coming from an educator. I see the teachers in my neighborhoods doing all they can to help students. I think if there is a crisis in education (and guess what I think this is way overplayed, most people that go through our schools turn out okay in life believe it or not), much of the blame lies in other areas such poor finances, lack of parental involvement, and other societal factors ranging from divorce to technological gadgets as really doing more harm to education then the teachers. Now I’ll duck and hide for cover…

    • Mary McConnell

      As I said, I don’t think that the Utah experience necessarily tracks with nationwide data. I searched the National Center for Education Statistics – the source of my data – but did not find a recent enough state-specific breakdown. Maybe someone else has these numbers?

      Of course I don’t think teachers can be blamed for all the ills of education. I’ve said this over and over, and I’ll say it again. Family dynamics, societal trends – both undoubtedly affect student performance more than teacher effectiveness. But it’s also true that teacher effectiveness is important, and that good teachers can make a big difference. So yes, we should look at ways to increase teachers’ effectiveness. (Not incidentally, I think this is also the best long run strategy for increasing teacher pay.)

      I’m skeptical of the recent

  8. Michelle

    I couldn’t agree with you more, Stephanie. As I sit here trying to help my son understand how to simplify square roots, how I wish his elementary teachers had drilled him on his multiplication facts!!

  9. Ze'ev Wurman


    If I were to accept your claim that teachers are blameless for the poor results and that much of the blame lies with lack of parental involvement and with social and societal factors, can you please explain how putting more money into education will help? After all, you argue that teachers are already doing all they can, and more money for teachers won’t improve parental involvement, or improve social factors. If at all, it will make the social factors worse — the extra money for education has to come from additional taxes.

  10. howard beale

    Ze’ev Wurman:

    Teachers could make a bigger impact if they had smaller classes, better technology, more support at home, more support from administrators, less top-down pedagogical strategies and curriculum that don’t work. Plus, teachers do more than teach curriculum. They are often the only role models these students have. They are often pillars of stability in the community. They teach citizenship and responsibility (if again not undermined by administrators, parents and society in general). I could argue that a high school history teacher teaching history might be down the list of the important things that teacher does for his/her students and school. So maybe with more support and better teaching morale, these parental and societal factors that seem to be eroding quality education could be fought more effectively.

    Further, I think revenues could be raised in other ways besides raising taxes but at some point you get what you pay for as taxpayers. I believe Utah is on the edge of an education abyss. I think the chickens are coming home to roost with these large classes Utah teachers face and the general assault on teachers certainly by our legislature and too many in the public at large. Members of the legislature have been trying to destroy public education to try to recreate it in some image I don’t completely understand. I hope it all works out (for the sake of our children). But I would like to see an investment in education, if it means raising taxes so be it. I think other things like a lottery, limiting tax deductions for parents with more than three children and taxing corporations higher might help. Many Utah local governments have sold the farm to many businesses with tax-free deals for 10 or 20 years. This has eroded the tax base nearly as much as decreasing property values. But I’m willing to look at anything because I think a crisis is at hand. There isn’t enough charters, let alone private schools, when Utah’s public schools coming crashing down to this assault. I think the citizens better realize that Utah schools in particular are near a crisis stage, mostly because of poor resources, overworked teachers and inordinate class sizes on top of an ever increasing diverse student base Utah is unequipped to help.

    I think teachers are the least of the problems, they need help, they need support, they need appreciation, they need resources. I wouldn’t mind even more federal commitment of dollars to help fund the schools if it came with local control and ended the silliness of these standardized tests. I maintain that teachers are doing amazing stuff not only teaching content but being great role models for our children.

    I don’t know if I answered the question like you wanted but so it goes…

  11. Suzie

    You people make it seem like doomsday. Yes, there are things that need fixing, but I hardly think we’re nearing a crisis. People who want to learn will learn. Actually, that’s probably the root of the problem. Not enough people want to learn, so things go down the tube. If we can make kids want to learn, I’m pretty sure most of the problems will go away. Unfortunately, that can’t happen. We’ve lead the horses to water, and many don’t want to drink, and there’s nothing we can do about it.

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