**Moderator**

Good afternoon and welcome to today’s chat with NCTM President Cathy Seeley. Here’s our first question, which is from Mexico City.

I teach calculus in several variables, and some of the activities my students work are manipulating real objects of revolution and then modeling the contour with a CAS, in order to find mass momenta, center of mass and momentum of inertia; surface integrals and volumes. They first take a picture of the real object and then use a grid over the picture to find some points in a coordinate system. They find these kinds of activities motivating and closer to reality than just the book exercises. I would be glad to share this experience with those who are interested.

**Cathy Seeley:**

This sounds like a great way to hook in advanced students to meaningful examples. Thanks for sharing these ideas.

**Question from**

Honolulu

What is teaching? Is it instruction? Is it pushing information into students? Brains? Or is it helping students learn to find answers for themselves?

I believe education is the process of guiding students, and giving them tools with which to explore and discover not only what the answer is, but why.

In my writing-intensive, exploration-based Geometry class, students use the Geometer Sketchpad software to explore and discover the properties of geometric figures. They discuss their discoveries in small groups and present them in writing assignments, explanatory papers and proofs done in and out of class, as well as projects and presentations. There are many opportunities for bonus projects related to the course work, often pursuing geometric topics in greater depth. Activities include individual and group work.

One of our projects is an exploration that I call Journey to the Center of a Triangle. We have found this project so interesting, and so successful that I created a set of Web pages about it: http://www.punahou.edu/acad/sanders/CenterTriangle.html

**Cathy Seeley:**

This sounds like it fits my notions of engagement to a ‘T.’ Reflection, involvement, discussion, communication… It would be hard to escape doing mathematics in this kind of classroom!

**Question from:**

Los Angeles

The teachers at a historically underperforming high school in Los Angeles (San Fernando High School) are embarking on a bold experiment, though not a new one. After years of student failure, the teachers are finally taking a hard look at where the students’ understanding intersects with misconceptions. The result is collaborative lesson planning that has led to activities that foster student engagement in conceptual understanding of mathematics. Both students AND teachers are radically changing their concepts about algebra. For example, the teachers have begun to plan lessons in context—with real life problems AND with attention to students’ cognitive development. One approach is to use students’ knowledge of arithmetic to develop understanding of equations by focusing on the notion of equality. This has led to the use of the “cover-up” method to solving equations. Further, various activities have been designed/adapted to provide foundations for future topics. Thus, topics such as combining like terms or distributive property are no longer taught as discrete units. Rather, they are embedded in bigger problems that require higher cognitive demand. Additionally, the methods for equation solving are selected to provide a natural and intuitively sensible transition to functions and linear relationships. All this is to say that the transformation on student motivation and achievement has been remarkable. Teachers are reporting student engagement like they have never experienced it before. Students, for the first time in many cases, are experiencing success. This is what happens when teachers truly attempt to implement the principles of NCTM.

**Cathy Seeley:**

It certainly does show what happens when teachers work together with an eye on how to engage students in relevant learning. There is power in collaboration (see next month’s President’s Message), and there is power in engaging students. Thanks for sharing this experience, and good luck as you close the gap!

**Question from**

Derby, Kansas

How would you communicate to parents, who feel the current instructional methods are adequate, that changes being made in mathematics instruction will benefit their children by providing a variety of strategies to solve problems or by allowing calculator use when they fear a possible loss of their child’s computational proficiency as a result and also don’t know how to help them with newer strategies?

**Cathy Seeley:**

I continue to believe that the best way to gain parent support is to let them see first-hand the kind of mathematics you are talking about. Whether by visiting a classroom, experiencing an activity on a Family Math Night or Back-to-School event, or whether through school television, letters home or other innovative ways of communicating, let’s show them the level their students can reach and the kind of mathematical conversation and thinking they can do! I’m convinced that most parents want their students to do the kind of mathematics we are aiming at.

**Question from**

Madison, Wisconsin

The need for computational proficiency is not because such questions will be on state tests, but as knowledge to build on for learning more mathematics.

**Cathy Seeley:**

Computational proficiency is one part of a balanced program in mathematics that also includes understanding mathematical ideas and having opportunities to use what students learn to solve complex problems. Often, students can learn higher-level mathematics even with some gaps in computational proficiency. We need to balance our time so that students can extend the areas of mathematics in which they do well and also continue their work on becoming proficient in areas where they have needs.

**Question from**

Brunswick, Ohio

I am currently co-op teaching a Core-Plus class at North Royalton High School. I find my students engaged in more memorable learning. They remember topics by what we did to learn them. That, to me, is active learning.

**Cathy Seeley:**

The most recent generation of curriculum programs funded by the National Science Foundation, of which Core-Plus is one example, often represent good examples of the kinds of engaging tasks we can ask students to tackle. Combined with classroom opportunities to deal with these problems in depth, students can indeed be actively engaged in their learning and can reach unprecedented levels of achievement. A Web site that has links to these programs and other related sites are: http://www2.edc.org/mcc.

**Question from**

Houston

Just a reminder from that, here in the United States at least, teaching through engagement started with Warren Colburn’s 1821 text “First Lessons in Arithmetic,” so we’ve had only 183 years to figure this stuff out!

**Cathy Seeley:**

Thanks for that reminder. I think it continues to be the case that students learn best when engaged. It seems that we keep learning improved ways to engage them. I have seen some really wonderful examples of classrooms lately where teachers structure lessons and do some large-group setup of a task, followed by small-group work, followed by large-group debriefing with an opportunity to cement the learning. We can learn a lot from history and from effective teachers, no matter what century!

**Questions from**

New York City

How are students going to learn on their own if they lack the basic skills?

**Lamesa, Texas**

How do we get students to engage in mathematics who do not have the prior math knowledge they need in order understand the math we are doing, and we do not have the classroom time to go that far back?

**Cathy Seeley:**

These are common questions that are important to address. We have dealt with this topic a bit in previous chats. At some point, I believe in giving students calculators to deal with computational gaps. Other gaps, including basic misunderstanding of number or operations (this is different from computation) may require extra time in order for students to gain the most benefit possible from their current math experience. But engaging them in doing challenging and interesting tasks may help them recognize a need and motivation to learn some of the things they are lacking. And we may even discover that some of these students can succeed well in the task at hand, even without all the prerequisite skills we wish they had.

**Question from**

Highlands Ranch, Colorado

Many teachers teach the way they were taught. What kinds of staff development do we need to provide to change this way of thinking?

**Cathy Seeley:**

Teachers definitely need in-depth professional development on how to teach in more engaging ways. Sometimes we can start by simply doing less talking and allowing students to do more. But the fundamental shift of having students wrestle with problems in small groups is when we see a tremendous change in how much students are engaged in the task vs. watching the teacher do it or having a whole class do the problem together. Look for professional development opportunities that involve modeling this technique or include videos of teachers using it. We should remember that a student-engaged classroom may be well organized and will likely include some periods of a teacher talking to the whole class, especially at the beginning and end of a task. As with every important improvement in teaching, professional development and teacher engagement in learning are key.

**Question from**

Easton, Pennsylvania

What role do you see math coaches playing in the improvement of math curricula?

**Cathy Seeley:**

Math coaches represent a new concept in some areas, usually with a teacher on special assignment or a master teacher available to work with other teachers. These individuals can be very helpful in assisting and encouraging teachers to try new approaches. Often, as we try something new, having someone to bounce ideas off of can be very helpful. And coaches can lead teachers to good resources specific to their needs.

**Question from**

Omak, Washington

I teach in a rural/remote community. I have a staff of four math teachers who are open to looking at changing teaching strategies. What is the best way to really light their fire so we can move ahead full speed?

**Cathy Seeley:**

You teach in a wonderful environment to implement schoolwide change. With four teachers who are open to new ideas, you can light each other’s fire! I would suggest taking frequent opportunities for small-scale professional learning through interacting around articles, NCTM President’s Messages, even recommendations from a math teacher’s manual. If you offer yourselves regular time to consider things like this, you may discover where your greatest needs and/or areas of interest lie. In terms of more in-depth opportunities, I encourage you to learn from each other, while also keeping your eyes open for workshops, conferences, institutes, courses, etc. that might meet your needs. Collaboration is a powerful motivator; so keep those communication opportunities coming!

**Question from**

Salem, Massachusetts

In response to your question, “Do you have some examples to share of student-engagement strategies that help all students, including those who have not been previously successful?”

Yes. First realize that it is a gradual thing. Students have to trust that they won't be nailed at the end.

I use cooperative group role cards for group responsibility sharing and change the roles either weekly or monthly or at other intervals as appropriate. In the beginning, I didn't worry about slackers. Gradually, the kids learned that there was some payoff (like points on the next test or similar rewards) to participating. Now I have a clipboard with some record cards pasted on that I got from bby Publications (elementary teachers who have MANY kid management ideas that even adapt to high school aged kids). I saw them use it and talk about how and when they write notes and I have used them ever since.

**Cathy Seeley:**

These are some nice strategies for managing groups. As we get better at these techniques, our students will benefit from richer and more active engagement in their own learning.

**Question from**

Washington, D.C.

What can informal learning organizations like science centers and museums—many of which are located in urban centers—do to promote engagement in mathematics? Are there specific topics or strategies that would be particularly effective in capturing the attention of children and adolescents in these out-of-school, science-rich settings?

**Cathy Seeley:**

Informal organizations like these can be powerful partners in engaging students and motivating their learning. When a resource right in the community can reach out and bring in students, we have a chance for them to see that mathematics and science are not just school subjects, but can be tools for dealing with other content areas and with everyday situations like weather, travel, groceries, astronomy, other cultures, and many, many more. We need to arm these centers with mathematically rich tools and examples to make the most of these opportunities. Partnerships with schools and teachers connected to informal centers can reap tremendous rewards for students and for the broader community.

**Question from**

St. Louis

Should a formal assessment of student understanding take place with every classroom activity?

**Cathy Seeley:**

More than a formal assessment, when students are engaged in rich learning tasks, the teacher can be informally assessing more often than in a large-group setting. Ideally, assessment should include both formal and informal information so that teachers can inform and guide student learning.

**Question from**

Ramsey, Minnesota

Twelve years ago I did a geometry lesson using the fabric of my necktie as the basis. I am still getting comments from former students who remember the necktie lesson as a math highlight of the course. We need to continually bring in problems from real-life to connect the mathematics of the textbook to the student's belief system about mathematics.

**Cathy Seeley:**

Wow! Nice example of an engaging context. When we combine this kind of real-life example with classroom strategies that allow students to deal with in-depth problems (often in small groups), we can really multiply the likelihood of their engagement and, therefore, their learning.

**Questions from**

Libertyville, Illinois

Engaging students in mathematics is a key issue, but achieving the standards as measured in Illinois by the ACT and ACT Work Keys is also a goal.

My concern is that the NCTM lesson plans/I-Math, etc. are seldom linked directly to what needs to be taught in the high school setting as measured by these standards. Can NCTM help provide engaging ideas and materials that we can actually use in our classrooms? (Especially for the core topics of Algebra I and Geometry)

**New York, New York**

How is this way of teaching going to fit in the new curriculum of NY State?

**Cathy Seeley:**

The challenge is that not all state standards are the same, and they match NCTM’s *Principles and Standards* to varying degrees. However, all students, regardless of their state, can benefit from active engagement in their learning. The key is to identify rich tasks that allow them to tackle problems representing their curriculum and then to structure the classroom so that students have opportunities to interact (usually in small groups) around those problems.

One of these questions raises a specific need—core topics of Algebra I and Geometry, whether taught in an integrated classroom or in courses with these titles. The Navigations series is one source, and the Illuminations Web site is another. In terms of algebra, I would also suggest looking at the NCTM Web site for items marked with a magnifying glass, signifying references relevant to the Professional Development Focus of the Year. For 2004-2005, this focus is on algebraic thinking from preschool through high school. At the high school level, there are good references available as designated by the icon.

Also, the recent generation of National Science Foundation-funded projects offers nice tasks, whether you implement an entire program or not. The K–12 dissemination site for these programs is http://www2.edc.org/mcc.

In addition to these resources, generating helpful teaching ideas and materials continues to be an important area for Council activities.

**Question from**

Waynesville, North Carolina

How do you begin a more student-centered approach to learning math? I just can’t seem to give up the idea that unless the students have received direct instruction, I haven't done my part.

**Cathy Seeley:**

Thanks for being so honest, and for representing the thinking of a lot of teachers out there, I’m sure. The best way to make the shift is to just try it one day. Choose a lesson that you would normally lecture about, and find a nice problem that will allow students to get into it. Instead of doing large-group direct instruction, try setting up the problem with a minimal amount of “telling.” Help students share what they already know about the content, and answer questions about what the problem is asking (not telling them what to do). Let them work on the problem in groups of three to five for a while and then let them share their solutions, not just their answers. One key to this type of teaching is to talk less and listen a lot more. Many teachers discover amazing amounts of knowledge, and also identify misconceptions, when they just listen. It’s fine to bring the lesson to closure, but you might try experimenting by telling a bit less each time you try it, to see what the students will tell you.

Another option is to visit classrooms in another school where this might be happening. Good luck in taking this risk for the benefit of students!

**Question from**

Nashville

What are your thoughts on the widespread idea that super-acceleration of advanced middle-school students in mathematics is what will best result in student success? In my school, students who are doing well in mathematics are being pushed into taking algebra I in seventh grade, which will result in them exhausting all available high school math courses before their senior year, if not sooner.

**Cathy Seeley:**

This is an important question. When we accelerate students into a formal algebra course, or a formal high school mathematics course before grade eight, we have to be very careful and take into account two important issues. First, middle school mathematics today has a lot of rich and important content, especially the idea of developing proportional reasoning. If students miss this, they are likely to have far more difficulty in algebra than if they have had this kind of experience.

Second, if there are not clear and excellent options for students at the other end—the end of high school—we must ask ourselves why we are accelerating them. The strong evidence is that far from all eighth-grade algebra students continue through calculus or a fifth year of high school mathematics. When we start the sequence before eighth grade, the danger is that some students will complete their mathematics requirement early and not take a math class for a year or two before entering college. This can have disastrous results for any student when they enter college having taken a year or more off of mathematics.

**Moderator**

Thank you all for your participation today. We couldn’t address all the questions submitted, but most will be included in the transcript, which will be posted on the NCTM Web site tomorrow.

The next chat with NCTM President Cathy Seeley will be at 3:00 p.m. EST, on Monday, December 13. The topic will be “Pockets of Wonderfulness,” Cathy’s President’s Message in the December *NCTM News Bulletin*.

**Cathy Seeley**

Thanks for joining me in this interactive exchange of ideas. I continue to appreciate your own active engagement in improving what we do for the benefit of helping all students receive a high-quality mathematics education.

I’m looking forward to our chat in December about how we can move beyond Pockets of Wonderfulness. Have a great Thanksgiving break.

**Moderator**

The following questions were submitted before or during the chat and could not be answered during the one-hour live chat.

**Question from**

Glendale, California

What is the impact of reform (sense-making) instructional strategies in the elementary grades on student performance in middle and high school? If a high degree of success can be established, that would go a long way toward convincing teachers to accept the call to change teaching practices.

**Cathy Seeley**

You are exactly right in that we need good studies of future success. The national dissemination centers should have some of this information, and other studies are still needed. NCTM has embraced a leadership role in connecting research with its influence on practice—look for future actions in this area from the Council over the next year or two.

**Question from**

Naperville, Illinois

Oh how we all look for the knight in shining armor—that magic answer to all that ails mathematics, or any other subject matter. Engagement has always been one of the tools to use with seeking students and those that aren’t. We need to wake up to the fact that all the fancy gadgets, new books, fancy labels, and federal programs (NCLB) will never replace a quality teacher in the classroom. We need to attract the more intelligent and people-oriented types to the classroom. We need to change our priorities and guarantee a quality teacher in the classroom. Nothing can replace that. The older federal programs were a start, but where are they now? Do our representatives really care?

Just a thought for you to ponder during the chat.

**Cathy Seeley:**

A professional teacher, committed to lifelong learning, is the most important factor for student learning. And starting with a teacher who knows and loves math is certainly a great way to help all students learn mathematics.

**Question from**

Scotch Plains, New Jersey

My (high school) colleagues and I are trying out different ideas for retesting students who have performed poorly on a test. We would like to, ultimately, find something that is equitable to all of the students, ensures that mastery of the material has been achieved, and that consumes as little of our (precious little) time as necessary. We teach academically gifted students. Any advice would be greatly appreciated.

Thank you.

**Cathy Seeley:**

It’s challenging to have gifted students who can move quickly through academic material and still deal with the needs of those who may have difficulty on some area. I know some teachers who make this kind of student responsible for his or her own remediation, giving them the opportunity to turn in written work that demonstrates they have learned what they missed on the test. In order to keep motivation high for doing well on tests the first time, they might offer students the possibility of making up half the points they missed, or they give them the option of making as high as a ‘B’ on any retest or on turning in demonstrations of learning. These students are sometimes motivated in different ways from less academically successful students. In any case, keeping students focused on being responsible for their learning is a tough but important challenge. And I would argue that, even for these students, getting them engaged in rich problems they have to wrestle with can yield very positive results for now and for their future studies.

**Question from**

Kansas City, Missouri

After many years of different teaching methods I now use an approach that incorporates a 10-minute lecture followed by the students working at the board in small groups to make sure they understand the ideas presented. This then leads to a handout that the students do in class or for the next day. Students enjoy being at the board and they learn a lot from working together, and it is always amazing how much of the lecture some of them did not understand until they do it themselves. This provides the teacher an opportunity to summarize the concepts and point out any pattern in errors made or concepts not understood.

**Cathy Seeley:**

It’s amazing to see how much we can learn when we shift the responsibility for talking and explaining to students instead of the teacher! For those students who enjoy being in front of the class, this sounds like a good approach. You might also consider having students work at their desks in small groups from time to time to see if there might be different stars who appear in class under a different structure.

**Question from**

Flagstaff, Arizona

How do you recommend teachers keep students with various levels of prior knowledge actively engaged during lessons?

Will a transcript of this chat be available later?

**Cathy Seeley:**

If we choose tasks that advance learning and if we make tools such as calculators available, we can open access to challenging mathematics to more students. As students become more comfortable and accustomed to working together and for being responsible for the group’s learning, they will learn to help each other when there are areas one student does not know well. It may also be that this type of structure can help a student recognize where they might need extra work or additional help outside class.

Yes, transcripts of all chats are kept on the NCTM Web site at http://www.nctm.org/news/chat_archive.htm.

**Question from:**

Providence, Rhode Island

Share time at the end of our investigation/activity is what most engages all my students. Because the expectations of listening for a purpose (to understand, to ask for clarification, to refute an idea, to expand or add to someone’s comments/ideas, or to question) are set up ahead of time, my class is fully engaged during this time all my students want the opportunity to comment, ask questions, or share their ideas because they’ve come to see that their ideas are valued in our class, not just by me, but by everyone in the room. Very often students who are struggling learners are willing to come to the group with something they don’t understand and have their peers “walk them through” where they’re confused. Students who used to struggle in mathematics now have an opportunity to “be incorrect” and still be successful because their difficulties allow for more discussion, which benefits everyone. Often new strategies will emerge as students work to try to help the struggling student. It’s exciting to see the kind of support and wonderful mathematical conversations that emerge when students share out ideas. The energy level and genuine interest in what is being presented is remarkable, and I love learning along with my students. Another success comes when the students start directing the questioning themselves. I love to watch them take over and lead the discussions. Sometimes they come up with great questions to extend thinking or clarify a point that I never would have thought of.

**Cathy Seeley:**

This is a very nice example of shifting responsibility to students for their learning, while the teacher continues to play an important role in helping guide, connect, and clarify what students have learned. It’s especially nice when you can create an atmosphere of constructive and positive learning from incorrect or confusing approaches. Thanks for sharing this beautiful description!

**Question from:**

Campbell, California

One of the most engaging strategies I have found for introducing a new manipulative is the “free exploration” time before I try to use them for a lesson. I have my students come up with uses for them and play with them in any way they find interesting. We then have a class discussion about what they have discovered.

As for computational fluency, I find that spending a few minutes a day on mental math strategies and practice pays great dividends.

**Cathy Seeley:**

These are great practices that create a classroom environment conducive for student engaging in rich tasks. Thanks for sharing!

**Question from**

Lynnwood, Washington

Engaging kindergartners in math can be challenging, but one technique I have used for 2+ years now is to include in my Student of the Day routine, an equation that the SOTD decides based on what they are wearing—it might be as simple as 1 ponytail + 2 shoes (I simply write it on the board as 1+2). I do this every day from the very first day and the children who are more advanced use bigger numbers (10 fingers + 25 stripes on a shirt + 10 toes). If it is easy enough to just count on the SOTD, we do that, larger numbers I build with unifix cubes, and we count them together. For the student of the day, the students all go back to their tables and write the SOTD’s name, their own name, the equation for the day, and whatever words we may have found on the SOTD’s clothing. It becomes a lengthy morning ritual, but I have seen such an increase in number sense. I also have added having the SOTD add two numbers to our 100 chart that we start building from day 1 (a first grade activity from a previous issue, but it works for kindergarten too). When we get it all filled in, I will remove a number and ask them to add or subtract 1 or 2 from the missing number, write that number on a Post-it note, and put it in a basket. We look at them later in the day, tally the amounts, and find out for sure. I give a lot of support for students who need it on the 100-chart and the SOTD has a great deal of control over how large or how small an equation they will use—I love how that works for everyone.

**Cathy Seeley:**

The hundreds chart, combined with skillful questioning, can be a very useful tool for helping students develop an understanding of and fluency with numbers.

**Question from**

Calistoga, California

As an elementary-level teacher of mathematics, I am constantly challenged to find ways of helping students connect words, numbers, and concrete experiences. About 10 years ago, I began trying to meet some of the needs of diverse groups of children by incorporating student-written riddles and puzzles into my classroom math program.

The process of creating new riddles provides an impetus for students to express themselves fully and clearly, since they want other children to solve—or at least attempt to solve—their riddles. Also, children working together tend to give editorial feedback in a constructive way, with peer comments ranging from comments about penmanship and spelling to more substantive discussions of riddle clues and solutions.

The process of writing and solving riddles provides a rich context for learning new vocabulary and phrases, from “It’s a trapezoid” to “three more black beans than white beans” to “there are equal numbers of dimes and pennies, and half as many nickels as dimes.” Using this type of language in conjunction with concrete materials helps reinforce meaning. Also, because the vocabulary required for a particular type of riddle is somewhat limited, children with special needs in reading, as well as those who are English learners, find most of the activities accessible.

**Cathy Seeley:**

Students seem to love creating problems or riddles, and they are often more motivated to solve them when they are created by other students. Nice idea!

**Question from**

Redding, Connecticut

Math teachers have often provided the impetus for technology in schools because we have seen from the beginning the power of technology to engage students. Programs such as Geometer’s Sketchpad, LOGO, and Excel engage students by making them active learners. The students produce the math, make the conjectures, and apply their knowledge. Technology enhances visualization of concepts for students and provides another approach to learning.

I have found that it is important to design lessons for a computer lab that are one-day, self-contained lessons. With the pressures of finishing/covering curriculum, we have to be careful not to let labs stretch over to multiple days. This can happen easily if we are not careful, and it often leads to reluctance to use the power of technology.

**Cathy Seeley:**

Technology can be a powerful tool, either through software such as you have mentioned or through graphing and statistical functions on calculators. It all comes down to how the teacher builds in the use of these tools into a balanced math program.

**Question from**

San Antonio, Texas

Just today in my statistics class, students were generating sampling distributions—drawing numbers from a hat, tossing coins. Each student put his/her sample statistic on a Post-it note and put that on the number line on the board to create the distribution. One of the students commented, “This is a lot better than taking notes.” Although I think the note taking is still an important part, students must have chances to actively engage in the mathematics. Only if they do this consistently will they be learning from the experiences rather than just “doing an activity” with no connections. I find that these activities are most effective as a vehicle for learning when there is a follow-up written piece that asks students to verbalize or apply what they have learned.

**Cathy Seeley:**

Yes, the engagement is the key, and the follow-up is critical!

**Question from**

Calgary, Alberta Canada

We have a new program in Canada called PRIME, which is Professional Resources and Instruction for Mathematics Educators published by Thomson Professional Learning. It offers a real breakthrough in Professional Learning because it centers on big ideas, phases of student learning—a key to differentiated instruction, and good pedagogy.

It focuses on mathematical concepts in a way that teachers in elementary grades can become more competent in their own understanding of mathematics. It is an invaluable resource for teachers.

**Cathy Seeley:**

Sounds like we can definitely take a cue from our Canadian colleagues in terms of focusing on professional learning as a key to student learning. Thanks for sharing.

**Question from**

Paterson, New Jersey

I am a new high school math teacher in an urban school in New Jersey. The students are not motivated and I found that offering a piece of candy for a correct answer has made a tremendous difference.

**Cathy Seeley:**

It’s amazing how simple rewards can be effective, even with older students (stickers may surprise you as well). In terms of motivation, I tend to believe that sometimes action precedes motivation, and that getting students involved in a challenging but interesting problem or task can work wonders for motivating them. Hang in there, and keep on exploring what works. There are no easy solutions, but I promise it gets a bit easier as time goes on, particularly if you keep on learning yourself. Good luck!

**Question from**

Rochester, New York

What a timely topic for New York State! I wholeheartedly agree that teaching mathematics for meaning, promoting understanding through problem-solving, and communicating understanding is the way to go. I know that by offering rich, meaningful instruction to students, they will become more flexible thinkers and stronger mathematicians. I wish our New York Education Department would join your chat! They have a lot to learn! Our newly proposed NYS Standards hide behind the powerful ideas of NCTM, but when you dig deeper, you see that they are asking us to go back to a skills-based approach to learning mathematics, and cramming a long list of skills down the throats of our children at each grade level. Welcome back to a mile wide and an inch deep. Welcome back to the world of “just memorize it” because it will be on a test. Thank you NCTM for keeping us on track with your Standards 2000 and for all the supplementary support you offer. Keep up the great work!

**Cathy Seeley:**

And thanks to all the teachers in all the states working to teach good mathematics programs to all students, whatever the direction of current state mandates. Thanks for all the good work that you do!

**Question from**

Urbana, Illinois

Each child in my first grade classroom generates his own math stories, writes equations using number to tell the same story, relates the story to the class, chooses others to act out each problem and read the equation to the whole group, then entertains comments, suggestions, and questions about the problem. Before telling the story to the class, the child draws the three pictures that tell the story on the chalkboard. The dialogue that follows each problem often results in rich discussions of mathematical ideas. Because mathematical discussion and numbers are generated by children and occur within an integrated curriculum, the children are engaged and more likely to spontaneously relate math ideas across the curriculum. A parent wrote, “After seeing the children act out their problems, I was truly impressed. This was a great way to make learning fun and math more personal.”

**Cathy Seeley:**

Wow! Nice example both of what students can do in the classroom and of how to generate parent support!

**Question from**

Lake City, Florida

Working in pairs to pre-read activities and short discussion before beginning work has improved student achievement in my classes. It helps students focus and analyze their task before attempting it. My students also reflect on this practice by writing in their journals. One of the most common comments is “I really hate prereading but it really helps!”

My students used pre-reading during their weekly visits to the computer lab. We used a word processor, spreadsheet, and Geometer’s Sketchpad.

I have video of my earliest practice of pre-reading in pairs on the Web in a digital edge exhibit “Gee I’m a Tree.”

Here is a quote from my reflection:

”My students were comfortable working in pairs taking turns reading and keyboarding. They worked well with each other. They were focused and engaged while learning new ways to construct and new problem solving strategies.”

**Cathy Seeley:**

Nice use of students talking with students. Thanks for sharing.

**Question from**

Hilo, Hawaii

I am interested in the meeting of art, writing, and math as a way to engage students in mathematical concepts. I am first-year teacher, fourth grade, at a school where writing and reading comprehension are very low testing scores for us. Any ideas about how to promote significant, thoughtful writing responses to mathematical problems of the day?

**Cathy Seeley:**

Whenever you can have students explain or justify their approach or solution, you contribute both to the development of language and the development of mathematical thinking. This is why communication is one of NCTM’s process standards in *Principles and Standards for School Mathematics*. We can indeed combine these content areas (and others) through art, visual representations, and written communication.

**Question from**

Helena Montana

This year I have had the chance to observe classrooms from grades K–12. I have noticed that the students’ level of engagement is inversely proportional to the grade of the student. Is this normal that older students prefer to be more independent in their learning and to sit back and absorb ideas? The older the student the more aloof they appear. Is this true, and should we worry about the level of engagement for older students?

**Cathy Seeley:**

I wonder if what you are seeing are students preferring to be independent or simply not being engaged in what is happening in class? Aloofness can be a sign of disinterest, and I come back to the notion of structuring classrooms so that students are expected to work in small groups on interesting and engaging problems that take a bit of effort. I think we absolutely should worry about the level of engagement for all students, and it is often much harder to accomplish with older students. But it can be done, and the rewards are tremendous.

**Question from**

Dayton, Ohio

I am teaching 6th graders right now and will be teaching 4th grade in January. Do you feel the strategies used to engage are the same throughout different grade levels? Using a real world context by far has been the most effective for me.

**Cathy Seeley:**

Using real-world contexts is always appropriate at any grade level. Strategies for 4 th and 6 th grade can be very similar, and you will quickly see how you might need to adapt. But students at all ages seem to respond well to being hooked into good problems if we give them both the opportunity and the expectation that they will do so.

**Question from**

Highlands Ranch, Colorado

Are there particular math videos that you would recommend that model this more holistic approach to teaching math?

**Cathy Seeley:**

There are an increasing number of videos of effective classrooms available. Some can be found on NCTM’s Reflections Web site. Others are available through materials developed at WestEd and some of the Regional Education Laboratories. Many universities have developed videos, especially where schools may be implementing some of the NSF-funded curricula. And there are also commercial videos that show teachers working with groups of students. Sometimes the videos are great models, and other times they can give you ideas about things that seem to work and things you might choose to do differently.

**Question from**

Cedar Park, Texas

Some of our classes here have taken a Challenge Course element and turned it into a math Challenge that works in probabilities; percents; graphing, and whatever else we can come up with.

We start by making a 10 by 10 grid on the floor with tape. The grid is big enough for the class to stand in and see from their desks. We can use the grid to figure out a maze to get us from one side to the other (which I have on a piece of paper, but the students have to figure out). We can set our X and Y coordinates and graph points on the floor. We have even used the grid to line out 100 numbers for our students to find Mystery numbers (by giving them clues).

The uses are really endless and can be student driven.

**Cathy Seeley:**

Thanks for sharing!

**Question from**

Richmond, Kentucky

I’m a middle school math major and will be graduating soon. Most of the classrooms that I have been able to observe and teach in seem to be using the same old methods that were used when I was in middle school 25 years ago. The texts they use aren’t usually set up for the kind of math teaching (relating math to the real world) that I believe is necessary if our students are going to excel in our technological society. Most of the students are apathetic at best about math. How do I, as a teacher, incorporate real world math situations into the curriculum?

**Cathy Seeley:**

Welcome to the challenges of being a math teacher in the 21 st century! You are correct that it is not as common as we might like to see the kind of teaching that engages students in their learning, rather than telling them what they are supposed to learn. I encourage you to keep looking for those good examples and role models. Ask your professors to help you identify where to find these programs and these teachers. They are out there!

**Question from**

New York City

Are the course materials that you mentioned available? And how do we obtain them?

**Cathy Seeley:**

I would suggest starting with the K-12 dissemination site mentioned earlier. Good luck!

**Question from**

Cedar Park, Texas

One of the fun ways that I like to talk about fractions *every day* is in the day of the six weeks. For example, my students know that today is day 7 of 28 days in this six weeks. I am a resource math teacher and it is important for the students to know where they are in time. So today is 7/28th into the six weeks. Or 1/4th.

I also have had the students make a huge mural on the wall that they enlarged on the overheard to teach enlargements. It is a giant hawk flying from the sun with the words. “Challenge Yourself!” This was an end-of-school year project that now hangs over the entryway to our school, Henry Middle School, LISD. These students are very proud to be a part of the school now and they made their mark in math class.

**Cathy Seeley:**

These are nice examples of getting kids hooked on math and hooked on their school. Great ideas!

**Question from**

Estill, South Carolina

(Cathy: I’m including this to you, but I’m a little uncomfortable including it online because it sounds like a plug for the TI-89.)

Math is like any other sport...if you’re going to PLAY math, you’ve got to learn the rules.

Students can *actively* learn to apply the rules with the application called SMG (Symbolic Math Guide) of the TI-89. This allows a student (Algebra I or higher) to enter an expression or equation; then select a transformation from an appropriate list. The calculator displays the result, which may or may not agree with the outcome predicted by the student. As the student works through each example step-by-step, SMG gives immediate feedback, constant reinforcement of good skills, and corrects misconceptions in algebraic thinking. If one approach doesn’t work, it’s easy to go back and try another, more efficient route. What is most important is that the student is DOING math instead of watching the teacher at the front of the class.

Telling isn’t teaching.

Listening isn’t learning.

**Cathy Seeley:**

Using technological tools such as those available today is a great way to let students get involved with mathematics, especially if they have the opportunity to discuss the mathematics they are learning with others. We must make sure, however, that all students have access to these tools, including outside of class, if we are going to use them as equity tools to help all students learn mathematics.

**Question from**

Ridgeley, West Virginia

Engaging students in cooperative learning activities has been very effective in the Algebra One Applied Collaborative classroom in which I co-teach. With two professional educators in the classroom for the 90-minute block, we are able to involve our students in problem solving tasks that require students to communicate the situations with their peers. We established the groups of three and four students based on class performance and ability to assist others—not students selected based on friendships. With team building and class building exercises planned to help foster working together, students have to be taught to work together effectively. By taking time to build these expectations into the class culture, we have witnessed remarkable interactions with the students.

**Cathy Seeley:**

Yay! Thanks for sharing a good success story that exemplifies the power of students working together and being engaged in learning mathematics.

**Question from**

Houston

An example of student engagement that I have used successfully with my Foundations students is the evaluation of their integer addition skills using a card game (standard 52 card deck) called Zero Out. The red cards are negative and the black cards are positive. (The rules and more details are available by request at leveillen@uhd.edu.) The students enjoy the game aspect and see (for many for the first time) that school math is something they can do, enjoy, and find useful. It does not take long to evaluate a class of students as you walk around and watch them play. It is faster than giving and grading boring quizzes.

**Cathy Seeley:**

Games can be a motivating way to approach computation. This is one aspect of a balanced and comprehensive program that includes development of skills, conceptual understanding, and problem solving or applications.

**Question from**

Redmond, Washington

The first day of class (secondary level), students are given a file folder and asked to write about where they see themselves ten years from now. They are asked to describe their lifestyle, such as where they will be living, the kind of car they will be driving, and how they will be supporting this lifestyle. This is their daily folder in which they hand in their class work. It serves a number of purposes. (1) It allows me to have a personal conversation with each student on a daily basis. The first few minutes of each class, they are asked to respond to my comments and questions. By the time the inside and outside are filled, I know each of my students much better than most of their teachers. (2) I have a very easy way of taking attendance, as I can see which file folders have not been picked up at the beginning of class. And, (3) the conversations provide a rich source of applications for the mathematics.

In the folder, I ask a student’s permission to share his or her comments or ideas when we discuss certain math concepts. For example, I identify students, through our folder conversation, who are interested in sports, music, automotive mechanics, dance, architecture, hairstyling, etc. asking them to be the “experts” when I present an application that fits their interest area.

Another way I use their responses is as the basis for a class project. The students research an aspect of their future vision. For example, if they say they will be living in an apartment in Los Angeles, they research the cost of apartments in Los Angeles for the past 20 years. Or if they say they will be a fashion designer, they research the income level for the past 20 years. They get to choose what they will be researching, with my approval, as I want to make sure it is something quantifiable and researchable. (I always give a heads-up to the librarian to have resources available, and I encourage students to talk to their parents, their parents’ friends, and relatives to get the info.) I usually put them into groups based on the type of research they are doing, such as a housing group, a profession group, a vehicle group, a university education group, etc. Then they graph their data and from their graph make a prediction of what it will cost 10 years from now. This information is displayed on posters, and they make a short presentation. I encourage family involvement by asking for the inclusion of comments from family members on the poster and presentation.

**Cathy Seeley:**

You have identified some great ways to personalize what students learn, which is surely a tool for engaging them in that learning. Thanks for sharing these ideas.

**Question from**

Bowen Island, B.C., Canada

My grade 4/5 students keep nature journals as a focus for engaging in mathematics that is directly connected to the real world. As they draw and write notes about their observations of such things as snails, spider webs, ferns and clouds, the children realize how much they can discover by simply slowing down to really look and wonder about the world around them. To make sure that math is included in these observations, I provide specific activities and guidelines. Recently, for instance, my students weighed, measured, drew, and described various gourds that I had brought into the classroom. Another job was comparing the areas and perimeters of two leaves: a medium-sized maple leaf with its jagged border and a huge skunk cabbage leaf with its smooth perimeter. I have also developed a simple template for creating a “mathematical portrait” of natural objects, with a section on the page for each of the four strands of our math curriculum: number, pattern, shape and space, and data and probability. Our nature journal activities take place both indoors and outdoors. I believe they connect mathematics with children’s strong affinity to nature and their excitement about making their own discoveries.

**Cathy Seeley:**

Nice way to connect math to other subjects and to motivate learners! Thanks for sharing.

**Question from**

West Chicago, Illinois

My wife and I team-teach a math methods course for prospective teachers. It is amazing to us how little math these adults know. All have B.A. or B.S. degrees on other areas. But they truly seem to understand when we point out the NCTM methods etc. Many of them have that elusive “Aha!” moment. Your thoughts?

**Cathy Seeley:**

We sure want more of those ‘Aha’s for both teachers and students. What we want to create is a new generation of adults who know a lot of mathematics. This must start with helping teachers really understand the power of math to solve problems. Focusing on the big ideas in math can be a great starting point (equivalence, proportionality, functional relationships, etc.).

**Question from**

Canton, Georgia

I use a smart board. The templates are wonderful, and the kids love teaching math on it.

The students also have a math log to jot down all the concepts and examples. The logs are tabbed with the main concept.

We have butcher paper to practice with buddies our progress on skills.

We have word walls and concept mapping displayed so the kids can see where they have been and where they are going.

Hey, it works.

**Cathy Seeley:**

Thanks for sharing these tips for developing mathematical communication. These can be great tools as part of a program where students are interacting around mathematics. Communication is key!

**Question from**

Lamesa, Texas

How can you justify the use of calculators when they are not allowed on standardized tests?

**Cathy Seeley:**

When teachers use calculators appropriately and help students learn when it is the right time and not the right time to use them, students can learn far more mathematics than if they are denied the use of these tools. Research has shown that when students have access to calculators as problem-solving tools, their computation skills are not hurt. More important, they outperform other students on understanding concepts and solving problems. Even when they don’t use calculators on the test, they are far more likely to be able to do well on the problem-solving parts of the test if they have had good experience with this content. Without calculators, many students would never have the opportunity to tackle rich problems in class, and they would be tremendously disadvantaged on their tests. If we limit their instruction based on the limitations of the test, we run the risk that they will not be well armed to succeed either on the test, or, worse, in future work in mathematics or other content areas.