mathlovergrowsup

Closing the Teach For America Blogging Gap
Jan 25 2012

Getting people to agree on a Common Core was just the start of the work: and I’m pessimisstic.

I went to a panel on Common Core Standards in DC this weekend led by Bill Schmidt at Michigan State University. He’s involved in Common Core assessment design and research on teacher’s implementation of Common Core. He was excited by the possibilities of collaboration around coherent standards but still the tone of the discussion he led was pessimistic.

Do I start with the good or the bad?
The good is that a lot of people are thinking about implementing the Common Core and providing resources and professional development to teachers. Also, there is a general positive feeling surrounding content of the standards by those high up in education with a lifetime of experience.

However, many in the discussion saw the same issues that I did-teachers don’t understand the math in the core deeply enough to teach it as intended and current resources are inadequate to support their attempts at change.(A Mewborn article summarizes some of the research backing that up: Teachers content knowledge, teacher education, and their effects on the preparation of elementary teachers in the United States) Those who have really tried to change teacher knowledge and practice have found that it can take more than 70 hours of professional development to start undoing the damage done by traditional math instruction. (Per usual, see the Teaching Gap for data and details.) And people have had difficult times linking teacher’s mathematical knowledge to student outcomes, probably because teachers need to be aware of student thinking to help students grapple with ideas that challenge them to advance their thinking. So even if we get these teachers up to speed on mathematics, we haven’t solved the problem of what they actually do in the classroom.

Dr Schmidt gave us some preliminary data reports from a study he is involved in that randomly sampled math teachers about their knowledge of Common Core. The good news was that 80% of them knew about the Common Core. The bad news was that 80% of them said that the Common Core was basically what they were already doing. See my post on a Teaching Channel video where a teacher is identified as teaching a standard that is only superficially present in her lesson. She is having kids memorize a rule related to proportion instead of interpreting and understanding proportional situations with math. If teachers see the Common Core as a way that states can write one big test instead of 50 and save some money they are not going to be perturbed enough to change their practice.

Since people in the discussion were so pessimistic about secondary curriculum aligned to the Common Core, I brought up the curriculum designed at Arizona State. Introducing this curriculum has perturbed some mathematics teachers to reevaluate and change their practice. It might be a tool to solve the 80% think Common Core is the same thing issue. An amazing woman, Marilyn Carlson wrote the curriculum Precalculus Pathways, aligned to many Common Core Standards, so that teachers could actually implement their new mathematical knowledge in the classroom. I hope that the curriculum goes somewhere-she included the topic she did because she knew that they were useful. She is faithful to the standards of mathematical practice because she believes in them. There is some beginning research that teachers using these materials are making shifts in instruction. But the shifts are slow. However, when I told the people at the discussion that there was actually materials that already were written and fit well with Common Core ideals no one had heard of it. This is where marketing and business seem to be a critical part of actually moving the world forward with your research.

Many textbook companies are just switching around the order of their books and slapping on “common core” to the cover. I worry that these superficial changes will just cause teachers more work but won’t result in student progress. I worry that the content and vision of the standards will be deemed a failure because they won’t actually be implemented. The Teaching Gap notes that time and again people adopt superficial features of reform. Japanese teachers use chalkboards instead of overheads in math. Math teachers use their whiteboards but do the same thing as they did on the overheads and nothing changes. Or we change the order of our standards and teach out of a new book but still teach proportions as cross multiplication and slope as rise over run. And nothing changes an we all wonder why education reforms don’t work.

So, I hope that a few brilliant people like Marilyn will be able to create resources and share them broadly that might actually make a difference. Common Core makes this more of a possibility, but certainly not a certainty.

5 Responses

  1. Andrew

    Can you give an example or two of something in the common core that isn’t something that people are expected to teach today?

  2. Ms. Math

    Andrew: Great Question-thanks for pushing back and making my post more clear.

    One main point of the Common Core is that teachers are supposed to teach math as a set of coherent ideas, not a set of disconnected procedures. It takes deeper understanding and more skill to teach connections between ideas than just procedures. Here is what they say on the website about what is new:

    “The standards stress not only procedural skill but also conceptual understanding, to make sure students are learning and absorbing the critical information they need to succeed at higher levels – rather than the current practices by which many students learn enough to get by on the next test, but forget it shortly thereafter, only to review again the following year.”

    An example might be the idea of slope. You can teach the formula and show kids how to find slopes. This is the old way. The Core wants teachers to connect the slope of an equation to quantitative interpretations of situations and rates of change. The change from “slope” to rate of change seems subtle, but to kids slopes often just refer to how slanty something is, not a way to measure how many times the change in one quantity is compared to the change in a related quantity.

    Here is the new standard:
    F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

    I’ve asked a number of University Calculus students what an average rate of change is and they don’t know. Some think you should add up and divide speeds. My definition is that the average rate of change is the constant speed a hypothetical object would have to travel to go the same distance in the same amount of time as the object in question. This involves the notion of constant speed and proportionality which are huge stumbling blocks for teachers and students.

    There is plenty of research on difficulties middle school teachers have with proportional reasoning(See Rational Number Project for papers). Here is one on how few teachers could solve problems requiring proportional thinking du/rationalnumberproject/95_2.html. It’s likely high school teachers will need to make up for some major deficiencies in proportional reasoning when they teach the standard above but few have been prepared to know how to do that.

    Additionally, there has been research showing that high school teachers with years of experience don’t have connected understandings of average rate of change, proportionality and division. So they can teach the old rules, but not follow standards of mathematical practice and connect these ideas together in meaningful ways. Here is a link to a study showing three high school teachers who struggled to talk about average rate of change and even got problems associated with this standard wrong. These were experienced teachers who were well respected at teaching traditionally. http://pat-thompson.net/ThesesDissertations.html See Ted Coe, “Modeling Teacher’s Ways of Thinking about Rates of Change.”

    Thanks so much for the question, keep them coming and feel free to disagree.
    Best,
    Cameron

  3. Ms. Math

    I feel a bit guilty for being critical of teacher knowledge-I don’t think it is their fault that they grew up in a culture of dysfunctional math education.
    I don’t think that I had the knowledge I needed to teach well-I saw many, but not all, of the math connections but I didn’t know how my kids understood math so I couldn’t really change anything either.

    I like teachers-but I just think the truth about what’s been discovered about teacher knowledge of math and teaching needs to be taken into account when thinking about Common Core implementation and it’s potential to improve education.

  4. Andrew

    Thanks for the thoughtful response.

    Slope IS different from average rate of change. Slope is in my mind a linear notion. Average rate of change is not. Linking the two is perhaps important, but quite an advanced notion and, I think, not that important. In fact, I would argue that people don’t think call rate of change “slope” in most cases – they call it proportionality and describe it with a linear function: f(x) = mx + b. Slope is just a name that we have for the constant of proportionality, “m”. But that’s not the problem that I think you’re really getting at.

    I’m sure that there are many teachers who fixate on formula for things like slope, but I think that you have to understand that this is a problem (mostly) with the knowledge of the teachers, not the pedagogy or the curriculum. Good teachers have always tried to make these connections – it’s not a matter of new versus old – it’s a matter of well-prepared and knowledgeable versus open-the-book and barely understanding. Your notion, one of speed and position, is already a little limiting, as it is just an example.

    If you say that your Calc students don’t understand this rate of change notion either, then they haven’t had good instruction or don’t care. I am quickly reminded of Denis Auroux’s MIT calc classes (YouTube) and the careful way in which he discusses instantaneous and average rates of change (dy/dx versus delta y / delta x). Or take a look at the old educational physics TV show “The Mechanical Universe”, which does a great job with this (and is a super introduction to what calculus is about, I think).

    It takes an immense amount of dedication to be prepared to do a great job with each topic – professors and TV producers have more time to think about how to cover this stuff. I had trouble keeping up just with doing a good job grading calc papers when I was in school – forget about preparing lectures, lessons or projects. If one has only a single prep or two, or has been teaching for a really long time, perhaps it can be well done every day, but that’s not reality for most – even for good teachers.

    When I was student teaching, I tried to do a good job preparing – and I only had one prep – but I always felt sorry for my first period class. I really would like to try my had at teaching now (I’m over 40), but I think that if I had started right out of school, my students would have suffered. Not just because of my age or my grasp of math, but of my warped perspective on the importance of school and mathematics and grades and test scores.

    No matter what teachers do, it’s the students that we’re asking to learn. If they don’t have an intrinsic interest in the material, it’s unlikely that they’ll retain much over time. They’re human beings. For some (many?) human beings, slope and rate of change isn’t intrinsically interesting — and that’s OK.

    P.S. – I take a dim view of almost all education research. The samples are usually tiny, the researchers necessarily biased, and the controls non-existent. I’m sure there is good research, but boy, it’s scarce. This isn’t to put down the researchers, but such endeavors are really, really hard. We’re talking about people — not chemicals in test tubes.

  5. Ms. Math

    Hi Andrew-
    Thanks for all of your thoughts. I agree that slope and rate of change are not the same, but I do think that some teachers will think they are basically the same thing and teach finding rates of change just as they would teach finding slopes.

    My students certainly suffered at my inexperience in the classroom-I don’t think grades and tests are the most important thing. I do feel bad for all the kids in math classrooms who are trying to memorize things that don’t make sense and feel stupid because of it. I wish they were doing something that might actually be interesting and relevant.

    I do have to admit, I got to teach Calculus as part of a team and it was our primary focus. We had way more time and space to plan than teachers-I don’t expect them to reinvent the entire class as we did. It happens sometimes, but not too often given all of the demands they are under.

    I think that education research can be valuable even if it doesn’t adhere to the methods of psychological research. However, if there is not theoretical foundations to what you are doing in small samples of people it won’t turn out much better than speculations about how people learn best. I don’t think that control groups are something that will work well in a lot of potentially valuable research. (Read Les Steffe and Pat Thompson’s explanation of the theory behind Teaching Experiments to see what I mean). And yes, I’m biased-I think Steffe and Thompson are some of the best educational researchers in the country even though they don’t use control groups.

    I do think that there are some good teachers doing good things-however the TIMMS study did a large random sample of classrooms around various countries and had experts rate the quality of lessons. 91% of lessons in the United States were scored low quality and the experts were looking at transcripts that didn’t reference the country. Japan had a much higher number of high quality lessons and almost no low quality lessons. There is something to the idea that most teachers in most schools are not teaching math with meaningful connections so most kids are not going to be able to learn math that way unless they happen to be curious and just figure it out.

    How did you take an interest in this post? You seem to have quite a few opinions on the matter :)
    Cameron

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