Equity and Quality in Mathematics Education

Talk at NMMATYC conference, May 1998
Dona Aña Community College, Las Cruces, NM


First, thanks to Benedict Nmah, Rita Gonzalez, and Kitty Berver for inviting me to speak at NMMATYC.

The title of my talk is Equity and quality in mathematics education. What does it mean, Equity and quality in mathematics education? It means providing every child with both an equal, and a quality, education in mathematics. Such an education can come from two sources: the school and the home. But some home environments are not as supportive in providing opportunities for learning as others. So schools have to take up the slack, to make up the difference when home environments are lacking, to provide equal opportunities for quality learning for all. This is not a new idea.

Dr. Carter G. Woodson (1875-1950), an African American, in his 1933 book The Mis-Education of the Negro, wrote:
"The African American children, as a rule, come from the homes of tenants... who have to migrate annually from plantation to plantation... The children from the homes of white planters and merchants live permanently in the midst of calculations, family budgets, and the like, which enable them sometimes to learn more by contact than the African American can acquire in school. Instead of teaching such African American children less arithmetic, they should be taught much more of it than the white children...."

Sixty five years ago, Dr. Woodson actually asserted that less privileged children should be provided with *more* opportunities for learning in school than their more privileged counterparts.

How can schools provide an excellent program of mathematics instruction, with many opportunities for learning quality mathematics, for every child?
There are three aspects.
• Teacher knowledge
• Content of curriculum
• Materials

(1) Teacher knowledge

It is a truism that kids can't learn what teachers don't know. So the highest priority should be given to teacher knowledge. This also is not a new idea.

George Polya (1888-1985), a Stanford University mathematician, wrote How to Solve It in 1945. While the book is more often quoted for its problem solving heuristics, he also wrote:
Rules of teaching:
• The first rule of teaching is to know what you are supposed to teach.
• The second rule of teaching is to know a little more than you are supposed to teach.

Our teachers need to know the content of what they are going to teach. and they need to know *a little more*. We need a strong program of teacher preparation and professional development in the United States. An opportunity for such a program rests in 2-year and 4-year colleges, and in universities, especially in math departments, that provide math content courses for preservice and inservice teachers.

Documents such as Everybody Counts (National Research Council, 1989), Crossroads for Mathematics: Standards for Introductory College Mathematics before Calculus (AMATYC 1995), and A Call for Change (MAA, 1991) urge strong, coordinated, and visible commitment to strengthening the mathematical preparation of teachers. Preservice teachers must have substantive mathematics course work as part of their preparation program. And inservice teachers must have opportunities for such course work on a continuing basis, to refresh and modernize their knowledge. I will focus here on teachers of grades K-8, since that is where my experience lies.

The minimum number of mathematics courses for preservice elementary teachers in grades K-8 varies in the US from one one-hour course to four three-hour courses (in Louisiana). At NMSU here in Las Cruces, the requirement is two three-hour courses, Math 111 and 112, Fundamentals of Elementary Mathematics I & II . (In New Mexico, there is not a separate credential for middle school. So you are certified to teach grades K-8, or you are certified to teach secondary school.) Needless to say, these six credit hours are not sufficient to prepare K-8 teachers adequately.

Some of us at NMSU, in the Math Dept. and in the College of Education, are working on trying to increase the number of required credit hours, in an already packed curriculum. But given this limited course requirement, how can we get more bang for our six-credit-hour buck?

There is a new experimental program occurring just across the street, in the Dept. of Mathematical Sciences at NMSU. Actually it is not so new. It began in Fall 1995 and has just finished its third year (sixth semester). In this experimental program, preservice and inservice teachers attend a joint course, Mathematics 111/112/501. Math 111/112 is Fundamentals of Elementary Mathematics I & II (required for elementary education majors). Math 501 is a special topics graduate course for teachers. The key word here is joint -- both preservice and inservice teachers attend the same class.

How can we do this, and how does it work?

When I came here three years ago, I asked the head of the NMSU Math Dept., Doug Kurtz, if I could run such a joint course. He said he thought it could be done. I asked Karin Matray, Director of professional development of the Las Cruces Public Schools, if she could locate about 15 K-8 teachers who would like to take such a course. Lucky for me, a few years earlier the LCPS had had an Exxon education foundation grant, and had run a number of workshops and special mathematical events for some K-8 teachers who were then designated as Exxon Math Specialists. Karin said, "No problem," and immediately got on the phone to these math specialists. Now, every semester, teachers are recruited through the Las Cruces Teachers' Center (with the help of Karin Matray).

Thus far (for six semesters) teachers have been given free tuition (provided by grants; right now, their tuition is paid by the NM Commission on Higher Education -- Dr. Bill Simpson is responsible for this). The math class meets in the late afternoon (4:30-6:10 MW) so teachers may attend after their regular school day.

• Teachers mentor preservice teachers.
• Preservice teachers make ten visits per semester (or more, if they want) to real classrooms of mentoring teachers. Teachers act in some sense as teaching assistants for the class.
• Preservice teachers observe, co-teach, and finally teach alone in their classrooms.
• Enrollment: about 35-40 undergraduates; 15-22 K-8 teachers per semester.
• In 6 semesters, over 100 teachers and about 200 undergraduates have enrolled.
• Some teachers continue to mentor students from the class after finishing the course.
• About 20 teachers have taken the course for two semesters, earning 6 hours of graduate credit.

A core of teachers is forming; teachers who are becoming permanent partners with the university, allowing undergraduates to visit (and teach in) their classes, even if they are no longer enrolled in the university. Some other sections of Math 111/112 have sent their preservice teachers to visit these teachers' classrooms. In the University class, students work in groups, with the requirement that at least one teacher and one preservice teacher is in each group.

There are lots of details I could add about how to run such a multi-faceted course. E.g.,
• What kinds of assignments are given to the two sets of students?
• How to grade preservice teachers? How to grade inservice teachers?
• How to manage 350-400 classroom visits per semester? (TB tests, transportation to schools, proper attire, proper classroom behavior, etc.)

Luckily I have a t.a. (or two half-time t.a.s) for the course. Last semester, one of the t.a.s was Sandra Nakamura, a first grade teacher from Mesilla Elementary, who was runner-up this year of LCPS teacher of the year. Mike Goar, from Mayfield High School, has also been t.a. Others are Dani Richardson and Gary Hartshorn. I thank them all!

Last semester (spring 1998) we offered a new partnership course, Algebra and geometry for K-8 teachers, sponsored by the Exxon Education Foundation. We came up with this course after discussions with the LCPS regarding which areas of mathematics they thought needed attention. (It was not probability and statistics; it was Algebra and geometry.) (This is a small example of working with those people who potentially are going to hire your graduates!) Some of the teachers who took the previous Math 111/112/501 courses were enrolled.

Our plans are to make the Algebra and geometry course, together with the others, a part of a new Master of Arts in Teaching with a specialty in mathematics for K-8 teachers. (Rick Scott of the NMSU College of Education is working with me on this.)

The main goals of the partnership courses:
• To increase the mathematics content knowledge of teachers and preservice teachers (we'll go into the content in a minute)
• To give preservice teachers, early in their career, a taste of what their future holds, by giving them real experience in classrooms, with teacher mentors. This experience comes several semesters earlier than their 'field experience,' which typically occurs in their final semester.

(2) Content.

What should the University math content courses contain? These courses are to prepare teachers. What math knowledge do teachers need in their profession? What should we be teaching them??

Has a student ever asked you, "Why do I have to learn this?" John Dewey (1859-1952) in The School and Society (1899) wrote: "While I was visiting in the city of Moline a few years ago, the superintendent told me that they found many children every year, who were surprised to learn that the Mississippi river in the text-book had anything to do with the stream of water flowing past their homes."

Alfred North Whitehead (1861-1947), philosopher, mathematician, and educator, born in England, in The Aims of Education (1929), argued against an education of 'inert ideas', i.e., "ideas that are merely received into the mind without being utilised, or tested, or thrown into fresh combinations." He wrote: "Your learning is useless to you till you have lost your text-books, burnt your lecture notes, and forgotten the minutiae which you learnt by heart for the examination..."

Another admonition from Whitehead: "Do not teach too many subjects. What you teach, teach thoroughly. Let the main ideas which are introduced into a child's education be few and important, and let them be thrown into every combination possible. The child should make them his own, and should understand their application here and now in the circumstances of his actual life. From the very beginning of his education, the child should experience the joy of discovery..."

Departments of Mathematics have been offering content courses to prepare K-8 teachers to teach mathematics for many many years. I think they haven't figured out yet what content they should be teaching.

Here is some evidence that we don't have the content right: the Third International Math and Science Study (TIMSS).
TIMSS is the largest international study of student achievement ever undertaken, involving more than 40 countries and 1/2 million students. The results here are reported from 1996-98.
Nov. 1996. 41 countries, 7th and 8th grades
June 1997. 26 countries, 3rd and 4th grades
Feb. 1998. 21 countries, 12th grade.
4th grade: US students scored above the international average.
8th grade: US students scored below the international average.
(The US was the only nation whose students scored above average in 4th grade and below average in 8th grade.)
12th grade: US students scored near the bottom.

William Schmidt, from Michigan State University, a spokesman for TIMSS, calls the US math curriculum a 'splintered vision,' 'a mile wide and an inch deep', containing scores of topics superficially treated, a piecemeal curriculum. He has also called US 5th-8th grade school mathematics an 'intellectual wasteland,' a rehash of K-4 mathematics. While other nations' curricula contain algebra and geometry in these grades, US curricula typically do not.

Mark Tucker and Judy Coddings, in their 1998 book, Educational standards for our schools: How to set them, Measure them, and Reach them, state: "The single most important obstacle to high student achievement in the United States is our low expectations for our students. ... Another factor affecting student performance is no less important: weak motivation to take tough courses and work hard in school."

So let's go back to the question, what math knowledge do teachers need in their profession that they do not know already?
The math should:
• be meaningful and worthwhile
• be solid mathematics
• be usable in practical situations (in schools)
• include technology
• not be a review of stuff they already know.
(If our university course is a review of math they already know, then this is the math that teachers will teach in schools. Then we are in a remedial spiral, hardly headed toward high achievement for our students.)
So, we don't want it piecemeal, we don't want it isolated, we don't want it remedial.
And there is a big gap between understanding a math concept and presenting to children a lesson that teaches the math concept.

(3) Materials.

You can make this task simpler if, IN THE UNIVERSITY CLASS, you provide, together with math concepts, actual tested materials that can be used in real classrooms.

Let’s take a look at the materials used in the partnership courses I have described. They are ‘classroom-ready’ units.
• Material is organized around hands-on tasks which require mathematics, and not around mathematical subjects.
• Use of calculators (4-operation in Math 111/112/501; 4-operation and scientific in the Algebra and geometry course) is an integral part of many units.
• Calculators allow significant changes in mathematics content and teaching methods in the elementary grades.
• Children can learn the arithmetic of real numbers from the very beginning.
• They do not have to follow the traditional sequence of whole numbers, fractions, negative numbers, and finally irrational numbers.
• The early introduction of real numbers allows one to base geometry on the concept of distance (metric spaces).
• Skill training can center on mental computation and use of computing technology, and not on paper and pencil algorithms for the 4 basic operations.
• Intellectually interesting and challenging problems can be introduced earlier.
• Materials can be organized around topics, not around skills; it can be modular and less dependent on a child’s pervious learning.. Modules do not have to be rigidly sequenced.
• Material is presented as ‘classroom-ready’ units.

• Materials come from Breaking Away from the Math Book: Creative Projects for Grades K-6 (Baggett & Ehrenfeucht, 1995), Breaking Away from the Math Book II: More Creative Projects for Grades K-8 (Baggett & Ehrenfeucht, 1998), and other units being developed and tested. We use these materials in the University courses, and then the preservice and inservice teachers use them in school classrooms with children.

Examples of materials.

Chambered nautilus (lesson plan is on this website)
We did cutting and we taped. We used math. We made triangles. Then we put them together. I learned that triangless are not boring. I learned about a new animal. It was a snail like thing. We measured. We also used adding. I also used geometry. I thought it was awesome. I loved it. I really liked the math alot .

Three congruent pyramids that form a cube (in Breaking Away II)
I liked making the Improper pyramid ... I liked the measuring and the improper pyramid house. It was neat. That is why I liked it.

(Other units that were shown:
Exploration of stars (in Breaking Away)
Sierpinski’s carpet (on this website as "Colorful triangles")
Fibonacci surprise
Number in between (in Breaking Away))


...When I get to college I would like to take the class that my teacher takes. My favorite words my math teacher speaks are "Last night at my math class..." When she says that, I know that we are going to do something fun!

You can see that an important aspect of the evaluation of the program is not about what teachers and preservice teachers learn. It's about what children learn, who are in classrooms of teachers taking the courses. We prepare tests and ask teachers to give them to children. And, what is more revealing and interesting, we ask teachers to ask children what they remember and what they learned from specific lessons and activities that they have gone through.

Children (and pre- and in-service teachers) are tremendously inventive in their problem solving, when they are presented with material that they see as challenging and worthwhile. They are capable of much more than previous research, educator beliefs, and the standard curriculum, have suggested.

Some responses of inservice and preservice teachers to Math 111/112/301/501
(taken from course evaluations):

The course materials are wonderful and the children enjoy the hands-on

experiences so much.

I would often look at a task presented to us and say, no way can I do that. But it was broken down to a series of doable steps. I learned that in this class, the impossible just takes a little longer.

I called it 'the evil n-pire' whenever we would try to generalize a finding to 'n'. But I learned n is not so evil. It is pretty neat when you can do that, when you see the pattern.

I loved being involved in the classrooms!

It is exactly the way mathematics for elementary and any grade level should be taught.

Structure of class was great; better than a pure lecture class. I really benefited from the exposure of co-teaching.

Our pairing up with a classroom teacher was extremely beneficial because after learning a lesson in class, we were able to go into the classroom and teach that lesson to some real students. Several Math 111 students had never been in a classroom before, so this was a great opportunity to get first hand experience as to what our future holds.

Going into the classroom really made the difference in this class. I feel a lot more confident about teaching (math especially). I've really enjoyed the hands-on experience; I've learned a lot more than I would have following a book.

Essays from children in first through fourth grade.

What are calculators used for?

first grade

I think calculators are for doing math. Thats what I think. I have more to tell about calculators. I think calculators are for adding and for times. I like calculators because they are nice to learn with.

second grade

You use a calculator becouse it make you lern math. You lern owe to do tecawae and plus.

You lern owe to do alls of thangs.

Calculators are used to do math. also they are used to figer out things.

I use a calculator for mathematics. I can add, subtract, meyltiply and dividing. I like to press the equals most of all to see what I come up with. I like calculators because they give you ansers to math Calculator are for questions.

To use in the right way. To figure out hard palns. You can cepe it in yor pocket.

Calculators can tell anssars to math plobalms. Calculators can be fun to yoous. Calculators can teawh you math plobims.

Calculators are useful. You can add and multiply. The nubers go up to nine from zero. I like them who ever invented it I like it when we do touf ones.

It is easier for math my mom uses one when she gos shopping it helps my mom a lot. have you ever used one before they are fun

third grade

Calculators can add, subtract, multiply, divid, and make almost any number and they can memoriz things, turn on and of and last but not lease that I know of count to verry high numbers. And they have other very complicated buttons.

It helpes you when you have lagre nuberes and you can't do tham in your head or on your fingeres and thats one of the reasn why you use a calculators.

When someone is dead you can find out how old they are. You can check your subtraion math.

They have brians in side the culeulators.

You can put some numbers together and have them to higher number like put 2 and 3 together then you have 23 or make it alot bigger nomber lits put 3 1 then yo will have one hundred and eleven they also have a button that says on/c thats for on clear you push it and what ever you have on the calculater that you want to erase will disapear. the End

Caulcators help you do math add your Bills up there are momery keys and to get the anser you push eqwals and it will give you the corect anser all the time an less you do something wrong

For realy hard problems. They are fun to play with. It helps you learn math.

They can help you lern. And they are fun to use. And if your stuck on a problem you can just punch the wright keys. They are used for finding out things. With a calculater it's giving you a break so you don't have to write it down.

fourth grade

I think people use calculators because they're fun. People experement things on them and they also learn new things. There are two kinds of calculators theres the regular calculator and the calculator with hair. I usually prefer the one with hair on it because its always good to exercise your brain. I always thought people use the calculor by thinking what they are going to push then they push and it shows on the little screne and it gives the answer.

Calculators are for helping you get throw stuff like Math, Science, Calculators will help you get throw your LIFE.

Problems with the approach.

• Teacher tuition is paid by grants.
• The courses require more preparation than a typical lecture course, and more work from the instructor and t.a.s (e.g., grading journals and artifacts, setting up supplies for use)
• We use many supplies, both permanent and expendable (e.g., calculators, measuring tools, posterboard, Scotch tape, Xeroxing), which are expensive. Similarly, such supplies need to be available in classrooms of participating teachers.
• Thus far, only one section of Math 111 or 112 per semester (out of 7 or 8 sections) is taught in this way. Other sections are taught more traditionally.

The promise of the approach.

• The program allows practicing K-8 teachers to increase and update their mathematical knowledge through a sustained university course.
• Preservice K-8 teachers typically get no experiences in real schools until their last semester. These courses give them this opportunity much earlier in their careers, under the mentorship of teachers.
• Both preservice and inservice teachers, having gone through materials together in a university setting, then try the materials together with children in classrooms. This seems to build a true partnership, different from a preservice teacher's field experience.
• The program has a presence in the LCPS and in the community of Las Cruces.
• Children enjoy the lessons and consider them worthwhile and challenging. They actually ask for math lessons.

So I have given one approach to providing children a mathematics education of both equity and quality.

Thank you.