DIMS Exemplar Set of Items 4 th -Grade Mathematics Wylie and Ciofalo (2008) identified a set of common misconceptions that cut across both mathematics and science, and provided a framework for teachers to use to identify other student misconceptions. The eight categories of misconceptions are listed and described below. Following this is a sample set of items taken from the larger set of 4 th -grade mathematics DIMS items that illustrate many of these categories. Categories of Common Student Misconceptions in Mathematics and Science Failing to recognize the limitation of diagrams, models, and other representations: Diagrams, models, and other types of representations used in scientific and mathematical discourse and materials can be inaccurate, incomplete, or poorly scaled, or the limitations are not articulated. While a teacher may understand limitations, students are more literal. For example, in science, some students may interpret a commonly-used poster of the solar system to mean that, since the Sun is located on the left with the planets arranged horizontally, Jupiter is the center of the solar system. In mathematics, younger students rarely see triangles that do not have a horizontal base and may describe them as upside-down triangles. Going too far with abstractions, generalizations, and simplifications: Characteristics and descriptions of mathematical procedures, expressions, or concepts (and scientific groups or concepts) can be too abstract, general, or simplified. For example, students need to understand when certain mathematical truths no longer universally apply; e.g., addition always results in a larger number (until negative numbers are involved). In science, students may think that all birds fly, all bacteria are harmful, and all metal/silvery objects are attracted to magnets. Confusing language and vocabulary: Words and phrases used in everyday communication can convey a confusing or completely different meaning in scientific or mathematical communication. For example, a student meeting the term continental drift for the first time may apply what he or she knows about the meaning of the word, and as a result, may think that the continents literally float upon the oceans. Similarly, mathematical terms such as mean, plane, and point have meanings that are different from everyday language. Accepting facts : Facts based solely on intuition or faulty reasoning can be misleading, inaccurate, or incomplete in relation to both scientific and mathematical concepts. For example, students sometimes incorrectly think that the first digit to the right of the decimal point is the oneths (trying to be symmetrical around the decimal point). In science, students may think that heavier objects fall faster, deserts are always hot, or gasses have no mass. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 1
Applying real-world experiences and perceptions which can contradict scientific learning: Everyday life and perceptual experiences can be misleading, inaccurate, or incomplete in relation to various scientific explorations, studies, or concepts. For example, the five human senses seem infallible to young students, yet human vision differs from certain insects that see in the UV portion of the spectrum; it is hard to believe that sounds exist that humans cannot hear; touch is deceptive and so wood feels warmer than metal in same room. Accepting common sayings, beliefs, and myths: Widely heard/communicated sayings, beliefs, or myths can be misleading, inaccurate, or incomplete in relation to various scientific explorations, studies, or concepts. For example, some students believe that humans coexisted with/killed the dinosaurs; meteors are falling stars, and camels store water in their humps. Applying metaphors and analogies that only partially explain a scientific concept: Metaphors and analogous comparisons of scientific objects or concepts with everyday terms can be misleading, inaccurate, or incomplete in relation to various scientific concepts. For example, the following analogies are partly helpful, but each has its limitations: electricity as water flowing; the heart as a pump; the eye as a camera. Accepting false equivalencies: Various mathematical procedures or concepts are sometimes viewed incorrectly as being equivalent. For example, students may try to manipulate fractions using whole-number reasoning, think that calculations involving time are the same as working with decimals, or assume that subtraction and division are commutative, like addition and multiplication. About the Sample Set of Items Presented Each item is presented along with the correct answer (answers) and the misconception category (categories) that is (are) addressed. For relevant answer choices, a longer explanation is provided regarding possible reasons that students may or may not select those choices. The explanations are tied to the common misconceptions that certain students bring to the classroom every year, along with suggested origins for those misconceptions. Also included are some caveats and suggestions for teachers to be aware of and think about as they teach the various concepts, procedures, and investigations each year. The items can be used as presented, or modified according to student needs. For example, an item could be read to the class or students could be asked to read it; answer choices could be revealed one at a time and students asked to think about each one; some answer choices could be removed (paying attention to how answer choices correspond to potential misconceptions) to simplify the item; students could work alone or in pairs, etc. The primary goal is to provide information that will be useful to inform next instructional steps. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 2
What s Parallel? Mathematics, Q4 55 01 Which of the following pairs of lines below is parallel? Correct answer: All Misconception category: Diagrams, Models, and Other Representations This 4 th -grade mathematics question addresses a common misconception regarding the concept of parallel lines. (B) and (E). Students who do not select these answer choices are failing to recognize the limitations of a conventional representation. While teachers may understand those limitations, students are often more literal. Since students likely see choice (A) the conventional form of parallel lines that is ubiquitous in text books and worksheets most often, they may not think of these choices as being parallel. Many students also harbor the misconception that orientation is a key attribute of geometric lines, angles, and shapes. Somehow tipping the vertical parallel lines or completely turning them to the horizontal transforms them into simply lines. (C) and (D). Students who do not select these answer choices mistakenly think that similar length is a key aspect of lines being parallel. Again, they rarely see forms of parallel lines where the lines are of unequal length. Teachers are generally surprised to realize how conventional representations of mathematical shapes and concepts they have displayed and used in their classrooms for years or that appear in textbooks and worksheets can often cause or reinforce common student misconceptions. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 3
Identifying Fractional Parts of a Whole II Mathematics, Q4 04 02 Which of the following circles has one third shaded? Correct answers: (B), (C), (D), (F) Misconception categories: Abstractions, Generalizations, and Simplifications Diagrams, Models, and Other Representations This 4 th -grade mathematics question addresses two different types of misconceptions regarding the concept of the fraction one third. (A) and (E). Students who select these answer choices are simplifying the concept of fractions. These students have observed, read about, or been told that the fraction one third or 1/3 means that an object or group of objects has been divided into three equal parts. They are simplifying that knowledge by thinking that, when dealing with any fraction, the parts do not have to be equal. They are basically ignoring a key aspect of the mathematical concept. (C), (D), and (F) Students who do not select these answer choices are failing to recognize the limitations of a conventional representation. While teachers may understand those limitations, students are often more literal. Since students likely see choice (B) a conventional form of representing one third, most often in textbooks and worksheets, they may not think of the other choices as representations of one third. In both cases, teachers can help these students build a deeper understanding of fractions by providing them with multiple representations, contexts, and opportunities with which to explore and expand their current thinking. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 4
What s a Right Angle? Mathematics, Q4 50 01 Which of the following is a right angle? Correct answer: All Misconception categories: Language and Vocabulary Diagrams, Models, and Other Representations This 4 th -grade mathematics question addresses two different types of misconceptions regarding the concept of a right angle. (C) Students who do not select this answer choice may be confusing language and vocabulary. In this case, the word right used in everyday communication conveys a meaning that is confusing or different from its use in the mathematical concept of a right angle. These students generally know that right either refers to being correct or to a specific side or direction. Therefore, they may think of choice (C) as a left angle. (A), (D), (E), and (F). Students who do not select these answer choices are failing to recognize the limitations of a conventional representation. While teachers may understand those limitations, students are often more literal. Since students likely see choice (B) the conventional form of a right angle that is ubiquitous in text books and worksheets most often, they may not think of the other choices as right angles. Many students also harbor the misconception that orientation and length of the lines are key attributes of geometric lines, angles, and shapes. Teachers need to be alert for words and terms used in everyday language that can convey a confusing or completely different meaning in mathematical communication and endeavor. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 5
Identifying Rectangles Mathematics, Q4 52 04 How many rectangles are in the picture? (A) 2 (B) 4 (C) 5 (D) 6 Correct answer: (D) 6 Misconception category: Commonly Accepted Facts This 4 th -grade mathematics question addresses a common misconception regarding the concepts of squares and rectangles. (B) 4. Students who select this answer choice are accepting a fact that has been told to them or that is based on intuition or faulty reasoning. These students think that squares are not rectangles. They often have been taught about squares and rectangles at different times, and so they think that they are completely different geometric shapes. The fact that the names square and rectangle are not related also adds to the idea that the two shapes must be unrelated. Unfortunately, this is the case with the entire family of quadrilaterals. (A) 2. Students who select this answer choice probably think that squares are not rectangles and that rectangles that have been oriented vertically (the ears) are also not rectangles. Teachers need to teach general concepts but then be careful to address special cases and relationships to both deepen and broaden students understanding of a concept. In this instance, students should learn enough about the key aspects of the various 4-sided shapes to recognize that squares are just a special case. This is similar to triangles ( 3-sided shapes ) and special case triangles, but the fact that squares and rectangles have different names further complicates student understanding. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 6
Calculating Length of Song Mathematics, Q4 42 01 The chart below shows the length of 3 of 4 songs on a CD. If the total time for all 4 songs is 20 minutes, and there are no gaps between the songs, what is the length of song 4? Song 1 2 3 4 Length (minutes:seconds) 5:14 4:57 6:03? (A) 3 minutes, 46 seconds (B) 3 minutes, 86 seconds (C) 4 minutes, 26 seconds Correct answer: (A) 3 minutes, 46 seconds Misconception category: Equivalencies This 4 th -grade mathematics question addresses a common misconception regarding the numerical relationship between hours and minutes. (B) 3 minutes, 86 seconds and (C) 4 minutes, 26 seconds. Students who select these answer choices are treating hours and minutes as if they are decimals related to each other by a factor of ten. They are incorrectly viewing the two mathematical concepts as being equivalent. This misconception is inadvertently reinforced for some students by the ubiquity of digital clocks in the world today. They are not consciously aware of the difference between the decimal point and the double dot. Teachers must be aware of the tendency of certain students to treat as equal various mathematical procedures and concepts. Because of our base-ten number system, it is difficult for some students to see any juxtaposed numbers, particularly those in the digital format of time, and to think of them as having a different relationship. Teachers should also relate to students the ancient connection of time and circular measurement with multiples of 12, long before base ten numbers were invented. Unpublished Work Copyright 2010 by Educational Testing Service. All Rights Reserved. 7