The impact of a state mathematics test on the structure and culture of a K-4 school.

Abstract

This study explored the impact of state mathematics testing on the structure and the culture of a K-4 school. We selected a school that was teaching students to the state mathematics test. Students at the school were doing well on the test. The school was ranked as an "excellent" in the state based on the result of the test. We examined the practices of the principal and two fourth grade teachers which were influenced by the state mandated tests. We also examined the perceptions of two fourth grade students in the school. The study raises some serious questions about the implications of such "high stakes", "end-of-the-line" testing on the structure and culture of this school and the reform movement in mathematics education.

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School change impacts school structure and culture and alters teacher/student interactions, attitudes, and beliefs towards mathematics. We investigated the practices of a principal and two fourth grade teachers which were influenced by the state mandated tests. We also examined the perceptions of two fourth grade students at the same school.

The state developed five categories for ranking schools. These five categories include: (1) excellent, (2) effective, (3) needs improvement, (4) academic watch, and (5) academic emergency. The state uses six points for determining school effectiveness. The six points consists of one point for student attendance on the average each day over the school year. For example, if a school has 95% or better students' attendance on the average each day over the school year, the school receives one point. Five additional points are given for the five sections of the test such as reading, writing, citizenship, mathematics, and science (one point for each). For example, if a school passage rate on each of the above five sections is 75% or better, the school receives one point for each of the sections.

If a school receives six points out of six points, then the school is categorized by the state as an "excellent" school. If a school gets four points out of six, then that puts the school in the "effective" category. Receiving three points out of six would mean that the school "needs improvement". Getting two points out of six means the school is in "academic watch." Lastly, obtaining one point or zero means the school is in "academic emergency."

The school that we studied was ranked by the state as an "excellent" school. Ironically, the teachers' philosophies and pedagogies were not compatible with current research on teaching and learning mathematics. The school administrator and teachers were not following the constructivist theory. They were not spending extended time for teaching mathematics. They were not using the Kumon mathematics program. The activities of the school educators were heavily focused on teaching to the mathematics test and preparing their students for the test. The students did well on the test. However, when we examined two of these students' conceptual understanding of mathematics, we found that they had limited knowledge for problem-solving and mathematical communication (cf. Martin et al., in this issue, section "dialogue with students").

To understand the above relationships between the state mandated mathematics test and the school structure and culture and the implications of these relationships on the school change, we conducted interviews with a school principal (four interviews), two fourth grade teachers (four interviews each), and two fourth grade African American students (10 interviews each) in the two teachers' classrooms for one school year. In addition, we incorporated extended participant observations (10 times in each) in the classrooms of the two student participants (one African American boy and one African American girl). We made field notes and examined some school documents such as the school's newsletters, bulletin boards, and student's written work.

Interview tapes were transcribed and all transcriptions, field notes and related documents were analyzed and interpreted. Furthermore, some participants were provided with opportunities to read and help revise transcriptions of the interviews. In what follows, we describe the setting and the participants.

Setting and Participants

The study was conducted in a suburb of a Mid-Western city. The K-4 school had 297 students enrolled. Fifty-nine percent of the students were white, 34% were African American, and 7% were others (I.e., Asian, Middle Eastern, Hispanic, etc.). The key participants in the study were Mr. Connor (principal), two fourth grade teachers (Mrs. Dickerson and Mrs. Stromberg), and two African American students (Beth and Carl). We changed the names of all participants to maintain anonymity.

The Principal.

Mr. Connor's first degree was in economics with a minor in geology but eventually, he decided to teach. "[I] sort of found where I felt I fit." He was a 6th grade teacher for 9 years. He worked in another state Department of Education as the science coordinator of curriculum and then came to this school district in 1982. About eight years ago, he became a principal at his present school "to calm hurt feelings and smooth ruffled feathers when race, class, and faculty issues got out of hand." He discussed his experiences and explained his thoughts about education and mathematics education:

     I was surprised at the ... tremendously high expectations for
     schools-[in this district] ... This school system and the community
     are its own worst critics. They are constantly evaluating
     themselves ... Do I think we have more great mathematicians? No, I
     don't think we have more great mathematicians. I don't think that's
     going to be a product of the schools. Do we have children with a
     better sense of numbers? Yes, I think so. We still have difficulty
     with children learning multiplication in the third grade and with
     long division. I've worked in integrated schools my entire
     educational career, and I have found in my own personal experience
     that the critical ingredient for all children's success really
     comes from the home. Nothing can change that ... In math, I just
     want to see [students] working at their potential. I can't ask for
     more than that.

From our long conversations with Mr. Connor, it became clear that he valued students' mastery of basic skills in mathematics (I.e. multiplication and long division). In addition, he thought that children's success in education was closely related to their home environment. The school structure and culture for providing educational opportunity was secondary. According to Mr. Connor, he was not out to 'change the world' or establish a new world order in mathematics education. He had some ideas about how "it could be good for students to work with more manipulative." It seemed, though, that he did not expect too much to happen quickly to change mathematics education.

Although he seemed interested and willing to consider the new awareness recommended by the National Council of Teachers of Mathematics (NCTM) Standards and other research documents in mathematics education, he did not seem to feel that the core of the mathematics program at his school would change. Tradition was still the rule. Nevertheless, he was very much aware of the importance of the state testing and made sure that his curriculum materials contained topics that students would be exposed to during the testing process and thus garner the highest degree of technical skill in all areas. He purchased many prepackaged mathematics and reading programs in preparation for the state tests. He encouraged his teachers to use these materials, setting clear objectives on a set schedule to make sure they "cover everything" that was indicated on the tests.

Mr. Connor emphasized the importance of students' scores on the state mandated tests. He believed that the school ought to be operated in a calm and predictable pattern. The school daily routines were very important to him. He called the people affiliated with his school a "family" and wanted him to fit the image of the 'papa' figure in his school.

The Teachers.

The two fourth grade teacher participants were Mrs. Dickerson and Mrs. Stromberg. The two teachers worked with groups of children that had been tracked into ability groups. One had the 'high' children (Mrs. Dickerson) and one had the 'average' children (Mrs. Stromberg).

Mrs. Dickerson was educated in a local suburban school district not far from the one under study, and received her college education at two state universities in the state where the suburb existed. She mentioned that her high school "did not have much diversity" and that she had only known "WASP's [White, Anglo-Saxon, Protestant] and Jewish people" until she went away to college. Her parents were both teachers. She earned an undergraduate degree in early childhood education and prepared to "open up my own preschool and then when I looked into it and realized ... no, it wouldn't be the thing for me to do." At that point she decided to become an elementary teacher. When asked about her teaching philosophy and methods she replied:

     The only thing I can say after 21 years of teaching is that there
     are a lot of great new ideas that are good for some of your class,
     but there isn't one idea that is great for everybody. So what
     happens is, I take a little bit of whatever I think will work from
     each of them and have a very eclectic mish-mash when it's done.
     There isn't one philosophy I think covers everything. I think it's
     true of life in general ... there were a lot of strands, and I just
     didn't want to cram it all in before March [when the state
     proficiency test occurred]. It just didn't happen and I wasn't
     going to lose hair over it ... I let them work in groups sometimes.

Mrs. Dickerson's main concern was to help her students pass the state mandated tests. When we observed her mathematics class on several occasions, her instruction heavily concentrated on preparing her students for the test.

The other teacher participant was Mrs. Stromberg. She worked with the "average" mathematics students. She was born and raised in the suburb where this study took place. She gives credit to the school system when she said that she "was well prepared for college, and in fact, found it easy by comparison." She mentioned one of her greatest moments was when she had been recently selected for service award by the teachers union:

     That was my proudest moment. It really didn't have anything to do
     with the classroom but, at leasy I got recognition from my
     colleagues for working hard for about the last twenty years! That
     felt good. We rarely got a lot of recognition from our peers or the
     administration.

When asked if she followed any particular theory of education or teaching method she replied:

     Not really. I just follow the lessons in the [mathematics series]
     and to through the activities ... this series has had a high impact
     on me. It's more visual, more literal and more tactile.

Mrs. Stromberg was very comfortable to follow her textbook routinely. Like Mrs. Dickerson, she wanted to prepare her students for the state test. She emphasized students' mastery of basic skills as one of her primary goals. Our observations of her mathematics class revealed her routine patterns of interaction with her students, starting with her short instruction, then teacher/student interaction, and students' assignment on presented instruction.

The norms of the school were based on predictability, certainty, linearity, order, and control. It was expected that children would fall into 'high,' 'average' and 'limited ability' groups. This tracking was understood and accepted by the school community.

The Students.

The two students (Beth and Carl) were observed in and out of their classrooms for one school year (10 times for each). We also conducted one-on-one interviews with each child ten times throughout the study. Beth was a fourth grader in Mrs. Dickerson's class. She had a very active life, being involved in church, the choir, and music in general. "I don't want to brag, but I'm a good singer!" She seemed to appreciate social gatherings. She liked mathematics and especially reading.

Carl was a fourth grader in Mrs. Stromberg's class. It seemed hard for him to open up and talk. It took some time to gain his confidence. He lived with his mother and two older brothers. One brother was especially close to Carl. This brother frequently helped him with his homework and "threw the ball" around with him. He seemed to have a quiet, yet fierce self-confidence. He was a person of definite likes and dislikes. He particularly enjoyed performing. He was in a performance in third grade that he really enjoyed. "I had a speech! I really liked saying it!"

The Cultures of the Two Mathematics Classrooms

Mrs. Dickerson's classroom was fairly long and narrow. The desks were arranged so that it seemed a long way from the desks at the front of the room to the desks at the back. The room was colorful and bright. There were beautiful health posters on the back wall exclaiming the virtues of good nutrition.

The teacher's desk was piled high with papers, posters, other small boxes and articles. The desk was in the back of the room by the windows, across from the door. The desk's position seemed to prohibit a good view of the students and vice-versa. Student desks were in clusters of four, arranged in asymmetrical groups toward the front of the room to the back. Everything seemed focused toward the front of the room. Directions were given and most talk was directed to the front.

During a lesson students began by having a 'pop quiz' on the material they had learned that week about lowest term fractions. They had to turn to a page in their books and complete the required set of problems. After about fifteen minutes they stopped. Papers were exchanged and answers were called out by the teacher. When the papers were returned to their owners, Mrs. Dickerson asked if there were any particular problems they wanted to see worked out. Several hands went up. As problems were brought up, Mrs. Dickerson worked them out and explained the correct answers. After all questions about the problems were answered, the papers were collected and students were given another page to complete in the mathematics textbook for seatwork. Students worked silently and separately for the remainder of the period. There was not further interaction among students or with the teacher. These patterns of interaction were observed consistently throughout the year.

In Mrs. Stromberg's class the first thing you noticed was all of the brightly colored 'animal kites' that were hanging from the ceiling. The room was bright, cheerful, and squarer than Mrs. Dickerson's, even through both classes were right next door to each other. There was a small computer lab on a raised platform on one side of the room. Student desks were arranged in four symmetrical groups of six. The overhead projector was the center of attention, but student desks were arranged so that most students seemed to have access to a fairly view of instruction. Mrs. Stromberg's desk was positioned near the center of the room, where she had a good view of all her students when she sat there.

During a lesson Mrs. Stromberg went over the previous night's homework. She called on students at random, asking them to give the answer. If there was some confusion, she taught on the spot, clearing up "misunderstandings" as they occurred. Students seemed free to quietly consult one another about their assignment, and also made corrections as they went along. After the homework was collected, students were given a problem about fractions. The problem was read and explained by the teacher before they began. Students worked on the problem individually or in pairs. Mrs. Stromberg went around to different groups to help as needed. Students seemed engaged and able to do the assignment.

Episodes of Mathematics Instructions

To illustrate the technical nature of curriculum development and teaching at this school, we present four lessons, two given in each of the two classes in this study. Evidence of the teachers' authority and control are clearly presented. The first lesson occurred in Mrs. Dickerson's class. The teacher was explaining how to divide two digit numbers by a one digit number.

T: Okay, class, most of you have learned your multiplication and division facts pretty well and you know that division is just multiplication backward, right? Now we are going to learn how to divide some bigger numbers. But before we begin I want to make sure we all know what these words mean. [She wrote: "dividend, divisor, and quotient" on the board.] Who remembers what the first word means?

S: It's the number that's being divided.

T: Correct. Who knows what divisor means?

S: That's what dividing the number.

T: Right! Now, what's the quotient?

S: [several voices] It's the answer!

T: Great! Now, let's look at this problem. [She wrote 56 divided by 7 on the board] Who can tell me what the 56 represents in this equation [sic]?

S: It's the number of things you have to split up.

T: That's right. What does the 7 stand for?

S: It's how many things are going to be in each group.

T: Yes! Can anybody tell me the answer to this problem?

S: [several voices] Eight!

T: Right! And this is called the ...?

S: Quotient!

T: Yes, it's the answer to the division problem. It is also the number of groups you will have. If you have 56 things, and you split them up so that you will have 7 things in each group, you will end up with 8 groups. Does everybody understand this? [Nodding of heads] Okay, now look, another way to write this is [She used the long division bracket symbol.] Inside the bracket is the number of things you start with, or the dividend. Outside the bracket is the number of things you want in each group, or the divisor. [She wrote 8 above the bracket.] And the 8 is the number of groups. Do you all see how this works? [More nodding] This one was easy because it is a basic fact that you all should know [She wrote on the board 7 times 8 equals 56 or 56 divided by 7 equals 8.] Now, I'm going to show you another probem where it won't be so easy to figure out the answer. [She wrote 72 divided by 4.] If we want to write this with a bracket where will the 72 go?

S: Inside!

T: Yes, and where does the 4 go?

S: Outside!

T: Now, since we can't use a multiplication fact to figure this out right away it is best to break up the problem into smaller parts. We can think of 72 as 7 tens and 2 ones. [She turned on the overhead and put down 7 base-10 strips.] Now, how many groups containing 4 tens can we get out of this group of 7 tens? [Looks of confusion on students' faces, no response, as a redirecting technique, she gathered four of the base-10 strips close together.] Can we get another set of 4 groups of ten from 7 tens?

S: No ...

T: Okay, now look at the problem [She wrote 1 above the division bracket over the seven.] So we know it's 1 group of 4 tens. Another way to say this is 10 groups of 4, do you see that? [Students nod their heads] Since there are no more sets of 4 tens we have to rearrange the rest to see how many more groups of 4 there are. [She changed the 3 remaining ten strips to 30 one pieces and added the 2 remaining ones.] How many ones do we have now?

S: [several voices] 32!

T: Okay, How many sets of 4 do we have now? [She circled them as the students counted them.]

S: Eight!

T: Yes! [She wrote an 8 above the bracket over the 2.] Since there are no remaining ones, our answer or quotient is 18. If we make 18 groups of 4 we should up with 72. Now, you and your partner use your base-10 blocks and make 18 groups of 4. See if your total is 72. [The pairs got busy counting and forming the groups. After they all came up with the correct answer she offered them several more problems to work out with the base-10 blocks as she walked among the groups and offered further instruction. The lesson lasted for about one hour and ten minutes.]

From the above episode we observed that Mrs. Dickerson regards mathematics as arithmetic skills to be followed through rote procedure "shown" by the teacher (I.e., adding, subtracting, multiplying, and dividing). Her actions in the classroom revealed that students learn those skills by repetition. The classroom observation indicated that she saw her role as the provider of information and expected her students to do the procedures given by the teacher. Although she used manipulative for mathematical modeling, their use was only to reinforce her pre-structured

algorithm to be followed. Her measure of assessing students' understanding was based on their correct responses to her questions. Her instructional objectives were highly influenced by the state mandated mathematics test.

The second episode involved learning to add fractions. Mrs. Dickerson presented a problem with the intention of helping the students to learn how to add fractions with different denominators.

T: We have learned how to add fractions that are alike. Now I would like to show you what to do if the fractions are not alike. [She wrote 3/4 plus 3/8 on the board.] What do you think will be the answer if you add these fractions?

S: [several voices] 6/4? 6/8? 6/12?

T: No, you can't add fractions that have different denominators. Before you can add them you have to do something to make the denominators alike. What can we do to make these two fractions have the same denominator? [Students seemed uncertain, one or two students raised their hands. The teacher called on one.]

S1: Add them together?

T: Well, that's what we're trying to do. Anybody have another idea? [No students responded.] Look at this ... [She pointed to 3/4 that she wrote on the board.] If we multiply this fraction by two over two it becomes 6/8. [She wrote 2/2 on the board and multiplied it by 3/4.] See, this is how we can make them alike [She was referring to 3/8.]. When you multiply something by a fraction where the denominator and the numerator are the same, you don't change the value of the fraction, right?

S: [Several voices] Yes. [Some students had puzzled looks.]

T: Now, since we have changed 3/4 to 6/8, by multiplying both the denominator and the numerator by 2, the two fractions we are trying to add are alike, see? [She is referring to 3/4 and 6/8.] So, now we have 6/8 plus 3/8. We can only add the top parts, the numerators when we are adding fractions, you will just add the 6 plus the 3. Who can add them now? What do you think the answer will be?

S2: 9/8? [Student was not confident of the answer.]

T: Yes! When you change the denominators so they are alike and add just the numerators, you will get 9/8. Who understands this? [Several hands went up.] Let me give you another example. Let's say we have 2/3 and 3/6. How are we going to add them?

S: [Several voices] Make them alike ...

T: How?

S3: You have to change the denominators?

T: Yes, you do. What can we multiply 2/3 by so it would have the same denominator as 3/6?

S2: If you multiply the 3 by 2 you'll get 6.

T: That's right! [She wrote 2/2 next to 2/3.] We multiply the 2/3 by 2/2 ... two times two is four, and two times three is six! Now we have changed the fraction. It is now 4/6. So, now we are able to add 4/6 to 3/6. What will we get if we add 4/6 to 3/6?

S4: 7/6.

T: Right! The answer will be 7/6 because 3 plus 4 equals 7. Remember, we can only add the numerators to do it correctly and get the right answer. We will do one more together. Look at this one. [She wrote 2/12 and 1/6 on the board.] How can we add these? Remember, the denominators have to be alike and you can't add the denominators together. We only want to add the numerators. What can we do to make the denominators alike? Think about multiplication. [She called on a student.]

S5: Make the twelfths into sixths?

T: That's one thing we could do! Could we do anything else? There is more that one way to do this. [She called on another student.]

S6: Multiply?

T: Yes. What?

S2: Could we multiply to make the sixths into twelfths?

T: Yes! Excellent! [That was the kind of response that she was expecting.] Sometimes you can change the fractions either way! Watch. [She wrote 2/2 next to 1/6.] Two times one is two and two times six is twelve! Now we have 2/12! Can we add them now! [More students responded.]

S7: 2/12 plus 2/12 is 4/12.

T: Right! It will be 4/12. Good thinking! Now remember, [S5] said she thought we could change the twelfths into sixths. We can do that by doing the opposite of multiplication. What is that?

S: [several voices] Division!

T: That's it! [She wrote the division symbol and 2/2 next to 2/12.] We can divide 2/12 by 2/2. Two divided by two is one and twelve divided by two is six! Now we have changed 2/12 into 1/6. As you can see, the fractions are now alike, but they are both sixths! Pretty cool, huh? [She looked very excited.] When we add them what do we get?

S: [Several voices] 2/6! 2/6!

T: Right! So look, from this we can see that 2/12 and 1/6 are the same! We will have a lesson about that part later. Right now I want to make sure you understand about changing the denominators to make the fractions alike, so I will pass out this sheet for independent practice. There are eight problems. I want you to work on these at your seat. If you have trouble raise your hand and I will come around to help you. [She passed out worksheet. Students began to work quietly. The second lesson lasted for about 55 minutes.]

An interesting part of the second episode was the teacher's excitement when S5 presented her solution for finding equal fractions. Mrs. Dickerson was not expecting to hear something that she did not "cover" in her lesson. Although she recognized the viability of S5's solution when the student used method of halfing for finding equal fractions by dividing the numerator and denominator of 2/12 by two to get 1/6, the teacher was more comfortable directing students back to her original procedure of using method of doubling 1/6 for making equal fractions.

The two episdes shown above illustrate the values and beliefs of Mrs. Dickerson regarding what mathematics is, how students learn mathematics, and the role of the teacher in the mathematics classroom. The social interactions were dominated by the teacher's questioning, the students' anticipated responses, and the teacher's evaluation of these responses. Multiple perspectives, students' invented algorithm, and students' dialogues were limited in Mrs. Dickerson's mathematics classroom.

Next, we describe two lessons in Mrs. Stromberg's class. The first lesson focused on number patterns and number relations. Although the concept was fairly sophisticated, the teacher's "objectives" wee apparent and the method was cleared geared for students to gain knowledge of key words in mathematics test and proficiency of arithmetic skills.

T: Who knows what a sequence is?

S1: When things go in order.

T: Give me some examples.

S2: Like a story, the beginning, middle and end.

T: Yes, but think math.

S3: Like numbers in a row, like 1, 2, 3 ...

T: Absolutely! That's called a simple sequence. Can you give me a sequence that doubles?

S; [several voices] 2, 4, 6, 8 ...!

T: Great! Sometimes we can have a sequence that uses two operations. What could happen if we combined the doubling sequence and the simple sequence ... look at this. [She wrote 4, 9, 19, 39, _ on the overhead.] What would be the next number? [Students worked to find out the pattern. Some used pencil and paper and some worked mentally.]

S: [Several voices] 79!

T: Wonderful! Now, we have been studying how to write things down in algebra sentences. Can you think of how you would express this sequence in an algebra sentence? Remember, we are multiplying a number by 2 and adding 1. You can let n stand for the number.

S4: Is it n times two plus one?

T: Yes, it is! Excellent! [She wrote this expression on the overhead.] Now, there is another knid of sequence that we will talk about today. It is a multiplication sequence. They are also called geometic progressions. When a number is multiplied by itself; 2 X 2, for example, it is called an exponent. Say exponent [Class repeated.] If I wanted to multiply 2 X 2 X 2, I would write it this way. [She wrote 2 to the power of 3 on the overhead.] The top number shows how many times I want to multiply the bottom number by itself. The top number is called the power. The bottom number is called the base number. What would be the answer to 2 to the power of three? [She pointed to the symbol she had written.]

S: [Many voices] Eight!

T: Great! Now look at this [She wrote 5 to the power of 2.] This means that five is being raised to the second power, or 5 X 5, or 25. Do you get it? [Several nods] What is this? [She wrote 5 to the power of 3 on the overhead.]

S5: Five to the third power?

T: Yes! It's 5 X 5 X 5. How much is it?

S: [Several voices] It's 125!

T: You've got it now! One more example, [She wrote 3 to the power of four.] Figure it out! [Students worked individually with pencil and paper to figure out the answer.]

S6: Is it ... 81?

T: You're right! Look. 3 X 3 = 9, 9 X 3 = 27, 27 X 3 = 81. Any questions? [No hands went up] Now, here is a worksheet. There is practice with sequences and exponents. Let's what you can do. [She passed out the worksheets for independent practice.] If you need help, raise your hand and I'll help you. [She went back to her desk while students worked quietly on their assignment. This lesson lasted for about 50 minutes.]

The above episode of classroom interaction was somewhat different from Mrs. Dickerson's class. For example, the teacher was interested in students' understanding of number patterns and number operations. The focus of the lesson was not mainly on students' mastery of algorithmic procedures presented by the teacher and followed by the students. It was geared more towards recognizing number relations. There was, however, a clear pre-set objective and a clear path by which the teacher guided the lesson toward that objective. There was little room for dialogue among the students, and the teacher retained control of the line of thought and response.

The second episode of Mrs. Stromberg's mathematics instruction demonstrates the culture of the classroom and the value placed on students' mastery of arithmetic skills for test taking. She was working on developing strategies that students would need to tackle the kinds of problems most likely to be presented on the state mathematics test.

T: Today, we are going to work on how to do some different kinds of story problems. We will work on three problems today. One will be about buying things, one about elapsed time, and one about doing a job. Here is the first problem. Listen and read with me as I read. Then, follow me as I work each one out on the overhead. You work the problems out on your paper just like you see me doing them, is that clear? [Students nodded their heads. She passed out a sheet with the first problem written and read it.] Your mother took you to the store to buy some things for school. She brought two notebooks for $1.49 each, a ruler for $1.25, three packages of pencils for $1.19 each, and six erasers that were three for $1.00. What was the total cost for these items? These are the steps you would follow to solve this problem. First, you need to look for the items where you are buying more than one of them. Which items are these?

S1: The notebooks and the pencils and the erasers.

T: Correct. You have to multiply each item by the number that is being bought. [She wrote on the overhead $1.49 X 2, $1.19 X 3, and $1.00 X 2.] Please notice that the erasers are three for one dollar so if you are buying six that will cost you ...? [She gestured for a response.]

S2: Two dollars!

T: Yes! Now we must multiply each one to find the totals for these items. First, let's do the notebooks. Two times nine is eighteen. Put down the eight and regroup the one into the tens column. Two times four is eight, plus the one is nine. Two times one is two. So the answer for the notebooks is two dollars and ninety-eight cents. Now we complete the multiplication for the other two items. [She went through the steps of the multiplication procedure. She worked out the other two problems on the overhead.] So the pencils cost a total of three dollars and fifty-seven cents and the erasers cost two dollars. Is that clear? To find out how much money you mother spent, you must take each of these two totals and add them up, along with the price of the ruler, so that is ... $2.98 plus $3.56, plus $2.00, plus $1.25. [She wrote the amounts in a column.] When we add them, what is our grand total? Remember, when you see the word total, it always means you are going to add something. You work it out as I do it here. [She walked the students through the addition procedure as she worked on the overhead.] What is the answer?

S3: Nine dollars and eighty cents!

T: Right. Do you all see how we did that? The important thing to remember is that if you have more than one of an item, you must multiply it first by how many you have before you can add it to the grand total. The word each is usually a good clue that you will need to multiply something. Good. Now, we will go on to the second example of a different kind of problem you may see on the test. [She passed out a second sheet and read it.] Mark has to be at his piano lesson at 4:15 p.m. If it takes 15 minutes to get ready and one hour to get there, when should he and his mother leave the house to arrive on time? To solve this problem we must think about the whole hour first. What time was it one hour before 4:15 p.m.?

S: [several voices] 3:15 p.m.

T: correct. Now what is fifteen minutes before that?

S4: 3:00!

T: That's right! So Mark and his mother must leave home at 3:00 p.m. to arrive on time. It is important for you to think about whole hours as much as possible. Any time you can add up minutes to make hours before you add or subtract them from a time, the easier it will be for you to find the correct answer. We will do some more of these kinds of problems in a few days, but I want to show you the last example before our time is up. [She passed out the last sheet and read it.] Michelle is trying to earn some money to buy a new CD. It costs $20.00. She got a job raking the neighbor's leaves. She will be paid $5.00 per hour. How long will it take her to earn enough money to buy the CD? The easiest way to do a problem like this is to make a table. [She made a table on the overhead. Students followed.] For one hour she earns $5.00, for two hours she earns $10.00, for three hours she earns $15.00, for four hours she earns $20.00. So how long does it take her to earn the $20.00?

S: [several voices] Four hours.

T: Yes. It will take her four hours. We can check our table by doing a simple multiplication problem. [She wrote 5X4 + 20.] You can easily see that your table was correct and it took four hours to earn the money for the CD. Good, boys and girls. I will give you homework for tonight that has problems like this for you to solve. We will talk about these problems and their solutions tomorrow.

Mrs. Stromberg's instruction was very direct. The methods were strictly formulated and she did not encourage any discussion or deviation from the prescribed techniques. She was clear and precise about specific matehematics language and how students could expect to encounter it on the state mathematics test. She presented each problem without providing opportunities for students to figure out solutions on their own. She solved the problems and insisted they follow her procedures exactly as she presented them. She wanted to be in full control of all aspects of the lesson.

Putting too much pressure on teacher accountability and the school's ranking based on the state mandated tests results (I.e., excellent, effective, needs improvement, academic watch, and academic emergency), from beginning of the school year until middle of March, before the state test begins, caused these teachers to feel obligated to spend most of their instructional time in mathematics for preparing the students for the test.

The Two Students' School Mathematics Experiences

The two student participants were questioned concerning how they felt about their experiences of mathematics learning at the school. They had somewhat different responses but in both cases it was evident that they experienced a direct instructional approach in a controlled environement. The following statement is from Beth. She described experiences in Mrs. Dickerson's class. She told us about what happens in her mathematics class, what she thought about mathematics, and her mathematics abilities.

     [I'm good at] addition, of course, subtraction, of course. Last
     year in 3rd grade, I learned my multiplication facts over the
     summer ... so, I'm good at multiplication, but sometimes I have to
     ask what's 4 times 7 or 4 times 8. Sometimes I remember, sometimes
     I forget ... We copy problems off the board and then we figure them
     out. Like when we were learning long division ... she taught us how
     we should go: divide, multiply, subtract, and bring down ...
     Actually my mom helped me. I took me about half a month to figure
     it out ... [She described some of her experiences in mathematics
     class.] ... We would look at our books first ... and find a certain
     page or something. She [Mrs. Dickerson] will say 'open your math
     spirals.' And we'll just write down the problems and then we'll do
     the answers together. Sometimes she lets us say the answers, and
     when it's my turn, sometimes I get stuck. Sometimes when I get
     stuck people will start laughing at me, like when I call out the
     answer--like if I say, "56" and the answer is 59, they'll say "56!"
     and start laughing. But I don't really think about it. I still
     think I'm good. On a scale of 1 to 10, I'd say I'm a 9.

It appeared that Beth thought mathematics consisted mainly of learning multiplication facts and division algorithms and little else. Her beliefs about what mathematics is (I.e., adding, subtracting, dividing, and knowing the multiplication table) were consistent with her classroom mathematics experiences. She valued gaining these skills above anything else. She was proud of her ability to "know facts" and complete number operations.

Carl, who was in Mrs. Stromberg's class, also shared his thoughts about his experiences and performance in mathematics class:

     I do good in math. I'm just good at it. I like making graphs. We
     did one about sports. We used baseball, basketball, football,
     volleyball, tennis, golf, and hockey. I liked putting everything on
     a graph. It was easy to see the favorites. You didn't have to read
     a lot of stuff to find out. Plus, I made it look good ... the
     favorite was basketball ... Sometimes I get nervous when I take a
     math test, especially multiplication tests or story problems. We
     only do story problems once in a while. We have practice problems,
     though. We had to work with a partner and explain how we did it and
     show our work. Usually, we work alone, though. I like to do both.
     If I compare myself to both math classes I would be a 5. If I
     compare myself to my math class I would be a 7. What makes you
     better is you do it quicker and you understand better.

Carl knew he was in the 'average' mathematics class and assumed that everyone in Mrs. Dickerson's mathematics class was gaining skills "quicker" and understood "better" than he. It seemed he gave himself a higher rating within his own setting, because he had a better understanding of the comparisons. He also had the idea that learning the multiplication and division algorithms were among the most important things to learn in fourth grade mathematics class. These ideas are consistent with the values and beliefs set forth at his school. The structure and the culture of the school placed high emphasis on technical ability and skill.

Discussion

Did the state mandated mathematics test impact the structure and culture of the school studied? Did mathematics instruction in the two classrooms and student/teacher interactions and values alter because of the test? From our observations of the two classrooms, talking with the principal, two teachers, and two students, and noting the school ranking based on the test results, we found that the state mathematics test had a profound impact on the schools' structure and culture. In addition, the test had a significant affect on mathematics teaching/learning, teacher expectations, and classroom climate.

From the beginning of the school year until the middle of March, before the state mandated tests began, the mathematics curriculum and instruction in both classrooms were heavily focused on preparing students to pass the mathematics test. As a result, Beth and Carl perceived mathematics as arithmetical rules and procedures (I.e., adding, subtracting, multiplying, and dividing) to be memorized. They believed that a good mathematics student was someone who computes quickly and correctly. Also, the state tests impacted the priorities and values of the teachers and the principal. Their main concern was getting the percentage of passage rate on the tests as high as they could.

The questions is not whether or not we should have standardized tests. Perhaps we should and we will. The point is, if the tests derive mathematics curriculum and instruction, if the tests impact the school's structure and culture in a drastic way (which we believe they do), then by moving the tests away from measuring mimicry mathematics and moving towards the kinds of tests that concentrate on assessing students' understanding and application of important mathematics, it may positively impact mathematics curriculum and instruction in our classrooms. Hopefully, then students like Beth and Carl would see mathematics as the study of patterns and relationships.

The current state mathematics test may lack assessment value for several specific reasons. First the majority of the questions on the state test are multiple-choice. The multiple-choice questions do not and can not adequately assess what students know and how they come to know it. With only three choices for each question, it is entirely possible to get a correct answer simply by guessing. Therefore, a correct answer does not necessarily mean that the students understand the mathematical concept. Conversely, an incorrect answer does not mean that a student did not understand the concept.

Second, the test is heavily grounded in basic arithmetic skills. The majority of the questions require students to employ various algorithms from addition, subtraction, multiplication, division, and rounding of numerals. The test questions actually tell students what to do and test whether or not they can carry out the algorithm. The test does not reflect the notion of coherence and openness supported by NCTM Standards (1995, 2000) and other research documents (Klein, Hamilton, McCaffrey, & Stecher, 2000; McNeil, 2000; Stigler & Hiebert, 1999; Whitford & Jones, 2000). As stated by Burns (1992) and Tsuruda (1994), students must be exposed to problems that require them to figure out a plan for solution.

Third, assessment should facilitiate student learning and improve instruction. So, there ought to be a recursive relationship between instruction and assessment practices. Ironically, the state mathematics test is conducted in mid-March and results are reported in early June. This prohibits any substantive alterations educators may make to improve student's achievement and learning. The test results are also reported numerically as "pass" or "fail" outcomes. What students missed or got right is not reported. This type of reporting provides little substantive feedback that educators could use to inform instructional program.

The political and authoritative pressures of this type of testing may create dilemmas and professional conflicts for teachers and administrators who are reforming instruction and assessment practices according to NCTM Standards and related research (Cohen & Hill, 2001; Confrey & Carrejo, 2002; Firestone & Mayrowetz, 2000; Firestone, Monfiles, & Camilli, 2001). The tests are "high stakes", "end-of-the-line" assessments encased in a highly political environment. Teachers know too well that they are often judged by the results of high stakes tests. As a result, many of them may continue to teach in a manner consistent with these multiple-choice tests.

From the researchers' perspectives, the two teachers and principal were caring professionals, however, when pressured by the state mandated mathematics test, when they were accountable for preparing their students to pass the test, and when the schools were publically ranked according to the test score (I.e., 75% or better passage rate on all five sections of the test such as reading, writing, citizenship, mathematics, and science), when the state put too many political and authoritative pressures on schools as "excellent" (six points out of six), "effective" (4 points out of six), "needs improvement" (three points out of six), "academic watch" (two points out of six), and "academic emergency" (one or zero point out of six), understandably, the school educators felt obligated to spend great amounts of time up to the middle of March, before the tests begin, teaching directly to the test. As a result, direct instruction was substituted for inquiry-based mathematics.

As researchers, we did not intend to question the professional integrity of the principal or the two teachers. These individuals took their administrative and teaching responsibilities seriously. They tried hard to provide their students with the best possible educational opportunities. We recognize that what the administrator and these teachers practiced is part of a larger national, state, and local political movement that emphasizes teacher accountability and school effectiveness based on the test results (Cohen & Hill, 2001; Confrey & Carrejo, 2002; Klein, Hamilton, McCaffrey & Stecher, 2000; McNeil, 2000; Schorr, Firestone & Monfils, 2003; Whitford & Jones, 2000). Also, we understand that the way the teachers and principal taught and "managed" their school is closely connected to the way they were taught or "trained." However, we questioned the viability of the mathematics test and implications of this kind of testing on reforming mathematics education.

The school was ranked as an "excellent" based on the state test results. The students did well on the test. However, when it came to problem-solving and mathematical communication, these students lacked mathematical understanding. Our observations of classroom mathematics instruction (I.e., teaching to the test) combined with the students' mimicry mathematics raise some serious questions about the credibility of the test results and schools' ranking by these results. Also, we have concerns regarding the direct impact of the state mathematics test on the school culture and structure. Furthermore, we are worried about the implications of this type of testing and ranking on reforming mathematics education.

Continuing to rely on current mathematics testing, from our perspective, will lead to even more children being left behind. Politicians, school administrators, teachers, parents, and community members ought to transform learning communities so that learning is an ongoing process for students and educators. They need to recognize school as a complex, living system where uncertainty, spontaneity, flexibility, creativity, as well as order, and chaos coexist (Fleener, 2002).

The current structure and culture of most schools in the U.S. with the existing testing in mathematics cannot and will not be able to provide most students the necessary conditions to succeed in our today's demanding world. Overall, we believe that the current testing is in sharp contrast with the reform movement in mathematics education and is incompatible with research on assessment. Changing the existing testing practices requires a radical shift. This shift includes political and instructional changes in perspectives as well as epistemological and pedagogical changes. Until then, the notion of "No Child Left Behind" remains a dream.

References

Burns, M. (1992). About teaching mathematics: A K-8 resource. Sausalito, CA: Math Solution Publications.

Cohen, D.K. & Hill, H.C. (2001). Learning and policy: When state education reform works. New Haven, CT: Yale University Press.

Confrey, J. & Carrejo, D. (2002). A content analysis of exit level mathematics on the Texas Assessment on Academic Skills: Addressing the issue of instructional decision-making in Texas. In D.S. Mewborn, P. Sztajn, D.Y. White, H.G. Wiegel, R.L. Bryant, & N. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 539-550). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Firestone, W.A. & Mayrowetz, D. (2000). Rethinking "high stakes": Lessons from the United-States and England and Wales. Teachers College Record, 102, 724-749.

Firestone, W.A., Monfils, L. & Famille, G. (2001, April). Pressure, support, and instructional change in the context of a state testing program. Paper presented at the annual meeting of the American Educational Research Association, Seattle, WA.

Fleener, M.J. (2002). Curriculum dynamics, recreating heart. New York: Peter Lang Publishing.

Klein, S.P., Hamilton, L.S., McCaffrey, D.F., & Stecher, B.M. (2000). What do test scores in Texas tell us? Santa Monica, CA: RAND.

McNeil, L.M. (2000). Contradictions of school reform: Educational costs of standardized testing. New York: Routledge.

National Council of Teachers of Mathematics (1995). Assessment standards for school mathematics._Reston, VA: Author.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

Schorr, R.Y., Firestone, W.A., & Monfils, L. (2003). State testing and mathematics teaching in New Jersey: The effects of a test without other supports. Journal for Research in Mathematics Education, 34,k 373-401.

Stigler, J.W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: The Free Press.

Tsuruda, G. (1994). Putting it together. Middle school math in transition. Portsmouth, NH: Heinemann Press.

Whitfort, B.L. & Jones, J. (2002). Accountability, assessment, and teacher commitment: Lesson's from Kentucky's reform efforts. Albany, NY: SUNY Press.

Roland G. Pourdavood, Cleveland State University (CSU)

Belvia K. Martin, Shaker Heights School District

Nicole Carignan, University of Quebec at Montreal (UQAM)