Significance Of Essay Type Test Items About Fractions

Forty-two states and the District of Columbia are now using the same math and English standards, but the tests they use to determine how well students have mastered them still vary significantly.

One of the goals of the Common Core State Standards was to be able to compare student performance from state to state on a yearly basis. Five years ago, it looked like that would happen. Nearly all Common Core adopters were in at least one of two national consortia that would be creating new exams to accompany the standards, the Smarter Balanced Assessment Consortium and Partnership for College and Career Readiness, known as PARCC.

Those numbers have dwindled. Just 20 states and the District of Columbia plan to give one of the two tests this spring. Others are back where they started: Using tests unique to their state. So even though, in theory, students in Connecticut, Wisconsin and Arizona are all learning the same thing, they’ll be measured differently.

The Common Core writers were very interested in improving how fractions are taught in U.S. classrooms, so we looked at six of these tests to compare how they deal with word problems involving fractions in the fifth grade: those from New York, Wyoming, Florida, as well as PARCC, Smarter Balanced and ACT Aspire, an exam made by the group that produces the college readiness exam. (So far, Aspire is only given in Alabama.)

Related: Common Core testing showdown in Massachusetts

As always, when we’re talking about testing, there are caveats. Even though all these questions deal with fractions, they may be testing different standards. They’re also a mix of actually tested items and sample questions. The sample questions never appeared on the tests but were published ahead of the exams to give teachers and students an idea of what to expect, and the actual test items were released after appearing on an exam last spring. In both cases, just because a type of question doesn’t show up here, doesn’t mean it wasn’t on the test – or won’t be on future tests.

In other words, without an army of undercover fifth-grade reporters spying for us, it’s impossible to do a comprehensive comparison of the exams.

Nevertheless, while we can’t draw conclusions about which test was best from this sample of questions, we can see some important differences in each one’s approach and how they differ – or not – from the old way of doing things.

Related: With all the tests scored, policymakers are grappling with what Common Core test results mean

Let’s start with the obvious: whether a question is multiple choice. Smarter Balanced and PARCC are computer based assessments, meaning it’s easy to go beyond multiple choice and require students to type in open-response answers. The contrast is seen most starkly when comparing these questions from ACT Aspire, a paper-based test, and PARCC.

It’s virtually the same question, but it’s much easier to guess the right answer on ACT Aspire than PARCC. That’s compounded by the fact that the correct answer on the ACT question, 36, is a clear outlier.

“You don’t want the correct answer to stand out,” said Andrew Latham, director of Assessment & Standards Development Services at WestEd, which developed test questions for both PARCC and Smarter Balanced. If students understand enough to know the answer must be greater than 9, they don’t have to do any math to get the right answer, so the question doesn’t necessarily test whether they’re able to divide by a fraction.

The test makers also made different choices about how many steps each problem would take to solve. Look at these two similar questions from Smarter Balanced and Wyoming’s state exam.

The Smarter Balanced question only requires one step to get the right answer, dividing 2 by 1/5. To get Wyoming’s test question right, students first need to be able to use the number line to figure out the shortest and longest distances before they can do the rest of the math. “If you can’t read a number line, it doesn’t matter if you can subtract fractions or not, you’ll get it wrong,” Latham said.

And that’s not a good thing, he added. “I prefer when you focus more on a given standard. If they got it wrong, I don’t know why they got it wrong.”

Phil Daro, one of the lead writers of the Common Core math standards, pointed out that students are asked to do multistep problems in the classroom, however. “You have to have them on the test,” he said.

Related: The surprising initial results from a new Common Core exam

Of course, paper-and-pencil tests don’t only rely on multiple-choice questions. They’ll also include open response questions, like the New York examples, where students are given a standard word problem and asked to show their work before writing down an answer.

These questions take more time and money to grade, and are more prone to human error while grading, but there are pluses. Like the write-in answer on the above PARCC example, short answer math questions eliminate students’ ability to guess with the added benefit of allowing students to get partial credit, if they set up the problem correctly but make an error adding two fractions, for instance.

PARCC sometimes mimics that process with a question like this, which requires students to first write out the expression they’d use to find the answer.

Regardless of whether a question is multiple choice or open response, clarity matters a lot. That’s a particularly important consideration on math tests, where you run the risk of a student getting a question wrong because of weak reading skills rather than weak math skills.

Daro pointed to this ACT Aspire question as an example of “inconsiderate” writing:

He suggested the phrasing could have been made better for a fifth grader by using consistent language and saying “Mario divided his circle into two equal sections.” And the instructions were particularly confusing. “Selecting a word that names the fraction?” Daro said. “That’s like a grammarian talking, not a fifth grader.”

The range of ambiguity in question phrasing is highlighted by these two PARCC items.

The first question’s wording is straightforward, the experts said. The second is more convoluted. In part it’s because the questions are attempting to assess different things. The first just checks if students can subtract fractions, while the second tries to measure students reasoning. (If you’re curious but all the fractions are making you cross-eyed, the correct answer is B and E.)

Daro criticized the test makers for using multiple choice at all to attempt to test how students think. “Multiple choice is the wrong genre for that,” he said. “Either have the kid produce the argument or show them a single argument and have them critique that.”

He also cautioned against blaming Common Core for poorly written test questions – particularly when many, if not all of these items could have been on old exams. Many people think that “whenever you see something that looks odd, it’s because of the Common Core, but that’s just not true,” he said. “Standards can differ in ways that don’t manifest in different items on a test.”

Read more about Common Core.

Learning about fractions is one of the most difficult tasks for middle and junior high school children. The results of the third National Assessment of Educational Progress (NAEP) show an apparent lack of understanding of fractions by nine-, thirteen-, and seventeen-year-olds. "The performance on fractional computation was low, and students seem to have done their computation with little understanding" (Lindquist et al. 1983, p. 16). Similar trends were observed in the first, the second, and the recently completed fourth National Assessments (Carpenter et al. 1978; 1980; Post 1981; Dossey et al. 1988). Even though operations on fractions are taught as early as grade 4, the second NAEP showed that only 35 percent of the thirteen- year-olds could correctly answer the test item 3/4 + 1/2.

The difficulty children have with fractions should not be surprising considering the complexity of the concepts involved. Children must adopt new rules for fractions that often conflict with well-established ideas about whole numbers. For example, when ordering fractions with like numerators, children learn that 1/3 is less than 1/2. With whole numbers, however, 3 is greater than 2. When comparing fractions of this type, children need to coordinate the inverse relationship between the size of the denominator and the size of the fraction. They need to realize that if a pie is divided into three equal parts, each piece will be smaller than when a pie of the same size is divided into two equal parts. With fractions, the more pieces, the smaller the size of each piece.

Another difficulty is that the rules for ordering fractions with like numerators do not apply to fractions with like denominators. In this situation, children can use their already learned ideas about counting. For example, a student might reason that the fraction 5/7 is greater than 2/7 because 5 is greater than 2.

When adding or subtracting fractions, children may have ideas about whole numbers that conflict with their ideas about fractions. An estimation item from the second NAEP reveals this phenomenon. Students were asked to estimate the answer to 12/13 + 7/8. The choices were 1, 2, 19, 21, and, "I don't know." Only 24 percent of the thirteen-year-olds responding chose the correct answer, 2. Fifty-five percent of the thirteen-year-olds selected 19 or 21-they added either the numerators or the denominators. These students seem to be operating on the fractions without any mental referents to aid their reasoning.

Clearly, the way fractions are taught must be improved. Because of the complexity of fraction concepts, more time should be allocated in the curriculum for developing students' understanding of fractions. But just more time is not sufficient to improve understanding; the emphasis of instruction should also shift from the development of algorithms for performing operations on fractions to the development of a quantitative understanding of fractions. For example, instruction should enable children to reason that the sum of 12/13 and 7/8 is about 2 because 12/13 is almost 1 and 7/8 is almost 1. Children should be able to reason that 1/2 + 1/3 cannot equal 2/ 5 because 2/5 is smaller than 1/2 and the sum must be bigger than either addend. Further, they should realize that 3/7 is less than 5/9 not because of a rule but because they know that 3/7 is less than 1/2 and 5/9 is greater than 1/2. Students who are able to reason in this way have a quantitative understanding of fractions.

To think quantitatively about fractions, students should know something about the relative size of fractions. They should be able to order fractions with the same denominators or same numerators as well as to judge if a fraction is greater than or less than 1/2. They should know the equivalents of 1/2 and other familiar fractions. The acquisition of a quantitative understanding of fractions is based on students' experiences with physical models and on instruction that emphasizes meaning rather than procedures.

This chapter presents suggestions for changes in the content and pace of instruction to help children develop both a quantitative understanding of, and skill in operating with, fractions. Specific suggestions are presented for what, when, and how fractions should be taught.

The preparation of this article was supported in part by the National Science Foundation (NSF DPE 84-70077). Any opinions, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation.

 

RECOMMENDATIONS FOR CHANGE

Many current textbooks introduce fraction concepts as early as grade 2, though the main work on them begins in grade 4. The scope and sequence of the topic in grades 4, 5, and 6 are remarkably similar. The naming of fractions and the ideas of order and equivalence are briefly presented at each grade level; the most instructional time is allocated to operations with fractions. Addition and subtraction of fractions with like and unlike denominators are initially taught in grade 4 and repeated in grades 5 and 6. Multiplication and division of fractions are introduced in grade 5 and repeated in grade 6.

The result of this repetitious scope and sequence is that none of the topics is taught well. We suggest that by postponing most operations with fractions at the symbolic level until grade 6 and using instructional time in grades 4 and 5 to develop fraction concepts and the ideas of order and equivalence, teachers will find that their students will be more successful with all aspects of operations with fractions and will have a stronger quantitative under- standing of them.

We shall present our recommendations in two ways: (a) general recommendations applicable to instruction at all grade levels and (b) specific changes for the primary and intermediate grades.

 

General Recommendations

  1. The use of manipulatives is crucial in developing students' understanding of fraction ideas. Manipulatives help students construct mental referents that enable them to perform fraction tasks meaningfully. Therefore, manipulatives should be used at each grade level to introduce all components of the curriculum on fractions. Manipulatives can include these models: fractional parts of circles, Cuisenaire rods, paper-folding activities, and counters (see fig. 13.1).
  2. The proper development of concepts and relationships among fractions is essential if students are to perform and understand operations on fractions. The majority of instructional time before grade 6 should be devoted to developing these important notions.
  3. Operations on fractions should be delayed until concepts and the ideas of the order and equivalence of fractions are firmly established. Delaying work with operations will allow the necessary time for work on concepts.
  4. The size of denominators used in computational exercises should be limited to the numbers 12 and below.
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