MATH 4100/Hazelwood Fraction Unit Post-test Analysis 1

Transcription

1 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 1 EFFECTS OF THE SMALL GROUP TEACHING EXPERIENCE: ARE APSU PRE- SERVICE TEACHERS MAKING A DIFFERENCE IN CHILDREN S UNDERSTANDING OF FRACTIONS? At APSU, the mathematics methods course, MATH 4100, has an associated field placement at two designated schools in the Clarksville-Montgomery County School System. For the past several years, MATH 4100 has included an extra experience at one of the two schools. The APSU faculty (Dr. Mary Lou Witherspoon and Dr. Andy Wilson) designed a six-day mathematics unit for our students to use in one of the schools. The administration works with us to determine the grade level, the topic, and the appropriate schedule. During the unit each pre-service teacher (PT) in our classes is assigned to no more than four children. Enough PT s are placed in each classroom so that all the children in that class are accommodated. During the academic year our PT s worked with all the fourth-grade pupils at Hazelwood Elementary School. Feedback from the classroom teachers, the PT s, and the pupils has been quite positive. However, the APSU faculty decided that we would like to have more information about the impact of this field placement on the children. Therefore, we administered a followup post-test from the most recent fraction unit to gain some indication of whether the children were gaining from the unit. For purposes of comparison, the post-test was also given to all the fourth-grade students at a second elementary school in the Clarksville Montgomery-County System where our students did not deliver the fraction unit. THE FRACTION UNIT For six days (February 8, 10, 11, 15, 17, and 18 of 005) the MATH 4100 PTs delivered a unit on fractions to all the children in the fourth grade at Hazelwood Elementary. The unit began with making sure that children could use the language of fractions and could translate that language into symbols. (For example, if the whole is cut into three pieces we call them thirds-- if four pieces, fourths, etc.) From there pupils explored equivalence of fractions using die-cut pieces and using a sharing scenario. Children also developed ways to determine whether a particular fractional amount was more or less than one-half. Finally, they solved word problems involving fractions and mixed numbers, most of which could be modeled. The PTs helped the children write the corresponding number sentences after the they had solved the problems. No explicit teaching of the fraction algorithms was done. Everything was approached through modeling and problem-solving. Improper fractions and mixed numbers were included in the unit alongside proper fractions, rather than being taught separately. Children were expected to explain their reasoning throughout the unit. THE FRACTION POST-TEST Three months after the unit was taught and after TCAP testing was completed, the APSU faculty asked the fourth-grade teachers at Hazelwood, where the unit WAS taught, and the fourth-grade teachers at another CMCSS school, where the unit was NOT taught, to give a seven-question fraction test to the children. Teachers were asked not to assist

2 MATH 4100/Hazelwood Fraction Unit Post-test Analysis the children in solving the problems. Neither APSU faculty nor PTs were present while the children took the post-test. After the post-tests were returned to APSU, a student worker was asked to assign identification numbers to the children s work. The id-number had no code indicating the school, and the id numbers were not ordered by school. She recorded which papers came from which school, but did not share that information until after the papers were scored. Tiffany Spicer, an experienced teacher who was also a graduate student, devised a rubric for scoring the post-tests with feedback from Dr. Witherspoon. After Ms. Spicer scored all the tests, she and Dr. Witherspoon re-examined them for consistency. Few changes were necessary. During the scoring process, neither Ms. Spicer nor Dr. Witherspoon knew which paper came from which school. After the work was scored, the papers were separated by school in order to look for differences. A chi-square analysis was used to compare the work from the children at the two schools on various aspects of each question. The analysis presented on the following pages is arranged by question. First the question is given, then the scoring rubric, then the chi-square table that shows the raw data and the statistic, and finally a brief summary. In the chi-square tables, the first column usually shows the score. The second and third columns show the numbers of students from each school who made that score. The fourth column shows the row totals. Hazelwood students are abbreviated as HS and students from the other school, XS.

3 MATH 4100/Hazelwood Fraction Unit Post-test Analysis Question 1 1. Draw lines to show how people could share these five cookies. Each person should get the same amount. Use all five cookies. Each person gets cookies. Question #1 Drawing 1 0 Drawing correct Explanation Correct Incomplete / Nonproportional but correct number of pieces (Close) Incorrect but close (reflects what was drawn, but drawing was nonexhaustive) Unequal or nonexhaustive Does not know how to name pieces (correct #of pieces based on drawing) Omit Incorrect or omit 65% of the HS and % of the XS answered this question correctly using proportional drawings. 9% of the HS students were able to name correctly the number of cookies each person should get. 15% of the XS named the fraction correctly. (Any fraction equivalent to 1 was accepted.)

4 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 4 Question. Draw a picture that shows what the fraction 4 8 means. Question # 1 0 Drawing Correct Close/ Nonproportional Incorrect Omit 85% of the HS drew a correct proportional picture, and 91% of the XS did so. This question, which asked for a drawing of a proper fraction did not show a statistically significant difference between the groups.

5 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 5 Question. Draw a picture that shows what the fraction 5 4 means. Question # Correct and used same unit for drawing whole number and fraction Correct, but used different units for drawing whole number and fraction Converted to with no drawing or incomplete drawing Incorrect Omit 49% of the HS and 5% of the XS were able to draw a correct representation of this improper fraction. (1 omit from XS) A common error on this problem was to draw a picture that represented 4 5 rather than 5. 60% of HS and 47% of XS who missed this 4 problem made that error.

6 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 6 Question 4 4. Draw a picture that shows what 1 4 means. Question # Correct and used same unit for drawing whole number and fraction Correct but used different units for drawing whole number and fraction Converted to 7 4 with no drawing or incomplete drawing Incorrect Omit 69% of HS and 5% of XS were able to draw a correct picture. (5 omits from XS) Note that in the rubric that to get the highest score (4), the student had to use the same 1 for the fractional part as used for the whole number. For example, if a student drew a circle to represent the whole number but used a square to illustrate the, that was scored a 4 rather than a 4.

7 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 7 Question 5 5. Circle all the fractions that are more than Question # Circled 6 8 Did not circle 8 6 Circled Did not circle Circled 5 Did not circle 5 Circled 1 4 Did not circle 1 4 5% of HS and 6% of XS were able to classify all four of the given fractions correctly. The tables on the following page shows the results for each of the four fraction choices.

8 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 8 1% of HS and 5% of XS indicated that 1 4 is greater than 1 8% of HS and % of XS indicated that. greater than 1. 5 is 15% of HS and % of XS did NOT indicate that 6 8 is greater than 1. 4% of HS and 8% of XS did NOT indicate that is greater than 1.

9 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 9 Question 6 6. = Question # Correct Rewrote Incorrect Omit 89% of HS and 77% of XS who answered the question gave an appropriate equivalent for. Most said either 1 or 1 whole. (HS--1 omit; XS--1 omits)

10 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 10 Question =. Tell how you figured this out. Question # Correct answer and provided an explanation Correct answer but stops at 5 4 ; or correct answer with no explanation Knows to add the whole numbers together, but cannot add the fractions Shows no understanding of what the fractions mean answer is unreasonable Omit 4% of HS and 14% of XS who answered this problem did so correctly in simplest form and clearly showed their thinking (scored 4). A score of indicated either an answer of 5 4 or an answer of 6 1 with no explanation. So technically these were correct answers. 6% of HS and 7% of XS had this less complete answer. (HS-1 omit; XS-8 omits) Conclusions There were statistically significant differences on six of the seven questions. In every case the difference favored Hazelwood, where the MATH 4100 students (PTs) had

11 MATH 4100/Hazelwood Fraction Unit Post-test Analysis 11 taught. This MIGHT suggest that the work done by the APSU students had some positive effect. However, there were no efforts made to gather baseline data on either group of students; therefore, any differences could have been attributed to the students previous background in fractions. Or differences could have been made by the classroom teachers rather than by the APSU students during the intervening months between the unit and the post-test. The APSU students did provide the following potential benefits during the unit: Each PT was responsible for no more than four children most had three. Therefore, they were able to individualize instruction. In fact, they were required to report on each child s progress at the end of each lesson and explain what modifications they would make in the next lesson. Each PT was equipped with manipulatives for the children to use. Each PT had received about two hours of instruction in how to use the manipulatives. Children were consistently expected to explain their thinking both orally and in writing. Each child s thinking was probed during each lesson so that the PT would know what modifications to make in instruction. Because children were working in small groups, they also were expected to listen to, learn from, and assist each other. As instructors of these pre-service teachers, we are encouraged by these results even though they do not prove the benefits to the children. In an on-going effort to evaluate the benefits of the unit experience to the children, this Fall Semester we would like to administer pre-tests as well as post-tests to the children. We are very grateful for the cooperation of the administration and the faculty from the two schools, and we look forward to working with both schools in the future. Respectfully submitted, Mary Lou Witherspoon Andy Wilson