Authentic assessment: a school's interpretation.

In the previous paper we discussed the challenges of instructional leadership for reforming mathematics education in a K-4 school. In this paper we describe how the school principals and teachers-leaders developed authentic assessment consistent with NCTM Standards (1989, 1991, 1995) and constructivist learning theory (Cobb & Yackel, 1996; von Glasersfeld, 1995). Furthermore, we explain the kinds of challenges and dilemmas the educators encountered when a state mandated mathematics test was initiated.

Early in the reform effort, few textbooks existed that addressed the NCTM Standards recommendations. Elementary teachers and principals researched and collaborated with university and secondary mathematics educators to understand mathematics content and develop a pedagogy grounded in constructivism. Within this collaborative structure, teachers invented and restructured mathematics curricula that centered on main mathematical ideas such as: unitized systems, zero/infinity, change, chance, dimensionality, location, and the key processes of combining, comparing and partitioning. It became necessary for educators to also develop mathematics lessons aligned with "restructured" curricula that valued problem-solving, reasoning, communication, modeling, illustrating, and student experiences (NCTM Standards, 2000).

Performance Tasks and Student Solutions

To evaluate instruction and student understanding of these mathematical skills and ideas, classroom teachers, principals, and two secondary mathematics teachers developed a set of performance tasks for each grade level K4 (Cowen, Alig, Bannon, Federer, Haas, Nader, Skitzki, Smith, Strachan, Svec & Thornton, 1996, 1997). The performance tasks were called Snapshots because each task provided a glimpse of student growth at certain times throughout the school year. The topics of time, money, area, length, volume and chance were used as a framework to create the performance tasks and guide daily instruction (see Appendix 1). Teachers were expected to select lessons that helped student performance on these tasks. These topics and their tasks ascended sophistication throughout K-4. In order to capture and assess student growth over time, students were given the identical performance tasks two to three times during the school year (September, February, May).

The Snapshot tasks connected instruction and assessment. Prior to each individual performance on the task, teachers and students read and discussed the task. Students determined the important information in the problem and suggested possible strategies for finding solutions. Often, during these mathematical dialogues, students solve the problem. Nevertheless, all students were still expected to solve the problem independently (i.e., defending their thinking and solutions with illustrations, words and calculations). What follow are some of the teacher-designed performance tasks and the students' solutions to these problems. For an entire set of performance tasks and some examples of students' solutions to the tasks, see Appendix 1.

LENGTH: (Performance Task)

Third Grade: Every day Sean took Rex, his grandmother's dog, for a long walk around the nearby high school. The total distance they walked each day was 3-1/2 kilometers. Erica, Sean's cousin, lived next door to their grandmother. Erica took her dog, Spot, to long walks four times a week. They walked 5-1/4 kilometers each time. Who walked the farthest by the end of each week?

Teachers noticed the following about one student's responses:

* October performance: Student constructs two calendars for each dog-walker. On one calendar seven days were marked off. Underneath each day was the number 3-1/2. The second calendar had four days with 5-1/4 listed under Monday, Tuesday, Wednesday and Friday. Student calculated the time on each calendar separately. Column addition was performed on whole numbers and then on fractions. Fractions were combined into whole numbers (1/2 + 1/2 = 1, 1/4 + 1/4 = 1/2). Student wrote at the bottom on the paper, "So Rex walked more km. than Spot. The difference is 3-1/2 km." Included with this written explanation was the equation: 24-1/2 - 21 = 3-1/2.

* March Performance: Important information from the text was underlined. The question was also underlined. Calendars were missing. Two large boxes labeled "Rex" and "Spot" contained column addition of mixed numbers (i.e., 5-1/4 + 5-1/4 + 5-1/4 + 5-1/4 = 21 km.) Student grouped halves into wholes but also grouped four-fourths into one whole before writing the solution. Student then added, "Sean and Rex walked the longest."

* May Performance: Teacher modified the distance in the problem to be 21/2 kilometers for "Rex" and 2-3/4 kilometers for "Spot." Teacher also included this extension: "For every 10 kilometers, each dog got a dog biscuit. How many dog biscuits did each dog get after two weeks?"

Student response: One line divided the paper in half. One side was labeled "Spot," the other, "Rex." Column addition of mixed numbers appeared. Interestingly, the student made several conversions horizontally before adding vertically. The students solved the problem this way:

2 3/4 1 1/2

2 3/4

2 3/4 1 1/2

+2 3/4

11 km

The assessment process was recursive. In many instances teachers who thoughtfully implemented this process were impressed with evidence of students' cognitive growth from September to May in grades 14 and from January to May in kindergarten. For instance, a third grade teacher noticed much progress with a student who demonstrated very little understanding about time and money on the third grade performance task at the beginning of the school year (September, 1999) but showed exceptional gains by May, 2000.

TIME AND MONEY: (Performance Task)

Third grade: Ellen earned money baby-sitting. She wanted to save her money to buy a portable CD player that costs $128.00. Ellen charged $4.50 per hour to baby-sit during the day and $5.50 per hour after 8:00 p.m. Mr. Holmes hired Ellen to watch his two grandchildren every Saturday in May from 3:30 p.m. until 11:00 p.m. How much money does Ellen earn in one Saturday? (see figure 1).

Figure 1

Third grade student's solution to time and money performance Task in
September

TIME   MONEY

3:30   $4:50
4:30   $4:50
5:30   $4.50
6:30   $4:50
7:30   $4.50
8:00   $5:00

The answer is 87 I go the answers becuse I add

In September, a third grade "underachiever" developed a chart to show the solution. In the first attempt, the problem solution is incomplete. The student did not understand elapsed time. She calculated 3:30-7:30 as five hours. She also was unable to calculate the amount of money for 1/2 hour (7:30-8:00). Besides these errors, it appears that the solution, evidenced by its lack of completion, was beyond the student's skills and knowledge even though the problem was relevant. This conjecture is somewhat supported by her limited and inadequate written response, "The answers is 87 I go the answers because I add."

The same student responded to the same problem significantly differently in May. The student's solution showed increased understanding and a more sophisticated strategy for solving the problem (see Figure 2, Appendix I).

Figure 2

The same third grade student's solution to time and money Performance
Task in May.

       TIME          MONEY

     3:30-4:30       $4.50
     4:30-5:30       $4.50
     5:30-6:30       $4.50
     6:30-7:30       $4.50
     7:30-8:00       $2.25
     8:00-9:00       $5.50
    9:00-10:00       $5.50
    10:00-11:00      $5.50

TOTAL: 7 hr. 30 min  $36.75

Most impressive of all was how articulate the student was when she defended her thinking and explained how she solved the problem.

I made a table for the time and the money. After that I put down the hours until 8:00. Then I put the money until 8:00. 3:30-4:30 was $4.50. 4:30-5:30 was $4.50. Then 5:30-6:30 was $4.50. 6:30-7:30 was $4.50. 7:30-8:00 was $2.25 because 7:30-8:00 was not a whole hour. Then after 8:00, she gets $5.50. 8:00-9:00 was $5.50. 9:00-10:00 was $5.50. 10:00-11:00 was $5.50. Then I added all the money and got $36.75.

Also evident was "improvement" in the student's understanding of elapsed time and her ability to calculate money for 1/2 hour. It appears the student grew in her ability to calculate time and money accurately and in her confidence to communicate her thinking process. Apparently, the solution reflects the social norms within this mathematics classroom. It seems the teacher and students may value the importance of connecting communication, illustration, and reasoning to solve problems and justify solutions.

CHANCE (Performance Task)

Kindergarten: Ms. Grieshop's kindergarten class decorated special shirts. Each child had to glue a red, blue, and yellow button straight down the front of each shirt. Ms. Grieshop did not want all the shirts to look exactly the same. How many different ways can the three buttons be arranged so that all the shirts are different?

Figure 3 in appendix 1 contains two solutions to the above performance task. These solutions are from one student who attempted the performance task in January and May of 1998. There was no prescribed procedure for solving this non-routine problem. The problem was relevant and reflective of the students' classroom experiences. In January, the student successfully solved the problem. Each shirt's colored buttons were displayed in six different orders. The order appears to be random and may indicate that the child used process of elimination to find all possible combinations. What also emerged was reflective writing which showed the student's pride in his/her creative solution ("I LIK MI PR"-translation: "I like my picture"). In May the student did not use process of elimination, but instead attempted to organize how she/he manipulated the buttons. For example, the yellow button (y) was used twice at the top, in the middle, and at the bottom of the order, indicating a degree of sophisticated thought about arra ngements. This level of thought might be construed as cognitive growth in mathematics. Words were used to solve the problem as well as pictures and numbers, ("R R 6 CHRS"-translation: "There are 6 shirts").

TIME AND MONEY (Performance Task)

Kindergarten: Robert saved $2.00 every day for six days. How much money did Robert save in six days? (January 1998). Robert saved $2.00 every day for six days. Does Robert have enough money to buy three fish that cost $3.00 each? (May 1998). (see Appendix 1, Figure 4).

In figure 4, the kindergarten solution demonstrates cognitive growth and emergence of student understanding about money and rate. In January, the student illustrated dollar-bills and grouped them together by two's. The student invented her/his own money system by labeling the bills to match the number of days. This invention was probably used to "keep track" of the passage of time. The solution is portrayed only through pictures of twelve-dollar bills-even though the invented denominations imply more than twelve. In May, the problem was modified to meet classroom learning needs. Students were asked to determine whether there was enough money to buy fish that cost $3.00 each. This modification demonstrates the interconnections between instruction and assessment. The teacher probably assumed that students could handle a more "difficult" problem since the problem went beyond the relationship of time and money to the relationship between money and the number of fish that could be purchased. Perhaps this modifica tion is attributed to the teacher's experience and recognition of students' understanding of time, money and how money is used.

The student's solution in May shows growth. Invented money is absent. Also, a detailed rendering of a dollar bill is absent. This lack of detail might be interpreted as a more sophisticated way of communicating. Included now are two illustrated functions-three dollars per fish and two dollars per day. Also included is a written response to the question. "Does he have enough money to buy three fish that cost $3 each?" The student response of "yes" indicates her/his-self-evaluation. The "self portrait" in the corner may be the student's self-evaluation indicating pride (smile) and mathematical power (muscles flexed).

Targeting the Performance Tasks: Teacher-Designed Instruction

The use of students' prior knowledge and experiences as the foundation for problem solving demonstrated the importance of relevant context. All mathematical calculations were done within a context to which children could easily relate. Rarely did students solve abstract algorithms without the context of money, volume, area, length, weight, chance, etc. Lessons were child-oriented-teachers designed lessons that reflected student experiences and how children used mathematics in their lives. Therefore, mathematics problems were centered on the home, school, and local shopping center. Problem-solving was often personalized. Frequently, teachers would use students as main characters in problem-solving situations. This attention to the mathematical activities children do and observe daily and the use of real student names engaged children. These methods were enthusiastically endorsed by parents. The examples that follow are teacher-created lessons designed to connect to the performance tasks (see Appendix 2).

Figure 5 in Appendix 2 is a fourth grade measurement lesson developed by a fourth grade teacher. Figure 6 in Appendix 2 represents a student's solution. The lesson is an example of how teachers coordinate instruction with performance tasks.

The format of Figure 6, Appendix 2 is different. All illustrations were originally included on one small poster. This mathematical artwork was displayed in the hallway. These classroom posters conveyed that mathematics learning could be relevant, contextual, and meaningful for young children. Measuring their room, their head size, height, arm span, food size, etc. allowed them the opportunity to graph relevant, personal data and reflect on the relationships among the data.

Teachers who understood and supported the instructional reform were challenged to create lessons that provided memorable problem-solving experiences aimed at the concepts and processes required for success on the performance tasks. Instruction needed to connect with assessment. Teachers wanted to prepare students to do the final performance (assessment) task in May.

For example, a kindergarten teacher who prepared a student to do the "chance" task about arranging three different colored buttons on a shirt had to create plenty of similar experiences for the children during the school year. The teacher might create the following types of problem for the whole class to solve together.

* There are three pieces of fruit in a box in the grocery store. There is one apple, one banana, and one pear. How many different ways can these fruits be lined in a box?

* Mary, John, and Tim were lined up to get a drink from the water fountain. How many different ways could they stand in line?

Likewise, a kindergarten teacher who prepared students to do the time and money performance task might create these problems for the whole class to solve together.

* If Jimmy saves 4 dimes in his piggy bank every day, how many dimes will he save in five days?

* If a pack of pencils cost 5 dimes, how many packs of pencils can Jimmy buy at the end of five days?

Teachers' Reactions to a Different Assessment Process

Teachers formed grade level teams to create problems for each of the five performance tasks at their grade level. Overall, teachers became creative designers of rich mathematical problems. To some teachers this was real professional growth. Other teachers were reluctant to create lessons in this framework for various reasons. Some teachers thought these lessons were not enough curricula. These teachers were used to a "laundry list" of skills, textbooks/workbook exercises, and a "coverage curriculum." Their traditional training taught them that "more were better." They could not adjust to a concentrated and focused approach to mathematical topics. Also, they could not see the importance of revisiting topics often during the school year. Other teachers were unsure of their ability to create meaningful mathematics problems and some were convinced that students needed to show mastery of basic skills before doing any problem-solving. Moreover, some teachers believed the performance tasks were "developmentally inap propriate" for young children and saw little value in developing mathematical communication skills. They thought the problems were too complicated and too advanced for young children in grades K-4. These teachers thought it inappropriate to ask kindergartners and first and second graders to experience multiplications, divisions, fractions and decimals at such an early age.

Despite complaints and resistance from some teachers, about 70% of the teachers moved toward reforming their mathematics instructions and assessment. There was a reflexive relationship between instruction and assessment.

Also, there was a close relationship between teachers' and students' mathematical empowerment when both the teacher and students created the need for instruction and assessed the worth of instruction (Grundy, 1987). These relationships seemed to produce a synergy among members of the classroom community. Students' voices echoed confidence when they did sophisticated mathematics in a risk-free environment where the teacher valued all students' prior knowledge and experiences (Cobb, Wood, & Yackel, 1990; Wheatley & Reynolds, 1999).

I need to express my voice more. By voice I mean my personal touch. For example, I am good in reading, but I need to express myself better in math, I need to organize my work more, and also I need to use more pictures to express my thinking (fourth grade student).

The interdependence of mathematics curriculum, instruction, and assessment was observed when teachers assessed student learning. Assessment of students' understanding of mathematical situations provided teacher-leaders with information about students' instructional needs. Teacher-leaders collected this information during classroom mathematics dialogues and when they reviewed students' assignments. The teaching and assessment system still continued to provoke concern and controversy.

Teacher: You mean I am to give the children the same problem three times during the year? What is that going to prove? (Fourth grade teacher)

Teacher: What happens when they get the answer right the first time around? What am Ito do on the second and third attempts at the same task? (Second grade teacher)

Solving the performance task with the whole class before expecting individual student performance was most controversial. Teachers' stated beliefs about assessment practices were perturbed.

Teacher: I'm not letting my class solve the problem. I will discuss it with them. But if we solve the problem together, they will all go back to their seats and just copy down the answer. (Fourth grade teacher)

Asst. Principal: Kids won't be able to do that unless the strategies and solutions make sense to them. Remember, they have to convince you that they know their solution 'works.' An equation is not convincing enough.

Overall, Snapshots produced dilemmas. How would teachers evaluate and communicate student growth over time? Teachers and parents were surprised by students' mathematical illustrations. The illustrations were meaningful and demonstrated that young children were capable of doing sophisticated mathematics.

Teachers saw student illustrations change throughout the year. Mathematical drawings evolved from detailed pictures to a more mathematical representation. For example, illustrations of people changed from a focus: (1) on minute detail ("eyelashes"), (2) to stick figures, (3) to tally marks. These changes showed teachers that students were becoming more sophisticated with mathematical concepts. Over time, students illustrated less and substituted numerical labels and calculations in their problem-solving situations.

Through the process of professional development and negotiated meaning, many teachers began to realize that mathematics could be learned through active engagement and dialogue (Bauersfeld, 1988; Pourdavood & Fleener, 1998). In most cases, this kind of mathematics instruction was completely absent from the teachers' own personal and professional experiences.

I love teaching math because it is so different from the way I learned it. But traditional math instruction has also been the hardest thing for me [to teach] because it is not what is best for children...To create something new everyday takes a lot of energy and a lot of work. (fourth grade teacher)

Transforming the Learning Community

This K-4 school cannot be understood as a learning community being transformed without accounting for various challenges that emerged from an instructional shift in mathematics. At the center of these challenges was a transformed assessment system. Teachers emerged as writers and creators of mathematics instruction and assessment. Teachers grew as action-researchers, connecting mathematical situations to students' prior experiences and valuing classroom interactions and dialogues. These experiences were shared, revisited, and collected into documents for the staff. "These math problems need revision this summer. But then, I guess we will always be rewriting them each year. Who would have guessed?" (fourth grade teacher)

Instructional and assessment changes were not smooth processes. They created perturbations and disequilibrium within the school community. "When this reform started, I did not agree with it because I did not understand it. I had to have a conversion I had to come to a place to understand it" (fourth grade teacher). As the reform evolved, many educators struggled to make mathematics relevant to students' personal experiences. This placed teachers in a role to which they were unaccustomed. Furthermore, the reform, in some cases, targeted teachers' limited understanding and knowledge about mathematics. Some teachers were encouraged and fulfilled with their new role as adult learners; other teachers were insecure and resistant (Cohen, 1990).

I would fight with the principal all the time. Every time he came into the room I would be using the math book and doing drills. I just couldn't accept that I couldn't do it [math reform]. It caused some tension and a breakdown of communication. He didn't convince me and I couldn't see anything else. I didn't try because I didn't believe in it. (fourth grade teacher)

However, some teachers did discard their dependency on mathematics textbooks and weekly pre-determined lesson plans. Instead, they wrote mathematics instruction and designed performance tasks to monitor student growth.

I enjoy the autonomy I have here. At other schools, lesson plans eliminate 'teachable moments.' I would have to stop and rethink. I usually have a general direction [ideas and processes] and I don't write lesson plans more than three days in advance. I change them so often. (second grade teacher)

Impact of the State Mandated Mathematics Tests on the School Reform

After the first three years (from 1996 to 1998) of mediocre scores on the state mathematics test, this school's principals and teacher-leaders concluded that mathematics instructional time would have to be doubled if students were to learn calculation skills and solve mathematics problems within pedagogical practices that valued dialogue, building concrete models, role playing, illustrations, and writing. The educators therefore decided that every student in grade K-4 should have 90 minutes of mathematics instruction each day. This was about a 45 to 60 minute increase over the time previously allotted to mathematics.

Moreover, the school's principals and teacher-leaders began Saturday morning school for fourth grade children who needed extra instruction about mathematics. In addition, the principals added after school tutoring for third and fourth grade students who needed help with basic facts and calculation skills. About 30 fourth graders attended Saturday morning school from September to mid-March. Starting in January of their third grade year about 30 to 40 students attended after school tutoring in mathematics calculation skills for three days each week. The same 30 to 40 students continued to attend this after school tutoring throughout their fourth grade year. In 1999, ninety percent of this school's fourth graders passed the state mathematics test. This was a 23% increase over the 1998 fourth grade scores. Ninety percent of the fourth graders passed the state mathematics test again in 2000.

These high scores on the state mathematics test attracted much attention to the school. The state awarded the school $25,000 for high scores on the 1999 state test. Obviously, parents' confidence was renewed. Throughout the school system there seemed to be much conversation about this school's high scores on the state mathematics test, and in particular, much interest about African-Americans students achieving an 80 to 90% passage rate on the state mathematics test for two years in a row. This impressive passage rate for African-American students is much higher than the average for African-American students in this school district, in the state, and in the nation.

African-American parents and students seem to be very proud of the school's performance in mathematics. At parent-teachers meetings, open-houses, and school socials, parents seem to be filled with pride and confidence about the school. At the fourth grade graduation party in June 2000, parents gave a "standing ovation" for the school principals.

Overshadowing Socio-Political Environment

Teachers and principals at this school continue to be influenced by certain socio-political factors. The two most prominent factors seem to be the constant media debate about mathematics instruction and the politics of the state testing. This school's efforts to reform instruction exist within a larger social and political debate about mathematics standards and mathematics instruction. For about ten years, and especially during the last school year, newspapers, news journals and internet sites devoted much space to debating what has been called "math wars." The New York Times, Wall Street Journal, Time, Newsweek, and other print media published many articles about new mathematics instruction according to NCTM Standards. Often the debate centered around basic mathematics calculation skills versus mathematics problem-solving within a constructivist learning theory. The media created two opponents: On one side stood basic mathematics instruction, on the other side was open-ended mathematics problem-solving that valued multiple solutions and creative thinking ("fuzzy math").

Educators focusing on reforming mathematics instruction at this school were often bombarded with media stories that distorted and confused the fundamental issue. Principals and teacher-leaders at the school had learned that constructivist theory was a theory beyond basic skills, not a replacement for basic calculation skills. For the principals and most teachers at this school there was no dichotomy between basic calculation skills and mathematics problem-solving. The real problem with traditional mathematics instruction was the acceptance of limited mathematics knowledge, the traditional belief that elementary students achieve mathematics knowledge when they can add, subtract, multiply and divide quickly and skillfully. Most educators and parents at the school learned that facility with calculation was only part of the overall goal for understanding mathematics. Put simply, children needed to learn both calculation and the use of calculation in the context of relevant mathematics problems. They also needed to understand that there are multiple strategies for solving mathematics problems. Students needed to experience mathematics as a creative study of patterns and relationships (Burns, 1992; Wheatley & Shumway, 1992).

Another factor that continued to put pressure on the principals and some teacher-leaders was the complexity of the state testing. Teachers and principals struggled with the effort to maintain a constructivist-learning environment while they were also required to prepare students for the kinds of questions asked on the state test. Many teachers addressed this challenge by maintaining creative, constructivist teaching and by adding extra instructional time to teach the type of questions asked on the state mathematics test. Extended time after school, before school, and on Saturdays helped many students and allowed teachers to teach more constructively during the regular school days. Despite recent success on the state mathematics test, some parents and educators at this school think that the teachers were much more creative with their mathematics instructional activities before the state test existed. However, no one can underestimate the impact of the state testing. The community realtors, local government le aders, and parents know the school choice decisions, property values, racial balance and the economic level of the community can rise and fall on the outcome of state test scores. With such high stakes, much competition exists among schools, which leads to jealousy, mistrust and accusations of cheating.

Conclusion

Efforts to implement the NCTM Standards must meet the challenges and complexities of designing more authentic assessment practices. Knowledge and understanding of mathematics may not be adequately assessed with specific answer or simple multiple-choice questions. If students are learning to apply mathematics to real life situations, they must learn to do problems that mirror the experiences they have now and the experiences to come as they learn more about mathematics.

New instructional theories and methods probably require new types of assessment approaches. Teaching and learning environments that respect misunderstandings and allow students the freedom to make mistakes, reflect on their mistakes, and reconstruct meaning out of mathematical situations must have an assessment process that is consistent, trustworthy, and open. New assessment systems will probably conflict with traditional high-stake testing procedures that are inclined to be closed, secretive, and final.

Educators at this school are trying to work with these assessment issues by combining traditional assessment with more authentic performance task assessment. They teach basic skills, and they also move beyond basic skills to teaching and learning experiences where students and teachers construct meaning of main mathematical concepts and processes by drawing, writing, dialoguing and building concrete models.

APPENDIX I

Snapshot Performance Tasks K-4

CHANCE

Kindergarten:

Ms. Grieshop's kindergarten class decorated special shirts. Each child had to glue a red, blue, and yellow button straight down the front of each shirt. Ms. Grieshop did not want all the shirts to look exactly the same. How many different ways can the three buttons be arranged so that all the shirts are different?

First Grade:

Brandon is going on vacation for one week. Each day he wants to wear a different outfit. He packed three T-shirts: a red, a blue and a yellow one. He packed black shorts and brown shorts. He also had two kinds of shoes, sandals and gym shoes. How many different outfits could he wear? What would they look like?

Second Grade:

In the Duck Pond game, twelve, large plastic ducks float around in the baby wading pool. Although all the ducks look the same from the top, their stomachs are painted red, blue or green. If a player picks a duck with a red stomach, then the player wins a book from the bookstore. How could the colors of the duck's stomachs be painted so that the player would?

--Be certain to win a book?

--Be likely to win a book?

--Equal chance to win a book?

--Not be likely to win a book?

--Be impossible to win a book?

Third Grade:

You were asked to design three spinners for the "Spin to Win" game at the Lomond Pumpkin Affair. Each spinner must have at least three different colored sections. Red must be one of the colors in all three spinners:

Spinner One--Design the spinner where red is most likely to win.

Spinner Two--Design the spinner where red is likely to win.

Spinner Three--Design the spinner where red is not likely to win. Test the three spinners. Collect and record the data Explain the results.

Fourth Grade:

For a mathematics project, Paula designed a game of chance. She challenged her class to figure Out two things. First, does everyone have the same chance to win? Second, is there a strategy for winning or is the winner just "lucky"--anyone can win. Here were the rules to Paula's game:

Materials: two dice, paper for recording numbers

1. Players select a number from 2-12.

2. Players take turns rolling the two dice. The sum of the two dice is recorded by one player.

For example, if a player rolls a "4" and a "6", the number recorded by the player is "10."

3. Players continue to take turns until the dice has been rolled fifty times.

4. The player whose sums (number) were rolled most wins the game.

LENGTH (distance)

Kindergarten

The beanstalk that Jack planted grew really fast! After five days the beanstalk grew 15 unifix cubes tall. How much did the beanstalk grow each day if it grew the same amount each day?

First Grade:

Jack's beanstalk is 16 units long. The giant's beanstalk is 32 units long. Jack's mother's beanstalk is 24 units long. Compare the height of all three beanstalks. What did you find out? Explain your solution using pictures, words and numbers.

Second Grade:

Randy bought a 72-inch long submarine sandwich for his sleepover party. He invited three friends to the party. Submarine sandwiches cost $2.50 for every twelve inches of length. If Randy divided the sandwich evenly between himself and his three friends, how long a sub sandwich will each person get to eat?

Third Grade:

Everyday Sean took Rex, his grandmother's dog, for a long walk around the nearby high school. The total distance Sean and Rex walked each day was 3.5 (3-1/2) kilometers. Erica, Sean's cousin who lived next door, took her dog Spot for 5.25 (5-1/4) kilometer walks four times a week. In one week, who walked the dogs farthest?

Fourth Grade:

Rapunzel, a heroine in a Grimm's fairy tale, never cut her golden hair. It was so long that when she was imprisoned in a castle's tower, she let her hair hang down outside the turret window so people on the ground could climb up to see her! Human hair grows about 1/2 inch each month. If the distance from the turret window to the ground is twenty feet, how old was Rapunzel in the fairy tale?

TIME AND MONEY (rate)

Kindergarten:

Robert saved $2.00 every day for six days. How much money did Robert save in six days?

First Grade:

Shenise "dog-sat" for her neighbor's dog for four hours. She "dog-sat" from 4:30 in the afternoon until 8:30 in the evening. Each hour Shenise dog-sat she earned 2 pennies, 1 dime and 1 nickel. How much money did Shenise earn dog-sitting?

Second Grade:

Tonya wanted to buy a bunny, case and a month's supply of bunny food. The total amount of these three things was $36.00. Tonya earns $4.50 each week doing chores around the house. Design three ways that Tonya could save her money to buy the bunny and the other items. Explain which of the three plans you would tell her to use and why.

Third Grade:

Ellen earns money baby-sitting. She wants to save her money to buy a portable CD player that costs $128.00. Ellen charges $4.50 per hour to baby-sit during the day and $5.50 per hour after 8:00 p.m. Mr. Holmes hires Ellen to watch his two grandchildren every Saturday in May from 3:30 p.m. until 11:00 p.m. How much money does Ellen earn in one Saturday?

Fourth Grade:

Every summer Jessica's father drives her to Chicago, Illinois to visit family and friends. Jessica and her father live in Shaker Heights, Ohio. The distance between Chicago and Shaker Heights is about 360 miles. Usually, her father drives about 65 miles an hour. The car can travel about 20 miles on one gallon of gasoline. Gasoline costs about $1.30 per gallon. How long will it take them to get to Chicago?

AREA (covering)

Kindergarten:

Baby Bear was going on a picnic and she wanted to take a lot of brownies with her. Baby Bear's mother had three different size trays *. One tray was a square and two trays were different size rectangles. How many brownies does each tray hold? Which tray holds the most, which holds the least brownies?

* Students are given "paper" trays and color tile "brownies" to solve the problem.

First Grade:

Little Red Riding Hood built a new cabin in the woods for her grandmother. The cabin had four rooms: a living room, a kitchen, a bedroom and a bathroom. The total area of the cabin was 24 square units. Design a floor plan that has 24 square units. Show where all the rooms are. How many square units are in each room?

Second Grade:

Mr. Kmitt's class bought a gerbil, Squeaky, for their classroom. Each weekend, a student took Squeaky home in a little gift box. The boys and girls decided to make a "paper carpet" for the bottom of the gift box. They measured the bottom of the box. Each side measured 7 units. What was the area of the bottom of the box?

Third Grade:

Jasmine bought carpet for her bedroom. Her room measured 10 feet x 15 feet. What was the area of the bedroom? How much floor space is left after Jasmine puts her furniture in her room? Design a scale model of the floor plan of Jasmine's bedroom. Include her furniture in your scale model.

Jasmine's Bedroom Furniture:

l desk        4 ft. X 2 ft.
1 bookcase    3 ft. X 1 ft.
1 nightstand  1 ft. X 2 ft.
1 dresser     3 ft. X 2 ft.
2 beds        7 ft. X 4 ft. (each bed)

Fourth Grade:

Mrs. MacGregor decided to build a large pen for her dog, Rex. Mrs. MacGregor bought the supplies at the hardware store. She purchased 36 meters of wire fencing. The fencing cost $9.00 per meter. She paid a 7% sales tax on the fencing. Draw and label some of the possible dimensions for Rex's pen. What pen will provide the maximum area for the dog?

VOLUME (filling)

Kindergarten:

Marcus had three different sized plastic glasses *. He wanted to find out which glass would hold the most amount of milk. How could Marcus find out which glass holds the most? Which glass holds the least? What glass holds a "middle" amount? Draw a picture of your solution.

* The shape of the glasses is deceiving so that the students can't tell by the height of the glass.

First Grade:

You are going to take fudge to a Cleveland Indians baseball game. Your mother has given you three boxes* for carrying the candy. You want to take the most amount of candy with you. How would you decide which box holds the most fudge? Compare the volume of all the boxes.

* Students are given a variety of jewelry gift boxes. Second Grade:

Cinderella's fairy godmother gave her a magic cube on her first birthday. Each edge of the cube was one unit long. When Cinderella woke up on her second birthday, the magic cube had grown. It now measured two units on each edge. On every birthday, the cube magically grew one unit longer on each edge. Record the dimensions and volume of Cinderella's magic cube for each of Cinderella's first four birthdays.

Third Grade:

Charlie, the owner of a chocolate factory, invented chocolate sugar cubes to make milk into hot chocolate. Each cube is one cubic centimeter. He packed the sugar cubes in boxes that measured 4 cm. X 5 cm. X 3cm. Charlie sold seven boxes of chocolate sugar cubes. He made $8.75. How many chocolate cubes fit into one box?

Fourth Grade:

Ryan used the following recipe to make punch for his birthday party.

Ingredients                  Cost of Ingredients

3/4 liter cranberry juice    $2.40 per liter
.5 liter apple juice         $1.60 per liter
1 liter gingerale            $1.00 per liter
250 ml. lemonade (optional)  $ .80 per 500 ml

Approximately how many people would this recipe serve, if everyone had 500 ml of punch to drink?

APPENDIX II

Targeting Performance Tasks: Teacher-Designed Instruction

* James has a piggy bank. Each day he puts two pennies in his bank. How much money will James have after one week? (Time and Money lesson- kindergarten)

* Henry earns $1.00 each hour for helping his grandmother plant vegetables in her garden. Henry works for 4 hours. How much money did Henry earn? (Time and Money lesson-first grade)

* Jamil baby-sits for his cousin, Tim. Jamin earns $3.00 each hour. Jamil babysat from 1:30 p.m. until 5:30 p.m. for Tim. How much money did Jamil earn babysitting his cousin? (Time and Money lesson-second grade)

* Ellen earns money baby-sitting. She wants to save her money to buy a portable CD player that costs $128.00 Ellen charges $4.50 per hour to baby-sit during the day and $5.00 per hour after 8:00 p.m. Mr. Homes hires Ellen to watch his two grandchildren every Saturday in May from 3:30 p.m. until 11:00 p.m. How much money does Ellen earn in one Saturday? How many days will it take Ellen to buy the CD player? (Time and Money lesson-third grade)

* Charlie, the owner of the Chocolate Factory, has invented chocolate sugar cubes that when dissolved in water, make hot chocolate. Each sugar cube is one cubic centimeter. Charlie packs the sugar cubes in boxes that measure 4 cm X 5 cm. X 2 cm. Charlie sold 7 boxes of chocolate sugar cubes for a total of $14.00. How many cubes will fit in one box? How much does one box of chocolate cubes cost? How much does one sugar cube cost? (Volume lesson-third grade)

* A plant grows 3 unifix cubes long every day. How tall will the plant be at the end of six days? (Length Lesson-kindergarten)

* Cindy's rope was 15 unifix cubes long. Tommy's rope was 7 unifix cubes long. How much longer was Cindy's rope than Tommy's? (Length Lesson-first grade)

* Mrs. Johnson, the art teacher, bought 60 inches of yarn for five students to use on their art projects. Mrs. Johnson wanted to give each student the same amount of yarn. How much yarn will each student get to use in his or her project? (Length lesson-second grade)

* Sally and Shawn were practicing for a race at school. Sally ran 4.5 kilometers each day for one week. Shawn ran 5.5 kilometers each day for six days. How many kilometers did each child run to practice for the race? (Length lesson-third grade)

* The MacGregors decided to build a pen for their dog, Rex. At the hardware store, Mrs. MacGregor purchased 36 meters of wire fencing. The fencing cost $9.00 per meter. To pay for the purchase, she handed the clerk 5 twenty-dollar bills, 1-one hundred-dollar bill, and 4 fifty-dollar bills. There was a 7% tax on the wire fencing. How much change will Mrs. McGregor receive back? Draw and label some of the possible dimensions for Rex's pen. What pen gives the maximum area for Rex to run? (Length lesson-fourth grade)

* Jasmine's parents bought carpeting for her bedroom. Her room measured 10 ft. X 15 ft. The carpet Jasmine chose cost $15 dollars a square yard. Carpet is sold in 12-foot widths. What is the area of the bedroom and how much carpet will Jasmine's parents need to buy to completely cover the bedroom floor'? Jasmine's bedroom furniture includes:

l desk         4 ft. X 2 ft.
1 bookshelf    3 ft. X 1 ft.
1 dresser      3 ft. X 2 ft.
1 small table  1 ft. X 2 ft.
1 bed          6 ft. X 3 ft.

Make a floor plan of Jasmine's bedroom. (Area lesson-third grade)

* Ms. Riley's fourth grade class is making candy bags for the Pumpkin Affair. The class bought a large box of candy bars, a large box of gummy worms, a large box of sourballs, and a large box of gum. All the candy was individually wrapped. Each candy bag had to have four pieces of candy. How many different combinations of candy can be put in each bag? After the students were finished, they placed all the candy bags in a large box. Marcia asked, "What are the chances of me reaching into the box and pulling out a bag that just had four candy bars in it?" Can Marcia do this? Justify your thinking. (Chance lesson-fourth grade)

One fourth grade teacher designed the following mathematics project to incorporate measurement, statistics and graphing.

Instructions for Students:

* Choose a unit of measure to measure several objects in our classroom. Collect, record, organize and display your data. Illustrate, write and compute your findings.

* Measure the circumference of heads of all students in this class...Collect, record, organize and display the data in a meaningful way.

* Measure other parts of the body. Make comparisons between head size and other measurements. What did you find out?

* Compare head sizes to the size of baseball cap sizes. Organize and display your findings to show: (1) head size and cap size, (2) the range between largest and smallest cap size, (3) the range between the largest and smallest head size, and (4) the median number for head and cap sizes.

* Self evaluation: On the back...include our class rubric and use the scale 1-4. [Rubric categories mutually agreed upon by the fourth grade class were: description of problem, methods used to solve problem (pictures, words, numbers/calculations), neat and colorful work, organization, accuracy, and justifying solution (reasoning)] (Pourdavood, Cowen, Svec, Skitzki & Grob, 1999, p. 40).

REFERENCES

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

Cobb, P., Wood, T., & Yackel, E. (1990). Classroom as learning environments for teachers and researchers. Journal for Research in Mathematics Education, 4, 25-146.

Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175-190.

Cohen, D.K. (1990). A revolution in one classroom: The case of Mrs. Oublier. Educational Evaluation and Policy Analysis, 12(3), 311-329.

Cowen, L., Alig, S., Bannon, P., Federer, J., Haas, R., Nader, J., Schlien, L., Skitzki, R., Smith, K., Strachan, T., Svec, L., & Thornton, R. (1996-1997). Snapshots: A mathematics assessment framework. Unpublished manuscript. Shaker Heights City School District.

Grundy, S. (1987). Curriculum: Product or praxis. Philadelphia: The Falmer Press.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

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

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

Pourdavood, R.G., & Fleener, J.M. (1998). The ecology of a dialogic community as a socially constructed process. Teaching Education, 9(2). HYPERLINK http://www.teachingeducation.com.vol9-2/pourdavood.htm.

von Glasersfeld, E. (1995). A constructivist approach to teaching. In L. Steffe & J. Gale (Eds.), Constructivism in education (pp. 3-16). Hillsdale, NJ: Lawrence Erlbaum Associates.

Wheatley, G.H., & Shumway, R. (1992). The potential for calculators to transform elementary school mathematics. In J.T. Frey & R. Hirsch (Eds.), Calculators in mathematics education, NCTM Yearbook, (pp. 1-8). Reston, VA: NCTM.

Wheatley, G.H., & Reynolds, A.M. (1999). Coming to know numbers. Mathematics Learning, Tallahassee, FL.