**Moderator**

Good afternoon and welcome to this month’s chat with NCTM President Cathy Seeley. Cathy has an opening statement to make before we begin answering questions.

**Cathy Seeley:**

Clearly many people have a lot of emotion about the issues raised in this month’s President’s Message. I’ll address as many of the questions as possible during our limited time, including a number of questions received ahead of time. But before that, I would like to make a general comment or two. First of all, there is no single ‘right’ way to teach mathematics. Effective teachers don’t all look alike or teach in the same way or use the same materials. There are no silver bullets, to use an overused phrase. The issues facing educators are complex and we need comprehensive solutions.

It’s important to engage in dialogue like this in order to know more about how others view issues and to see the range of opinions and perspectives about the challenges we face. I learn something from every point of view I hear. I appreciate the diversity of opinions expressed in the questions and comments received, and to the best of my ability I will reflect the Council’s position in my responses. I offer my best thinking about these comments, recognizing that, like each of you, I come from my own perspective.

Thanks in advance for being part of a professional dialogue about how we can improve the teaching and learning of mathematics for all students.

Question from

**Rehoboth Beach, Delaware**

What happens when a student enters your classroom 3 to 4 grade levels behind? Sometimes I can help them if they are hard workers, come to school everyday, and do the assignments. But these are not usually the type of students displaying those poor skills. I think the student should stay in my classroom and get additional help during the school day because these are the students who often cannot participate in the after-school programs for a variety of reasons. Does anyone have any other ideas?

**Cathy Seeley:**

This is one of the hardest questions we have to deal with, so it’s good that we are posting it early in the chat to see what others may have to offer. Many schools have come to look at additional time during the day for a second math class that can focus on students’ lack of number sense as well as additional work on skills. The most successful programs I have seen make this extra work as engaging as possible, including opportunities to make sense of the operations and use the skills in a variety of contexts. Suggestions from other chat participants?

Question from

**Austin, Texas**

I loved your article and could not agree more! My favorite comment from your article -- ‘I question whether doing more of the same is the answer for the challenges we face.’ The ‘back to basics’ movement has, unfortunately, resulted in methodologies associated with nostalgia versus vision, and has placed mathematics as an experience for children on the ‘who cares/why should I learn this’ list.

**Cathy Seeley:**

Thanks for your comments. We definitely don’t want students to be turned off to mathematics and lack the confidence or ability to use it well! This is why I’m convinced that engaging students actively in their learning is so important.

Question from

**Bettendorf, Iowa**

The number-one change that needs to be made is the attitude that only some students can learn challenging mathematics. By having multiple course offerings at the high school, many of them low level, we deny many students the opportunity to learn the mathematics they need to succeed. Changes are difficult to make when teachers, parents, and administrators believe students can’t learn. This assumption is built on past experience using traditional methods. The understanding of cognitive science and how people learn has not reached enough people.

**Cathy Seeley:**

I couldn’t agree with you more about the importance of what we believe and what we expect of our students. The main limitation of our traditional lecture approach to teaching is that it worked for only some students, especially those students with good support from home, good attendance, etc. We have seen enough examples to know that some of our students whom we have believed could not learn turn out to be strong math students, sometimes even gifted students, when given the opportunity, resources, expectations and support to learn. When we commit to more student engagement in our teaching, we enable far more students to excel.

Question from

**Alameda, California**

My own automatic way of teaching tends to be very didactic, teacher centered—even though I know better. The problem is that, given the current state of the California Standards (and the push from our district that every one should be on the same page on the same day), it is very hard for me to find the time and energy to be creative with the curriculum (not to mention the ongoing additional discipline I need to get out of my normal automatic drive to talk and explain way too much). I find it to be the height of irony that a political party that idealizes the efforts of the individual has crammed a lock-step, central planning committee form of education down the throats of its citizens. I guess that in this moment I am a little bitter because I watched a number of my sixth graders rather dispiritedly trying to find the magic formula to get the “right” answer (never mind that it made no sense to them)- how much harder do I have to try?

**Cathy Seeley:**

My heart goes out to you and to so many other teachers who are indeed trying harder than any other group of professionals I know. Each state and school system brings its own challenges, some greater than others. As you observe, overcoming our own natural reluctance to change our practice is already difficult, even if we know that improvement requires change. Combined with external pressures to divert a teacher from what you believe (or know) is right for students, it may seem nearly impossible to teach a balanced program of skills built on understanding and focused on applying those skills to solve problems. All I can say is that we have to keep focused on improving how we teach and advocating for what we professionally believe to be the way to help students learn. Surely making sense of mathematics and being able to use it with confidence are goals we can all agree on. Maybe establishing a positive trajectory of improvement and student learning might be more important than reacting to the details of new mandates and requirements.

Question from

**North Caldwell, New Jersey**

School districts must improve communication between teachers. If teaching the same subject, teachers need to be given time together to plan. But the administration can’t simply say “plan together,” there needs to be structured, formalized, or at the very least scheduled time to work together to help share resources and make the best of one another. So many math teachers are excellent in their own right. I’ve noticed in my 7 years as a math teacher that often teachers are territorial lest their “secrets” get out. This is so wrong. Encouraging teachers to share ideas but will put forth an image that the teachers and administration work together as a group to help students and are united in the effort. And students will see this and will react well to it because we all know that consistency helps. This is not to say that all teachers need to do the same thing in the classroom, because we all have our own teaching styles, just as students have their own learning styles.

**Cathy Seeley:**

Collaboration and articulation are key to building sustained improvement in student learning. Without this kind of cooperation, we may create ‘pockets of wonderfulness’ where we lose the opportunity for students to build on their previous learning to the maximum extent possible. I’m becoming increasingly convinced that communities of professional educators can accomplish small (or large!) miracles.

Question from

**Fort Dodge, Iowa**

The most fundamental change needed is TIME. Time for teachers to collaborate and plan together—to use strategies like lesson planning. As long as our culture continues to think of teaching as babysitting, teachers who are not doing direct teaching will be considered by administrators and the public as having “free time.”

**Cathy Seeley:**

Time is the most precious resource we have! When I taught in a French educational system in Burkina Faso, Africa a few years ago, we teachers had as much time to plan as we had scheduled to teach. This is common in other countries. On this side of the globe, teachers continue to tell me that time is the biggest impediment or enabler in their own improvement efforts and in efforts to improve mathematics teaching and learning in their school.

Question from

**Helena, Montana**

Lately, I’ve seen lots of articles on the depth of the dropout problem in America. What suggestions do you have for teachers and administrators on what strategies will encourage students to stay in school?

**Cathy Seeley:**

I’d like to see a real survey of students who have made the decision to drop out of school. Based on students I’ve known, I suspect that we would see some who had to get a job for family reasons. But even for those students, and for many others, I’m convinced that they might choose to stay in school if their school experience were interesting and if they perceived that they were learning useful information. I think it’s fair to say that in general, students don’t drop out of school because their courses get too hard. More likely, they drop out of school because their classes are too boring and too irrelevant. I still think that engaging students directly in their learning, and challenging them with interesting and important mathematics that calls for a bit of effort, serves students well and may even keep them in school longer. Meanwhile, let’s look for that dropout survey data to shed more light on why students are leaving.

Question from

**Beaufort, South Carolina**

After teaching high school math for 26 years, there is no doubt in my mind that the most important change that we need to make is in curriculum. Today’s math needs to be relevant to today’s needs. Not all students need to know how to factor, rather they need to know how to problem solve. The less successful students get a “watered down” version of what we think is needed for our top students. We need to teach “real math.”

**Cathy Seeley:**

I couldn’t agree with you more that today’s math needs to be relevant. We need to constantly be examining our choices of what mathematics topics students will really need in their future. It seems clear that in this global economy, where students are competing for jobs with well-educated citizens in India and China, for example, our students need deep and broad skills that can transfer to new jobs and new expectations. For this reason, the importance of problem solving, as you note, is absolutely essential.

By the way, as a side note, in nearly every other country in the world, students solve quadratic equations using the quadratic formula (and now, a calculator), rather than solving by factoring.

Question from

**Manhasset, New York**

I am an Elementary Mathematics Specialist and I have the opportunity to work with teachers from Kindergarten to Grade 6. I have found that many teachers are math phobic and have great difficulty teaching math to students. We need “math teachers” to teach math to elementary students, like in high school. In many instances there is a subliminal message to young students that math is difficult and must be learned by rote rather than understanding.

**Cathy Seeley:**

Every student needs to be taught at every level by a teacher who is knowledgeable about mathematics and how to teach it. You are absolutely right that the attitude and knowledge of the teacher has a tremendous impact on students’ learning.

A few schools have been successful reaching this goal with all of their elementary teachers. Far more schools have found that the way to accomplish this goal is by using elementary mathematics specialists. Mathematics specialists might be used to work with other teachers or might have the assignment of teaching mathematics to several classes (much as we used to send in specialists to teach art or music).

Question from

**Norfolk, Virginia**

My school system has placed mathematics specialists in all elementary schools, much like reading specialists that have had a place in schools for years. I think this is a step forward. Having someone with specific, in-depth training in teaching mathematics overseeing and guiding mathematics instruction in classrooms is a good start.

**Cathy Seeley:**

Yes, it is a great start. Teachers who love math and know it well can have a tremendously positive influence on students.

Question from

**Billings, Montana**

We are trying a second math class in our high schools for students who are behind in mathematics. So far it has helped some students succeed in mathematics who may have failed again. However, the planning time needed for teachers to design a rich, activity-based curriculum for these students is currently unavailable. So some teachers are reteaching the same way with same textbook that the students failed in the first place. Then the class seems like a punishment to the students rather than helpful. I strongly believe that if teachers are given the time to collaborate and plan, this approach would benefit all students.

**Cathy Seeley:**

More of the same rarely motivates students or helps them learn well. I absolutely agree with you that collaboration and planning together are the key to making any program work. Otherwise, we operate in isolation and keep missing opportunities to build on what students have already done or learned. More time can be helpful, but only if it is purposeful and connected to the rest of the student’s mathematics learning.

Question from

**Marlton, New Jersey**

As teachers of mathematics we need to look at our curricula that currently cover topics a mile wide and an inch deep. We don’t allow enough time for students to internalize concepts nor do we factor in the academic maturity of our students.

If we want to change student achievement, raising the bar is only the first step.

**Cathy Seeley:**

I think the next wave of curriculum attention needs to be identifying the most important focal points for instruction at each grade level. We have to make some hard decisions about what is most important, and we have to be willing to live with not doing every topic every year. It is possible to have a rich and diverse curriculum with a focus on key ideas that are really important for students to learn. NCTM will be looking at this notion in more depth over the next year or two.

Question from

**Washington, D.C.**

You state in your essay “‘Try Harder’ is Not the Answer” that [administrators, policymakers and communities] “call for a return to ‘traditional’ methods, emphasizing skill development and calling for teacher lecture as the primary means to accomplish it. In fact, we have considerable evidence from national and international assessments that the traditional approach has not served most of our students.”

“Traditional” methods does not always mean something that is bad for students, and “skill development” does not always mean rote memorization. Textbooks used in countries such as Singapore that achieve the highest scores in international math tests rely on careful sequencing of topics, with arithmetic at the heart of everything.

The following problems are taken from a 5th grade math textbook currently used in Singapore:

Express 75 cm as a fraction of 3m.

There are 50 oranges in a box, 3/10 of them are rotten. How many oranges are not rotten?

A tank is 4/5 full of water. If 40 or more litres of water are needed to fill the tank completely, find the “capacity” of the tank.

Also, there are these:

Multiply: 26 x 49 [there is a pictured hint here showing that 26 x 49 = 26 x 50 - 26];

89 x 29 =___; 59 x 36 = ___. (The hint is not repeated for these.)

Such problems have been in math textbooks at the 5 th grade level (or earlier) for a century. I think we all agree that teaching materials and techniques should challenge students to think and discover approaches that work, that develop basic skills, and foster the understanding of the underlying mathematical principles. The Singapore math texts do just that. The scores of the Singapore students on international tests such as TIMMS and PISA also provide an indication that such traditional approaches have served these students well. In fact, some mathematicians (including Hung-I Wu of U.C. Berkeley, Yoram Sagher of University of Chicago, and Tom Parker of Michigan State University) are intrigued with the approach used in the Singapore texts and are providing instruction to teachers to enhance their mathematical knowledge as well as to help them to work with the Singapore text materials. As such, they are realizing your goal of providing teachers “high-quality professional learning opportunities that help them continue to develop their mathematical knowledge and their understanding of teaching and learning mathematics.”

**Cathy Seeley:**

The examples you mention from the Singapore books are good problems that we want students to solve. I want to wholeheartedly agree with your statement: “I think we all agree that teaching materials and techniques should challenge students to think and discover approaches that work, that develop basic skills, and foster the understanding of the underlying mathematical principles.” Unfortunately, these components are not always reflected in commonly used teaching methods, whether labeled as ‘traditional’ or labeled in some other way. Too often, we see classrooms characterized by teacher presentation of rules, participation of only some students, and limited types of problems for students to consider, usually related to the method just presented. In the Singapore program, the nature of the accountability system and the preparation of teachers, among other factors, drive teaching so that students are expected to deal with a range of problems.

There is no single right way to teach mathematics. My greatest concern is that many of us know only one way, or at least we use only one way to teach—by telling students how to solve a certain kind of problem and then asking them to solve several like the example provided. Given the increasing complexity of the world outside of school, I would hope that we would also all agree that we need all of our students to be willing and able to tackle tough problems and work them through to solutions.

We can definitely learn lessons from Singapore, Japan and China. But we have to look beyond their textbooks to determine what these lessons are. The more we learn about what is being done in the Asian Rim countries, the more it appears that mathematics is not taught in the same way in which we have traditionally taught mathematics in this country. Of even more importance is the fact that all three of these countries are engaging in updating their mathematics programs to reflect a stronger emphasis on problem solving and thinking.

Question from

**Queens, New York**

One way we can equip teachers as an organization is to lead the pathway in defining mathematical vocabulary. For example, one of the largest misconceptions is the term “cancel.” Some teachers present it to mean “one” as in: 4x/4x cancels to one. Some teachers present it to mean “zero” as in: 2x+3=11, cancel the 3 by subtracting 3 from both sides. Too many teachers present both concepts to mean “cancel.” It is mathematically incorrect to present a word to mean two different numbers! Cancel in the accounting world means zero (as in a cancelled check). NCTM should take the lead in defining mathematical terms so that a precise meaning is presented no matter which math teacher a student has or where in the country they preside. ‘Cancel’ the wrong definitions!

**Cathy Seeley:**

Teaching appropriate mathematical vocabulary well and modeling good mathematical vocabulary on the part of teachers is tremendously important in teaching our students for lasting understanding. This comes with expanding and solidifying teachers’ mathematical knowledge, which is an important part of NCTM’s professional development priority. The Council will continue to explore ways to get at this important idea of mathematical words used in appropriate vs. potentially confusing ways. Thanks for raising this issue.

Question from

**Springfield, Illinois**

I agree with everything you said. Math instruction needs to change. We can no longer have students sitting in math classes and listening to teachers tell them HOW to do the math. Students should be engaged in their learning. They should be in groups and given time to discover and talk with their peers. Teachers need to take on more of a facilitator role. They should not be the focus of the class. Teachers should listen and observe. But for this to happen math teachers need training on using a workshop model where students work in cooperative groups to discover math concepts and talk through the process with one another. This has not been part of the teachers’ math training. Many of the math teachers today teach the way they were taught: sitting in rows listening to a teacher and memorizing rules, never being told why they work, just that this is the way to do it. More money is needed for professional development to take the fear of trying something new. I don’t think that most teachers are against the change because they feel it’s not good, I think they fear the change because they don’t know how to implement it. Taking away this fear is the biggest challenge we face

**Cathy Seeley:**

Investing in the professional learning of teachers is surely one of society’s greatest needs. Education has to improve, and in order to do so, we all have to learn new things. Making change is difficult, and it is accomplished best within communities of committed professionals. Many projects around the country have created such communities (West Virginia and Colorado come to mind, among many others). Fear is much easier to face if you’re not alone.

Question from

**New York City**

This is my first year teaching 8th grade Math in NYC. I decided to leave a high paying corporate job to take on this challenge. I am amazed at how HARD I am working. The number of hours I put in is so unreasonable that they don’t even equate to minimum wage. So asking me to work harder or care more is humanly and physically impossible.

There are however three things I think a school system can do to improve student achievement in math.

1) Pay more! And I’m not saying raise 5% as is currently being negotiated by my union. I mean pay to attract and keep the kind of people that become good qualified math teachers . . . period!

2) Reduce class size! Trying to teach high-level math skills to underperforming students in a class of 30 is physically impossible even with reform methods.

3) Use ONLY math teachers to teach math from grades 4 and higher!

I am confused as to why so many elementary school teachers have a difficult time teaching math. Several have mentioned to me it’s not a priority since they must ensure kids can read and write. Okay that’s important too, but then who will?

Thanks for continuing to make math such a focus and priority and for giving us such wonderful resources.

**Cathy Seeley:**

Thanks for your commitment to education by becoming a teacher! And thank you for sharing how hard you are working. Even if non-teachers in the community just spend a few hours a week volunteering in schools, it can help them understand a bit of the tremendous challenges and pressures (and also moments of pure joy) that teachers face.

Question from

**Eugene, Oregon**

University desires seem to drive what we high school teachers teach. You said, “Most of all, we know that the kinds of problems that employers, workers, and professionals now handle are often far more complex than those that were common during the agricultural and industrial times that generated our traditional educational system. Simply stated, today’s citizens and workers need a far deeper knowledge of mathematics and greater quantitative abilities than at any time in the past.” My question is:

I am not convinced that there is an agreement at the university level of what this mathematics should look like beyond preparing students to be a math major. So... the question that we at high school seem to get bogged down on is: Are we preparing students to be successful in their lives or are we supposed to be making the University math professors happy. Are we all swimming down the same stream?

**Cathy Seeley:**

There is definitely not a consistent message coming to schools or communities from higher education. One thing I would like to call for is ‘truth in advertising’ from universities and colleges about their philosophy of teaching mathematics (including the use of technology) and their expectations from and for students. If they were to publish a statement such as this, graduating high school students could make informed decisions about where they choose to continue their mathematics study. In some communities, students come to university mathematics study armed with a wide range of tools and approaches, including the use of technology, only to find that they are not permitted to use these tools. In other cases, students may come from schools where the focus was on pencil and paper procedures and enter a university setting where they are expected to have worked extensively with graphing calculators. Technology is just one area where we have unclear expectations across these levels. More fundamentally, we need discussions about the nature of calculus, pre-calculus and statistics courses that may be expected for college entry. I encourage and invite mathematics faculty in local post-secondary institutions to work with school educators in opening discussions about what students can and should do, with an emphasis on the need to broaden the population of students who can use mathematics in their future.

Question from

**Tampa, Florida**

As a mathematics subject area leader trained to use one of the National Science Foundation-funded project-centered middle school math programs, I attempted to have the program implemented at my school. The school principal was very supportive, in theory, but as we sat down to discuss all the issues involved in making such a fundamental change, we found what was to be our major stumbling block: our state standards.

The program was designed to probe deeply into concepts, covering fewer topics per academic year. BUT, our state standards, which are tested at grades 6, 7, and 8, include many topics at every grade level. To implement the program raised a strong possibility that our 6th- and 7th-graders would perform poorly on the state test. Undoubtedly, our 8th-graders would be well prepared, but in a state that grades schools based on performance on the FCAT (and ties monetary incentives to that test), we considered it too great a risk. Our teachers try to include problem-centered approaches within the more traditional curriculum with some success, but I am certain that we could be more successful in the long run were we not tied to YEARLY testing.

**Cathy Seeley:**

Unfortunately, some of our states still base their mathematics programs on long lists of expectations for students. This makes it challenging, to say the least, to try to teach fewer topics in greater depth. But I still think our students would learn (and remember) more if we did just that. At the very least, we need to continue to explore ways to cluster expectations together around bigger ideas, prioritizing which are the biggest and most important ideas and skills at each grade level. This is a new way of thinking about curriculum, and one that NCTM is doing some work to develop over the next year or two.

Question from

**Johnson City, Tennessee**

I think some of the articles that appear in *Mathematics Teacher* are excellent. Why not bring out a manual of some of these gems and give it to the high school math teachers to incorporate in the curriculum wherever appropriate and possible?

**Cathy Seeley:**

From time to time, selected articles from the *Mathematics Teacher* are published in book form. Today, we also have the wonderful opportunity to be able to search the archives from several years online as part of a person’s subscription to the *Mathematics Teacher*. The Council is continuing to refine ways of searching these archives for resources focused on teacher interests and needs. Also, keep your eyes open for various activities, including publication of selected articles from the past, as part of the 100th anniversary celebration of the *Mathematics Teacher* to be celebrated in 2007.

**Moderator:**

Thank you all for your participation in this afternoon’s chat. The transcript that will be posted on the NCTM Web site will include additional questions on today’s topic that were submitted and answered by Cathy.

The next online chat with the President will be at 3:00 p.m. EDT on Tuesday, July 26.

**Cathy Seeley:**

Thanks so much for your enthusiastic participation in this professional dialogue. I’ll look forward to other opportunities in the future to exchange ideas, either online or in person.

**Moderator:**

Cathy has provided answers to the following additional questions, which could not be answered during the one-hour online chat.

Question from

**Edinburg, Texas**

I believe one of the most important changes that needs to be made is using instructional time more effectively. When schools cut 5 minutes for this and 5 minutes for that, it makes a big difference in the math classroom. When I go from having 50 minutes for each class to 45 or sometimes 40, it hurts my students and my teaching. Sure we get the time back for “tutorial” but I would much rather teach the concepts correctly during class time instead of trying to tutor test prep during the “extra” 30 minutes.

Another challenge stems from the “test prep” mentality that we can teach the mathematics through the test prep, multiple-choice questions. Math cannot be learned by answering multiple-choice questions. Yet, many times when I have been teaching a hands-on lesson and the students are “getting it” I am asked, “How are the students being prepared for ‘the test’ if they are talking and playing?” It frustrates me when an administrator acknowledges good teaching as using an overhead projector and students copying what they see instead of helping students explore and make discoveries on their own. I know what I am suggesting takes longer to see results, but I truly believe that the students will win in the end.

**Cathy Seeley:**

I truly believe this also. The most important thing for a professional mathematics educator to do is to teach students in the way that you know they are learning with understanding and developing proficiency. And yes, time is absolutely critical, and we have to allocate adequate time for students to get into the mathematics they are studying. Short periods of 40 or 45 minutes—even 50 minutes—are challenging and students rarely get to study anything in depth. Thanks for your perseverance!

Question from

**New York City**

I firmly believe that in order to improve students’ achievement in mathematics efforts should be made to lessen the number of students in each class. A 45-minute class with 34 students is ludicrous. That doesn’t even give a teacher 2 minutes to spend with each student! In addition, resources such as textbooks, geoboards, calculators, and manipulatives should be made available to teachers and students right from the start of the course. Resources for students such as tutoring and homework help should be made available as well and should be free for students. Teachers should not be expected to give up lunch periods and time after school for tutoring without being paid for their time and services since taking the extra time to run tutoring, as many teachers in my school do, places a strain on teachers who aside from teaching have households to lead, or (as in my case) courses to take in the evening and so forth. I am sure there are other things that would help give better experiences for students. Certainly, societal changes that would ensure all students have adequate health care, nutrition, and housing as well as sound jobs for their parents and guardians would help also.

**Cathy Seeley:**

Resources, especially time, are critical to allow students to learn. If we learn to teach with more focus on what each student is doing, we can clearly benefit from smaller class size. Meanwhile, we’ll keep working on those societal issues… Thanks for your efforts!

Question from

**Santa Maria, California**

The challenge in improving learning is motivating students. I have heard more “I don’t care,” “I’m lazy,” and “I’m bored” this year. Teachers are competing in a video game era, and learning takes time, practice, and thought. Students expect immediate gratification. In order to develop higher-level thinking skills they need to contemplate and struggle. I don’t see them wanting to do that.

**Cathy Seeley:**

These are challenging times to teach mathematics. Students do indeed need to contemplate and struggle. Sometimes, if we can just find the right ‘hook,’ we can involve students in a challenging math problem that is engaging and important. I’m convinced that sometimes action precedes motivation, rather than the other way around.

Question from

**Kermit, Texas**

I feel that in Texas we are really at a disadvantage right now because of outdated resources. We have not adopted new textbooks in many years. We are expected to prepare high school students for the new TAKS test, but we have received very little training in how to accomplish this.

I have been teaching secondary mathematics for 27 years and I have never felt so frustrated as I have this year. I am beginning to think retirement might be a really good idea. In our district, some of our teachers fear losing their jobs if their students do not perform well on the TAKS.

It is unrealistic to believe that one math program will fit all students. When I was in college many years ago, we were taught that the ability to think abstractly is required for understanding topics such as algebra and geometry. Not everyone acquires the ability for abstract thought at the same age. Some children are ready for algebra much sooner than others, yet all are expected to be ready for Algebra 1 by 9th grade. No other option exists for children who are not ready.

**Cathy Seeley:**

I agree that students reach a developmental level for abstraction at different times. At the same time, in other countries, students study algebra topics earlier than the equivalent of Grade 9. If we develop algebraic thinking throughout the grades, this abstraction is far more likely to develop, and we can minimize the difficulties in high school. In the past, when we have allowed high school students to be placed in lower-level courses, too many students became increasingly bored and unmotivated to learn. I’m convinced that expecting more of our students, especially at high school is necessary. I know this presents special challenges to both teachers and students. Meanwhile, yes, it’s absolutely essential to have appropriate instructional resources for all students.

Question from

**Pittsfield, Massachusetts**

Fractional computations and manipulations are the most difficult concepts for students to master. Students need more hands-on learning to break fractions apart in order to visualize mathematical sequences within equations that make sense.

**Cathy Seeley:**

Students need to get deeply into the understanding of fractions using models (including physical, contextual and mathematical). We want students to make sense of fractions and to be able to use them fluently to solve problems.

Question from

**Independence, Missouri**

In response to our state’s new Grade Level Expectations, which were written from the NCTM Principles and Standards, we are offering a Core Plus pathway in 2005 in addition to our traditional path of Algebra 1, Geometry, and Algebra 2, as well as dropping courses that were below grade level. We recognize that many of our students have gaps in their learning and we are enrolling them in an additional class, which is a math resource class to help fill in the gaps. Our elementary and middle schools are all teaching NSF integrated series and we at the high school level have heard nothing but positive remarks from the teachers about the depth of understanding and the way the kids are learning to think for themselves and becoming problem solvers. At enrollment time we were hopeful that we’d have at least 40 percent of incoming freshmen enroll in Core Plus and 60 percent in Algebra 1. What we got was 97 percent enrolled in Core Plus and 3 percent enrolled in Algebra 1. We were stunned, but parents have seen what their kids are learning in the lower grades and they want it to continue. Our Central Office has been extremely supportive by making sure we get all of the training we need this summer, for we recognize that teaching Core is completely different from what we do currently in the classroom. We have all the technology needed, our administrator has made class sizes no larger than 25, and we, the teachers, are committed to the program. Like you said, we have been trying harder and harder and have not seen the results. So we researched, observed, questioned and decided on a new approach. Time will tell if we succeed.

**Cathy Seeley:**

Wow. What a wonderful story of a professional community of teachers working together to improve the teaching and learning of mathematics. We will all be watching to see the results continue. It takes the kind of ongoing support that you are describing to help teachers make such a transition. Keep up the good work, and keep me informed of your progress!

Question from

**Grand Rapids, Minnesota**

At my high school we have implemented an NSF reform curriculum (Core-Plus) and have had to deal with a few but nasty attacks on the curriculum. If a student struggles the curriculum is to blame, not the lack of responsibility of the student. Also, about half of our math staff prefers lecture over activity/investigation and thinks skill development is the most important element in math instruction. So we are a divided staff that makes for friction and at times, an uncomfortable teaching environment. Those of us who want to be more progressive are being worn down and are about to give in to the traditionalists, this despite our standardized test scores being very high with record numbers of students taking more math than is required. I am looking forward to this discussion to give me strength to continue to advocate with as much passion as I can curriculum reform that is good for kids, not for ease and comfort of teaching. I would also like to be armed with data to show that the reform curriculums really are effective.

**Cathy Seeley:**

You present a picture that I suspect is common in many secondary schools. Your ‘traditional’ colleagues may be effective in presenting mathematics to some students. It is unlikely, however, that many of these teachers are successful in reaching all or nearly all of their students, especially if their students represent a range of success in previous levels of mathematics. The major shift that is called for is not unlike that seen in many other countries. In Japan, for example, the most common lesson model is for teachers to present students with a problem that the teacher does not show how to solve in advance. After students work for a while on how to approach the problem, the teacher helps students compare their methods, discuss which are effective and which are not, and, most importantly, bring to summary the explicit connection between the problem and the underlying mathematics being targeted. In the United States, we tend to try to show students in advance how to solve most problems. This lecture or presentation constitutes the majority of the work we see in many classrooms, especially at the secondary level. We need to expect more of our students than to listen, watch, and practice. We need to expect them to think. Obviously, sometimes there will be class periods that focus primarily on practice or on skills; these will certainly go better for students if they have an understanding of what they are practicing. But this type of lesson needs to be balanced with a lot of time on students being expected to think and work complex mathematics problems (appropriate for their age and development).

I continue to wonder at how success can be criticized. If your students are doing well, and if you recognize that more students are engaged with mathematics and learning mathematics, how can a system let your efforts be set aside? There can’t be an argument that it doesn’t work, if in fact it does. If, as you say, it’s a matter of how much harder it is to teach mathematics today than it might have been 30 years ago, this is inevitable. We are preparing more of our students to do more with their mathematics than ever before in our history. We can’t be sidetracked by resisting change just because it is challenging. Hang in there!

Question from

**Toronto, Ontario, Canada**

I am a transplanted American. As an educator I have watched with interest the dilemmas faced by my fellow educators in the United States. We here in the Great White North also struggle with the problem of educating varied and disparate socio-economic and ethnic groups. In the not too distant past we also felt the sting of a government bent on placing blame and not on solving problems. Finally, we also care for those left in our charge. We want the best for our students. That is why I commend your challenge to stay the course. NCLB is a wonderful, inspired ideal that should not be sullied by politics and interest groups. It should be left up to the teachers (the individuals who have to sit across from those children, work with them, struggle with them, get frustrated along with them, and comfort them) to decide how best to help their students.

Government should have only one goal…to help teachers become the best educators that they can become. To borrow a phrase from the military, the government should help teachers “be all that they can be.”

**Cathy Seeley:**

The United States shares many issues and ideals in common with our colleagues in Canada. Regardless of the mandates we all face, and regardless of political swings, I absolutely agree with you that the job of professional teachers is to continue to make informed decisions about the best ways to teach their students. Mathematicians, engineers, policy makers, employers and parents all have something to offer discussions in schools about how to help students. But a knowledgeable professional teacher has to be central in these discussions and in the decision-making process if we are to make a difference in the learning of students.

Question from

**Scott, Louisiana**

I read an e-mail given to me the other day about how teachers were the babysitters. I hear parents complain that their child has too much homework at night to do. I have not been a teacher for long, but have been out in the workforce and with what I have seen since I’ve started teaching, I think politicians and the public at large should follow a teacher for a few weeks. What they would see is a teacher overwhelmed.

When people leave their jobs and go home, they have left work behind. But a teacher is grading papers, preparing the next test, and planning the next lesson. A teacher sees that too many students missed something on a test and then re-introduces the topic another way so they get it. A teacher has to make plans for the Saturday trip for the math tournaments, leaving a family at home that they only get to spend a little time with, because the kids in school come first. I had never begun to realize the work involved in being a teacher.

Most people think of a teacher sitting behind a desk with students just working in books. This is just not the way it is, and it’s time for politicians and the public at large to see this. Teachers are grossly underpaid for their work, caring, and time. Some people say a teacher gets paid for 12 months of service and works only nine of them. If they would see the whole picture, they’d see that the teacher teaches for nine months, but fits more than 12 months of work into nine. Teachers do this because they are dedicated to the future. The future is what walks off the field of every high school after 12 years of learning, and with it is they hope that they are the solutions to the unknowns—hunger, racism, technology, cures of the diseases, maybe even the ones who may bring peace as a solution in war-torn countries. For all a teacher has given, this is the biggest reward that a teacher can ever reap.

It is time for education to be left alone, because teachers are the biggest civil servants. Teachers do not tarnish the seats of chairs, do not blow smoke or hot air across a blackboard, and do not take their students on trips to the Bahamas. But they take students to school tournaments and they find ways to raise money for their students to go. I’m still trying to figure out how the teacher is the babysitter!

**Cathy Seeley:**

There is no doubt in my mind that education is the most important profession on the planet. The reason there is so much discussion about how to teach mathematics is that more and more people are realizing the importance of our students’ education. Especially in today’s world with the pressures put on teachers, there are not many who have chosen this profession in order to have time off. As you point out, committed teachers work well beyond the regular workday throughout the school year. It is the responsibility of every community and of society as a whole to find the best ways to support our teachers who are making a difference in the lives of students. Students are, as you point out, the hope of the society’s future. Our teachers open doors to enable students to achieve that hope.

Question from

**Jackson, Mississippi**

I think the single most important element is parents who care about education and realize their attitude matters. Without this you have difficulty getting through to the child.

**Cathy Seeley:**

Parents play an extremely important role in the educational process. We know that students do best whose parents are involved with and supportive of the educational process. Not all of our students come to us with parent support, however. We have to find ways to work with these students, too. It really does take a village to raise a child, and the more communities focus on the importance of this task, the better we will do it.

Question from

**Reno, Nevada**

We have been trying to focus in on specific essential skills to teach these more in depth at the high school level. The accusation is that we are “dumbing down” the curriculum and not challenging the “brighter” students. How do we encourage teachers to make math available to all students and combat against the notion of dumbing down the curriculum?

**Cathy Seeley:**

As challenging as it may sound, I think we need to both make math more accessible to all students and also challenge our most successful students. I firmly believe that when we do the former well, we will discover more students who fall into the latter category. Focusing the curriculum makes so much sense, especially when we realize how wide and thin our curriculum has become over the years. When we have good data on student learning that shows how students extend deep understanding better than learning many things superficially, we can more readily justify this shift. Unfortunately, it takes time to see results. But I come back to the importance of a lasting understanding of key ideas by more students as an essential component of a strong mathematics program.

Question from

**Ventura, California**

I have been hanging out with schools for almost 50 years and I just took a class from a true math NCTM teacher. One who showed us simple things like the magic multiplication box, the hart box, partitioning, scaffolding, and several other tricks to make our students more involved. I have two boys that were both math students and they are amazed at the things I showed them. They might have gone farther if only they had been given the right tools. I worked in a first grade classroom where the use of “incredible equations” gave them a chance to find as many ways as possible to arrive at a number. They were impressed and excited as they came up with more and more ways to get the number. We do need to teach more than skills. In this society thinking outside the box is going to be the only way to be successful. Thank You NCTM for fighting for a better tomorrow.

**Cathy Seeley:**

Yes, we need to teach a lot of mathematics. And yes, there are people out there doing good things. Thanks to all the teachers ‘just doing it.’