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Untapped Potential

 chat archive 

Moderator
Good afternoon and welcome to this month's chat with NCTM President Cathy Seeley. The subject of today's chat is how to address the achievement gap, or in Cathy's words, untapped potential. We'll take as many questions as we can in our hour. Our first question is from

Highland, California

Is there a greater advantage in teaching the way of the rest of the world? If so, why aren’t we considering teaching this way? What advantages are there to teaching Algebra I, Geometry, and Algebra II as separate courses?

Cathy Seeley:
The United States appears to be the only country teaching high school mathematics according to separate courses, rather than in a continuous, integrated program. I can’t tell you any advantages of teaching by courses like this, except that it appears that it is very hard to make a change to structure high school differently. Regardless of how the program is organized, the main thing is that we teach good, challenging mathematics appropriate for high school level students. Perhaps some day the United States will be ready to make this change on a national level. For now, only a small number of districts and the State of Georgia seem to be willing to tackle this challenge.

Question from
Philadelphia

Do you think AP math classes should be given a pre-test to qualify their prior mathematical readiness to take the course being offered?

Cathy Seeley:
Pre-testing is an issue at every grade level. Sometimes a pre-test gives useful information, and sometimes it does not accurately reflect what students know. Part of the problem in doing pre-testing at the beginning of the school year is that many students forget over the summer some skills and understandings that may quickly come back to them as they move ahead. Also, if a pre-test is not well designed, it may not accurately point to areas of misunderstanding. It could provide both false positives (indicating that a student knows the skills, when he or she may in fact have an important misunderstanding) or false negatives (showing students don’t know the material when they may simply not show well what they know on a test, especially under certain testing circumstances). I find it preferable whenever possible, to interview students by talking with them individually. Obviously this isn’t always possible. If pre-tests are used, they should be used in combination with other factors such as a student’s previous work and the previous teacher’s recommendation. And a brief conversation with the student can still provide additional information.

Question from
Alexandria, Virginia

I taught a special education class of highly defiant students. These were the kids no one else wanted to teach and they were all grouped in one class. I tried to incorporate much of what the NCTM touts. It was impossible. When the parents are enablers and the families dysfunctional, it's incredibly difficult to implement NCTM Standards. But I didn't quit trying.

Cathy Seeley:
We received a couple of questions with the same theme, so I’ll respond to them together.

Question from
Tuscumbia, Alabama 

I am teaching an Algebra I, part 1 class of 18 special ed students who are seeking a regular high school diploma. SAIs from eighth grade Stanfords are in the 70s. Some students will probably change to an occupational diploma curriculum, but until changes are made in IEPs we have to follow the state course of study for Algebra I. I have a special ed teacher and an aide to help, but this is an overwhelming challenge. I will appreciate any suggestions that can help me make this situation >0. Thanks.

Cathy Seeley: 

This is truly a challenge. From conversations with effective teachers, here are a few thoughts. First, these students can make good use of multiple representations, especially concrete and pictorial representations. Thus, the use of graphing calculators can be very helpful. Many special education students can learn symbolic skills, but using a visual approach can make this more likely. It can also help to use hands-on models, such as blocks or tiles, to model algebraic ideas. Many equation-solving skills can be modeled using cups and chips or tiles. It’s important that you, your aide and the special education teacher have appropriate professional development in order to use these tools to help students learn. As with all tools and activities, making the connection to the mathematical ideas and skills is critical. One teacher I know prepares special pages of notes and tips for students such as these. Group work can be very effective, so that students work together and talk about their approaches. Obviously, your role as a facilitator, asking appropriate questions at the appropriate times, is very important.

It may also be that for the long term, your school might look into scheduling a double block of time for algebra for some students or a study/tutorial period just after math class. Also, students from higher grades might be called on to serve as tutors either during or outside of class.

I have talked with teachers who tell me that they are sometimes surprised at how much their lowest students can achieve when given certain kinds of activities, structure, guidance and encouragement. Expectations again play the most important role of all.

Question from
Santa Monica, California 

As the lead teacher of the Mike's Math Club Program, I have encountered some students with surprising math ability in “special Education’ classes within the Los Angeles Unified School District over the past 15 years. One child in particular proved to be extremely gifted in mathematics, but he was placed in a special ed class because he was still unable to read in fifth grade. Unfortunately, there didn’t seem to be a program in place within the school to support the student’s self confidence and nurture their interest in mathematics while providing remediation for his lack of reading skills.

Other special ed students I’ve worked with have proven to be very intelligent (often with math skills superior to their mainstream counterparts) but they have been relegated to special ed classes due to emotional and behavioral problems. These students have the ability to be very successful in school, but their teachers face special challenges in encouraging their academic and personal growth.

Cathy Seeley: 

Thanks for these additional insights into working with students who carry labels that may or may not be helpful.

Question from
Overland Park, Kansas

I think some expectations are unrealistic. If a person has an IQ of 80, for example, maybe they can never learn to do elaborate word problems. But they could maybe learn to calculate, without a calculator, that 10 gallons of gasoline at $2.00 a gallon cost $20.00. What good are elaborate word problems and calculators for a person who thinks 1/3 + 1/3 = 2/6? I think the schools should set goals of minimal competency for everybody, higher goals for the majority. I oppose “exposing” students to ideas they cannot understand. Everything taught should be taught thoroughly and well. Skip the two weeks on statistics, where students learn to look up values of the normal distribution with no understanding of how statistics are used. Instead, make sure people with 3 years of high school mathematics know that the sine curve goes up and down and can graph it accurately. College teachers can teach statistics in statistics courses, but they usually cannot take the time to teach trigonometry in calculus.

I have taught for years students of all levels of ability—high school and college. The most important single thing a teacher can do for students is make good assignments, collect them, grade them, return them. Promptly, and discuss them. Mathematics is not a spectator sport and does not suffer fools.

Cathy Seeley: 

I certainly agree that students should be taught so that they understand, not so they are just exposed to content. We may differ in terms of what we think should be expected for everybody and what should be expected for more successful students. There is much evidence that many of our students can accomplish much more than we have previously expected of them. We clearly agree that teachers should have clear expectations and provide appropriate feedback. You are right on target that mathematics is not a spectator sport and that it calls for effort on the part of both teachers and students.

Question from
Flushing, New York

I am a member of NCTM. I looked through the high school curriculum and standards, but I still can’t find out whether or not natural logarithms and change of base is taught. Also, determinants. I am thinking of submitting pedagogical articles on these.

Cathy Seeley: 

The purpose of Principles and Standards for School Mathematics (PSSM) was not to outline specific topics at a detailed level. However, the topics that you raise, if not directly, are indirectly referred to in PSSM. Work with matrices likely entails the use of determinants. Work with logistic curves likely entails work with the base e and most likely the use of natural logarithms when considering inverse functions involving logistic curves. Work with manipulatives such as multi-base blocks leads naturally to different bases. Much of this is left to state and local standards as it is not specifically addressed in PSSM.

The Council would welcome articles about these topics for consideration in the Mathematics Teacher. You might also want to look at weaving in the standards, especially the process standards, as appropriate, although this is by no means a requirement. Manuscripts are evaluated by reviewers and an editorial panel, taking into account the usefulness of each article to the readers of the journal.

Question from
Noname, USA

When helping children, especially underachievers, to develop an initial understanding of division, I have found that arrays have been a most valuable tool, and I am pleased with their return in popularity. When we start with arrays in developing multiplication concepts, I always draw the top horizontal and left vertical frame for the array. Children are responsible for putting the number of rows to the left and the number of columns at the top. The total number is superimposed onto the array. As children become comfortable with the multiplication arrays with given numbers of rows and columns, they love to play the “Sneaky Game” in which the product and either the number of rows are given and the student has to tell the number of columns. (Alternate missing rows and columns.)

The transition to division is then a given when students are told they have been dividing all along since the division sign is simply an array frame with a curved left vertical line. It just seems to make so much more sense to them.

Cathy Seeley: 

Thanks for sharing this idea. Arrays are a wonderful tool. They help connect many ideas in mathematics and they help students make sense of what they are learning.

Question from
Vashon, Washington

Reaching all students is the challenge! In the end, mathematical achievement is individual, and most public middle school math teachers work with 130+ individuals! I have been very successful in working with 8th graders. My secret? High demands, regular communication with parents, after-school help sessions for students, regular volunteer Saturday morning math help sessions, flexibility on assignments, test retakes, individualizing assignments, and requirements for some students, flexible grading, fun creative projects, and more. Middle school math teachers work harder and longer hours than anyone else! I work 65-70 hours a week, and I don’t think I could do my job in less time...because middle school math requires much one-on-one individual time ...and my case load is huge! Add other teacher responsibilities, and the result is long days...6+ days a week. The kids are great though, and the work is fun... so life is good!

Cathy Seeley: 

Wow! Middle school teachers really do work hard, and you can easily end up spending long hours to try to meet the needs of all students. Thanks for sharing your ideas; I know your students will appreciate your efforts, if not now, then eventually. Meanwhile, please don’t ruin your health in the pursuit of excellence. We can probably never do everything we would like to do in support of our students’ learning.

Question from
Fort Lauderdale, Florida

I set up a Web page that helps my students see that there are different ways of perceiving, different ways of learning, different ways of expressing what they've learned (Howard Gardner’s work). To make it accessible to the kids, I have a video that shows hunters and gatherers, Rob Becker’s work with the caveman. Then I pull out fractions and talk to the kids about “How do we divide $5 among 4 people?” “How much does each person get,” then 5 people, then 7 people, etc.

Then I ask the kids to go to the challenging problems on the nctm.org Web site and just look at situations, rather than at the abstract equations that the text book asks the kids to solve.

The kids love the Web site with Figure This! situations. The colors help them break through. I have placed some of these problems on a CD that they can use when they are not on the Internet and the breakthrough comes because the kids are dealing with a variety of problems and are not constrained to do the problems in order or in sequence. I also make it clear that my class has a relaxed attitude... If you don’t get the math today, you'll get it eventually. Rita Rudner, the comic, doesn’t like math but eventually got it so she has a great standup routine (I tell students this)... it reduces anxiety and the “test at the end of the chapter” tension. I tell students that they can retake any quiz (with different questions) if they feel they have mastered the material. The re-do policy helps kids because they learn where their weakness is and then work to bolster themselves. I follow the BigPicture.org and I believe like Dennis Littky that na=atives are better than grades. I share this info with my students and they work harder because of my relaxed attitude, knowing that they can improve things, and poor performance at the beginning of the course can be overcome.

Cathy Seeley: 

I think we lost a small part of your question, but your idea of a Web site is great. Increasingly teachers are using the Internet in support of their teaching. We need to make sure, of course, that all students have a way to get online, even if it means opening the school or the public library after school hours for computer access. Meanwhile, you are a great example of the importance and impact of the teacher’s attitude on student learning.

Question from
Georgetown, Texas 

For many years, Texas has had an exit level math exam that students were required to PASS in order to graduate. When I first started working with seniors that were not going to graduate—because they had not passed this test—I began asking myself “How can they have been in school for at least 12 years and not know enough to at least pass?” Well I refused to believe they could not pass—but I had to find a way to make them realize that they could pass­­­­––that they had the knowledge! So what was something that they respected but didn’t do? Mental math was my answer. The first time I suggested that we do a little mental math the students almost laughed me out of the room. They reminded me that they were the DUMB kids and that they didn’t do mental anything!

With a little coaxing, they agreed to give it a try. We did mental math with the aid of a 100 chart. I wish I had a video of the visible change that came over the students as they were successful in doing something that they believed only SMART kids could do. So I suppose that my message is that we can find the untapped potential, if we are willing to open a few untapped doors!

Cathy Seeley: 

This is a great story. Thanks for sharing your experience. I have long believed that mental math needs a much stronger place in our mathematics programs. This is something about which pretty much everyone can agree, and it turns out, as you found with these students, that many students can learn mental math skills that can serve them well forever.

Question from
Los Angeles

We need to figure out a way to teach proofs. I know there must be someone out there who can share dynamite lessons and materials to implement them.

Cathy Seeley:

Yes, proofs continue to challenge both teachers and students. It’s as important as ever that students understand the role of formal proof in mathematics, especially as they go on to study higher level mathematics. I would also suggest that even more important is the need for students from Pre-kindergarten through high school to learn how to reason and how to justify mathematically. We need to place a strong emphasis on reasoning of all kinds across the grades and across mathematical topics, as called for in Principles and Standards for School Mathematics.

As you suggest, maybe someone out there has a dynamite idea to share about this need.

Question from
Los Angeles 

How about matching students with teachers who teach in the students’ learning style?

Cathy Seeley: 

It’s an interesting idea to match students’ learning styles with teachers’ teaching styles. Unfortunately, in a world where there are not enough qualified math teachers and where students end up in classrooms with many other students who may not all learn in the same way, it’s not very practical. Maybe we should instead advocate teachers using a variety of teaching approaches so that they can engage more students in their mathematical learning.

Question from
Thomson, Georgia

I have succeeded in helping unsuccessful students in mathematics by not lowering my expectations but by making them more attainable. I think students feel more confident once they have achieved the required goal. After the goal has been mastered and the student’s confidence has been captured, the bar of expectations raises. I believe that every child can learn. However, we must adjust our teaching style to meet each child's individual learning style. I continuously study and research to gain more knowledge to be an effective teacher that generates multiple paths for my students. As educators, if we continue to work to improve “self,” we will discover more stars.

Creating communities is the most difficult challenge I discovered in teaching “all” students. Yes, all children can learn, but I must create an environment that allows each child to learn at his or her pace. Utilizing small group instructions and peer tutors has afforded me the opportunity to direct a little more attention on the “untapped potential.”

Entering this profession after 15 years in corporate America, I realize that teaching comes from the heart. Yet, there are some who no longer teach from the heart. We expect so much from our students. Who has asked what our students expect from us?

Cathy Seeley: 

Thanks for sharing both your approach and your thinking about the teaching and learning process. I firmly believe that what a teacher expects a student to learn is perhaps the single greatest factor in what the student will actually learn.

Thanks for being part of the most important profession on the planet!

Question from
Geneva, Switzerland 

Working with children in the early years of grades one to four, I have often had students who had previously been unsuccessful in mathematics, didn’t like math, and thought that they were ‘no good’ at it. Why is it that our society and education systems allow people (of any age) to say, “I’m not good at math”? I know of MANY otherwise good classroom teachers who say this about themselves; it is no wonder they think this all right for their students to say!

When I have students with any learning difficulties in mathematics, as with any subject, I believe it is my responsibility to help them improve their understanding and reasoning skills. I have found that for these children, math had often previously become too abstract (at the ‘symbolic’ or ‘abstract’ level), too quickly; they had not been able to grasp an understanding of the concepts as easily or as quickly as other children.
As children have different learning styles and modes, I try to use a multi-sensory sequence in my instruction. These include:

1. CONCRETE level: Using manipulatives to explore math concepts in simple but meaningful contexts. Kinesthetic and visual representation of concepts is very important for all children and even more so for children with learning difficulties. Verbal explanations and justifying their thinking in class discussions are also needed for all children to understand that there are many ways to solve a problem. For ‘weaker’ children, I scaffold their learning much more, giving them additional experiences at this level and making sure that they truly UNDERSTAND before moving on to the next level.

2. CONNECTING level: once children are able to demonstrate their understanding by explaining/justifying their thinking, then it is important to connect the concepts with symbols such as numerals to show how their ‘method’ looks on paper. Allowing children to devise their own algorithms and methods along with having them explain these to their peers in class discussions shows that we trust them to solve their own problems. This is also a good opportunity for me to model representing their thinking using conventional symbols. I have colleagues who believe that allowing for and discussing multiple algorithms and solutions confuses children, especially those with learning problems. However, I have found this to be untrue, as long as I take time to help these children develop algorithms that make sense to THEM. This may mean guiding weaker children towards using one or two ‘methods,’ but always in a meaningful or real world context with understanding as our goal. I think also that for these children, we need to take much more time to really help them connect their understanding with symbols before asking them to work at the ABSTRACT level, that is, using mental math or doing practice-type worksheets.

Helping all children to achieve understanding of math concepts must be our foremost goal, and then helping them connect their understanding with conventional math symbols of numerals, operation signs, etc. Different children can have different learning problems even with the same concept, so we must be sensitive to the learning styles of each child and help them get to the ‘Yes!’ stage. In the end, we want them to have confidence in their own learning so that they will say “I CAN do math!”

Cathy Seeley: 

Thanks for sharing this well-founded approach to learning. I certainly agree that students need to understand the mathematics behind the procedures they learn and that the step of connecting understanding to proficiency is critical (and often not well addressed). Helping students make these connections is one of the most important jobs of a mathematics teacher at any grade level.

Question from
Fairfax, Virginia

Do you believe there is as much untapped potential in China, Japan, Russia, Finland and Singapore as in the United States?

Cathy Seeley: 

This is a fascinating question. Data from the TIMSS and PISA international studies show that the United States has one of the highest relationships between student math achievement and wealth. This relationship highlights our achievement gap—the untapped potential in too many classrooms.

We are not the only nation that shows this relationship. Beyond the relationship to socioeconomic status, many countries have fixed and sometimes rigid tracking systems that direct students toward a particular path early in their school experience. However, the United States operates from a base of democracy where we proclaim equal educational opportunities for all. Yet, we have inequities in our system. Unfortunately, many states are currently facing the challenge of funding schools equitably, and none to my knowledge have accomplished this goal to the level we might wish. In my opinion, one of the greatest possibilities in the United States is the possibility that we might actually achieve the goal of a high-quality education, including mathematics, for all students. This is also our greatest challenge.

Question from
Saida, Lebanon 

I want to thank you first, and ask for your help. My question is: What are the newest and most successful ways to increase my elementary students’ abilities to solve reasoning problems? Problems of indirect questions? What are the helping materials that I have to use? Note: We can’t buy electronic materials because we don’t have enough money for that or use a computer for each student.

Cathy Seeley: 

I don’t know what age/grade level your students are, so it is hard to respond specifically. However, regardless of the materials used, there are some instructional techniques that can use many standard word problems to begin the development of reasoning skills. One of the most important things we can do is expect students to solve problems and give them opportunities to do so. Students should be expected to think and reason about problems and should be regularly expected to explain and justify their approaches and solutions. This should be a regular part of teaching and also of assessment, beginning at the earliest grades. It takes time for students to become accustomed to having to justify their work, but eventually this can become the expected requirement.

One good source that I like from several years ago continues to be useful today. It is a thin book published by NCTM called: “Mathematics Assessment: Myths, Models, Good Questions, and Practical Suggestions.” Also, even though you don’t have a computer for each student (most American schools do not have this luxury either), if you have access to the Internet for this chat, you have at least some opportunities to find good problems to stimulate your students’ thinking. One possible source for such problems is the Math Forum (http://mathforum.org) produced through Drexel University. I’m sure there are other sites as well where you might find good problems to supplement your program. Good luck in your efforts to help your students learn the important ability to reason and think mathematically.

Question from
Boardman, Oregon 

As a student all through my elementary, middle, and high school years math was difficult for me. It was obvious that my teachers understood the concepts and did not know how to explain in terms of the mathematically challenged student. The tutors they had for me had the same unawareness the teachers had, which obviously did me no good. I remember in high school one geometry class I had with a teacher that was a coach and placed to teach that one class of math in order to keep his job. This usually does damage to your students, and may even harm those that understood in the first place. This class opened my eyes, but also taught me that I was different in my learning approach to math.

I started my teaching in the elementary classroom in 1988. My experience as a math student has been the best training in helping to create my teaching style. I teach to the type of student I was, and how I needed subject matter presented in order to make sense. What I have found out is that there are more students like me than there are students that naturally get math concepts. I teach with hands-on manipulatives, drawing everything out if I can, and not starting with the formulas that have been passed down to teach from for years.

I have become a better math teacher because of my inabilities to see things mathematically in the norm. It is nice to see research that validates this approach to teaching. It feels good to put my frustrating experience as student into a positive teaching method for my students.

Cathy Seeley: 

Sometimes it takes a personal experience to help us see the challenges our students face. We each bring our particular perspective to teaching, and we probably are better at meeting the needs of some students than others. Teaching for meaning and understanding should help all students, even those who get mathematical ideas easily. Our continuing challenge is to teach in a way that helps students who struggle and also challenges students who are successful. Thanks for sharing your very personal story and for using your own negative experiences to create positive experiences for your students.

Question from
Stockbridge Georgia

What type of curriculum should I see in my child’s elementary school that will prepare her for prealgebra and hopefully, college?

Cathy Seeley: 

Thanks for this question. There are many indicators of a strong elementary program. Students should be engaged in learning, not just listening to a teacher. This means that when you ask your child what he or she is learning in math, the child should be able to explain it to you with some level of understanding (what the number stands for; what addition means). It’s also reasonable to expect that the teacher will use a variety of ways to assess students’ learning. It might be having students complete a written test, participate in a one-on-one interview, turn in a project, write a story or draw pictures about numbers. The key is that there should be progression from understanding with concrete objects to drawing sketches or pictures to writing number sentences or doing mental math. It is reasonable to look for early algebraic thinking that might take the form of describing patterns or looking for missing numbers. There should also be a clear progression from grade to grade into higher-level concepts, not just the same operations with longer numbers. Also, while there are differences of opinion about the importance of speed in mathematics, some level of automaticity or recall is eventually helpful. However, it is not a necessary element for all students, since some students are thoughtful problem solvers and not fast at calculation. And it certainly isn’t necessary to use timed tests on facts as the major indicator of a student’s mathematics success. I’ve seen more students end their mathematics progress under this kind of program than be motivated by it.

NCTM has a few resources for families, including a recently published Family’s Guide and linked from the Family section on the NCTM Web site. Over the next several months, as part of a new effort to reach out to families, we hope to provide additional resources for families in support of their students’ mathematics learning.

Question from
Omaha, Nebraska 

At the community college we do reach students who did not achieve well in their earlier years. Often it is simply a matter of maturity and desire. However, why do some students pass high school classes, even pre-calculus math analysis classes, and then assess at the community college with little to no knowledge of basic mathematics? It is not simply a matter of faulty assessment, either—if they attempt to take classes above their placement they tend to fail. Are we declaring students to be successful in mathematics when they truly aren’t? How do we justify our assessment of student achievement?

Cathy Seeley: 

There are indeed situations where students seem to receive grades that are higher than their lasting learning might reflect. Sometimes this is a matter of learning at a surface level only what is necessary for the next test, sometimes it may be a matter of a grading policy. This is not that different, however, from the problem faced at every grade level where it appears sometimes that students have not learned what their grades show from the previous grade level. Sometimes students have learned material but have forgotten it. Our historically shallow and repetitive curriculum actually has encouraged this. Recent emphasis on accountability has increased expectations on some level. Unfortunately, in many states it has unintentionally led to just the opposite by encouraging instruction on only that mathematics that can be tested by multiple choice items. In any case, our expectations need to be clear and high, accompanied by teaching for lasting learning. This comes in a balanced program where proficiency is built on understanding and anchored in problem solving and applications.

Question from
Kailua Kona, Hawaii 

Are you aware of any assessment tool that can address a student’s conceptual knowledge (an equivalent to DRA in reading)? The only ones I have found are skill based.

Cathy Seeley: 

I don’t know of widely available tools for measuring conceptual understanding. I think that we have a great challenge in mathematics assessment to come up with assessments that can be administered on a large scale and economically scored and that reflect the depth of mathematical understanding and thinking we say is important. We do have good tools for assessment on a classroom level, such as the document mentioned earlier: Mathematics Assessment: Myths, Models, Good Questions, and Practical Suggestions. 

Question from
Umatilla, Oregon

I noticed that there were no math instruction books for those students who are English language learners in the catalog that was recently sent to me. That rather surprised me since we have such a high percentage of students from Mexico. Some of them have a strong math background, but due to the cap between teacher-Spanish capability and the child’s English capability, these students are sometimes misdiagnosed as being at a low math level. Does the NCTM have any books to help elementary teachers who work with ELL students?

Cathy Seeley: 

You are absolutely right that this is a tremendous need and that there simply are not adequate resources available. I know that our Educational Materials Committee has discussed this, but I will pass on your message to them to more urgently see how we might address this issue. We will also continue to look for effective mathematics materials that may already exist from other sources in support of English language learners. Thanks for bringing up this important need.

You might find the following NCTM publication helpful: Changing the Faces of Mathematics: Perspectives on Latinos. It's one of a series of six in the Changing the Faces of Mathematics series.

Question from
Orem, Utah 

Teachers who seriously implement reform-oriented pedagogy such as that espoused by NCTM are consistently discovering a large number of students who are very capable in mathematics but would otherwise fail in more traditional instructional settings. In addition, they are also discovering that this same pedagogy is a natural means of challenging those who are gifted in mathematics.

Cathy Seeley: 

This is the premise behind many of the recently developed programs—that when we engage students in challenging problems with the teacher playing a critical role as guide, questioner and facilitator, students of all levels can be more successful and can learn important mathematical knowledge and skills. Implementing this kind of program well is challenging, and it calls for a lot of professional development and a lot of ongoing support.

Moderator 

Thank you for your questions today. We were able to answer almost all of them. The next online chat will be on next year's NCTM Focus of the Year "Assessment: Assessing to Learn, Learning to Assess," at 4:00 p.m. EDT on Wednesday, September 21.

Cathy Seeley: 

Thanks for sharing your great ideas and for participating in this chat. Most of all, thanks for what you expect of all students and for how you support them in meeting those expectations. Together we may be able to realize some of that untapped potential out there.

I’ll look forward to our next chat in September.

Your feedback is important! Comments or concerns regarding the content of this page may be sent to nctm@nctm.org. Thank you.