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What Should Secondary School Mathematics Teachers Know?

Moderator
Welcome to today's online chat with NCTM President Johnny Lott. We'll take as many questions as we can in the next hour. Here's our first question:

Question from:
Fredericksburg, Texas

Just so you know, I don't have a math "degree." I've taken through Calculus 3 and one semester of Probability and Statistics. My ninth-grade kids last year (my first year teaching math) scored an average 98% passing on TAKS with 29% receiving state commendation for superior scores. Students of the other teachers with math "degrees" teaching the same course scored quite a bit below mine. I'm thinking you might want to take that into consideration next time you suggest that "current high school mathematics curricula can be taught successfully only by teachers who have a deep knowledge of mathematics." I don't mean it rudely, but I think your statement should be modified. Or am I the exception to the rule? Do you care to comment?

Johnny Lott:

Deep understanding of mathematics comes to teachers in many ways. A major question for certification agencies and administrators is how to measure this or to predict whether or not teacher either have it or may attain it. One measure may be a degree in a content area. This is not the sole way to attain deep understanding. There have been many famous mathematicians without degrees. However, there have been more with them. You've mentioned test scores as a criterion for students. Could you assess deep understanding of mathematics for teachers with tests?

Question from:
Beverly, Massachusetts

I teach at a school for students with language-based learning disabilities. We hire two or three new math teachers every year. How can I help new teachers develop their "bag of tricks?" (different ways of presenting the material, concrete representations of the concepts, problems students typically have...) Are there any resources that you can recommend?

Johnny Lott:

One resource that was designed for middle school students and their families is the Figure This! set of materials available online at www.figurethis.org. The first set of materials was translated into Spanish. Those materials have different solutions presented to the same set of problems and could be a start. Though they were designed for middle school, they have been used in different places. The first booklet of 15 of the entire set of 80 challenges has been translated.

Another set of resources from NCTM is the Changing the Faces of Mathematics series. With references there, you have a wealth of information.

Question from:
Indio, California

A significant problem that seems to exist in this area is the fact that we do not employ enough teachers who actually have a degree with mathematics as a major. In many of the mathematics departments, the majority are non-math majors. This majority has an overwhelming influence in very critical issues regarding policies set in the math department. As a result, the suggestions put forth by the minority of math faculty with well-established degrees are not welcomed and almost appear intimidating to those who do not understand the extent of a full mathematics curriculum. The opinions and actions of this majority of non-math teachers are supported by the administration. Basically, during the year multiple-choice tests are strongly encouraged, and practice test booklets are given out a few months prior to the state tests. Teachers are told to teach to the test.
How can the math-major faculty encourage both the administration and non-math major faculty to teach math in a more productive way and allow students to become better problem solvers? Currently the students are taught how to eliminate some of the multiple-choice answers and then given repetitive questions so that students identify answers by the type of questions.

Johnny Lott:

You raise many issues. Let me try to respond to some of them. With a mix of math majors and non-math majors in a department, it behooves members to find ways and times to discuss issues. If discussion is limited to policy issues, then it may be very difficult to find a common ground. If teachers talk and share lessons, visit each other's classes—even in a limited way—and communicate on small issues, this may help on the bigger ones that you identify.

Another issue is teaching how to take a test. This is not a bad skill for students to learn for a variety of reasons, including it being a means to talk about reasoning beyond the classroom. However, if this overwhelms the really big mathematical ideas that we are to teach, then there are many other potential issues.

Question from:
Exeter, Pennsylvania

What high school curriculum do you specifically recommend for what used to be "basic math students?" New state standards have left us no "basic math" option. Unfortunately, some students cannot function proficiently beyond a "basic math" curriculum. Please help!

Johnny Lott:

There may be a very small subset of students with specific challenges who need special curricula. However, it sounds like your "new state standards" may align with the No Child Left Behind (NCLB) Act in promoting mathematics for all students. The phrase "all students" is not used lightly either by NCLB or NCTM. In general, ALL students can learn mathematics and ALL can typically benefit from the same curriculum. For a variety of reasons, not the least of which is inequity, some students were not allowed to study important mathematics in the past. One important aspect of NCLB has been to make us as teachers think very carefully about what is being done with students. Having this challenge may also pose other issues for us, including what additional tools we may need to add to our toolkit to make sure that all students can succeed.

Message from:
Ashdown, Arizona

My students are mostly the lower math students and don't necessarily want or need upper high school math. What can I do to motivate them to higher math achievement?

Johnny Lott:

A part of this question deals with that is "upper high school math." One basic question that I would ask and I think that Principles and Standards for School Mathematics addresses is what is "upper high school math?" Based on the work of the Conference Board of Mathematical Sciences, we know that far less than 20% of students who go to postsecondary mathematics ever take calculus. We must think about what "upper high school math" is for the majority of students.

One typical reason that many high school students think that they may never need or should study more mathematics in high school is that they see few applications. Teaching math in context has many benefits for these students. Look at some Math for Liberal Arts Students college texts for ideas that you might want to consider as topics for high school students who may never go to college. My belief is that all need math.

Question from:
Mahopac, New York

How can we "encourage" Schools of Education to increase the math requirements for their students, especially in elementary education? Also, in my experience with student teachers, many, if not most, of their methods teachers do not have a math background, and yet they are responsible for these students.

Johnny Lott:

In general, Schools of Education are not solely responsible for requirements for prospective teachers. This is usually a shared responsibility for Colleges of Arts and Sciences and the Schools of Education. Prospective teachers need both mathematics and pedagogy. Neither stands alone. Also, just as we expect precollegiate teachers to have a solid grasp on mathematics and a deep understanding of it, it is at least as important for college and university personnel who are teaching the content and methods classes to have a solid grasp on both content and methods.

Question from:
Philadelphia

All secondary school mathematics teachers should know the state math standards and benchmarks for the grades they are teaching, and teach to proficiency in their classroom daily.

Johnny Lott:

I doubt that anyone would disagree with your statement. However, secondary teachers also need to know not only what they teach daily, but what students have studied before and what comes after. The daily lessons cannot stand alone.

Question from:
Corpus Christi, Texas

In our alternative discipline high school we are being force fed "schools to watch" middle school reform. Despite some similarities, we're trying to teach math skills for high school kids and I'm trying to do that and teach in special-ed learning disabled math (exceptional learners). Any ideas on how to get the kids what they need and keep inexperienced, unrealistic, and immature administrators off our necks and out of our classrooms? If we have to defend ourselves, documenting on "instructional drive bys" (instructional walks) responding to "scoldings," abusive and unrealistic administrators' knee jerks, we're out of energy to be very creative with the kids and their math needs.

Johnny Lott:

I hate to admit that I'm not positive what some of the terms that you have used mean. I can guess, but that puts me in the mode of being a student and trying to answer what I think my teacher has asked.

However, you have pointed out something that we as math teachers have a major responsibility for that is in no job description—that is, educating our administrators about mathematics education. With those administrators with little math background, we may have to begin and continue a professional development effort that is subtle and persistent. Copying exciting lessons or lesson results to administrators, inviting them to participate in the class when they are doing a "walk-by," and being proactive about what is happening in the classroom are all small ways to start.

A very effective tool for helping get kids what they need is to have the kids help outline and document what they need for administrators and school boards. Having a student present class needs has proven very effective in many parts of the country.

A new NCTM resource that might be helpful is "Administrator's Guide: How to Support and Improve Mathematics Education in Your School." You might consider giving it as a gift.

Question from:
Reston, Virginia

How much should teachers know about proof for teaching geometry and for teaching other courses? In what college courses should they be getting the proper perspective on what secondary school students should know?

Johnny Lott:

If we are to reach the Principles and Standards for School Mathematics standard for reasoning, teachers have to be knowledgeable about reasoning and proof. Being mathematically mature about reasoning and proof is probably not something that is learned in a single class, but is something that typically develops over time. Many teacher preparation programs have gone to a class called "Introduction to Higher Mathematics" or "Introduction to Proof" or it could be in a "baby analysis class" where different styles and types of proof are studied.

In the classrooms, reasoning and proof should be a continual objective—stated or unstated, and it should start as soon as a student starts to school.

The second question about a "proper" perspective on what secondary school students should know is loaded. With our variety of school management structures and independent school districts designing standards, it is a virtually impossible task to prepare a prospective teacher for all that may be faced in the schools. However, we cannot ignore the approaches to mathematics that are being used in schools in the teacher preparation programs. Many other countries bring school materials into the content and methods classes for prospective teachers at all levels. The Mathematics Education of Teachers document suggests some of this as well. This is a lesson that we should take to heart.

However, because of all the different systems and requirements across the country, no teacher preparation program can prepare prospective teachers for everything they will be faced with in the classroom. It is the responsibility of the school systems and the fellow math teachers in schools to develop a mentoring program and a professional development program to help the inexperienced teachers with the school culture. It cannot be done effectively from the outside.

Question from:
Coatesville, Pennsylvania

In addition to all the appropriate mathematics that should be known, secondary teachers, especially new teachers, should have better classroom management skills. Without good classroom control very little effective teaching can happen. It should not be simply on-the-job training.

Johnny Lott:

And classroom management skills are not easily taught to prospective teachers. It can be done in theory, but of all the issues for prospective teachers, this one is almost impossible to simulate on a campus. Even in intern programs for prospective teachers, there is still a
"real" teacher who is in charge. Good mentoring of new teachers can be very helpful here.

Question from:
Libertyville, Illinois

What can teachers do to increase their ability to make connections between the different math subject areas? Most of our own study was disconnected, as are many of the texts that are utilized in high schools.

Johnny Lott:

You asked about teachers increasing their ability to make connections between different math subject areas. This is a place that a deep understanding of mathematics is important. It also emphasizes a reason why teachers of different math subjects must talk to each other. Perhaps the best long-term solution in a school is for teachers to plan and work together to find the connections. Sometimes the best connections may start outside mathematics and then the mathematics connections inside can be made.

There are many good texts that make connections outside mathematics. One book with examples at different levels and from many fields from a very long time ago was "A Sourcebook of Applications of School Mathematics," written by a Joint Committee of the Mathematical Association of America and NCTM. It still has many good problems and ideas that can provide the start that might be needed.

Another more recent resource from the Council is "Integrated Mathematics Choices and Challenges." It has many good ideas.

Question from:
Reston, Virginia

How much discrete mathematics should a teacher know?

Johnny Lott:

Teachers must know some discrete mathematics topics even if they have never had a discrete math course. Most curricula have data analysis sections. Knowing when tools are appropriate for discrete data versus when tools are appropriate for continuous data is a necessity.

We really do not need to re-do minor graph theory topics every year with students. How much a specific teacher needs depends somewhat on what and how much is in a school's curriculum. Note that in Principles and Standards for School Mathematics, discrete math is not a separate standard but is woven throughout the curriculum. There will be a Navigations book coming out on discrete mathematics.

Question from:
Denver

Is it possible to set a simple criterion for middle school math teachers on how much math they should know? For example, would it be a good criterion to say that they should understand secondary school mathematics through a proofs-based geometry course? Should the goal be set higher or lower? Or is using secondary math courses not even a good way to measure the best background for middle school math teachers?

Johnny Lott:

Perhaps a better answer might be to say that middle school mathematics teachers have to know what comes before and what comes after the mathematics that they teach. I'm not sure that picking out a single high school course is the answer. If we teach the topics recommended in Principles and Standards for School Mathematics, there will be some data analysis topics that students will see throughout high school. Middle school teachers have a large load in terms of what they need to know. Note that high school teachers do as well because they need to know how mathematics is used or applies in postsecondary education and the workforce.

A single criterion might be far too simplistic.

Question from:
Moorhead, Minnesota

In addition to the things listed in your article, I believe teachers need to understand the history of mathematics. In a recent study of college freshmen taking math courses, more than half of the students could not even name one famous mathematician. Putting a human face on math would help to change the view that math is just a bunch of abstract symbols. Including a brief historical setting when introducing a new concept helps students see why the concept was developed, how it was and is used, and why it is important to know.

Johnny Lott:

As someone who loves history of mathematics, it is very difficult to disagree with you. However, if we want to use history of mathematics as a motivator for students, we have to recognize that math history did not stop at about 1900. Using math history effectively requires that we know current mathematicians and their work as well. This gets more difficult because the work is more specialized and much deeper. However it is with current history that you find major examples of diverse peoples who are still alive working in math.

With more "modern" history, you might also consider inviting to your school people who are actively working in mathematics. That too can be helpful.

Question from:
Alpharetta, Georgia

I think that the current suggestion that mathematic educators be prepared as a math major with a 6-hour capstone class to bridge high school to college level mathematics is a good idea with respect to knowing theory. Since NCTM supports the idea of connections and context, what support or recommendations exist to encourage teachers and future teachers to engage in applied mathematics courses, such as engineering, science, and business? Do you see this as efficient means for teachers to bring "real-world" meaning to the classroom, perhaps demonstrating the importance of mathematics to the teenage population?

Johnny Lott:

The capstone course also provides an opportunity for more than theory. I personally am always for teaching mathematics from the real world as contexts. Note that I also believe that mathematics itself is a context that we should use in the classroom. It should not be the sole context as often happens.

The "in-development" Committee on Undergraduate Preparation in Mathematics document by the Mathematical Association of America suggests different ways of thinking about connections and contexts for college freshmen that go far beyond engineering, science, and business. Those areas are not ignored, but neither do they provide the only important contexts for mathematics. Some examples might be found in this forthcoming document.

Question from:
Bloomfield, New Jersey

Are we EVER going to do anything about teaching to the test? The HSPA in New Jersey is the test that gets students out of high school, and that is the obsession with all administrators. Getting kids into college after that is secondary in everyone's minds-at least in urban schools. The problem is, the more we teach to the test, the fewer students PASS the test. Why? Because there is no consistency in teaching the subject. There must be a hundred HSPA prep manuals out there and administrators are spinning wheels and spending money on this material instead of having teachers JUST TEACH MATHEMATICS in an orderly, understandable fashion. Can someone please address this problem?

Johnny Lott:

In our test-driven society, you have addressed a major problem that touches many areas. NCTM is on record in an official position statement that opposes high-stakes testing. The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) has an excellent chart on tests, when they might be used, and for what purpose. It is as valid today as it was when it was written.

If tests are being used by school systems as the sole criterion that limit a student's future accessibility to mathematics and careers that use mathematics, then the school system is in trouble.

Note that NCTM is not opposed to testing—reasonable testing. What is reasonable may have to be determined by different school systems. The testing mania of today is in large part a response of the public to a wariness or mistrust of school mathematics and teaching. We are going to have to rebuild the trust, and that will take time.

Yes, we are all interested in teaching mathematics more than testing. Our new emphasis has to be on students learning mathematics more than on "teaching." Note that the emphasis is on students where it always should have been.

Efforts to help teachers with assessment that are underway at NCTM include three assessment literacy guides that should be available during 2004, a continuing assessment column in the NCTM News Bulletin and a set of planned assessment samplers being written now at the different grade bands with items that could be used in the classroom. We hope that these will be helpful.

Question from:
Milwaukee

I use the Connected Mathematics curriculum and love it. The kids get involved and learn to understand the math, as well as to do it. The reports I have from students who come back after 2-3 years are great—math is easier for them because they understand it and don't just do it by rote. But some teachers don't like this curriculum and want to change to something that is easier for them to teach and to understand. This takes constant inservicing of current and new teachers, understanding of algebraic thinking, a willingness to understand student thinking, and the flexibility to see more than one way of working a problem and seeing a situation. It is also very active teaching for the teacher—not always a quiet classroom. It is difficult to change the mindset of many current math teachers. Actually, those who are 1-8 trained are often more open to it because they are more used to hands-on types of teaching.

Johnny Lott:

As teachers, we have to know that if a curriculum is static, it is dying. To expect any less is unrealistic. Whether a teacher is in a reform curriculum or in a traditional curriculum, if teaching methods and content are not continually evolving, the discipline is in trouble.

Your comments emphasize the continuing need for professional development for all teachers. We cannot let up on efforts to upgrade what we know and teaching strategies.

A plan that you might consider is to have your very new inexperienced teachers share content knowledge with experienced teachers while experienced teachers can share knowledge about teaching strategies. Both of these give opportunities for discussion and dialogues among peers. The students are the winners.

Question from:
Morrisville, North Carolina

About 2 or 3 years ago, I attended an algebra conference where I found manipulatives for teaching algebra. I have not been able to find this since I began teaching Algebra 1. Could you help me locate some manipulatives for making algebra come alive to learning disabled students?

Johnny Lott:

NCTM does not endorse any specific commercial products, but in the journals, both Mathematics Teacher and Mathematics Teaching in the Middle School, products are discussed in articles and in reviews. Those are good places to begin a search.

Question from:
Surabaya, Indonesia

I am a physics teacher from Indonesia but I like mathematics. I want to know how I could transfer mathematics principles or mathematics lessons more easily to my students.

Johnny Lott:

My general answer to questions like this is to show the students how the mathematics is used in interesting problems and contexts beyond mathematics. Physics provides many opportunities, but don't rely solely on science. Look for resources in the community where mathematics is used. In the United States, fingerprint analysis, jury selection and census data are a few examples where the mathematics involved is really interesting and is accessible.

It's great to hear from colleagues everywhere, but thanks especially to you for participating when you are so far away physically. This use of Internet resources has really allowed us to cooperate worldwide. Thanks again.

Question from:
Portage, Michigan

"Best Practices" and use of technology to enhance learning should be the goal of education, not which skills to teach. Multiple representation should continue to be a goal.

Johnny Lott:

I don't disagree that you have listed valid goals. However, skills cannot and should not be ignored.

Question from:
El Paso, Texas

I think they should know more teaching strategies to engage students and assess students' progress.

Johnny Lott:

Because students learn differently, we as teachers cannot rely on single-shot teaching strategies. The best teachers are those with multiple strategies for teaching at their command.

Assessing student progress should also rely on more than single-shot tests. By using a variety of assessment techniques in the classroom, we can use best practice with students in assessment on a daily basis. Here we truly have the opportunity to use tools with which we teach and techniques with which we teach in the assessment process.

Question from:
Peoria, Illinois

Teaching is the only profession I can think of where a young person can move from graduation to total responsibility with no significant job-specific training. Should schools be more accountable to developing teachers as professionals? And if not schools, who?

Johnny Lott:

Though most teacher preparation programs do have some job-specific training in internship programs, field experiences, and student teaching, none of these are typically specific to an individual school. The specific job skills that are needed in individual schools MUST be the responsibility of the experienced teachers, administrators, and the district officials in the schools. For example, it is important for prospective teachers to experience working with diverse students, yet few pre-teaching experiences can provide the specifics of an individual school. This is one of the cases where the mean or median of teacher preparation programs may or may not prepare students for the real school experience. The programs cannot do this across the board. This is a frequently neglected responsibility of the schools and teachers in the schools. As teachers in the specific schools, we must do everything that we can to mentor and help inexperienced teachers. With the growing shortages, we cannot afford to do less.

Question from:
Knoxville, Tennessee

Many preservice secondary teachers have been successful in "school mathematics." That is, they know what they need to do to pass the test. Understanding is not a desired goal, just a passing grade.

Is there a culture that we need to confront to really change the development of secondary school teachers?

Johnny Lott:

I'm not totally sure that I agree with your first paragraph. I believe that in most programs the desired goal is understanding. At least it is the goal of most professors I know. It might or might not be for individual students. In a grade-conscious world, no one should downplay the passing grade, but students have to understand that it is only one component of what they need for the future.

The components may, like other things, not be appreciated at the moment, but we must still strive to make the prospective teachers aware that true understanding as well as grades are important.

This is one of those places that university professors can also use good practice in testing. If we are only asking simplistic questions that require little understanding of content to assign grades, maybe we need to rethink what we do.

Question from:
Sterling, Virginia

What is your view on the trend of more students taking advanced math courses at the high school level? It seems that we are trying to push students to learn certain mathematics topics at an earlier and earlier age, and the result is a superficial understanding.

Johnny Lott:
Interesting question. The bigger issue to having more students taking advanced mathematics courses at any level is what the plan is for the students beyond the particular level. For example, for those states, schools, and districts that are requiring algebra in seventh or eighth grade, if there is no plan for solid mathematics courses or studies for students who reach the upper grades in high school, then the idea is really not good at all. The same can be said of many of the high school courses that are "pushed down" to lower grades. There must be a plan for ALL students that allows them access to good mathematics at the upper grades. What we do not need is students completing all requirements at the tenth or eleventh grade and taking no math in their junior or senior years.

There is a wealth of good mathematics topics that could require creation of yet more math courses if curriculum is pushed to lower grades.

I've not touched on whether or not it is reasonable for students to study more advanced topics at lower grades. Just as research is needed to back up practice, any school doing this needs to look at learning theory or cognition research to be informed when making serious decisions like these.

Moderator

Thank you all for participation today. An announcement of the time of the next online chat will be posted on the NCTM.org homepage.

We have received several messages from teachers who wanted to participate in these monthly online chats but were unable to because they were scheduled when many teachers are in class. Can you suggest a time when you would participate in the live online chats? Please send your suggestions to chat@nctm.org.

Thank you again.
Johnny Lott:

Thank you very much for participating in the chat this month. I appreciate those of you taking your time to do this. This gives me a chance to know what you're thinking and it also gives an additional opportunity to talk with you about what NCTM is doing. Thanks again. Talk to you in a chat next month, November 17, on political advocacy.

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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