**Moderator**

Good evening everyone, we're glad to have you join President Johnny Lott for this special late-night (for some of us) chat. We've received a particularly interesting set of questions in advance. We look forward to receiving more over the course of the next hour.

Our first question is from Florence, Texas

Should number reversal be counted completely wrong at a kindergarten level as well as issuing zeros for class participation?

**Johnny Lott**:

This is almost impossible to answer not knowing the situation. By number reversal, I'm guessing that you are talking about writing an E for example for a 3, or something like that. This is a common error for children when they are learning to write symbols. It is the reason that different schemes have been developed for writing such symbols. Touch-point is one of the schemes that some have used though it is not the only one.

Whether or not this should be counted completely wrong would in my mind depend on when this was done, how extensively this had been studied, and so on. Also, heavily dependent on this answer will be how the kindergarten gives or doesn't give grades. In the 37th NCTM Yearbook on Mathematics Learning in Early Childhood, it was suggested that most educators delay formal writing of symbols until age 5 or 6.

In addition, you asked about class participation. There could very well be a cultural issue here. In some cultures, children are taught to think very carefully about something before responding. I would have to know more about the class and its culture to answer a question about class participation.

**Question from:**

Springfield, Missouri

Our district has a goal of getting 75 percent of our 8th graders through Algebra I. I recently read where California is trying to do something similar. I know NCTM doesn't advocate a "formal course" in algebra before high school. What is the hurry? Why are school districts trying to push students through Algebra I before they are ready? I am working on a research paper on abstract thinking and, specifically, are 8th graders ready for Algebra I? I am having some difficulty finding the research on this issue. I find as an eighth-grade teacher that some of my students are ready, but the majority are not. What does the research show? I will be investigating NCTM's resources, but if you have any suggestions for me on where to find the information, I would appreciate it. Both for and against.

**Johnny Lott:**

There are many reasons why schools are pushing students to complete Algebra I early in the students' careers. Perhaps the better question is whether the school is prepared to continue to educate students mathematically through their high school careers in an intellectually honest way. If there is no plan for solid high school mathematics students in each of their four years, and I do mean for all students and not just those who plan to be mathematicians or scientists, then we are doing students a big disservice.

There are many people who argue for algebra early in an effort to open doors for students later in their careers. You might want to read the work of Robert Moses and his Algebra Project as you think through the research.

**Question from:**

Cherry Hill, New Jersey

Has the NCTM ever considered publishing a K-12 math textbook series? This reminds me of the phrase "put your money where your mouth is."

**Johnny Lott:**

NCTM has not considered publishing a K-12 mathematics textbooks series in recent history. It did consider whether or not it could become a "consumer reports" type agency for materials that were published.

In both cases, one of the primary issues was money. A publishing venture literally costs millions of dollars, and to conduct the venture in the way that NCTM would do that would require even more money. Right now even for the small Navigations booklets that NCTM publishes, it works with an author team in conjunction with an editorial group to establish the outlines and how the work will be done. The authors write and try out the materials with students; it is reviewed extensively; rewritten, and sometimes re-reviewed and edited before it is released for publishing. Along the way, answers have to be written, applets have to be developed and eventually there has to be a merger of a print booklet and a CD. This process takes at least a year and a half, and it is for a 64-page booklet. Consider how long this process could take the organization for an entire series.

At least one of the reform mathematics high school series was written over an 8-year period with over 75 authors, testing in many schools, and in at least three states with teacher materials and supplements written in the process all at a cost of about $13 million. And this was only for grades 9-12. As a nonprofit organization, NCTM does not have the dollars to do the type of work that you describe. It is a nice idea however.

**Question from:**

Port Coquitlam, British Columbia

Teachers at different levels often say that they would like students coming to them from a certain grade level band (i.e., from primary to intermediate, or intermediate to secondary etc.) are lacking in certain knowledge or skills in mathematics. Rather than blaming any particular area or grade level for not doing its job, in your time as president of NCTM, is there a grade level band that does seem to be making particularly forward strides in the general move toward improving mathematics learning and education? Also, on the reverse of this, is there a grade level band that could be more "alert"?

**Johnny Lott:**

First, I believe that teachers have to work with all students regardless of the background with which they reach their current classes. We cannot use as an excuse, "They did not know this before they got here." At the same time, we cannot re-teach everything at any given grade level. To accomplish what is needed, there has to be communications across grade bands—and this goes to all levels, including high school and university.

Now beyond that, are there areas where there may be more mathematical issues than others? I don't know that there are more issues, but as a general statement, because of the variety of certification levels, there are probably more teacher knowledge issues at middle school. In today's world, a single K-8 certificate probably does not provide adequate mathematics knowledge for today's middle school teachers. Having said this, I would also be a proponent of specialization in mathematics at earlier grades as well; high school teachers really must know about the mathematics taught in middle school and university mathematicians really must know something about high school mathematics beyond what they remember. The message: we must talk beyond normal borders in mathematics education.

**Question from:**

Detroit, Michigan

Hello, I am a new member to NCTM and have recently finished an MA in education. My undergrad was not in education. I have taken two "Math for Elementary Teachers" courses and one math methods course. This was never a strong subject for me as a child, however, I have overcome my math phobia and tutor kids online in math on a daily basis and have been quite successful. However, I am sure there are many who are going through quick programs similar to mine who do not feel comfortable with math instruction. I thought NCLB dictated that you had to hold a major to teach math. Does this not apply to elementary teachers? Don't you think this might contribute to some poor mathematics instruction?

**Johnny Lott:**

NCLB talks about highly qualified teachers at the middle school and high school having mathematics majors or a masters plus a demonstrated knowledge of the content. Elementary teachers are in a different category. If you are asking me if I think that it is time we had mathematics specialists for elementary school, the answer is yes. Those don't necessarily need a mathematics major but could be specialists. I believe that such a program is in place in some states and has been proposed recently in Virginia. I could be mistaken here, but I think that this is true.

**Question from:**

Erie, Pennsylvania

I am a student at Edinboro University of Pennsylvania and my major is Elementary/Early childhood education. I have to incorporate children's literature into a math lesson for children. Do know of any good quality children's books that I could use?

**Johnny Lott:**

I do! Mathematics and children's literature is a favorite of mine. E-mail me separately and I'll try to respond. If you choose not to do that, check out Teaching Children Mathematics, the elementary NCTM journal. There are book reviews and frequently articles on children's literature and mathematics. For a specific answer, I need to know grade level and topic of interest. NCTM just published a book on this topic edited by Diane Thiessen, University of Northern Iowa. There are also books either written or edited by David and Phyllis Whitin, copublished by NCTM and NCTE. Check them out. If they are not in your school library, they should be.

**Question from:**

Los Angeles, California

What strategy is effective for students who are not interested to learn? They are just there to sit and wait for the end of the period.

**Johnny Lott:**

My first response is that there are few who are not interested in learning. They may not be immediately aware of their interest in learning. I am a firm believer in using materials and problems around us to make the points that we want to make. For example, as a teacher, collect everything you can get your hands on that talks about math in the real world. A cheap source is *USA Today.* Pick out different topics that can lead you to the math that you are trying to teach and ask the leading questions of students. Some simply need to see that they need math to survive in a good way. One might survive without it, but scams are more prevalent today than ever. Are your students prepared to deal with scams from a mathematical standpoint?

**Question from:**

Louisville, Kentucky

My school has decided to align its curriculum, including mathematics, with the ACT. How is this compatible with NCTM standards? Is there a framework to work within both realms?

**Johnny Lott:**

More and more schools are trying to align curriculum to various tests. In my mind, this is almost exactly backward UNLESS the test contains all that you want students to learn. Tests should be used to inform teaching showing us as teachers what we need to do and what we have completed. Most tests weren't designed to align with NCTM standards. They were designed to measure student achievement, progress, etc., on some body of knowledge or on some specific topics. The real question here is does the specific test cover all that you want every student to know? If not, then administrators and school boards need to be made aware that is not the case and why not.

Let's not use tests for more than that for which they were designed. To use a phrase I heard from Dr. James Lewis, "Corn doesn't get heavier because you weigh it." I don't think that students get more knowledgeable only because of testing regardless of the test. It could happen if teachers are provided feedback and that is used to influence teaching. Is that the case in Louisville?

**Question from:**

Memphis, Tennessee

NCTM standards de-emphasize factoring. I would like to de-emphasize them in my classroom. But what about in my students' future classes where a teacher may require them to do a lot of factoring? If I don't have them do a lot of drill and practice now on factoring, then down the road, they may be at a disadvantage.

**Johnny Lott:**

It is the case that the 1989 standards suggested that factoring be de-emphasized. Those standards did not say that it should not be taught. Consider what we know about factoring. All quadratics can be factored if students are taught the quadratic formula. With that tool in their hands, all students can factor. Some cubics can be factored easily; many are more difficult. Similarly, some different types of higher order algebraic expressions can be factored, but most cannot. In the whole scheme of algebra, factoring is not a big idea. It is a specific skill that is useful in some places. With your curriculum, what are the big ideas of algebra that you want students to learn? Those concepts are where you want to spend the majority of your time. Do you drop factoring? Probably not. How much time you spend may depend on how fast you want students to be able to factor quadratics, the sum and difference of cubes, or odd powers. You are probably not going to go beyond that. If you want speed, you will spend more time on drill and practice. If you want efficiency in the big sense for factoring quadratics, you may spend more time on the quadratic formula. If you want understanding, you will probably do a combination including the use of graphing calculator and graphing to help students understand the relation of roots of equations and factors on expressions used to write those equations.

The bottom line is that if students understand the concepts, they won't be at a disadvantage UNLESS speed is the issue, and it could be for some teachers.

**Question from:**

Austin, Texas

Isn't NCTM the greatest organization around?

**Johnny Lott:**

I'm guessing that the question is from NCTM President-Elect Cathy Seeley, and she is absolutely correct. If not Cathy, you are still correct!

**Question from:**

Butler, Pennsylvania

Why are secondary education mathematics teacher training programs so focused on upper level mathematics courses and lacking in the areas of teaching techniques and practices?

**Johnny Lott:**

Many of the pure content courses for secondary mathematics teachers are taught in mathematics departments by mathematicians. While many of these professors are excellent teachers of content for collegiate students, they may have little knowledge of teaching techniques and practices for use in secondary schools. See the April 2004 Notices of the American Mathematical Society and look for an article by Anthony Ralston.

**Question from:**

Gurnee, Illinois

I am an adult student at Barat College of DePaul University. In my observations, I have seen many teachers turn to "gloom and doom" when the math books come out. Young students pick up on this and their attitude toward mathematics is shaped before it ever had a chance. How can we future teachers help to change the negativity in the classrooms?

**Johnny Lott:**

This is more than a teacher problem; this is a cultural problem in the United States. To use a really old line, "Have you ever heard an adult say, ‘I hate reading'?" But how many times have you heard, "I hate mathematics."? For a good article about this by Brunetti, see "Why Do They Teach? A Study of Job Satisfaction among Long-Term High School Teachers" (SOURCE: Teacher Education Quarterly 28 no.3 49-74 Summer 2001)

**Question from:**

New Jersey

This probably seems very silly, but... I'm in the middle of a certification program, and everything is going fine. But when am I supposed to take my Praxis? Either part. No one even mentions it.

**Johnny Lott:**

This varies with the school. Some have students take parts of Praxis as an entrance exam; others take parts in the process; and yet others take it as an "exit exam." Quick: go ask your adviser or the certification officer. Here at the University of Montana, it is an exit exam.

**Question from:**

Indianapolis, Indiana

Hello Mr. Lott! It's great to finally e-meet you! I will begin teaching in a township that is changing its math textbook from a "drill-and-kill" approach to the problem-solving method. I'm already nervous about teaching math anyway, but I feel better because I was taught to teach math through manipulatives and problem solving. Can you recommend some strategies or basics that will help me deal with this crucial task of teaching math well?

**Johnny Lott:**

Thanks for the note. Good places to look are NCTM publications: 1980 Yearbook on Problem Solving, Teaching Mathematics through Problem Solving Prekindergarten-Grade 6 and Teaching Mathematics through Problem Solving Grades 6-12, both edited by Frank Lester and Randy Charles, and Children Are Mathematical Problem Solvers by Sakshaug, Olson and Olson. These with the references in them are great places to start.

**Question from:**

Rapid City, South Dakota

How important are instructional materials in the overall picture of mathematics instruction? Is it possible to teach a standards-based curriculum with traditional instructional materials?

**Johnny Lott:**

To me and I don't mean to be flip, this is like asking a carpenter, how important are your tools; could you build a house without a hammer? As a longtime teacher from middle school through college, I don't know how people teach any mathematics class without instructional materials. In today's world to me, technology is a must as an available tool. I won't say that mathematics cannot be taught without instructional materials, but frankly, I cannot imagine how effective the learning would be or how long the retention would be for students.

**Question from:**

Salisbury, Maryland

Hello, President Lott! I am a second-year math education professor at Salisbury University. Some of my students and I are following this conversation tonight. I wondered if you have any general advice for me as a relatively new prof or for my students who are preparing to teach secondary math?

**Johnny Lott:**

As a new professor in mathematics education, I would strongly encourage you to reach out to teachers in your area. You need to get to know them and them you. You will want to place your students with them, and you will want to work with their students at times. As well, you will want your students to work with their students if you teach methods classes.

Join your local math organization; become active as a part of your service load at the college. Encourage your students to become student NCTM members. We have great benefits for them. I'm advertising here, but go look at Illuminations and Figure This! as examples of NCTM freebies. The benefits include those and more. Attend any conference that you can and encourage your students to become mentors for each other. I'll stop now, but good luck.

**Question from:**

Charleston, Illinois

What are your views, and NCTM's view. of inclusion in math classes? Do you feel that one student can really change the learning of an entire class?

**Johnny Lott:**

All students deserve the chance to learn mathematics in an intellectually honest way. They learn from teachers AND from students. Inclusion is the law in most places, so a view is unnecessary. I'm not sure what you mean by one student changing the learning of an entire class, but we all learn from each other regardless of socioeconomic status or special challenges. We not only learn mathematics in the classroom; we also learn that not all people are alike but we share the same world. This is a very important lesson for all of us.

**Question from:**

Clinton Township, Michigan

I recently did a research paper on Japanese science instruction. Instead of covering 100 concepts throughout the year, they chose to cover 10 concepts, THOROUGHLY. This is something I would like to do in my elementary classroom in the subjects of math and science. I have heard of school districts that so closely dictate curriculum that teachers are required to cover a large mass of material during an allotted timeframe. (I suspect, this is due to an alignment with standardized tests.) How does one balance thorough mathematical instruction with such rigid curriculum constraints? I will have my own classroom next year, and this is my biggest worry.

**Johnny Lott:**

One of your biggest concerns is how will covering 10 topics in your class affect the students in the coming classes? It appears to me that this cannot just be one teacher working alone. That is very unlike the Japanese lesson study model. To do what you are describing means that teachers need to be working together. I don't think that you can do this alone.

Whether one teaches many concepts in a year and revisits later or concentrates on a few concepts in given years demands much communication among teachers, administrators and parents. Don't try this as a lone wolf is my advice. It is not necessarily good or bad, but you must worry first about the whole career of your students in mathematics.

**Question from:**

Wheelersburg, Ohio

Next year, thanks to No Child Left Behind, I will be teaching a couple of remedial math courses to students who aren't up to par with their math skills. I told my principal that if a new class is introduced to help students with lower ability levels, then a new class should be introduced to help challenge students who are bored with the regular curriculum. The principal understood my view, but said that No Child Left Behind focuses primarily on kids with low ability levels. Do you have any recommendations for challenging my academically gifted students when so much of my time will be focused my lower-ability students?

**Johnny Lott:**

Some of the reform mathematics curriculum materials developed under National Science Foundation grants have research stating that they work well for both types of students. I'm not sure what grades you are talking about, but check out these materials online through the centers that promote them. A middle school site is at the University of Missouri; a high school site is at Ithaca College; and an elementary site is at COMAP. You may also find some opposing views. You need to read both views and make your own decisions for your school. Depending on the grade level, NCTM has a wealth of materials for use with students at all grade levels.

**Question from:**

New Hudson, Michigan

To what extent do you feel that the teaching of math history and historical figures in mathematics helps in generating more interest in math, particularly in secondary education? How much of this is taught now, how much do you think should be taught?

**Johnny Lott:**

I am a mathematics history fan and regularly teach the course as a senior level writing course at the University of Montana. This course is required for prospective secondary teachers but is taken by many regular math majors. History can be used to generate interest—and not just in secondary mathematics. More important than whether this is taught specifically in schools is that the teacher knows how some of the mathematics developed. For example, our traditional algorithms for number operations did not just appear from heaven. It took years for those to develop; now some expect them to be automatic for children. Consider the book by Swetz, "Capitalism and Arithmetic." How much should we teach? I'm not sure; should we use topics? Yes.

**Question from:**

Wheelersburg, Ohio

What are your views on the Saxon Math Series?

**Johnny Lott:**

NCTM does not promote specific series; nor does it speak against them. There is a body of research that can be examined on the use of the Saxon Math Series. It might take a while to search this out to get different views.

**Question from:**

Fullerton, California

As an incoming teacher, I have not developed a lot of good strategies or methods to teach mathematics. What is your advice? Where is the best place to find excellent resources, besides our master teachers, credential program, NCTM, etc.?

**Johnny Lott:**

Don't be too quick to bypass all the resources that you just listed. But do include the people with whom you have been studying and working. Those fellow students can turn out to be great mentors for you. As far as strategies for teaching go, you will need to use all of your resources to find the best ways to reach the students. Join local math organizations, go to conferences; read journals; read; read; read. Go to the Math Forum online as a different resource—not so much for strategies as for content help, and look at resources still available in ERIC.

**Question from:**

Chicago, Illinois

I started student teaching today after spending a good part of the weekend doing lesson plans and preparing. It seems like it is absolutely impossible to have enough time to do a thorough enough job of planning and preparing in order to be a very good teacher. Am I missing something, or will experience reduce this amount of time from infinite to only a couple hours a day? Do you have any other time-cutting suggestions in the meantime? Thank you.

**Johnny Lott:**

My guess is that as a conscientious teacher with some experience you will cut the number of hours needed for planning, preparing, and grading. However, my guess is that it will never be minimal. At least it was not for me. It still isn't. In the fall I expect to be teaching a mathematics-modeling course with technology and the number of hours needed to prepare is huge. I've been at teaching for a long time, and it still takes time. My suggestion is that whatever you are doing; find some way to have some time to do non-school things most days. Even if it is only 30 minutes at night for reading non-math material, it will help you relax and probably make you a better teacher. Even teachers need a non-school life. Good luck! You couldn't have chosen a better profession.

**Question from:**

Charleston, Illinois

How would you encourage parents to become more involved with their child's homework?

**Johnny Lott:**

My very first suggestion is to visit the school and talk with the teacher. I don't mean just at a parents night, but make an appointment. Ask about books that are non-text that can be used with students; check out the www.figurethis.org Web site of NCTM for middle school kids. Though all parents want to help, teaching independence is also important along the way. Do math things together that aren't necessarily homework. My message is get involved with your child's math, not just the homework.

**Question from:**

Eastern Illinois University

What are some ways that we as teachers can get students motivated about mathematics when they feel that they can't do it?

**Johnny Lott:**

There are no magic bullets. Look for math in things that interest them. Try surveys at school. Examine data that interests them to look for patterns and suggestions. Consider reading books that have math in them and look for math in books that aren't math books per se. Read newspapers including comic strips looking for math. Most importantly may be to try and find out why they think that they can't do math.

**Question from:**

Charleston, Illinois

Dr. Lott,

Do you think that the idea of using calculators in the classroom a good idea, or is it not as good as the students solving the problems by hand?

**Johnny Lott:**

Look at calculator research. One suggestion is to find the articles by Hembree and Dessart to start. I believe that the articles were in 1984 and 1992. Those studies looked at large numbers of calculator studies. After reading that research, draw your own conclusions. I think that the evidence shows that calculators can be good tools for teaching math. Are they the only tools to be used? No! Neither are pencils and paper alone.

**Question from:**

Charleston, Illinois

What are some manipulatives that you recommend that first-year teachers buy immediately?

**Johnny Lott:**

The question is too broad, and with no grade levels I have a hard time answering it. Also, you need to know what the school has before you put lots of your salary dollars into manipulatives. You can be talking lots of money for new teachers. But here are some suggestions: calculators that are grade level appropriate; software and a computer for yourself (I don't know how you will survive without one—even if it is only to communicate with fellow teachers in different districts or states.); counting cubes for young students; but more than this, consider what types of books and support materials you need for yourself. Many simple, easy-to-obtain, everyday materials are good manipulatives. Dan Dolan and Mari Muri of Connecticut have taken pictures of windows and their panes to use to teach fractions for years. Think about what you need as a teacher. That may lead you to manipulatives you may need. Also in a school, ask other teachers to share school manipulatives. I can think of nothing worse than having a closet full of unused manipulatives.

**Question from:**

Charleston, Illinois

Have you ever had to teach any math concepts that you personally struggle with? If so, how did you overcome that?

**Johnny Lott:**

My answer is absolutely. I still do. I truly believe that new teachers and more experienced teachers can mentor each other. You may know more "modern" mathematics, like statistics, for example, which may not have been in more experienced teachers' required courses. They may know more ways to consider and think about some mathematical topics. Become each other's mentors. Ask. You do not have to do this alone. The Math Forum (online) is a good site to ask questions and to seek some help. I also firmly believe that you and students can learn together; this requires very careful work and should not be a crutch for not preparing, but you will never stop learning if you are to be a good teacher. You can learn from students as well as other teachers or professors.

I have to add one other thing if you are still in college. Right now form study groups and mentorships with each other that you can continue with regardless of where you teach. E-mail is a really good tool to help.

**Question from:**

Charleston, Illinois

Probability seems to be lost in math programs in the early elementary grades. Should probability be introduced in the early elementary grades? And if yes, what are some activities?

**Johnny Lott:**

Probability is tough for young children as well as adults. Think about why and look at the standards for the early grades in *Principles and Standards for School Mathematics.* Very young students don't know about fractions. They may have some knowledge of likely or unlikely at an early age, but this is a topic that has to develop. See the NCTM Navigations books on data and probability at the early grades for some specific thoughts.

**Question from:**

Eastern Illinois University

At what age should students start learning about division?

**Johnny Lott:**

This depends on what you mean by division. If algorithms, then postpone until the students can understand why you are doing what you do in division. If the beginning notions of division, consider even the book "The Doorbell Rang" by Houtchens, I think. It has beginning notions of factors and division for very young children. As students learn multiplication facts, they should also begin to understand how to "undo" multiplication, that is, division.

**Question from:**

Charleston, Illinois

What made you decide to become part of the NCTM? I am sure this isn't what you always wanted to be. What were your future plans when you were younger?

**Johnny Lott:**

Yes, I was a nerd (still am). In college long before I knew about NCTM but as a math major, I used to sit in the stacks of the library and study. It turns out one of my favorite places to study was in the stacks where the Mathematics Teacher was shelved. I started reading articles long before I knew about NCTM. My adviser when I was a senior told me that if I was to be a mathematics teacher, I had to join NCTM. Imagine my surprise when "my membership journal" was one of the things I liked to read. Told you I was a nerd.

In the early years, I thought about being an engineer, and that lasted one year and a part of one semester. Then I discovered that I really did not like some of the required classes for pre-engineering. That ended that. Then I was a math major adrift for some time. I took political science, selected history courses, and yes, 15th century English lit as I considered what I wanted to be when I grew up. I still have those interests, but teaching and working with students at all ages captured my interest and love from the time I started student teaching. Nothing can challenge you more than students, and I mean in both good and not so good ways. I would change little at this point in my life.

**Question from:**

University Park, Illinois

No Child Left Behind has left us at a crossroads. Many, like myself, are looking forward to the advancement of mathematics integrated into all areas of education. The problem with No Child Left Behind is a lot of teachers are forced to use older teaching methods; such as teaching to the test using skill and drill. As a future educator, what can we do to reverse this process of teaching to the test because of No Child Left Behind? The government only wants to see results in whole numbers, how about the individual progress of each student?

**Johnny Lott:**

While I am not the biggest proponent of all aspects and implementations of NCLB, we must reach all students with mathematics. I sincerely do not believe that this forces us to rigidly teach to a test. That may be a school's interpretation, but it does force us to think what is important. If we are teaching the important mathematics that we must, and students are learning that mathematics, won't they survive (and we in turn) the tests? I know that this is simplistic, but we as teachers have to keep hammering away at the important mathematics AND keep the obligation to make sure that all students are learning it. Our real responsibility is to the students.

**Question from:**

Charleston, Illinois

I really enjoyed reading your article in the NCTM News Bulletin. I student teach next fall and was wondering if you would give me some advice for a successful student teaching experience. Thanks!

**Johnny Lott:**

Don't do what I did! About the third day I was in student teaching, my supervisor was sick and the school provided no substitute. It was truly trial by fire. I struggled.

Listen to students. Hear what they are saying and try to think through their answers instead of having a single expectation. Listen to the supervising teacher; get to know the books being used before you have to go in the classroom. Listen to and observe teachers who you think are "good." Be a sponge before you start, but retain only what appears that it could work for you. Good luck. Primarily, enjoy the students and the work. As a start before you are there, think about "interactive" bulletin boards that you could use to have students think about math. Changing them frequently and using them as a teaching tool add a dimension that they sometimes do not have. It takes planning but is worth it.

**Question from:**

Charleston, Illinois

Have you ever had tremendous amounts of trouble teaching math to students? If so, what types of strategies did you discover that helped you to reach the students?

**Johnny Lott:**

There are a few students whom I would like to find and apologize to. They were primarily students who were in my classes when I was very young and I did not do a good job with them. I think specifically of Brenda and Russell. I would like to re-do that teaching and working with them. They were great kids, but I fear that I didn't help them. Then I had no strategies that seemed to help me reach them. I asked for help from mentors and they tried. My basic message is to listen. Russell could solve algebra problems but never in a way that I wanted him to. Now I would keep asking questions until I understood what he was doing. Then I didn't. Brenda basically claimed she hated math and refused to do any regardless of the consequences. Now there would have been regular conferences with Brenda and her family to try and figure out what was going on and why. It could have made both of our lives easier.

Again, LISTEN to the students. By listening, you may get your best clues for the strategies needed to help.

**Question from:**

Charleston, Illinois

What do you feel is the most important aspect of teaching mathematics to students?

**Johnny Lott:**

I'm not sure that there is a single most important aspect. However, that doesn't stop me from answering. The single most important aspect of teaching mathematics to students is to know, not just think, but to know that all students can learn mathematics and intellectually honest mathematics. I truly believe this and I hope that you do as well.

Moderator:

Thank you all for your participation and questions this evening. We received many more questions than we could address in this hour, but we'll include other questions and answers in the transcript that will be posted on the NCTM Web site within the next two days. Thank you again, and good night.

**Johnny Lott:**

I want to thank our moderator, NCTM Director of Communications Ken Krehbiel, for helping with this late chat. Also to all of you, thanks for contributing some really challenging questions. We will post them.

To those of you who are student teaching, be a sponge, enjoy and learn every day you teach. To those of you who are in preparation to be teachers, learn all that you can about everything that you can. It won't be enough, but it will set a tone for you.

Thanks again and good luck.

Johnny

The following questions were submitted but could not be included in the one-hour online chat:

**Question from:**

Winchester, Virginia

I would like to have hands-on lesson plans that allow a child to conceptually understand fractions.

**Johnny Lott:**

As would everyone, but we all know that there are no silver bullets for teaching and learning. There have been huge strides in the development of lesson plans on teaching of different concepts, but I'm probably very safe in saying that there is no single plan that will always work. A good resource is *Making Sense of Fractions, Ratios and Proportions,* NCTM 2002 Yearbook.

**Question from:**

Pico Rivera, California

I am a Calstate-LA student and I am interested in attending the conference. However, due to a tight budget, are there any grants or vendors from Southern California that will pay my way up there? It does not hurt to ask. Thank You.

**Johnny Lott:**

I'm not sure which conference you refer to. NCTM's annual conference will be in Anaheim in April 2005. We work with student volunteers to allow them to come to meetings. If you e-mail me, I will try to put you in contact with the person in charge of volunteers for that meeting. It at least is close to you. The volunteers for Philadelphia have already been selected. It is worth checking out the NCTM Web site, www.nctm.org, and the Affiliate section in your area to determine if the local Affiliates have any way to help support student participation in meetings. At some universities, there are small grants available to students to attend meetings. You should check out your school resources. These are usually limited and often are on a first come-first serve basis. It's worth asking.

**Question from:**

Charleston, Illinois

Dr. Lott,

While teaching third graders about money, my class used manipulatives to recognize the various forms of money—pennies, nickels, dimes, and quarters—and their worth. They also participated in centers counting and exchanging money for various items. The next day they were doing worksheets that had the pictures of the different forms, and they did poorly taking what they had learned the day before in centers and putting it on paper. What is a way to be able to retain what was learned kinestically and reproduce it visually on a worksheet, how do you help them make that connection?

**Johnny Lott:**

Think about using overheads with actual pictures of coins as you construct activities. Use the actual size coin and not reduced ones for a while. A question is why take away the coins used early on and revert to paper sheets. Let them use the money until they are comfortable. For example, if students use pennies for a while and have to keep counting them when they know that nickels and dimes are available, they will ask you for other coins. This is likely a place where the jump to paper was too fast.

**Question from:**

Charleston, Illinois

When did you become interested in teaching mathematics, and what were some methods you found worked well in the classroom?

**Johnny Lott:**

Long before using real-world examples was super fashionable, I resorted to using newspapers and catalogs to work with students who struggled with math. These were high school students who for different reasons had not been allowed to pursue traditional courses leading to college, but some of these were students who could (and did) succeed in college. They needed to know that they could do math, sometimes even basic arithmetic. Sorry, but this was before calculators were widely available. We studied math that affected them: shopping, jobs and their wages, taxes, cars and insurance, and the list went on. I also fell into using cooperative groups as a way to have students help each other.

**Question from:**

Charleston, Illinois

In an 8th-grade math class where I was doing my Practicum they were prepping for the ISAT test using a green workbook. The material seemed straightforward and dry. I found difficulties pepping it up while remaining "on schedule" with the timeline for the test. Do you have any suggestions for making it more exciting, and for reviewing old lessons while learning new ones?

**Johnny Lott:**

Sorry but I do not know the ISAT. My guess is that it is similar to many tests being used around the country. Basically, I don't believe in prepping for tests, but reality sometimes has forced the issue. If you have to do it, think about brief reviews at the beginning or end of class. Review is usually a good thing. See if you can find the connections from what is being reviewed to what students are about to study. You may have to look carefully, but most topics you will teach are connected to something that students have studied. Use this to your advantage if you can. A review can be an intro to new material.

**Question from:**

Charleston, Illinois

Can you suggest a way to introduce a math concept in an elementary classroom? Do you think an activity should be done before the concept is completely introduced, or do you feel that the concept should be mentioned and discussed before an activity is done?

**Johnny Lott:**

I'm not totally sure what you are asking. Can you preview a concept before explaining it? Absolutely. Should you? Probably. Think about how you learn and I will think about how I learn. Rarely do I "get" something the first time I see it. I have to think through things. I'm better off to have a teaser to think about a few days before with more clues leading me up to the concept. Your students might be like me. Thus, think about preview, clues, connections to past, clues, teasers, and then concepts as a model. I probably left out some needed steps but you get the idea.

**Question from:**

Charleston, Illinois

I am going to be student teaching in the fall. I will be in a first grade classroom. I was wondering what manipulatives you would suggest. I would like to start to build my collection and I would like to know where to start.

**Johnny Lott:**
Know that what grade you student teach may be very different from the one you teach in real life. My first suggestion is to go see what the school and classroom has available, talk to the teacher about the manipulatives you may be using, and then think carefully about the support materials that you may need instead of the manipulatives themselves. Good luck.

**Question from:**

Bensalem, Pennsylvania

Mr. Lott, I wanted to inquire in your opinion and your expertise if you think that statistical methods for large or small sample sizes are useful for evaluating and promoting particular educational objectives?

**Johnny Lott:**

Without context, how do I guess what is being asked? When you ask about promoting particular educational objectives, I wonder if you are questioning methods of research and how conclusions are reached based on the research. If that is the case, then consider how some science is studied. One typically tries experiments on very small samples, maybe randomly chosen subjects but probably not. If one can detect some patterns from these samples, then it may be time to consider some conjectures that need more testing. To test the conjectures may require different samples with some degrees of randomness. The degrees of randomness of course depend on subjects, availability, type of experiment and so on. Only through looks at previous evidence, and small studies would most of us launch into a "full-fledged study." Know that with my early samples, I cannot generalize beyond the sample. To have true generalization, you may have to go the randomization route to a "scientifically based study." Or you may find that such a study is impossible and have to base conjectures on smaller samples until the conjectures are refuted.

I just re-read this quickly and decided that I need you in my research methods class where we can really discuss what you are asking.

**Question from:**

Seattle, Washington

I am a first-year math teacher at a private K-8 school. Our principal wants us to improve our math program. I have been assigned the task of finding the best math textbook to use. Help! I am currently looking at CPM for 6th through 8th grade and Everyday Math for K - 5. Any suggestions?

**Johnny Lott:**

Go online to COMAP, EDC, and the Show-Me Center in Missouri to look for the research that is available on the reform curriculum projects. That will help you start. Look for all views of the curriculum to be as informed as you can. AND talk to teachers that are using the materials. See if the schools are comparable to yours. These are ways to start. Good luck. Who is helping you? Also discuss with the high schools you feed to see if there are recommendations to help advise you.

**Question from:**

University Park, Illinois

There are many new ways of teaching math. The problem I see as a future educator is the need for seasoned teachers to accept new practices. Many of the new methods are put aside because of the judgment of veteran teachers. When I get in the field how do I meet the common ground with a veteran teacher concerning math?

**Johnny Lott:**

You will see this response in other places but "new" teachers and "veteran" teachers need to mentor each other. The newbies can teach the veterans new tricks so to speak while the "old-timers" can be great mentors for current students, background, materials, the school situations, and so on. Talk and build these bridges; you both need them and it is worth the effort.

**Question from:**

Charleston, Illinois

As a pre-service teacher, I have seen more teachers reading than teaching mathematics. In many schools, the teachers are reading from prepared lesson plans in programs like Saxon, not teaching. How can we break this cycle so that teachers are teaching math? When teachers feel like they must conform to the traditional ways of a school so that they avoid conflict, is it possible to break this cycle? And with added pressure from new legislature like NCLB, can teachers afford to?

**Johnny Lott:**

I'm not sure what you mean by "more teachers reading than teaching mathematics." My real question is are the students learning independent of what the teacher is doing? If so, then the real goal is achieved. If not, then I need to know more about what you mean by "teachers are teaching math." Do you mean, "lecturing?" There are great teachers who lecture though few probably do it all the time, just as few do cooperative work all the time. The bottom line is that teachers need many strategies to help students learn. Don't reject one in the name of "traditionalism" or "reform." The kids come before ideology.

**Question from:**

Greensboro, North Carolina

I am student teaching high school math (honors advanced math) right now. I would appreciate some suggestions about how to get my very bright students involved in class when I am having to move slowly in order to be on the level of most of my students. My really bright students sit bored and playing calculator games or doing their math homework ahead of time in class unless I catch them and ask them questions or put the class in groups to work on some project.

**Johnny Lott:**

Think about how students may help each other. One way is a homework check pairing students. With that you may not need to spend as much time reviewing. Students who have done things correctly may be the demonstrators and not necessarily you.

Without specific topics, it is hard to know what all the issues are, but consider ways to group students. At times, you may want those who are "ahead" to be grouped together on questions of theory or harder "real" problems. Try to cut across the knowledge lines to keep students involved. That may be your best bet. I don't have easy answers. What do your colleagues do in the school with similar problems?

**Question from:**

Charleston, Illinois

Why is it so important to test the students with multiple standardized tests?

**Johnny Lott:**

This assumes that it is important to test students with multiple standardized tests. It is important to assess students with different methods but not all have to be (or even should be) standardized tests.

**Question from:**

Charleston, Illinois

What do you say to a child who has a question or an answer that is way over your head? The child is at a very high level for their age and is always curious. Should you be looking all this information up for them or having them do it? What is your opinion on this topic?

**Johnny Lott:**

It is probably impossible for a teacher to have ready answers to all questions. You see me trying to violate that tonight! Suggestions are to say that you will have to check if it is important for you to answer. If you want the child to find the answer, you may have to have resources for the student to use. You need the resources not automatically ready answers. One suggestion is to have students write really hard questions that can have answers or sources for further investigation provided the next day. If you do that, make sure that the suggestions are there for the next day. No procrastinating if you use this.

**Question from:**

Charleston, Illinois

With the focus so much on standardized testing, do you think that children are losing out on creative and memorable ways of learning math?

**Johnny Lott:**

Students can take standardized tests and still be creative and learn math in memorable ways. Tests do not have to overwhelm schools and classes. Think about what is really important for students to learn. That should come first. Note that may be defined in a school system by tests, in which case you may have to adapt and try to educate administrators. But for the students, think about the really important grade level mathematics. Help them identify the big ideas that they need. Those can be creative and memorable.

**Question from:**

Charleston, Illinois

Dr. Lott,

How do you feel about the Saxon Math program in the Elementary Schools?

What do you think about using the Math4Today program? I have been in two different school systems, one using the Saxon Math and the other using Math4Today and I'm not sure what to think about using those programs.

**Johnny Lott:**

NCTM does not endorse any program. Look for research on both programs and let that help answer your questions. Look for both the pros and cons to programs as you weigh the options.

**Question from:**

Charleston, Illinois

Mr. Lott,

I recently finished my practicum experience in a 2nd grade classroom. The school district uses the Saxon Math Program. I feel that this program may move too quickly for many of the students. The class would learn a new concept practically every day. It worries me that many of the students are just moving along and not fully comprehending the concept. What are your feelings on the Saxon program?

**Johnny Lott:**

NCTM does not endorse any program. Look for research on the program and let that help answer your questions. Look for both the pros and cons to a program as you weigh the options.

**Question from:**

Philo, Illinois

Do you think that the No Child Left Behind law is helping or hindering the teaching of mathematics in elementary schools?

**Johnny Lott:**

The basic tenets of providing a good mathematics education to every child and making sure that teachers are qualified to deliver the mathematics is a must. These are the basic tenets of the *Principles and Standards for School Mathematics.* However, rigid implementation may overwhelm some schools as different states have recognized. We are truly in a situation where though the tenets are the best, we are not a nation where one size fits all. The problems of the inner city school are not the problems of rural America. Adjustments may have to be made, but we should stick to providing a good mathematics education to every child by qualified teachers. That cannot change. Let's not destroy public education while trying to save it though.

**Question from:**

Yale, Illinois

How do you feel about students getting placed with certain teachers and in certain classes depending on the grades they made the previous year?

**Johnny Lott:**

This sounds suspiciously like tracking and I firmly believe that tracking students very early in their math careers is not a good thing. It might work for the early "bloomers" but could be a disaster for those who take more time. We cannot close future options by today's tracking early on.

**Question from:**

Charleston, Illinois

Do you think that standardized testing will be soon be considered out of date...and teachers will finally be able to teach the basics without feeling the pressures of teaching only what is on the tests so that students will achieve high scores?

**Johnny Lott:**

It is doubtful that standardized testing will go out of date. It serves a purpose for evaluating programs. What it may or may not do (probably does not do) is assess an individual student's work. One note bothers me a bit. The basics are continually changing or at least they should be. Teachers must be continually willing to change curriculum as the world changes. Curriculum is not static or should not be if math is to survive.

**Question from:**

Charleston, Illinois

Do you feel leveled mathematics is a healthy instructional strategy for the elementary classroom?

**Johnny Lott:**

If by leveled mathematics you mean tracking by another name, then I do not think that this is a healthy strategy for elementary schools. If not that, then I'm not sure what you mean.

**Question from:**

Charleston, Illinois

I just recently returned from practicum and I was curious about 3rd graders and their assessment testing. The students make a table into 4 quadrants and label each one.

1. What do I do?

2. What to I need to find out?

3. Solve

4. How and Why?

Do you think this is important to test kids over? Furthermore, do you think that standardized testing is necessary?

**Johnny Lott:**

I am truly unsure what you are asking. Is the test item how to make and label four quadrants or something else?

You asked also about standardized testing. See pages 200-201 in the 1989 *Curriculum and Evaluation Standards for School Mathematics* for a great guide to the uses of different kinds of tests.

**Question from:**

Charleston, Illinois

What is your stand on Saxon Math? How can you convince the administration that perhaps a change is needed?

**Johnny Lott:**

Look up the research pro and con on the use of Saxon Math. You will find no endorsement or non-endorsement of the product by NCTM. Use the research and build a deductive argument as we teach students to do based on whatever conclusions you reach.

**Question from:**

Charleston, Illinois

Why do teachers have to teach to standardized tests? Or any tests for that matter?

**Johnny Lott:**

See pages 200-201 in the 1989 *Curriculum and Evaluation Standards for School Mathematics* for a great guide to the uses of different kinds of tests. One should teach to a test only if you are positive that the test includes all that is important for students to learn.

**Question:**

Is giving a reward for getting right answers good or is it not?

**Johnny Lott:**

Some use rewards very effectively. Others find them offensive. How do you feel? Think about why and analyze whether or not this is appropriate for you. If you use them, you should be able to defend their use.

**Question:**

Do you believe that new teachers should have to pay for the Basic Skills Test? Some of us college students do not have that kind of money. What should we do to solve this problem? Should the school pay for it?

**Johnny Lott:**

If by the Basic Skills Test, you mean an exam for certification, then you probably have no choice. Tests as a method for demonstrating qualifications for a career are common. Think about the bar exam, the medical exams, the engineering exams, and the list goes on. It is one form of demonstrating your qualifications. Few schools will pay to show that you are qualified before you are hired.

Editor's Note: NCTM moderators retain editorial control over online discussions and choose the most relevant questions for guests and hosts. The moderator and host may decline to answer questions.

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