Quantity, Fractions, and Assessment
6/26/08
This article is a review of articles regarding teaching numeracy and quantity, including fractions, and the use of assessment generally.
The first part is a review of an article titled “Young Children’s Representations of Groups of Objects: The Relationship Between Abstraction and Representation”. The authors of the article are Yasuhiko Kato, Constance Kamii, Kyoko Ozaki and Mariko Nagahiro. The article was published in the Journal for Research in Mathematics Education, Volume 33, No. 1, on pages 30-45.
The article’s primary focus is comparing each subject’s level of abstraction and level of representation in mathematics. The purpose of the article seems to be to compare these two concepts and further define a child’s development in each. In the author’s words, “we decided to study more specifically the relationship between children’s construction of number through constructive abstraction and their development in representing numerical quantities.” In a sense the authors are trying to reconcile the difference in development between number sense and numerical representation.
The study used 60 Japanese children between 3 and 7 years old to determine their ability to represent quantity and their ability to understand quantity without the benefits of a picture or graphic. The authors developed two different scales, one to measure the development of representation and the other to measure the development of the ability to create mathematical abstractions.
The concept of representation was very clearly defined in the article as written symbols that either represent quantity (like 5, five, etc.) or represent operations on quantities (like +, -, etc.). The first step in the survey was to create a spectrum of types of notation to represent a group of objects, like 3 balls or 5 houses. At the first level, children can only make a mark for each of the objects (3 balls = ), and at the final level, children can write the quantity and the description (3 houses = 3 houses). The authors then compared the children’s representation scale to their ability to create mental abstractions of quantities.
In order to create a similar scale for the children’s ability to create mental abstractions of quantities, the authors created a scale, where the lowest level is the child’s inability to create a one-to-one correspondence between a graphically recreate a given group of objects (i.e., 5 given circles is drawn as 4 circles). The highest level is when a child creates a one-to-one correspondence, and additionally is able to conserve numbers (i.e., “o o o o” has the same number of o’s as “oooo”).
The study found that children cannot create abstract concepts before they are able to represent those concepts (e.g., numerical).
Thus, the information provided in this article implies assessing students’ ability to represent quantities and operations, and then assessing their ability to understand the abstract concept of quantity. This could be done by creating a variety of methods assessing whether students are “seeing” any taught mathematical concept, whether it is multiplication, fractions, or even geometric figures. This would correspond to the “abstraction” concept, and that would create a “ceiling” at the level that a child can represent a concept or quantity mathematically. It would be silly, according to the article, to expect a student to write 21 + 12 = 33, if they can’t grasp the abstract concepts of addition, and the quantity of the addends.
Now, onward to learning fractions. This is a review of an important article regarding Math Education, titled “Initial Fraction Learning by Fourth- and Fifth-Grade Students: A Comparison of the Effects of Using Commercial Curricula With the Effects of Using the Rational Number Project Curriculum”. The authors of the article are Kathleen A. Cramer, Thomas R. Post, and Robert C delMas, all of the graduate school of the University of Minnesota. The article was published in the Journal for Research in Mathematics Education, Volume 33, No. 2, on pages 111-144.
The article’s primary focus is comparing two different approaches for the initial teaching of fractions. One method, called the Commercial Curriculum (“CC”) method, essentially is the more traditional, rote and procedure-based method. The other method is called the Rational Number Project (“RNP”), which is a newer method, relying more on “multiple physical models and translations within and between modes of representation – pictorial, manipulative, verbal, real-world, and symbolic.”
In this article the authors appear to be unbiased in their comparison of the two methods of teaching fractions. However, it is important to note that the study was partly funded by the RNP itself, and this could impact their desire to publish negative results of RNP.
The study used a sample of 66 fourth and fifth grade classrooms in a suburban school district in Minnesota. The teachers were randomly assigned to 3 groups, either the CC Group, the RNP group, or a control group. The teachers then taught their classes using either method for 30 days. The instruments to measure student achievement at the end of this period consisted of written tests and student interviews. In addition, 4 weeks after the study, teachers administered retention tests. The authors used Factor Analysis and MANOVA as statistical methods to compare results from the 3 groups.
The study found that overall the RNP Group performed significantly better than the other groups. The RNP Group outperformed the other groups both in the written tests and in the student interviews, where students were asked to discuss their thought process behind their answers, whether correct or incorrect.
The chief impact this article makes on me is the importance of using manipulatives, and more specifically a variety of manipulatives and other physical representations. The RNP method uses many exercises and lessons based on translating among these different representations. Again, this makes sense, since each type of physical representation gives a different perspective of the concept of fractions. The ability of translate freely among them would seem to enhance the overall concept, much like the old tale of the blind men and the elephant trying to figure out what they are feeling. (It is interesting to note that the total amount teaching time was lower with the RNP method than the CC method, and the article also states that teacher preparation is easier using the RNP method rather than the CC method.)
The RNP method is also aligned with the NCTM standards, which makes its teaching more defensible in terms of other state-wide standardized tests.
I recommend using the information provided in this article by showing students are variety of methods of “seeing” any mathematical concept, whether it is multiplication, fractions, or even geometric figures. Perhaps even developing worksheets that ask students to translate among the representations would be useful even if our school does not use the RNP methods.
Assessment is absolutely vital to teaching math, but all other subjects as well. It is probably more important than pedagogy, if I can be so bold. The problem occurs daily, and it’s painful. A student looks at his grade and asks a question about it. I look at the work, and the grade, and quickly find a “rationale” for the grade. The student learns nothing along the way, gets no ability to justify or argue his case, and lastly probably sees that the grade and comments are not well thought out. Often the student walks away with confirmation that the system in general and I specifically failed her. And that hits a nerve.
The reason this problem is painful for me is that I have come to feel that many of the teachers in the SURR schools are “passing the buck” so to speak. I often hear teachers complain about (a) the students’ intellectual ability, (b) the students’ work ethic and general attitude, (c) the lack of interest of the students’ parent/guardians, (d) the administration, (e) the Board of Education, etc. “We didn’t know things would be this bad…I can’t believe all the fighting in class...no respect for authority…etc.”
I think the blame lies solely within ourselves. Why? Because there ARE teachers that can do it. They exist in my school, and most probably in others. In these classrooms, on-task children are working in groups in a quiet classroom. They are learning, and there is no screaming. The teachers I have in mind arrive at 6:45am and leave at 4:30pm every day. The work during their prep, and often their lunch. Yes, while it is painful to admit, I think the problem lies in our abilities, and it’s simply incorrect to blame others.
In some ways I think that when I grade poorly, I am cheating the children, and have completely lost my right to complain.
“I have become what I sought to replace.”
THE ACTION PLAN
As stated above, the most significant problem I face currently in the classroom is assessment. While I am able to get the students to focus and generally do their class work and homework, I would like to give them comprehensive, transparent and ultimately useful assessment. Primarily, I would like to use this assessment for the students themselves; however, I obviously need to “test” the students so that the administration can properly collect an objective collection of measures for students’ long-term history. Ideally, though, I would like to give students meaningful feedback about their work without discouraging them, or with very clear methods of improvement. Also, I would like to grade fairly and consistently across students.
My action plan will involve (1) developing clear rubrics for students and parents to use to “verify” my grading rationale, (2) provide students with clear and useful grades and comments on their work in a timely manner, so that they can use these comments to build even better understanding of the material, and provide them a time (perhaps after school) to discuss/argue any issues, and (3) develop methods for creative or poor-testing students to demonstrate their mastery of the material without compromising the assessment process.
Rubrics: I will provide students with an example of “excellent”, “average”, and “poor” work, as well as a rubric describing the grading. Student will be given these rubrics, and I will also have one posted on the wall, so that in some instances all students will know almost exactly what grade they will get before they even hand anything in. This concept also helps me, the teacher, become a more consistent grader, and I will average out the grades I give to ensure that, generally, students are graded consistently over time, rather than being dictated by my mood or subconscious.
The use of a rubric is supported by research in the text and in the handouts given in class. The following are some citations from a handout regarding the use of rubrics:
“Students become more involved in their work and more interested in their progress as they become more accountable to themselves. In addition, parents seemed better able to accept evaluation at the end of the term.” (Marchall, 1960) (This points to the fact that students should be given a clear explanation of exactly what is expected of them, I.e., a rubric.)
In another study, teachers graded the same paper with widely varying results, often as much as 25 points. And in yet another study, “180 geometry teachers, each scoring the same test paper, gave grades from 38 to 87.” A rubric would give teachers a better ability to grade fairly and consistently.
Useful Comments: I want the papers that I return to have meaningful and useful comments on them. Students should learn from those comments. Often teachers simply put smilely faces or “good job” words of encouragement on the paper, without instead providing thoughtful remarks on improvement as well. I also think that students should be given time to actually read those comments rather than stuffing them into a desk (or in the garbage, depending on the grade given).
Importantly, I would lay clear goals for the grades to be given to parents on the report card. The reasons for this are that students often don’t know exactly what goes into their final report card grade, and while this gives teachers a more mystical power, it also creates unneeded anxiety in our students. Instead, students will learn how averages work, and their grades will for the most part be a composite of the grade they achieve during the year. This also creates more accountability for the students and their parents, as the lack of clear grading puts more subjective and nebulous power into the teacher’s hands, whether or not they want it.
Providing meaningful comments has also been the subject of studies, and the following as just a few mentioned in the text and in the handouts:
“Students who had the closest relationship to the teacher achieved greater academic success than students who were not as close.” (Schmuck and Van Egmund 1965) (This shows that meaningful comments, which constitutes some relationship, can help students academically.)
In a large study student papers were given grades and either no comments, specific predetermined comments list “let’s raise that grade” or free comments which the teacher felt would be helpful. Students who received free comments …showed a significant improvement…” (Page, 1958) (This shows very clearly that there is utility in providing comments.)
Anxiety raised the grades of high ability students and lowered the grades of middle ability student.” (Phillips, 1962) (Again, here we see clearly that clarity of expectations, which would reduce anxiety, will help those that I teach.)
Alternative Testing: I will develop alternative types of tests that students can use, after they have demonstrated that the tests typically given are not best. In other words, if a student clearly doesn’t perform well on a test, rather than providing him with an alternative I would meet with parents and with previous teachers of the student to determine whether the issue is truly with the test. This is because I don’t think tests are biased in any meaningful manner, nor are they “confusing” if one really knows the material.
The tests I would develop would be based on the material, but would use more creative ways for students to demonstrate their knowledge. For example, students could write essays that would contain the same facts and explanations as a multiple choice test, without the confusing “wrong” answers included. This, to me, is fine. Lastly, some students may be able to develop even more imaginative ways to demonstrate knowledge (plays, songs, poems, etc.) However, I would try to keep this top a minimum, since this could create confusion and poor behavior management in the classroom.
Alternative testing, while potentially cumbersome, can create self-esteem in exactly those students who need it most. The following quotes from the test and the handouts show that, sometimes, the tests themselves are the problem:
Traits which describe individuals characterized as “creative” tend to be significantly different from the traits recognized in “achievers”. The grade-conscious achiever tends to be less willing to take risks, more subject to group pressures, less dominating, more persistent and has a stronger superego.” (Holland, 1960) It seems clear that the grading system, at all levels including the graduate one, tends to penalize the imaginative student who is likely to make a significant contribution to nearly any field.” (Miller, 1967) (So here we see that teachers need to address the most difficult question: What am I assessing with this test? I have no problem with children demonstrating mastery in unorthodox ways.)
CONCLUSION
In conclusion, quality feedback to students requires:
(1) Increased interest in their performance;
(2) Increased performance;
(3) Decreased arguments and bad attitudes (from students and parents) upon reading their grades and feedback; and
(4) Increased participation from students that otherwise do not exhibit interest in school in general, or a particular subject.
I would expect that these measures could be assessed and reviewed at least every marking period, as I as the teacher must justify my grades and opinions on each student to the student himself, the student’s parents, and the administration.