In Teaching Math, What’s the Right Mix of Content and Context?

| January 31, 2013 | 3 Comments
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“Polynomial functions!”

“Trig identities!”

“How about the properties? Commutative, associative, distributive.”

So unfolded a laundry list of what a group of math teachers considered the more painful and less necessary concepts covered in the average high school math curriculum.

The laments, aired at EduCon 2.5 in Philadelphia at Science Leadership Academy last weekend, were part of a discussion around how to rebuild math instruction under the radically different—and admittedly unlikely—parameters posed by moderator Mike Thayer, a math teacher at Summit Public Schools in New Jersey.

Thayer, who also has a background teaching high school physics, proposed a scenario in which high school freshmen would take a one-year course (or a one-semester course in a block scheduling system) that covered the essentials of Algebra 1 and 2, Geometry, and possibly parts of Trigonometry. Any additional math concepts might be learned in a cross-disciplinary fashion through other courses. For example, chemistry teachers would be responsible for teaching

“I’d like to delete polynomial functions, but I’d like my students to see a roller coaster and think, ‘There must be math involved in that,’ and to go online and try and figure that out.”

students the basics of logarithms while covering the pH scale. Biology teachers would explain concepts of exponential growth to their students when discussing species population and reproduction.

The rationale of such a course, Thayer said, would be to create a version of math instruction that more fully lives with the inquiry-based learning approach embraced by the Science Leadership Academy, the public magnet high school where the conference took place. His vision—which hinges on what he concedes is a large assumption that students would enter high school competent in basic computational thinking—is for a course that would both streamline a high school student’s general math experience, and empower and encourage them to learn additional math skills to solve real-world problems of their own interest.

As one teacher at the discussion put it: “I’d like to delete polynomial functions, but I’d like my students to see a roller coaster and think, ‘There must be math involved in that,’ and to go online and try and figure that out.”

Thayer asked the teachers to consider four questions as they imagined the hypothetical course:

  1. WHAT STAYS AND WHAT GOES? Consider both what concepts would get more or less emphasis, as well as what method of learning (lectures, work sheets, group work, collaborative projects, etc.) would work best.
  2. NEXT STEP FOR STUDENTS? Options could include more advanced mathematics courses, independent mathematics projects, courses in other subjects that included applicable advanced math concepts, or some combination.
  3. HOW WOULD TEACHING CHANGE? Choose which lessons you’d save and which lessons you’d skip. Envision whether you’d use the same kinds of exercises to develop students skills, and whether you’d structure class time in the same manner, or perhaps utilize it differently.
  4. WHAT WOULD YOU ASSESS? Tests should reflect the purpose of the course, to develop students’ understanding of the theoretical and practical purposes of math.

Most of the discussion during the 90-minute talk focused on the first two points, and the group generally agreed the course would need to focus on changing student thought processes.

“What I am hearing is that if we would like to really make math meaningful for our students, we need to do things to create the ability for them to be truly mathematical thinkers,” Thayer said at one point after hearing a few responses.

There were, however, disagreements over the relative importance of concepts. And a couple of teachers even asked whether geometry would fit within the parameters of such a course.

The group also questioned whether a focus on real-world math applications would be the most likely way to spur students into independent investigation, and whether that focus could create an unintended bias in the kind of material covered. As an example, teachers noted that using tools like 101 Questions, a website that asks users to think of a question related to a displayed image, could result in an excessive focus on proportionality.

Thayer encouraged such discourse, suggesting it would be essential in his new model.

“I think the first thing for us, in order to be masters in context as well as content, is to recognize our strengths and weaknesses,” he said. “I would love for somebody else to be able to come into my classroom and explain why [a concept] is important.”

In a speech at EduCon earlier that morning, Philadelphia public schools Superintendent William Hite stressed the need for teachers to move from content to context expertise. And in a later discussion, Science Leadership Academy founding Principal Chris Lehmann conceded such an approach could be more difficult in a math classroom, but not impossible.

Thayer, meanwhile, warned that if math teachers didn’t find a way to make that difficult shift, they could be marginalized.

“Most of the stuff we teach” in traditional courses, he said, “the Khan Academy does it for free.”

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  • James Street

    I liked this article, for the most part, except for the fact that it clearly misses one of the key goals of teaching mathematics: the metacognitive processes associated with conducting mathematics. As a math teacher, I love everything that I teach, but I think that in-depth treatment and deep understandings ought to be reserved for college-bound STEM students. I use my math classes to try to teach students to be persistent in problem-solving; I try to produce students who can proceed with confidence into a problem despite being unclear as to their path.

    “Thayer, meanwhile, warned that if math teachers didn’t find a way to make that difficult shift, they could be marginalized.

    “Most of the stuff we teach” in traditional courses, he said, “the Khan Academy does it for free.””

    What a bold, bold statement from a well-intentioned but ignorant person. To say that a robot can teach a whole child the art of problem solving clearly shows the level of misunderstanding not only the general public but also the powers that be have about teaching mathematics. I do hope this guy knows what he is saying.

  • Chris Brownell

    As a person who is working in the field of educating Math teachers, both pre-service and in-service, I have some perspective to comment on this. A significant difficulty is going to be that the vast majority of teachers have had a very broad background education in mathematics. One that focused on properties of the Real numbers, simple geometric relationships, general number theory, and an abstract introduction to Calculus. Few of these foci actually provide opportunity for the students/now teachers, to contextualize the mathematics they study. In short they lack real applications for the math they learn. A symptom of this is the pat answer most math teachers will give when asked the question, “When am I ever going to use this?” is, “Next year when you take (insert name of next math course here) you will need this so you can… Few Math teachers have even cursory experiences in the Sciences where there may be some application, and far fewer in Engineering where the Science teachers also have a lack of experience, but is a field rife with opportunity for cross-disciplined thinking.

    So assuming this (the type of reform discussed here) becomes a shared goal in education. To make it a reality many hurdles will need to be crossed. Math teachers and Science teachers alike will have to cross them, albeit in different directions. STEM education is at a cross-roads in America. The introduction of the Common Core State Standards in Math and Language Arts in the next year or so, followed by the Next Generation Science Standards a few years later can provide an impetus for change, if we really want it. Past experience though with reforms, reconstruction projects, and implementations of new ideas would suggest that success will require a great deal of Force, to overcome the Inertia of the status quo. What is the average citizen willing to do?

  • Ken Jensen

    I have placed comments from the article in brackets. My responses follow each bracketed statement.

    [Thayer, who also has a background teaching high school physics, proposed a scenario in which high school freshmen would take a one-year course (or a one-semester course in a block scheduling system) that covered the essentials of Algebra 1 and 2, Geometry, and possibly parts of Trigonometry…
    Thayer believes that additional math concepts such as
    logarithms and exponential function should be reserved for science class.]

    I am not convinced that the “essentials” of math are covered in Algebra 1 and 2, or Geometry. Learning additional math skills will not necessarily encourage the solving of real-world problems. The skills must hang on a conceptual framework
    for them to be useful.

    Thayer seems to be implying that the “additional math concepts” and the context that goes with them should not be part of a school’s math program. James Heibert in
    his books The Teaching Gap and Making Sense tells us that bringing in context, such as what is done in a science class, is essential towards creating a conceptual framework where math makes sense.

    [WHAT STAYS AND WHAT GOES? Consider both what concepts would get more
    or less emphasis, as well as what method of learning (lectures, work sheets,
    group work, collaborative projects, etc.) would work best.]

    To determine what stays and what goes one must consider what it means to mathematically literate. The National Council of Teachers of Mathematics has already done a great job of defining what should be taught. The National Research Council in their land mark publications Adding It Up, and How Students Learn Math as well as other researchers such as Yackel and Cobb, Jo Boaler, Margaret Smith, and John Van de Walle have written extensively on the instructional practices necessary for kids to learn the content.

    [NEXT STEP FOR STUDENTS? Options could include more advanced mathematics courses, independent mathematics projects, courses in other subjects that included applicable advanced math concepts, or some combination.]

    Next Steps beyond a combined Algebra 1/Algebra 2/Geometry course assumes that students need to understand the isolated skills before they can apply them. The National Research Council as well as the National Council of Teachers of Mathematics both state otherwise. Skills must be taught in conjunction with both models and real-world problems for them to make sense. One of the primary criteria for mathematical literacy is representing real-world problems with graphs, tables, and equations, as well as converting between and solving through graphs, tables, and equations. According to Shelly Kriegler, UCLA mathematics department, Algebra 1 and 2 resources as well as the lecture style instructional model that is use to deliver them does not allow for this kind of Algebraic Thinking.

    [HOW WOULD TEACHING CHANGE? Choose which lessons you’d save and which lessons you’d skip. Envision whether you’d use the same kinds of exercises to develop students skills, and whether you’d structure class time in the same manner, or perhaps utilize it differently.]

    Choosing which lessons would be skipped or which exercises would be used has nothing to do with how teaching needs to change. Inquiry is about choosing
    problems with high cognitive demand and then utilizing instructional practices that cause the rigor to remain in the lesson.

    However, choosing a resource that has demonstrated the ability to create the conceptual framework necessary for math to make sense means that the user must adhere to the trajectory or “story line” of the resource.

    [WHAT WOULD YOU ASSESS? Tests should reflect the purpose of the course, to develop students’ understanding of the theoretical and practical purposes of math.]

    The theoretical and practical purposes of math are based on the conceptual framework that a student has created around the meaning of the math they are learning. When math does not make sense then it is non-sense.

    Marzano tells us that assessments must be designed with the proper level of cognitive demand to determine a student’s proficiency based the standards.

    [“What I am hearing is that if we would like to really make math meaningful for our students, we need to do
    things to create the ability for them to be truly mathematical thinkers,”]
    [There were, however, disagreements over the relative importance of concepts. And a couple of teachers even asked whether geometry would fit within the parameters of such a course.]

    Conceptual understanding around proportional reasoning is decidedly another critical component to being mathematically literate. This of course must be developed in algebra though an understanding of the differences between y=mx and y=mx + b. However, it must also be developed through the understandings associated with similarity. Understanding similarity and the proportions associate them is the foundation of trigonometry. Without an understanding of proportionality trig ratios are seen as tricks to get an answer rather than ratios that are used to set up proportions. Also, a comparison of direct and inverse proportionality is critical to understanding the patterns associated with 3-varible equations such as F=ma and V=IR

    […teachers noted that using tools like 101 Questions, a website that asks users to
    think of a question related to a displayed image, could result in an excessive
    focus on proportionality.]

    American mathematics courses, K-12 and beyond, need more emphasis on proportionality than is currently delivered. However, this needs to be tempered with being able to represent, convert, and solve with graphs, tables, and equations.

    [“I think the first thing for us, in order to be masters in context as well as content, is to recognize our strengths and weaknesses,” he said. “I would love for somebody else to be able to come into my classroom and explain why [a concept] is important.”

    Wow! This is exactly what the National Research Council has done. Students need a conceptual framework to hang the skills and procedures on so that they make sense and are therefore useful. This is also supported by the mathematical practices as presented in The Common Core State Standards for Math, and well as Phil Daro the principle author of the CCSS for math.

    [Science Leadership Academy founding Principal Chris Lehmann conceded such an approach could be more difficult in a math classroom, but not impossible.]

    This shift is very difficult for both math and science teachers. Jo Boaler has said that teaching is a cultural endeavor and as such teachers have a tendency to teach the way they were taught. It’s really the only model they have for how to run a lesson. The need to create inquiry in the classroom is causing American educators to work against the model they were exposed to as students. The hardest thing you can ever ask a teacher to do is to teach in a way they were not taught themselves.

    Finally, to answer the guiding question, the right mix of content and context needs to be developed through the research around what kids need to know and be able to do to be mathematically literate. Our traditional math content and the instruction associated with it have driven people away from math, and many who have learned math have learned it in spite of their formal mathematics training. If we cannot learn how to create students mathematicians who learn math because of what we do as math educators then we should be marginalized.