Another high school math debate September 25, 2009
Posted by Matt in Making Change, process standards.add a comment
In a follow-up to the last post, I came across an article in the Salt Lake Tribune (math education in Utah is a particularly fascinating topic). It seems that a district superintendent sparked a debate with the state superintendent when he sent a tweet that called the state’s position on high school math standards, “curious.”
The disagreement comes not from whether students should take more math in high school, but rather from what math they should take. The state superintendent believes that all students should take math through Algebra 2, and then have options for further study. His critic believes that all students should take calculus.
I agree with the first idea, for a few reasons. Calculus has been inappropriately crowned the king of math. Calculus is merely a doorway to further studies in math or a related field. Students considering a career that is rich in mathematics (pure math, math education, engineering, physics, etc.) should plan to take calculus, preferably in high school.
Many college-bound students will benefit more from a statistics course (required if they choose to attend graduate school) than a calculus course. Most students, regardless of their career plans, would benefit from a course in discrete math, although most schools and districts are slow to consider this path.
The danger of the argument is that these options are being labeled “tracks,” a negative term that implies that students that take statistics are not as smart or capable as students that take calculus. The responsibility lies with the schools and teachers to ensure that this ability grouping doesn’t happen, and that students are given every opportunity to follow the path of their choosing beyond Algebra 2.
…And the Counterpoint September 14, 2009
Posted by Matt in Making Change, process standards.2 comments
A few days ago, I posted a link and some commentary about reform math in Washington. Today, I came across this post, which is specific to Everyday Math.
The author notes that, “Reform math has dominated our schools for more than 15 years. Over this period, our international ranking has plummeted.” It seems that the article in the Seattle paper directly refuted this claim. At any rate…
The author basically degrades Everyday Math, citing several states that have banned or failed to adopt the program for various reasons. Here’s what might be my favorite paragraph:
Everyday Math has been described as a “mile wide and an inch deep.” U.S. Secretary of Education Arne Duncan is calling for “more depth and less breadth” in education. States like Connecticut are heavily invested in reform programs like Everyday Math. The Hartford Courant newspaper recently reported that 40 percent of incoming college freshmen require non-credit “remedial” mathematics.
Mile Wide, Inch Deep: Show me a core basal program that isn’t. It’s a symptom of over 50 different sets of standards and a long-running debate over what students really need to know.
More Depth, Less Breadth: This should be the goal of every teacher. Figure out what your students know, what the “kinda” know, and what they don’t know, and then adjust your teaching to fit. I’m a big fan of Texas Instruments and what they are dong for education, but stories like the one I received in a TI email today send shivers up my spine: “Imagine having your whole year planned out before stepping foot in your classroom.”
Remedial Math: Only 40 percent? Seems low. Again, this is a symptom of more than the program. It’s about outdated standards, outdated teaching, and a refusal to move away from the teacher’s comfort zone.
So we’re back to the same place: It’s about instruction.
(Note that nashworld does a great job of highlighting the need for quality instruction-through his own experience-in a recent post.)
Related:
How many times do I have to tell you…
What did you expect?
Wait. I Can MAKE Kids Learn? September 12, 2009
Posted by Matt in General Information, process standards.add a comment
I came across something interesting today. I ventured (for the first time) onto Google Books to see what they had to offer. I made my way to K-12 Mathematics – no big surprise there. And I came across this:
The Equation for Excellence: How to Make Your Child Excel at Math by Arvin Vohra.
Perhaps I’m naive, but I’m not familiar with Arvin or this book. I’m not going to talk about the author – I’ll let you form your own opinions based on his website.
I looked at the table of contents, and Chapter 11: The Calculator Fallacy caught my eye. So I started reading. I will admit that some of the points are valid and made me stop to think, but there is a general theme of “calculators make students lazy” and “teachers are misinformed.”
Then we get to this: “A student solving a complicated problem spends very little time doing actual calculations. Most of the time is spent examining relationships and determining what concepts apply.”
Wait. Didn’t he just make the case for calculators? I used graphing calculators to help students examine relationships and link concepts. If they used the calculator to multiply six and four, so be it.
The author then supports his statement: “The student who does math by hand has these concepts ingrained in his mind, and is adept at using them.”
Again, wait. Did he just tell us how students gain conceptual knowledge? Wow. We’ve been trying to figure that out for a while, and here was the answer all along. Make them do the work by hand. (Nobody’s ever tried that one before.)
Doing math by hand does not build a solid conceptual foundation for learning. Models help students build this foundation. Rich activities that apply learning help build this foundation. Regurgitating facts and working everything out by hand do not build conceptual understanding.
Finally, this assumption: “Thus, he rapidly sees relationships between various formulas and concepts, and can quickly figure out how to do the problem.”
I can count on one hand the number of students who made connections between formulas and concepts by simply doing problems by hand. I agree with the idea that a calculator in the hands of a less effective teacher is a dangerous thing. But the author discounts the role that a calculator can play in discovering patterns and understanding relationships, and the role of an effective teacher in promoting this kind of calculator use.
Washington’s Miniature War September 10, 2009
Posted by Matt in General Information, Making Change, process standards.add a comment
If you follow the ongoing saga of the “math wars” at all, you are likely familiar with the long-running debate in the state of Washington. Many school districts in Washington were early adopters of NSF-funded “reform” mathematics curricula, and much of the debate surrounding these programs has come out of Washington. (If you don’t believe me, do a search on YouTube for math.)
Even given this background, I was a bit surprised to see this guest editorial in the Seattle Times regarding discovery-based math. Of particular interest to me were the comments.
I think we’d like to believe the math wars are over. This article, and the related comments, bring us back into reality. It begs the question, “Will the math wars ever end?”
Why STEM teachers leave… April 2, 2009
Posted by Matt in process standards.1 comment so far
From a recent article in EdWeek. I’m not finding fault, I really enjoy their content. See if you can find the problem with this graph (the fact that the percents don’t add up to more than 100% doesn’t count).
Why Teachers Leave
Why do I care? March 9, 2009
Posted by Matt in General Information, process standards.add a comment
You might ask, “Why does he care about math education so much?”
And then I come across something like this:
If you see what’s wrong with this, you get my point. If you don’t see what’s wrong with this, you’ve made my point.
If you need a hint, let me know. I’m still a teacher at heart.
(Image from eggindustry.com)
Talking the Talk August 11, 2008
Posted by Matt in process standards.Tags: communication, vocabulary
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We had an opportunity to welcome Dr. Kevin Feldman to our school district last week. He presented a day-long session on “Narrowing the Lexical Divide: The Critical Role of Vocabulary & Academic Language in Improving Secondary Literacy Across the Curriculum.” His focus on academic vocabulary was of great benefit to the teachers in attendance.
One thing that really caught my attention in his presentation was the discussion about where we find Academic English – that Hayes and Ahrens (1988 ) used a measure of “rare words per 1,000″ to evaluate the frequency of word use. They found that the everyday adult speech of college graduates is at approximately the same level as preschool books, and that most informational texts are at a level comparable to newspapers and magazines.
This reinforced my belief that we have to talk about math before we write about it, and also supports the notion of developing formal spoken language as one path to formal written language (see Pimm (1991)). It also made me wonder about the level of spoken English in math classrooms, both by teachers and by students. Then Dr. Feldman showed us this website, which will analyze passages to determine rare words per 1,000.
Those who know me will likely guess what I’m thinking: research. This should be fun!
What did you expect? May 20, 2008
Posted by Matt in Making Change, process standards.Tags: algebra, ASCD, high school, instruction
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From the ASCD Blog:
“More math, particularly Algebra, in California high schools has yet to pay off. Last week’s most-clicked Smartbrief story reported that enrollment in remedial math courses still remains high in California Universities, leading many to question why high school reforms are not transferring to college-preparedness.”
Are we surprised? This is much less an issue of what we teach than an issue of how it is taught.
Standards are not the problem. Expectations for student learning and the pervasive “sit and get” culture of high school mathematics teaching are the culprits. Our focus needs to extend beyond the curriculum to include sound instructional strategies.
Gratuitous Irrelevance March 28, 2008
Posted by Matt in process standards.Tags: BSCS, David Thornburg, inquiry, instruction, NCTM, NSTA
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While I may be somewhat critical of the lack of new information in Foundations for Success, I do agree with the findings and recommendations of the National Math Advisory Panel. Which is why I’m so troubled by this article by David Thornburg. The basis for Thornburg’s argument is this:
Recent pronouncements from Washington regarding math education have suggested that pedagogical points of view don’t matter in the teaching of mathematics. For example: “There is no basis in research for favoring teacher-based or student-centered instruction,” Dr. Larry R. Faulkner, the chairman of the panel, said at a briefing last Wednesday. “People may retain their strongly held philosophical inclinations, but the research does not show that either is better than the other.”
Thornburg goes on to cite two “counterexamples” to refute this claim, both from “Rising Above the Gathering Storm“:
- Statewide specialty high schools (e.g., IMSA ), and
- Inquiry-driven project-based learning.
This is a wonderful example of the misconception of inquiry as being something totally student-centered, with little or no teacher input. Granted, part of the ownership for this misconception lies with the math education community – we do not often enough discuss the concept of inquiry using the word “inquiry.” Instead, we use terms like “problem solving,” “reasoning and proof,” or “connections.”1 It is the stubborn insistence of some educators that math is math and science is science and never the two shall meet.
The science education community, on the other hand, gets it. They understand inquiry.2 It’s part of their standards. In fact, the National Science Teachers Association (NSTA) describes scientific inquiry as
a powerful way of understanding science content. Students learn how to ask questions and use evidence to answer them. In the process of learning the strategies of scientific inquiry, students learn to conduct an investigation and collect evidence from a variety of sources, develop an explanation from the data, and communicate and defend their conclusions.3
Sounds to me a lot like the process standards:
Reasoning and Proof:
Instructional programs from prekindergarten through grade 12 should enable all students to–
- recognize reasoning and proof as fundamental aspects of mathematics;
- make and investigate mathematical conjectures;
- develop and evaluate mathematical arguments and proofs;
- select and use various types of reasoning and methods of proof.
Instructional programs from prekindergarten through grade 12 should enable all students to–
- build new mathematical knowledge through problem solving;
- solve problems that arise in mathematics and in other contexts;
- apply and adapt a variety of appropriate strategies to solve problems;
- monitor and reflect on the process of mathematical problem solving.
Instructional programs from prekindergarten through grade 12 should enable all students to–
- recognize and use connections among mathematical ideas;
- understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
- recognize and apply mathematics in contexts outside of mathematics.
Not to mention the communication and representation standards.
But this isn’t the issue at hand; the real issue is how to teach it.If we approach this from a logical perspective, then we understand that students will not develop these skills of scientific inquiry without some direction from the teacher. Inquiry is developed along a continuum, beginning with structured or directed inquiry, moving to the broad category of guided inquiry, and finally – often after much support and scaffolding – to open or student-initiated inquiry. One can also think of this in terms of the Gradual Release of Responsibility model for literacy instruction.
In other words, Thornburg’s argument is entirely irrelevant. His counterexamples fail miserably to disprove the findings of the panel with regard to student-centered v. teacher-directed instruction. What we know is that a balance of both is critical so that students have the opportunity to develop a solid conceptual foundation of school mathematics.
—
1See the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics, with particular attention to the process standards. [back]
2Some science resources and organizations that discuss inquiry:
- BSCS
- Northwest Regional Education Laboratory’s Science Inquiry Model. They also have a great problem-solving model for math.
- National Science Education Standards.
- Inquiry and the national science education standards: A guide for teaching and learning.
- The Valle Imperial Project in Science. [back]
3From the NSTA Position Paper on Inquiry. [back]
The Way We Do Business January 23, 2008
Posted by Matt in process standards.Tags: high school, instruction, math, pedagogy, reform, resources, systems
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I’ve been thinking lately about math in high school. This is not unusual, by any means, especially for someone in my position. Nationally, we are struggling with math in high school; some schools, districts, and even states are getting it right, but most are struggling.
A word if caution: if you’re looking for solutions in this post, you will be disappointed.
One thing that frustrates me most about the work we (as a nation) have to do at the high school level is the lack of concern about instructional practices. Most high school math teachers have a solid grounding in the mathematics that they teach. This is often incorrectly equated to a solid understanding of how to teach that content. Unfortunately, content knowledge does not imply pedagogical knowledge. (I would pose the question: is the converse true? My jury of one is still out on this, although I thought I knew the answer until about 30 seconds ago.)
So how do we change instruction? I’m not even going to attempt to answer that today. What I know for certain is that if we want the change to happen in the future, we have to start now. That sounds pretty obvious, but here’s why:
- Current high school students who are thinking about becoming math teachers are learning how to teach even now. They watch their teachers. If all of that student’s teachers are teaching they way that they were taught, then that student is going to someday teach the same way. The cycle continues because…
- Secondary teacher preparation programs at the university level do not do enough to promote a change in instructional practice. Those that do are often ineffective in reforming students’ attitudes and beliefs about pedagogy that were learned in high school. The result is…
- More teachers, teaching the same way they were taught, and influencing the next generation of teachers.
You’re probably thinking that I’m the world’s biggest pessimist right now. I really believe that we can change. I also believe that we will not wholly change the way that high school math is taught in the future unless we start the change now. In summary,
- Systems matter – they need fixing sometimes.
- Resources matter – they need to be of a high quality and aligned to appropriate benchmarks.
- Instruction matters – it’s the critical third element that we too often overlook.
Changing the way we do business in high school means taking a close look at all three areas. Until we do, we aren’t going to have the overall effect that our nation so desperately needs.
