William G. McCallum accepted an offer about five years ago to lead a team in developing the Common Core State Standards for mathematics. The Sydney, Australia-born mathematics professor, who has worked at the University of Arizona in Tucson since 1987, chaired the 51-person work group, which included researchers, mathematicians, state education department representatives, and teachers. Along with Phil Daro, a senior fellow at America鈥檚 Choice, a school-improvement program owned by Pearson Inc., and Jason Zimba, a math and physics professor who later co-founded the nonprofit Student Achievement Partners, Mr. McCallum then served as a lead writer of the college- and career-ready standards, which the majority of states are now implementing.
In an interview with 澳门跑狗论坛, Mr. McCallum, who is currently collecting activities and diagrams for each of the standards through a project called Illustrative Mathematics, discussed the thinking behind various components of the math standards, and addressed some criticisms of the framework. (The interview has been edited for length and clarity.)
You started the process of developing standards by writing 鈥減rogressions documents,鈥� which describe the order in which particular topics should be taught. From there, you broke the progressions into grade-level standards. But you and many others say the progressions documents are key to understanding the standards. Why and how should teachers be using them?
We wanted the standards to be coherent across grade levels. We wanted them to tell a story that made sense about mathematics and we thought that was a good design to start out.
The progressions documents tell the story of how the mathematics in a particular domain, let鈥檚 say the fractions domain, progress. You can detect that from reading the standards鈥攆or fractions, for example, you can look at the grade 3 standards, you can look at the grade 4 standards, and you can see how they fit together. But the progressions documents have examples, and they have more narrative explanation. So those are useful documents if you want to understand this idea that the mathematics does progress, and that what you鈥檙e teaching in grade 3 you鈥檙e teaching for a reason so that kids going into grade 4 will be able to do what they鈥檙e supposed to do there.
Unlike most state standards, the math common core includes, in addition to content standards, eight Standards for Mathematical Practice. What exactly are those and why did you all decide to include them?
They鈥檙e various mathematical habits of mind that we wanted to codify. They鈥檙e an evolution of the National Council of Teachers of Mathematics鈥� process standards and a National Research Council document called 鈥淎dding It Up鈥� that talks about the strands of mathematical proficiency.
There have always been attempts to describe not only the knowledge and skills that kids have to have, but how they approach doing the mathematics鈥攄o they have persistence and perseverance solving problems? Do they use language precisely? It wasn鈥檛 an entirely new idea to try and describe what mathematical practices look like, but it was new to make that part of the standards.
So will students be assessed on these? How will that work?
I know both assessment consortia are trying to figure out how to do that, but of course it鈥檚 limited to what you can do with a large-scale computer-administered assessment.
But I also think assessment of the Standards for Mathematical Practice will happen in the classroom. Not all assessment is about the tests that kids take at the end of the year. Teachers observe students, they can give performance tasks, they can see how they do. It might be more in the form of formative assessment than summative assessment, but that鈥檚 an important location for assessment of the practice standards.
In your view, what鈥檚 the biggest misconception people have about the math standards?
I think probably the biggest one is they think standards are the same thing as curriculum. When their kids bring home a homework problem that鈥檚 a quote 鈥渃ommon-core problem"鈥攊t isn鈥檛. It鈥檚 a homework problem from a curriculum, and the curriculum may or may not be aligned to the common core. But the common core never said the kid had to do that problem. But I think people don鈥檛 see that.
Standards are just expectations for what we want students to know and understand and be able to do by the end of the year or a particular course. Curriculum is how you get them there. There can be good or bad curriculum implementations for the same standards.
What do you think is the biggest specific change in the standards, as compared with most previous state standards, for math teachers?
I鈥檇 say the level of focus is probably the biggest change. Every teacher is used to teaching a certain amount of stuff in their grade level, and the common standards basically rearrange the material so that at any given grade level, you鈥檙e focused on a smaller set of things.
So for every teacher that used to teach topic X, and it鈥檚 no longer there, they have to make that adjustment. And I think there鈥檚 a temptation probably to say, 鈥淚鈥檒l just do it anyway even though it鈥檚 not in the curriculum anymore.鈥�
Which of the new standards would you say teachers have found most challenging to implement?
Different things for different grade levels. I think the fraction standards are challenging for grades 3 through 5 teachers. I think the thinking about the Standards for Mathematical Practice is hard.
If you put the question a little bit differently, which is, 鈥淲here do I start,鈥� I tell everyone don鈥檛 try do everything at once; pick your battles. If you鈥檙e a school district deciding what professional development to give to elementary teachers, I鈥檇 say fractions. In middle school, I would say ratios and proportional relationships and how they鈥檙e different in the common core.
Are there any changes you wish you could make to the standards?
I have a bunch of tweaks. But I鈥檇 advocate stability for a while. Not that the standards are perfect by any means, but my changes might not be the right ones.
That said what kind of changes would you hope to see in the future?
I think the geometry progression could be evened out a bit in elementary school. I think in high school there could be more focus. High school was difficult because everybody has their pet topic, and it was difficult to resist those pressures.
Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich., recently wrote in The Wall Street Journal that math teachers are focusing too much on students鈥� conceptual understanding in math, which is a key tenet of the common standards. The key to expertise in math, she said, is not conceptual understanding but practice. What鈥檚 your response to that?
I think she鈥檚 completely right that people are overemphasizing conceptual understanding. But the common core balances conceptual understanding and procedural fluency.
What鈥檚 interesting to me is that both the supporters and the critics of the common core, I think, are overemphasizing conceptual understanding鈥攁nd understandably because everybody鈥檚 always demanded procedural fluency, and the conceptual understanding is what鈥檚 new. But that doesn鈥檛 make the other requirement go away.
In some sense, previous waves of reform have swung back and forth between one or the other, but the common core strikes a balance.
What鈥檚 your response to the criticism that the math standards don鈥檛 go far enough? That they don鈥檛 prepare students who want to pursue a career in math?
That鈥檚 a complete red herring. Standards never did that. It鈥檚 always been true that if you wanted a career in math and science, you take more math than kids who don鈥檛. It鈥檚 always been true that the standards for college acceptance for a general wide variety of majors is Algebra 2, and it鈥檚 always been true that if you want go into a STEM career or math major or physics major, you take calculus in addition. Or if you want to get into Harvard or Stanford, you take calculus.
There鈥檚 nothing new there. The standards don鈥檛 describe calculus because there鈥檚 already a national standard for that, which is the AP [Advanced Placement] curricula.
Fundamentally, this is a confusion between college readiness and STEM readiness. Some people want to define college readiness to be the same as STEM readiness. That鈥檚 fine if that鈥檚 their opinion, but it鈥檚 not traditionally the definition that most people have used. And it鈥檚 not the one that we use. We use the definition of college readiness as ready for credit-bearing courses in college.
How do you hope teachers and schools will use the 鈥減lus鈥� standards in the high school sections, which introduce advanced material?
There were places where we wanted to have the plus standards around for the sake of coherence. You don鈥檛 want to just teach a bit of trigonometry without teaching the rest. We made decisions about what should be required for college readiness, but that鈥檚 not quite the same thing as what makes a coherent course in high school. So I hope that educators will use the plus standards in places where they see that they fit successfully into the coherent structure of a course.
In the intro to the high school standards, it says you can put plus standards in courses that are required for all students. But the college- and career-ready assessments might not assess those standards.
We were supposed to come up with some threshold beyond which you鈥檙e college and career ready. If you鈥檙e going to describe a threshold, you need to describe what鈥檚 on the other side of it. The plus standards help delineate that threshold.
What would you say is the biggest promise of the common standards for math teachers and their students?
Just having focused and coherent standards helps teachers. If you have standards that are focused, that means you have more time to cover each topic at each grade level, you have more time to make sure kids really learn it.
If the standards are coherent, you can understand the purpose of what you鈥檙e doing because you know where it goes in the next grade level, and you know where your kids came from. So focus and coherence help teachers as well as students.
But I also think, if you have standards between different states, and one state develops a really cool resource, it鈥檚 no longer a local innovation鈥攊t鈥檚 a national innovation. That resource is available to any teacher who is teaching based on the same standards. That鈥檚 useful. That means if you go online and you鈥檙e looking for resources, you鈥檒l find things that are exactly what you鈥檙e looking for.
