Good afternoon and welcome to this month’s chat with NCTM President Cathy Seeley. The subject of today’s chat is A Flattening World, Cathy’s President’s Message in the October NCTM News Bulletin, which borrows from the title of Thomas L. Friedman’s popular book “The World Is Flat.” Our first question is from
While there are parts of your comments that warrant merit, I am very troubled by your “us” versus “them” sentiments as you express the need for U.S. students to develop mathematics literacy. I find these comments deplorable and divisive. The underlying sentiment seems to be that economic and workforce development in “other” nations is a threat to the United States; we should be aware of the danger on the horizon.
Furthermore, in stating “For some time, those of us involved in mathematics education have joined with the business sector to advocate for a more mathematically literate population and for programs that prepare more students for careers in science, technology, engineering and mathematics” ignores the fact that these same business partners have been primary exploiters of workers in many of the countries that you portray as a threat to our nation’s standing. I question your assertion that the purpose of mathematics education should be to help students develop the kind of “mathematics literacy” that allows them to be employed by these same business partners. Is that what you are suggesting? Are there not other forms of math literacy that would empower students to change the conditions of their lives?
Finally, you state, “Suppose we succeed. Suppose we create a generation of citizens who know mathematics well, with many who can conduct basic research or work in programming or design. Even then, it is possible that citizens prepared in these areas may not be employable in the United States or Canada—their salary requirements may price them out of jobs.” I would submit that salary demands are not the only factors that would prevent these employers from hiring workers. Research data are very clear that other factors like racism and discrimination because of gender are equally at play in hiring decisions. Mathematics literacy may be a necessary condition for participation in the workplace, but it is by no means sufficient, given the impact of these other factors.
My ultimate question is what exactly are you advocating for students in terms of mathematics literacy and whose interests does your version of mathematics literacy serve?
I am a university mathematics educator and know the importance of mathematics literacy, but I find your framing of its importance to be troubling. This applies to your appeal to U.S. nationalism and to your flawed assumptions about being in partnership with the business sector.
I am very much interested in your comments on the issues I raise. Please do not avoid this because of any discomfort or uneasiness that this may cause. NCTM needs to address the tough issues and not promote rhetoric that goes unchallenged.
I sincerely appreciate this question that we received in advance. It raises some important issues and it has caused me to reflect on where I/we stand.
On a personal note, I am a strong supporter of advances in other nations, particularly in the developing world. I recently taught in Burkina Faso ( West Africa). I have friends and students struggling to find work there, and I do what I can in support of their particular efforts as well as broader development efforts.
At the same time, I cannot watch the United States jeopardize the future of our citizens on this side of the globe. You are absolutely correct that mathematical literacy is not enough to ensure that students will find jobs; but to neglect to equip our students with broad mathematical skills would be foolish and short-sighted.
I welcome this nation’s participation in the global economic arena, including participants from nations both established and developing. What I don’t want is for the United States to be watching the game from the sidelines as more and more other nations carve out a future for their citizens while our citizens fall ever further behind.
In my Message, I noted that if the playing field is indeed being leveled, then it opens up the world for producing more competitors, but also for connecting colleagues. I hope that as a global community, we find a way to capitalize on this latter aspect—connecting colleagues—as a positive force for the future of all of the world’s citizens, including those in the United States. The part of this picture that we can most directly influence is ensuring that the students we teach have the best mathematics education we can give them.
Many people agree with your recommendation that students be given challenging problems rather than a step-by-step instruction of how to go about it is well taken. But what passes for challenging problems really amounts to nothing more than busy work a lot of the time. For example, let's look at how pi and area of circles is commonly taught in today's classrooms. Students may be asked to measure circumferences and diameters and calculate the ratio, and then construct spreadsheets from the data. Constructing bar charts and spreadsheets provides an excuse for an engaging activity that everyone in the classroom can enjoy, regardless of their prior math skills. Together, they waste an enormous amount of instructional time. Maybe spending 10 minutes on such activity is enough to show the general pattern, and then move on to more substantial instruction. Also, such activity tends to leave students with the impression that math is largely empirical and that pi is derived by observation.
Many American mathematics classrooms could benefit from more challenging problems than what we currently expect of students at any level. The activity you describe above can be an important way to help students understand a significant and long-standing mathematical concept. Learning how to represent mathematical relationships with models, graphs and tables is an important piece of students’ developing mathematical knowledge. But I agree with you that the focus should be on the mathematical outcome, and not on the activity itself. The critical piece in the value of such an activity lies in how well the teacher helps students connect what they have done with mathematical outcomes.
We’re going to combine two questions, which Cathy will answer together:
Hampton, New Jersey
One of our biggest difficulties in American education and specifically math is that we try to “cover” the curriculum, prepare students for state tests, and our children are in so many activities and subjects that we overextend them. We introduce, practice, and test and move on. There is little mastery of information that can be applied in new ways. Has anyone figured out a way to stop teaching “1 inch deep and 100 miles long” and “to the tests” yet? No wonder so many students say “I hate math.” The learning never becomes part of them, it only covers them.
As you mentioned in your article, our colleagues in other countries often teach with less “telling.” There is no doubt in my mind that when students are given the opportunity to construct their own meaning, learning occurs. As a result, kids are able to apply mathematical concepts to solving problems. However, this constructivist form of learning takes time.
If you look at curriculum (K–12) in Japan, you will see that kids are given few concepts to master at an early age, and each year a few more are added. This design provides for kids to be able to focus and master material. In Ohio, we are given a set of standards that is a mile long at each grade level. If we “teach” every standard we are only able to dig an inch deep in order to go a mile long. Teachers feel that in order to cover everything, they simply must “tell” kids what to do at times. By the end of the school year, they have “taught” their curriculums and we send kids to the next grade level. My question to you is this: If we desire to be more like the countries that provide successful mathematics programs where students construct their own meaning and truly “learn” mathematics, how do we accomplish it when our state mandates we teach a curriculum that is a mile long at each grade level?
These are important questions, and they point toward a dubious distinction for the United States. We are among the countries with the most topics addressed per grade level of any country in the world. These questions point out the most common and best-deserved criticism of the American mathematics curriculum, often characterized as “a mile wide and an inch deep.” Depending on the state in which they live, some teachers face lists of 40 to 80 or 90 things for students to learn at a particular grade.
NCTM is initiating what we hope will be the next round of discussions in mathematics curriculum with a new effort around Curriculum Focal Points. Currently, a writing group is working on identifying and describing three or four major focal points at each grade level. This document will be available for review during the next year or so, providing a basis for discussions among teachers, curriculum developers, mathematicians, teacher educators, and others. Watch the NCTM News Bulletin for updates as this effort advances.
Why do college students not know the division algorithm from elementary school? What are we doing to cause this?
There can be many causes for this, not the least of which is our very crowded curriculum that tries to address too many topics at every grade level, as described in the previous question. I think we need to focus our attention on teaching fewer topics in greater depth to counteract this ‘mile-wide-inch-deep’ phenomenon. Unfortunately, I think some of today’s young adults do not have a strong sense of what division means, nor have they learned the algorithm. Elementary school mathematics needs to include a strong commitment to teaching the base-ten structure of the number system, the meaning of the operations, and the concepts of numbers. But to do this, we have to focus our instruction.
Each day that we will cover new material, my students bring to class an index card on which is written a sentence about the new material, or a question about it. I grade it from 0 to 4 points, and don’t react well to “In this section I learned how to find symmetries.” I am after some thought, as opposed to mere copying of one or more section objectives. Then while I take roll and hand back papers, they write some simple thing from the book (e.g., on p. 256, what is the definition of a function). In Intermediate Algebra and College Algebra, however, the day I was planning to review long division of polynomials and do synthetic division, I had them divide long hand 4026 by 37. Half of each class could not do this, as they did not know the division algorithm. This made it much harder to teach, because they had no foundation that I could easily use. I believe that our elementary schools are causing barriers to higher education by inappropriate use of calculators. What do you think?
First, thanks for sharing this technique. Second, the subject of calculators is one we could certainly discuss at length. Perhaps a future chat will focus on calculators. But let me share that I do not believe the use of calculators is the main problem we face. Certainly, as you note, ‘inappropriate’ use of calculators can be a negative influence on student learning, just as inappropriately teaching rote procedures of any kind has been a negative influence for some students in the past. I would argue more for the development of a strong understanding of numbers and operations than for less use of calculators. For years in my own teaching, I saw students make mistakes in pencil-and-paper computations that showed that they were not using basic ‘number sense.’ Today we see this reflected, perhaps even more strongly, with the ready access to a quick answer on a calculator and the willingness for some students to accept what they see on the screen, even if it doesn’t make sense for the original problem. We might all agree that we need to build a strong understanding of numbers and operations at the elementary level as a foundation for future work.
Hi, I love the show NUMB3RS, and at my school library they have a poster promoting the show. I was wondering if you know where I can get one.
The “We all use math every day” poster is wonderful, isn't it? It is part of a joint project between Texas Instruments (TI), NCTM, and CBS around the new CBS crime show, NUMB3RS. The project is based on teaching activities posted early each week in support of the mathematics addressed in that week’s episode of NUMB3RS. TI has many components of the initiative, including the posters and a teacher’s kit, available for free by signing up at the NUMB3RS Web page. You can access all this information from the NCTM Web site at http://www.nctm.org/news/2005_09numb3rs.htm#poster.
The NCTM conference (in Hartford) was one of the best conferences that I have attended in a long, long time. I particularly liked the Learn and Reflect Strand. Dr. Timothy D. Kanold was an incredible speaker who not only talked about math strategies, but the systems that we, as teachers, have in place that can turn students off to math and to learning. He talked about the Carnegie Foundation and it says we can no longer have systems in place that fail students in math, i.e., allowing zeros to stay in the rank book, not able to make up tests and quizzes, and not giving full credit. I would like to have the sources of this so that I may bring this information to teachers. I believe in what Dr. Kanold says...it is not okay to allow students to fail.
We received the following information from Dr. Kanold with respect to this reference:
The main piece of research is from a piece by Lynn Canady (at the time a professor at the University of Virginia) titled " Grading Practices which Decrease the Odds for Student Success” He presented this paper in 1990, based upon his article written in the Phi Delta Kappan and titled, "It’s a Good Score, Just a Bad Grade!" September, 1989. In this talk he referenced the Carnegie Foundation report regarding the behaviors that I have placed into a more modern day context.
I think that it is very important to teach students to see things from many perspectives. The ability to identify and then start to solve problems depends on seeing the world from various perspectives. I think that our best hope of economic success in the future is to give every child the ability to see different perspectives, and this should be a goal of every discipline, not just mathematics.
I certainly agree that students need to be able to see things from different perspectives, both in mathematics and beyond.
Las Cruces, New Mexico
Should we let our kids at any age figure out their own problems without instruction? What is the role of the teacher?
The role of the teacher is to find and pose good problems and tasks that challenge students, and then to facilitate students’ work on those problems. This doesn’t mean telling them everything first, nor does it mean letting students work on problems without instruction. Rather, the teacher’s role is to set the stage for students' work by talking about the problem with the class, then to push students’ thinking, asking questions that cause students to think ever deeper about the problem and keeping them focused on the problem. In particular, the teacher plays a critical role in helping students’ connect their work on a problem to the mathematics they need to learn for the future.
Hoboken, New Jersey
I was very inspired by your questions. They relate to the very thing that we forget to do in the classroom. As we educate students in mathematics and other subject areas we must see the student. We must hear the student. We must respect their physical and metaphysical development. By the third grade most students (primarily amongst minorities and or children from lower economic homes) will say they hate math. This notion is the portal as to whether we as teachers can inspire students to embrace, love, and feel successful in the classroom or close them off to the one subject that helps them to comprehend the universe in which they live and their relationship to it. When students say they hate math, I think they are saying they hate the universe. How can they hate something so extraordinary? How are we teaching that we are making our students feel this way? There are three main branches of mathematics. Is the secular branch of mathematics the only thing that is important? From my experience and leadership skills, if we don’t embrace the whole child while teaching mathematics, then we will continue to see the same cycles that are preventing our children from succeeding. NCTM should develop K–12 schools around the country. We should begin with the very notion that schools around the world have a longer school year than U.S. schools. Why? The structure of the U.S. school year and number of school days raise the question of how serious we are in enlightening our children in the field of mathematics/science and literacy. Does our school year show we value mathematics education/ or education in general? Thank you for listening.
Thanks for your description of what we should value. Certainly, spending time on mathematics, both in terms of minutes per day or week and also in terms of days per year, can make a difference. It is interesting that some countries that outperform the United States don’t have significantly longer school years. At the same time, other countries may not lose as many days of instruction as we do in this country. In any case, we need to spend significant time on mathematics and we need to help students become proficient as well as confident. And the American idea that it’s ok to dislike or even hate mathematics is something we should work hard to not pass on to another generation. Mathematics can sometimes be challenging, but it is something most people can do if provided the right experiences, and it is something that is critical to a citizen’s future.
In order to help our students rise to the challenge of being competitive mathematically with other students and other countries, we understand having all students achieve certain standards. How do we meet the needs of an individual student who is not developmentally ready to take Algebra I as an 8th grader but that is the only option available to them in the curriculum? Cooperative grouping helps...but pre-testing them doesn’t bring them into different groups for each concept because usually your low students are low in most math concepts. Any thoughts on how to challenge and yet meet needs of individuals in a 40-minute time period? Or any books, speakers, or workshops that you would recommend?
You have raised one of the most difficult and prevalent questions that result from our recent push to provide a high-quality mathematics education for every student. First, a 40-minute time period for mathematics instruction at any grade level, especially for algebra, is woefully inadequate. Second, I have shared my views before about whether all students should be enrolled in algebra in grade 8. But regardless of the grade level, this issue presents itself.
Students who are being challenged to perform at a level higher than we have expected in the past need some support. It is not enough to “expect” more, we have to provide a way to help students actually achieve more. This support might come in the form of back-to-back math periods (especially when class time is so short), before- or after-school help sessions, or summer programs. Meanwhile, in class, it seems reasonable that you may want to provide such students varied experiences and groups of students with whom to interact, as you have mentioned.
NCTM has some print resources for supporting teaching algebra to all students. You might want to go to the NCTM Web site and look at resources for the Professional Development Focus of the Year. This year, the focus is on Assessing to Learn and Learning to Assess. But you can also access last year's Focus on Algebraic Thinking from that part of the Web site. Meanwhile, the Council is offering E-workshops online, including some on algebra. You can find out information about these workshops and future offerings from the Professional Development page of the NCTM Web site, and they are often advertised in the electronic member updates we send out. You might also check with neighboring universities and any NSF-funded Math/Science Partnership projects in your area.
Las Cruces, New Mexico
What is so wrong with showing the student how to perform a problem if you then let them work out similar problems on their own? Wouldn't this reinforce the subject because they would be forced to apply the same methods to new and varying problems?
The problem comes with expecting students only to watch and repeat, rather than having to think about how to approach a problem. As described in “The Teaching Gap” (Stigler and Hiebert, 1999), this is one area in which the United States tends to have low expectations for our students compared to other countries. This model of showing and asking students to repeat has been prevalent for many years in this country. Yet our performance has not backed up this method as effective.
There is nothing wrong with the teacher telling students some things they need to know, as long as we also expect the students to do some of the work. Repeating what they are shown is not the same as asking students to figure out how to solve a problem. Some mathematicians argue that problem solving is the most important goal of school mathematics. Too many students who know the procedures fail to realize which procedure goes with which problem. This difficulty is accentuated when students are presented with problems that may not look like other problems they have solved. I believe it is possible for students to learn procedures and also learn how to think and solve problems.
Can non-mathematics majors and education degreed graduates teach math with the same inspiration to their students as those math majors and science majors who clearly love their field of interest? In other words, would it be a good idea to eliminate the BA Education degree in American universities in favor of requiring specialist teachers?
Many, if not most, states today require a degree in the discipline for a teaching credential at the secondary level. This is true in many institutions that offer education degrees, often calling for a dual major. As described in NCTM's position statement on highly qualified teachers, strong mathematics knowledge, as well as pedagogical knowledge, are both essential in order for a teacher to be well prepared to teach mathematics. Consequently, we need to look for programs that bring together strong programs in mathematics with a strong foundation in how students learn.
Windsor , California
We use a mathematics program in our high school just like the one you describe with less “telling” and more hands-on and guiding student inquiry. Because this is not the way their parents learned math, we get a small but vocal group of parents who want their student to have the “old” way, especially if the student struggles a bit or gets poor grades.
What are some ideas for helping parents understand the benefits of this method of teaching and learning?
Involving parents is a critically important aspect of improving any mathematics program. Ideally, parents should be involved before and during the implementation of a new program. I have found that the best way to help parents understand any quality program is to invite them into the classrooms of effective teachers. Many parents become strong supporters when they see the kind of high-quality mathematics you are expecting of your students.
I wholeheartedly agree with your message. I am a masters student at Clark University, whose philosophies coincide with the movement to help foster higher-order thinking from students, setting high expectations for every one of them. I just began taking over one class at South High (where I am a student teacher) and I want to set up an environment that promotes learning and responsibility for each and every student. However, the task of overcoming the students’ past conditioning in mathematics is daunting me. I feel like starting over at the early elementary level, where the students haven’t been through 8 years of “spoon-feeding” as you put it and are convinced that they cannot think for themselves. They are set in their idea of what education is; in particular, my students, who are tracked into my class because they failed the grade 8 MCAS, and believe that math is just not accessible to them. They come into my class expecting me to show them how to find answers. Math has never made sense to them, so they don’t even imagine relying in their ability to reason. I think the fact that they were taught procedures (which may not have made sense to them) is what got them into the math rut they are in now. I think if they had just been able to ask why instead of hear HOW, they might have gotten it. My students question every move I make. They find clever ways of pointing out contradictions in things not related to math. I believe if they only knew that they COULD do it, they would be the cream of the crop. THEY are the ones with the different perspective that just may result in some innovative break-through! But they resist being asked to think. And, unfortunately, without the teaching experience behind me, I find myself unable to balance the setting of high expectations, the development of enriching relationships, the lesson-planning, and class maintenance. It seems I’m trying to learn it all while at the same time swimming upstream (the “conditioning” I referred to earlier) in a river that only gets deeper every day.
My heart is with you as you swim upstream. Overcoming students' past conditioning is a huge challenge, and it gets more difficult the higher the grade level because you are dealing with more years of conditioning. You are right that these challenges become a bit more manageable with some experience. But on the positive side, at least you aren't dealing with your own years of conditioning as a teacher.
I would focus on choosing those engaging tasks that will allow students to think, and then standing as firm as you can on not giving in to their first reaction of “What do I do next?” It's fine to do some telling to students, and you will get better at judging when to do what. But as you have already observed, the more they have to do for themselves, the less likely they will be to give up in the future. You might talk with them about why you are reluctant to tell them everything without them having to think.
Learning how to ask students questions that push their thinking is one of the most important things you can do for your future as a teacher. "If you did know what to do, what might you do next?" "Does this situation seem like anything you have seen before?" "How is this problem like the one we worked on yesterday? What's different about it?" I would also encourage you to identify one or more teachers who seem to accomplish the kind of teaching you would like to do. See if any of them are willing to serve as a mentor or buddy as you enter into this arena.
You have chosen the most important job on the planet. Your attitude toward students and toward mathematics will help a lot. Continue to expect a lot of your students and yourself. You may not always succeed, but I am certain your students will benefit from both your attitude and your high expectations.
I agree wholeheartedly with your statement that we must, "...go beyond teaching basic skills, beyond requiring students to know how to perform procedures, beyond offering recipes for solving problems that look alike." However, I received a somewhat different message from a representative of our U.S. Department of Education when I participated Saturday in a National Math Symposium at Skywalker Ranch in northern California. The speaker's emphasis was that we need to do the same thing in elementary mathematics education that we have done in reading instruction, that is to say, "emphasize phonemic awareness and phonics." The inference was that these similar areas in math would be basic skills and memorization of procedures. Is NCTM involved in communications with the Department of Education, especially as it involves their attempts to influence math curriculum and instruction methods nationwide? The next thing we know may be that "No Child Left Behind" will be dictating to us the "low-level" math curriculum that you speak against.
I have heard similar comments, usually not from those directly working with mathematics. There is a desire to find something that will be easy for folks to do and will make a big difference. However, there is not a good parallel between reading and mathematics. What we know about how students learn mathematics is that they need a balance of skills, concepts, and problem solving. This kind of balance is difficult to package in a single packaged program; there are schools making gains using different types of programs. The National Assessment of Educational Progress (NAEP) results released this week show that many schools are moving in the right direction in mathematics.
Thank you all for your participation this afternoon and for the many questions submitted in advance. Unfortunately, we had more questions than we could address in one hour today.
The next chat with President Cathy Seeley will be at 4:00 p.m. EST on Wednesday, November 16. Cathy’s President’s Message, “Thank You, Mr. Bender,” reflects on teachers who were important influences in her own education.
Thanks to all for your participation. I'll look forward to our next chat, when we will talk about teachers who make a difference.
A Flattening World (October 2005)
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