Pre-Assessment Analysis
I had my students take Pre-Assessment test on November 1st. Table 1 shows the results of all the students by topic. Chart 14 shows how well they did as a class on each topic by taking averages. Charts from 1 to 10 show what an individual student already knew or what he did not know.
The first question asked to find the least common multiple (LCM) of 8 and 10, and nine out of ten students answered correctly. TD answered the greatest common factor (GCF) instead of the LCM. The second question asked to find the least common denominator (LCD) of 5/6 and 1/9, and only three students wrote correct answers. Two of them wrote the GCF and two of them did not answer. Another two of them listed several multiples of two numbers correctly but did not choose the least one. The third question was 1/6+3/4, which asked to find the sum of fractions with unlike denominators. Four students answered correctly. SH wrote the GCF for her denominator instead of the LCD. TD simply used the denominator of the first fraction 6. AL and SM did not answer. JJ wrote just a random number. RS found the correct the LCD 12, but she did not find the equivalent fractions correctly. She wrote the same numerators 1 and 3, and rewrote 1/12+3/12 instead of 2/12 and 9/12.
The fourth question was 5/6-1/6, which asked to find the difference of fractions with like denominators. Five students answered correctly. EM and SS did not answer.TD subtracted both the denominator and the numerator of the second fraction from the first fraction and wrote 4/0. SH did not simplify his answer correctly. JJ wrote a random number again. The fifth question was 4 5/6+2 2/3, which asked to add mixed numbers with unlike denominators. Four students answered correctly. EM and TD did not answer. SH wrote the GCF instead of the LCM for her denominator. WT added correctly as 6 9/6 but did not simplify. Instead, he tried to write the mixed number as an improper fraction. He multiplied the whole number and the denominator to get the new denominator by writing 9/36. JJ wrote a random number. RS found the LCD correctly but did not add two fractions correctly and wrote 4 5/6+2 2/6=7 1/6. The sixth question was 3 5/8-1 1/8, which asked to subtract a mixed number with like denominators. Four students answered correctly. TD, SM and EM did not answer. RS found the difference correctly but made a simple mistake when she simplified it. She wrote writing 3 5/8-1 1/8 =2 2/4= 2 2÷2/4÷2=2 ¼. SH and AL found the correct difference but did not simplify their answers.
The seventh question asked to find the perimeter of a rectangle with the sides of 2 ¼ in and 4 1/16 in. Only two students found it correctly. SH and WT added only two sides, and SH used the GCF for his LCD when he added all sides. TD found the perimeter as 4 ½+8 2/16 but added the denominators instead of finding the LCD when he added the two numbers. EM also found the perimeter correctly as 4 ½+8 1/8. When she rewrote the formula, she wrote 1/6 instead of 1/8 and did not get the correct answer. AH rounded up 2 ¼ as 3 for one side instead of using the given number although she used the given number for the other side. JJ wrote a random number of 6. MM added two sides together twice but did not add them together to complete his work. RC found the area instead of the perimeter. SM did not write anything. RS added 2 1/16 twice instead of 2 ¼.
The eighth question asked to find two fractions that have unlike denominators that added to 5/18. Five students answered correctly. Two of them did not answer. AH found two equivalent fractions. JJ and RS separated the denominator into 8 and 10, and also separated the numerator into 3 and 2. Then they wrote 3/8 and 2/10 as their answers. The ninth question asked to measure a printed pencil. Five students measured correctly. SH did not answer. WT wrote something, but his handwriting was not readable. Three students measured wrong.
Overall, I decided that I should teach the students all the topics that I included in the Pre-Assessment because only about a half of the class understood each topic at most except the LCM as shown in Chart 14. I thought that they needed to learn what exactly the LCM, the LCD and the GCF were. Even though 90% of the students knew the LCM, some of them could not use it to find the LCD. Some of them also confused with the LCD and the GCF. Secondly, they needed to learn equivalent fractions about when and how they should use. Some students did not know that they needed to find equivalent fractions when they added and subtracted fractions. Thirdly, they needed to learn how to add and subtract fractions. Some of them added denominators as well as numerators. Fourth, they needed to learn how to add and subtract mixed numbers. Some of them subtracted the first fraction from the second one in subtractions. Others did not add whole numbers. Then, they needed to learn how to find the perimeters. Some of them added only two sides. They also needed to learn how to read measurements on a ruler. Some of them were not sure how to read smaller measurements than an inch.
Formative assessments through homework
First, I taught how to estimate the sum and the difference in fractions and mixed numbers. Table 2 and Chart 11 show how the students did on their homework. I found that they seemed to have understood how to estimate them fairly well by checking their homework except some fractions such as 1/3. One student rounded it down to 0 by paying attention to the numerator 1. Another student drew a number line and found it closer to ½ resulting in rounding it up to ½. TD found the exact values instead of finding estimations. SM did only a half of the homework.
The students also did pretty good jobs on adding and subtracting fractions except a couple of small mistakes. I kept telling them to remember to simplify fractions after finding the sums or the differences, but some of them still forgot to do so. SH and AL made many simple mistakes such as copying wrong numbers, writing wrong quotients when simplifying and misreading own handwriting. SM added denominators as well as numerators in additions. I tutored her after school to correct her misunderstanding, and she understood it.
The students started making more mistakes when they worked on adding and subtracting mixed numbers. The answers of WT were totally wrong. He added two denominators to get a new denominator and also simply added two numerators to get a new numerator without finding equivalent fractions. He also made mistakes in subtraction by subtracting the second denominator from the first to get a new denominator and subtracting the second numerator from the first one to get a new numerator. After I tutored him after school, he understood and redid them all correctly. Some students subtracted the first fraction from the second fraction when they found the first fraction was too small to subtract the second one. In the next lesson, I reviewed addition and subtraction of mixed numbers. I told the class to start with adding or subtracting fractions because the whole number could be changed after adding or subtracting fractions. I also told them that they must borrow 1 from the first whole number in case the first fraction was too small to subtract the second one. I added that they must remember to add 1 to the whole number if the sum of the fractions was greater than 1.
I thought that most of the students understood how to find the perimeters when I checked their homework. However, some students only added two sides of the polygons and forgot to add the rest. They made more mistakes when they had to add six sides of hexagons by forgetting to add some of the sides. Another type of the mistakes they made was not measuring sides correctly. I saw JJ using a yard stick to measure half an inch and did not read it correctly because he could not handle the ruler properly. I suggested them to use a ruler of an appropriate size to make measurement more accurate. I also suggested them to write marks on the numbers of the side lengths after they added so that they could avoid missing them out.
Formative assessments through games
We played math games to practice adding and subtracting fractions and mixed numbers for twenty minutes on November 14th and thirty-five minutes on November 15th. Table 3 shows the number of the problems they solved. Because the time played on the both days differed, I converted the data of the first day into corresponding values to the second day of 35 minutes-long to compare them as shown in Chart 12 and Table 4.
Two students scored slightly lower in the second game than the first game, and one student scored the same for both of the games. Seven students improved their scores by 0.3 to 6.9 points. JJ solved 1.5 more questions, MM and SC about three more questions, RC and WT five more questions, and RS seven more questions. Each individual solved two more questions in average. SH improved to 294%, RC and RS to 235%, and WT to 176% from the first game to the second game. Each individual improved to 135% in average. Most of the students showed their improvements in their understanding in adding and subtracting fractions and mixed numbers.
I think one of the major reasons why the scores on the second day increased was because the practice on the first day helped the students gain understanding the materials. Another reason might be the practice on the first day helped them gain skills in the task resulting in reducing mistakes and saving time for working with more questions. Third reason may be that familiarizing how to play the game helped them focus on solving questions. The fourth reason may be rewards. I gave rewards to a winning pair after they won on the first day. On the other hand, I announced that a winning pair would get an “arigato” before we started a game on the second day. The students might have gotten more motivated to win because of the rewards.
I found that repetition helped students gain knowledge and skills in adding and subtracting fractions and mixed numbers through the games. I want to continue to have students practice math problems in my teaching in the future. I also noticed that given them rewards is very effective to motivate their learning. I may keep using rewards to motivate them to learn math.
Post-Assessment Analysis
I had my students take the Post-Assessment on November 17th after they learned how to estimate, add and subtract fractions and mixed numbers. Table 5 and Chart 15 show the comparison between the Pre-Assessment and the Post-Assessment by topic. Table 6 and Chart 13 show the comparison among individual. Table 6 also shows the evaluations of their learning whether they exceeded, achieved or did not meet the goals.
Six students found the least common multiples (LCM) correctly in both questions 1 and 2. Two students wrote the greatest common factors (GCF) instead of LCM. One student did his multiplication incorrectly. One student answered only one of the questions. I think the reason why fewer students answered correctly compared with Pre-Assessment lies in the numbers I used in the Post-Assessment. The two sets of numbers12 and 18, and 6 and 27 were more difficult to find the LCM than a set of numbers 8 and 10 that was used in Pre-Assessment. Seven students found the least common denominators (LCD) in question 3 and 4 in the Post-Assessment. One student did not find LCD of 13 and 3. Two students multiplied incorrectly and wrote as 3x13=35 or 3x13=36. I think more students found the LCD more correctly than the Pre-Assessment because they have learned the LCD in the lessons.
Eight students added 5/3 and 2/3 correctly in question 6, which was an addition of fractions with like denominators. One student got a right answer as an improper fraction but changed into a wrong mixed number. Another student also got a right answer but did not simplify it. They seemed to have learned adding fractions with like denominators very well. All students also have learned how to find difference of fractions with like denominators and got the right answer for 5/7-3/7 in question 5. Questions 11 and 12 asked to add mixed numbers with unlike denominators, and five students answered both of them correctly. The rest of the students answered only one of them correctly. One student simplified her answer incorrectly after she added them correctly. Two students tried to find equivalent fractions but wrote wrong equivalent fractions. Two other students added fractions correctly but forgot to add a whole number together.
Questions 8 and 9 asked to add or subtract fractions with unlike denominators. Nine students were able to find the LCD of 5 and 4. On the other hand, seven students were not able to find the LCD of 18 and 8. I think certain pairs of numbers seem to make students more difficult to find the LCD. Nine students found the perimeter of a triangle correctly in question 16. One student knew how to find the perimeter but wrote a wrong equivalent fraction when she added the sides. I think the students did well on this question because they practiced calculating the perimeters a lot when they drew a house plan.
Three students correctly found two fractions or mixed numbers with unlike denominators that added up to 3 5/12 in question 17. Two students found two mixed numbers that added to 3 5/12 but left the denominators without simplifying them. Five students did totally wrong. Students would understand a whole number can be written as a composition of two different whole numbers. For instance, they could easily come up with the idea that 5 is made up with 2 and 3 or 1 and 4. However, they seemed to be confused when it comes to fractions. Many students could not think 5/12 could be written as a composition of two fractions such as 2/12 and 3/12 or 1/12 and 4/12. Three students measured a printed pencil correctly to the nearest eighth of an inch in question 15. One student forgot to write a whole number of an inch. Another student measured to the nearest sixteenth of an inch. The other student wrote three measurements including the right answer. Two of them measured wrong. One student answered as 2 inches. Many students were still confused with reading scales on a ruler because there are many ways to read them to the nearest of an inch including a quarter of an inch, an eighth of an inch and a sixteenth of an inch.
I compared the outcomes of Pre-Assessment and Post-Assessment by matching each topic. As a whole class, the students exceeded in learning how to find the LCD and showed the great learning growth from 30% in Pre-Assessment to 78% in Post-Assessment. Their learning also exceeded in adding and subtracting fractions with like denominators resulting in the growth from 55% to 86%. They exceeded in finding the perimeters as well by showing the gain from 25% to 89%. They achieved in learning how to add or subtract fractions with unlike denominators by showing a slight growth from 30% to 50%. They also showed the growth in adding or subtracting mixed numbers from 48% to 55%. Their learning did not meet in finding two fractions with unlike denominators by resulting in a decrease from 40% to 22%.
Table 5 and Chart 6 show the comparisons of the results of Pre and Post Assessments by individual. I decided to say that a student exceeded the learning goals if he earned either 75 points or 200% at least in the growth. EH, MM, RS, RC, SH and SM exceeded by showing their learning growth from 42.5 points to 82.5 points, from 74.5 points to 100points, from 58.5 points to 75 points, from 83 points to 87.5 points, from 16 points to 61 points, from 26 points from 68.5 points, respectively. Next, I determined that a student achieved the goals if he earned 60 points or 150%. JJ and TD achieved the goals by showing the growth to 162% and 151%, respectively. I came to a decision that a student did not meet the goals if he showed a regress. AL and WT did not meet the learning goals because they resulted in the decreases from 46 points to 40 points and 71 points to 66 points, respectively.
AL and WT usually did as well in class and on their homework as the group of the students who exceeded. However, both of them tended to hasten to get right answers when they solved problems. They wanted to know answers quickly, and they were likely to spend little time in thinking or reasoning. The group of the students who exceeded answered at least two higher order thinking questions correctly whereas AL and WT did only one of them correctly. I want to make sure that they would take more time in thinking and reasoning when they solve problems in the future to support them to increase their learning.
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