Comparing Fraction Operations The Comparing Fraction Operations assessment is designed to elicit information about a common misconception that students have when comparing estimated results of different operations with fractions: Misconception 1 (M1): Multiplication Always Makes Bigger / Division Always Makes Smaller Although you can access the assessment here at any time, we strongly recommend that you reference the information below to learn more about this misconception, including how it appears in student work, and how to score pre-­‐ and post-­‐assessments once you have given them to students. Contents Topic Background: Learn about comparing fraction operations ................................................................................... 2 Student Misconceptions: Learn about student misconceptions related to the topic. .............................................4 Administering the Pre-­‐Assessment: Learn how to introduce the pre-­‐assessment to your students................6 Scoring: Learn about the scoring process by reviewing the Scoring Guide. ................................................................8 Sample Student Responses: Review examples of student responses to assessment items............................... 36 Administering the Post-­‐Assessment: Learn how to introduce the post-­‐assessment to your students..........46 Comparing Fraction Operations Topic Background
Learn about comparing fraction operations. Operations with fractions, particularly with multiplication and division, can be challenging for students if they do not yet have a solid understanding of fractions in relation to whole numbers. Students come into their work with fractions with a familiarity with whole-­‐number operations, yet those same operations with fractions can appear to follow a completely different set of rules than what students already know with whole numbers. For example, with whole numbers, multiplication “makes bigger” yet multiplying by a fraction less than one makes the result smaller. And the opposite is true for division. If students understand how fractions fit within the number system, they are better positioned to make sense of the seeming contradictions in fraction operations, and instead see them as a logical extension of the rules they already know. As students work with operations with fractions, they need ample opportunity to practice estimating the relative size of the result. Rather than focusing exclusively on what the answer will be (a focus on the numbers in the problem), students need equal opportunity to pay attention to the operations in the problem, estimating and discussing why sums, differences, products and quotients of fraction problems come out the way they do. Different representations of fractions can be useful in helping students make sense of these ideas about fraction operations. Giving students opportunities to explore and to create their own examples of situations that would correspond to different fraction operations can help solidify this understanding. Connections to Common Core Standards in Mathematics (CCSS) The CCSS outline specific understandings that students should be able to meet at each grade level. At grade 5, students should be able to: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.B.4a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) Interpret multiplication as scaling (resizing) 5.NF.B.5a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5.NF.B.5b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1. 2 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5.NF.B.7a. Interpret division of a unit fraction by a non-­‐zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. 5.NF.B.7b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. 5.NF.B.7c. Solve real world problems involving division of unit fractions by non-­‐zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-­‐cup servings are in 2 cups of raisins? 3 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Student Misconceptions Learn about student misconceptions related to the topic. When students are developing the understandings described above (see Topic Background), they can develop flawed understanding leading to misconceptions about various fraction operations. The following misconception about estimating and comparing the results of operations with fractions is targeted in the Comparing Fraction Operations assessments: Misconception 1: Multiplication Always Makes Bigger / Division Always Makes Smaller Students whose responses suggest misconception 1 overgeneralize from early work with whole numbers in which “multiplication always makes bigger” or “division always makes smaller.” They incorrectly assume this to be true in all cases of multiplication or division, regardless of the sizes of the fractions involved. Watch the brief video clip for a fuller description of this misconception. See included files. To see additional examples of student work illustrating this misconception, refer to page 36 of this document. 4 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Student Misconceptions Resources Bell, A., Greer, B., Grimison, L., & Mangan, C. (1989). Children's performance on multiplicative word problems: Elements of a descriptive theory. Journal For Research In Mathematics Education, 20(5), 434-­‐449. doi:10.2307/749419 Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics (draft): 3–5 Number and Operations—Fractions. Retrieved from http://ime.math.arizona.edu/progressions/#products Guiler, W. (1945). Difficulties in Fractions Encountered by Ninth-­‐Grade Pupils. In Elementary School Journal (pp. 146-­‐156). Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., & … What Works Clearinghouse, (2010). Developing Effective Fractions Instruction for Kindergarten through 8th Grade. IES Practice Guide. NCEE 2010-­‐
4039. Tirosh, D. (2000). Enhancing Prospective Teachers' Knowledge of Children's Conceptions: The Case of Division of Fractions. Journal For Research In Mathematics Education, 31(1), 5-­‐25. Van de Walle, J. A. (2007). Elementary and middle school mathematics (6th ed.). Boston: Pearson. 5 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Administering the Pre-­‐Assessment Learn how to introduce the pre-­‐assessment to your students. About This Assessment These EM2 diagnostic, formative pre-­‐ and post-­‐assessments are composed of items with specific attributes associated with student conceptions that are specific to estimating and comparing the results of operations with fractions. Each item within any EM2 assessment includes a selected response (multiple choice) and an explanation component. The learning target for the Representing Fraction II assessment is as follows: The learner will use estimation and/or mental math to compare the results of a pair of fraction operations, without having to complete the actual calculations. Prior to Giving the Pre-­‐Assessment •
Arrange for 15 minutes of class time to complete the administration process, including discussing instructions and student work time. Since the pre-­‐assessment is designed to elicit misconceptions before instruction, you do not need to do any special review of this topic before administering the assessment. Administering the Pre-­‐Assessment •
Inform students about the assessment by reading the following: Today you will complete a short individual activity, which is designed to help me understand how you think comparing the results of different fraction operation problems using estimation and/or mental math. •
Distribute the assessment and read the following: The activity includes six problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about estimating and comparing fraction operations, it’s important for you to include some kind of explanation in the space provided. This can be a picture or words or something else that shows how you chose your answer. You will have about 15 minutes to complete all the problems. When you are finished, please place the paper on your desk and quietly [read, work on ____] until everyone is finished. 6 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Administering the Pre-­‐Assessment Monitor the students as they work on the assessment, making sure that they understand the directions. Although this is not a strictly timed assessment, it is designed to be completed within a 15-­‐minute timeframe. Students may have more time if needed. When a few minutes remain, say: •
You have a few minutes left to finish the activity. Please use this time to make sure that all of your answers are as complete as possible. When you are done, please place the paper face down on your desk. Thank you for working on this activity today. •
Collect the assessments. After Administering the Pre-­‐Assessment Use the analysis process (found in the Scoring Guide PDF document under the Scoring Process section and found on page 8 of this document) to analyze whether your students have the misconception: Misconception 1 (M1): Multiplication Always Makes Bigger / Division Always Makes Smaller 7 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Learn about the scoring process by reviewing the Scoring Guide. The Comparing Fraction Operations assessment is composed of six items with specific attributes associated with a particular misconception that is directly related to comparing estimated results of different operations with fractions. We encourage you to carefully read the Scoring Guide to understand these specific attributes and to find information about analyzing your students’ responses. This Scoring Guide is intended for use with both the pre-­‐assessment and the post-­‐assessment for Comparing Fraction Operations. To use this guide, we recommend following these steps: • Read the Misconception Descriptions below, and be sure you understand what the misconceptions are. You may want to view the videos included. Numerous examples of student work illustrating the misconceptions are included in this guide, but you may also want to refer to the additional examples of student work found in the “Sample Student Responses” section. • Familiarize yourself with the six assessment items and what they assess. • Consider completing the optional scoring practice items and checking your scoring against the answer key. • Score your students’ work using the Pre-­‐/Post-­‐Assessment Analysis Process described below. • Refer to the various examples found here and in the “Sample Student Responses” section for guidance when you are unsure about the scoring. 8 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring PRE-­‐ASSESSMENT The assessment is composed of four items with specific attributes associated with understandings and misunderstandings related to an area model representation of fractions. Each item may elicit information about the students’ understanding of the equivalence of fractional parts that have equal area but may not be (or appear to be) the same shape or may be positioned differently within the figure. Item Understandings and Misconceptions •
Students with misconception 1 will tend to choose less than, reasoning that multiplication makes answers much bigger than addition does •
Students with misconception 1 will tend to choose greater than, reasoning that division makes answers much smaller than subtraction does. Item 1 Correct response: > Greater than Item 2 Correct response: < Less than Students with misconception 1 will tend to choose greater than, reasoning that multiplication makes answers bigger while subtraction makes answers smaller. •
Item 3 Correct response: < Less than 9 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Students with misconception 1 will tend to choose greater than, reasoning that multiplication makes answers bigger while division makes answers smaller. •
Item 4 Correct response: < Less than Students with misconception 1 will tend to choose less than, reasoning that division makes answers smaller while multiplication makes answers bigger. •
Item 5 Correct response: > Greater than NOTE: Students with misconception 1 will also tend to choose greater than, but by reasoning that multiplication makes answers bigger while division makes answers smaller. It’s important to consider the student’s explanation carefully for this item. •
Item 6 Correct response: > Greater 10 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Pre-­‐Assessment Analysis Process Some important things to know about the analysis process for this diagnostic assessment: •
This diagnostic assessment has been validated to reliably predict the likelihood that a student has Misconception 1. You can weigh the relative likelihood that your student has this misconception by considering whether the student’s written response provides “Strong Evidence” or “Weak Evidence” of the misconception. •
If a student is determined to show evidence of the misconception on even just one item, the student is likely to have that misconception. •
For each item, you need to look at both the selected response choice and the explanation. Students will show evidence of the misconception only if they select the corresponding response choice and have an explanation that supports the misconception. To learn more about how to tell whether an explanation supports a particular misconception, go to “Student Misconceptions” on page 36 and watch the videos provided. •
An optional scoring guide template is provided for your use when you score your own students’ diagnostic assessments. In each row of the assessment, write the name of one of your students. Then circle the appropriate information for each item on the pre-­‐assessment (shaded) and later the post-­‐assessment (in white). HOW TO DETERMINE IF A STUDENT HAS THE MISCONCEPTION 1. Identify any items for which the student has selected the M1 response choice. Table 1. Response Patterns for the Pre-­‐Assessment Ite
m # Correct M1 Likely Responses 1 > < 2 < > 3 < > 4 < > 5 > < 6 > > Note that for item 6, a response of “greater than” is correct and can also indicate the possibility of M1. Therefore, it is particularly important to also consider the student’s explanation in order to determine whether the misconception is present. 11 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring What if there’s no multiple-­‐choice response selected? In that case, carefully consider the explanation the student gives. If the explanation leaves no room for doubt about how the student is reasoning, you can code it Correct, M1, or M2 with “Strong Evidence” of the appropriate misconception. (For additional guidance on determining the strength of the evidence, see the “What counts…” information provided in Step 2 below.) However, if there’s no selected response chosen, and the explanation leaves some question about what the student was thinking, code it as “Other” and move on to the next question. 2. For each item with the M1 response choice, note whether the evidence of M1 from the explanation is strong or weak. If the student provides a response on any item that aligns with M1, look next at their explanation to determine whether it supports Misconception 1. An explanation can be categorized as Strong Evidence of M1, Weak Evidence of M1 or No Evidence of M1. A Caution! Table 1 shows most responses indicate either correct thinking or M1 reasoning. However, there are a number of other student difficulties that often are made apparent as students take this assessment. We provide more about these other difficulties in the Sample Student Responses document under in Sample Student Responses on page 36. So it is still necessary to check the student’s explanation to confirm evidence that the student’s thinking matches what the response indicates. It is not unusual for a student to choose a response that appears to indicate one thing but then to provide an explanation that does not necessarily support their choice. The upshot: Always check both the explanation and the selected response. An explanation can be categorized as “Strong Evidence” of a misconception, “Weak Evidence” of a misconception, or “Other” if it does not fit the typical reasoning indicative of those misconceptions. What counts as “Strong Evidence” of a misconception in the pre-­‐assessment? In general, responses with strong evidence of a misconception include a clear indication that the student is exhibiting the reasoning typical for that misconception. There is no need to make inferences about what the student is thinking; the thinking is quite clear from the combination of the selected response and the explanation. Below are two examples of student responses with strong evidence of the misconception, using pre-­‐assessment items. To see additional examples of student responses that illustrate these misconceptions, go to “Sample Student Responses” on page 36. Example A: Strong Evidence of M1 For students with M1, the explanation will include clear evidence that the student believes that multiplication always makes the results bigger, or that division always makes the results smaller. (For a more detailed description of this misconception, see the video found in the “Student Misconceptions” section.) 12 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring “Multiplying gets you bigger numbers then [than] when you divide.” For item 6, this student chooses “greater than,” which could indicate either correct thinking or the possibility of M1 thinking (see Table 1). However, the student’s explanation states that multiplying gets you bigger number than when you divide. So this is considered strong evidence of M1. Example B: Strong Evidence of M1 “I think the answer is less than because even though adding and multiplying are both m aking the numbers go up, when you add, you are “plusing” the 10. But when you multiply it’s making them even larger. For item 1, this student chooses “less than,” which indicates the possibility of M1 thinking (see Table 1). The explanation also clearly states that when you multiply, it makes the answer get even larger. This is strong evidence of M1. Can a correct response be considered to have “Strong Evidence”? Yes, a correct response can also have “Strong Evidence,” “Weak Evidence,” or no supporting evidence as well. While it is not necessary to categorize correct responses as strong, weak, or non-­‐existent for the purposes of this diagnostic assessment, you may want to note this on your scoring template for your own purposes. What counts as “Weak Evidence” of a misconception in the pre-­‐assessment? Responses with weak evidence of a misconception include some indication that the student is exhibiting the reasoning typical for that misconception. However, these responses also generally require making more inferences about what the student was thinking, or they leave some question or doubt as to whether the misconception is present or to what degree it is present. Below are two examples of student responses with weak evidence of the misconception, using pre-­‐assessment items. To see additional examples of student responses that illustrate these misconceptions, go to the “Sample Student Responses” on page 36. 13 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Example A: Weak Evidence of M1 “because one is multiplying and one is dividing and the fractions are the same.” For item 4, this student selects “greater than,” which indicates the possibility of M1 thinking. However, the student’s explanation doesn’t provide clear evidence that the student believes that “multiplication makes larger” and “division makes smaller.” This makes it “Weak Evidence” of M1. Example B: Weak Evidence of M1 “2/5 x 6/7 is the correct answer.” For item 1, this student chooses “less than,” which indicates the possibility of M1 thinking. However, it is unclear why the student chose this response, so it is considered “Weak Evidence” of M1. Example C: Weak Evidence of M1 “I think the answer is less than because multiplying and dividing are basicly [basically] the exact opposite.” For item 5, this student chooses “less than,” which indicates the possibility of M1 thinking. However, it is unclear whether the student believes that “multiplication makes bigger”/”division makes smaller,” so it is considered “Weak Evidence” of M1. 14 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring What if the student selects one of the choices, but provides no explanation? If a student selects a correct response choice or an M1 response choice but provides no explanation at all, then there is no evidence to support the choice, and should be scored as “Other” on the scoring template. What if the student’s choice matches a misconception and the student provides an explanation, but the explanation does not reflect the type of thinking typical of that misconception? If a student’s response choice suggests a possible misconception, but the student’s explanation does not support it, then the item is not considered to be indicative of the misconception, and again can be scored as “Other.” What other difficulties are likely to show up in students’ responses? While many possible difficulties related to operations with fractions may arise, there are several that appear frequently. For more information on these difficulties, go to “Sample Student Responses” on page 36. 3. After you have analyzed each item for a student, use the guidelines below to determine whether the student has Misconception 1. This diagnostic assessment has been validated to predict the possible presence of Misconception 1 for a student. That means that if a student shows evidence of the misconception on even just one item, the student is likely to have that misconception, regardless of whether the evidence is coded as “Strong” or “Weak.” The relative number of items with weak or strong evidence gives you information about how strongly the misconception may be present for the student. What if my student has only one item coded as M1 with “Weak Evidence,” and the rest are correct? Even if your student has only one item with “Weak Evidence” of a misconception, this diagnostic assessment is validated to predict that it is likely your student has that misconception. However, the presence of only one item with “Weak Evidence” of the misconception suggests that the misconception may not be very deeply rooted in this student’s thinking. You may want to keep an eye on this student during regular classwork to watch for other evidence of this misconception. 15 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring (OPTIONAL) SCORING PRACTICE ITEMS—PRE-­‐ASSESSMENT The following sample student responses are provided as an optional practice set. If you would like to practice scoring several items to further clarify your understanding of the scoring process, you may try scoring the following 10 items. We recommend scoring one or two at a time, checking your scoring as you go against our key, found on page 19. Practice Example 1 “If you multiply, you will get more than adding.” Practice Example 2 “When you multiply a fraction less than one, it will always come out smaller than it originally was.” Practice Example 3 “If you look 8/9 is bigger, and usually when you do division, you swap the numbers.” 16 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 4 “Its equivalent because you have to turn the divition into m ultiplacation.” Practice Example 5 “Dividing is like um well making it into parts, subtraction is just loseing numbers.” Practice Example 6 Practice Example 7 “Multiplying a fraction lowers a number.” 17 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 8 “When you m ultiply, the numbers get larger. When you divide, the numbers get smaller.” Practice Example 9 “Because of x and ÷ they are the same thing and that now are equivalent.” Practice Example 10 “I chose this answer because when you multiply 2 and 6, you get 12 and when you multiply 5 and 7, you get 35, so 12/35.” 18 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring SCORING PRACTICE ITEMS ANSWER KEY—PRE-­‐ASSESSMENT Practice Example 1 “If you multiply, you will get more than adding.” This is an example of M1 with “Strong Evidence.” The student selects “less than,” indicating possible M1 thinking. The student’s explanation adds that if you multiply, you will get more than adding, clear evidence of M1 reasoning. Practice Example 2 “When you multiply a fraction less than one, it will always come out smaller than it originally was.” This is an example of a correct response with “Strong Evidence” (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment; it simply gives you more information about your student). The student selects “greater than,” and accurately describes the effect of multiplying by a proper fraction. 19 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 3 “If you look 8/9 is bigger, and usually when you do division, you swap the numbers.” This is an example of M1 with “Weak Evidence.” The student selects “greater than,” indicating possible M1 thinking. However, the student’s explanation leaves it unclear why the student is comparing the size of 8/9 and 3/5, and why the product will be greater than the quotient. This lack of clarity makes it “Weak Evidence” of M2. Practice Example 4 “Its equivalent because you have to turn the divition into m ultiplacation.” This is an example of a response that would be coded as “Other.” The student selects “equivalent,” which is not correct nor does it indicate possible M1 thinking. It might indicate possible confusion between multiplication and division of fractions (see Sample Student Responses on page 36 for more discussion of this difficulty). Practice Example 5 “Dividing is like um well making it into parts, subtraction is just loseing numbers.” This is an example of a correct response with “Weak Evidence” (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment; it simply gives you more information about your student). The student selects “less than,” indicating possible correct thinking. However, the explanation leaves it unclear how the student is comparing division to subtraction. This makes it “Weak Evidence” of a correct response. 20 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 6 This is an example of a response that could be coded either as “Correct Weak” or as “Other.” The student selects “less than,” which is correct. However, the explanation shows that the student completed both problems to compare them. This inability to estimate to compare the results is one of the difficulties that students have, so you could code it as “Other.” Or you could choose to code this as “Correct Weak,” since the student chooses the correct selected response, but shows no evidence that he or she can estimate to compare the results. Practice Example 7 “Multiplying a fraction lowers a number.” This is an example of a correct response with “Weak Evidence” (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment; it simply gives you more information about your student). The student selects “less than,” indicating possible correct thinking. However, the explanation leaves it unclear how the student is comparing multiplication to subtraction. This makes it “Weak Evidence” of a correct response. Practice Example 8 “When you multiply, the numbers get larger. When you divide, the numbers get smaller.” 21 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring This is an example of M1 with “Strong Evidence.” The student selects “greater than,” indicating possible M1 thinking. The student’s explanation clearly states that multiplying makes bigger and dividing makes smaller, clear evidence of M1 reasoning. Practice Example 9 “Because of x and ÷ they are the same thing and that now are equivalent.” This is an example of a response that would be coded as “Other.” The student selects “equivalent,” which is not correct nor does it indicate possible M1 thinking. There is strong evidence that it indicates confusion between multiplication and division of fractions (see the Sample Student Responses on page 36 for more discussion of this difficulty). Practice Example 10 “I chose this answer because when you multiply 2 and 6, you get 12 and when you multiply 5 and 7, you get 35, so 12/35.” This is an example of M1 with “Weak Evidence.” The student selects “less than,” indicating possible M1 thinking. However, the student’s explanation leaves it unclear how the student is comparing the operations; the explanation merely explains how to multiply the fractions. This lack of clarity makes it “Weak Evidence” of M2. 22 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring P OST -A SSESSMENT I TEMS
The post-­‐assessment is structured exactly the same as the pre-­‐assessment, comprising four items with specific attributes associated with understandings and misunderstandings related to the area model representation of fractions. Each item may elicit information about the students’ understanding of the equivalence of fractional parts that have equal area but may not be the same shape. Item Understandings and Misconceptions Students with misconception 1 will tend to choose less than, reasoning that multiplication makes answers much bigger than addition does. •
Item 1 Correct response: > Greater than •
•
Students with misconception 1 will tend to choose greater than, reasoning that division makes answers much smaller than subtraction does. Item 2 Correct response: < Less than Students with misconception 1 will tend to choose greater than, reasoning that multiplication makes answers bigger while subtraction makes answers smaller. •
Item 3 Correct response: < Less than 23 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Students with misconception 1 will tend to choose less than, reasoning that multiplication makes answers much bigger than addition does. •
Item 4 Correct response: < Less than Students with misconception 1 will tend to choose less than, reasoning that division makes answers smaller while multiplication makes answers bigger. •
Item 5 Correct response: > Greater than NOTE: Students with misconception 1 will also tend to choose greater than, but by reasoning that multiplication makes answers bigger while division makes answers smaller. It’s important to consider the student’s explanation carefully for this item. •
Item 6 Correct response: > Greater 24 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring P OST -A SSESSMENT A NALYSIS P ROCESS
Some important things to know about the analysis process for this diagnostic assessment: •
This diagnostic assessment has been validated to reliably predict the likelihood that a student has Misconception 1. You can weigh the relative likelihood that your student has this misconception by looking at the number of responses coded as either “Strong Evidence” or “Weak Evidence” of Misconception 1. •
If a student is determined to show evidence of the misconception on even just one of items, the student is likely to have Misconception 1. •
For each item, you need to look at both the selected response choice and the explanation. Students will show evidence of Misconception 1 only if they select the M2 response choice and have an explanation that supports Misconception 1. To learn more about how to tell whether an explanation supports Misconception 2, go to the “Student Misconceptions” on page 4 and watch the video explanation provided. HOW TO DETERMINE IF A STUDENT HAS THE MISCONCEPTION :
The post-­‐assessment uses the same scoring process as the pre-­‐assessment. If you are not already familiar with the steps for scoring the assessment, please review that section starting on page 4. 1. Identify any items for which the student has selected the M1 response choice. Table 2. Response Patterns for the Post-­‐Assessment Ite
m # Correct M1 Likely Responses 1 > < 2 < > 3 < > 4 < > 5 > < 6 > > Note that for item 6, a response of “greater than” is correct and can also indicate the possibility of M1. Therefore, it is particularly important to also consider the student’s explanation in order to determine whether the misconception is present. What if there’s no multiple-­‐choice response selected? In that case, carefully consider the explanation the student gives. If the explanation leaves no room for doubt about how the student is reasoning, you can code it Correct, M1, or M2 with “Strong Evidence” of the appropriate misconception. (For additional guidance on determining the strength of the evidence, see the “What counts…” information provided in Step 2 below.) However, if there’s no selected response chosen, and the explanation leaves some question about what the student was thinking, code it as “Other” and move on to the next question. 25 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring 2. For each item with the M1 response choice, note whether the evidence of M1 from the explanation is strong or weak. If the student provides a response on any item that aligns with M1, look next at their explanation to determine whether it supports Misconception 1. An explanation can be categorized as Strong Evidence of M1, Weak Evidence of M1 or No Evidence of M1. A Caution! Table 1 shows most responses indicate either correct thinking or M1 reasoning. However, there are a number of other student difficulties that often are made apparent as students take this assessment. We provide more about these other difficulties in the Sample Student Responses section. So it is still necessary to check the student’s explanation to confirm evidence that the student’s thinking matches what the response indicates. It is not unusual for a student to choose a response that appears to indicate one thing but then to provide an explanation that does not necessarily support their choice. The upshot: Always check both the explanation and the selected response. An explanation can be categorized as “Strong Evidence” of a misconception, “Weak Evidence” of a misconception, or “Other” if it does not fit the typical reasoning indicative of those misconceptions. What counts as “Strong Evidence” of a misconception in the post-­‐assessment? In general, responses with strong evidence of a misconception include a clear indication that the student is exhibiting the reasoning typical for that misconception. There is no need to make inferences about what the student is thinking; the thinking is quite clear from the combination of the selected response and the explanation. Below are two examples of student responses with strong evidence of the misconception, using pre-­‐assessment items. To see additional examples of student responses that illustrate these misconceptions, go to “Sample Student Responses” on page 36. Example A: Strong Evidence of M1 For students with M1, the explanation will include clear evidence that the student believes that multiplication always makes the results bigger, or that division always makes the results smaller. (For a more detailed description of this misconception, see the video found under “Student Misconceptions” on page 4.) “I know this because multiplication will usually have a bigger answer than addition.” For item 1, this student chooses “less than,” which indicates the possibility of M1 thinking (see Table 1). The explanation also clearly states that multiplication will usually have a bigger answer than addition. This is considered strong evidence of M1. 26 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Example B: Strong Evidence of M1 “Dividing fractions makes the fraction more divided.” For item 2, this student chooses “greater than,” which indicates the possibility of M1 thinking (see Table 1). The explanation provides convincing evidence that the student sees division as making the answer smaller. This is strong evidence of M1. Can a correct response be considered to have “Strong Evidence”? Yes, a correct response can also have “Strong Evidence,” “Weak Evidence,” or no supporting evidence as well. While it is not necessary to categorize correct responses as strong, weak, or non-­‐existent for the purposes of this diagnostic assessment, you may want to note this on your scoring template for your own purposes. What counts as “Weak Evidence” of a misconception in the post-­‐assessment? Responses with weak evidence of a misconception include some indication that the student is exhibiting the reasoning typical for that misconception. However, these responses also generally require making more inferences about what the student was thinking, or they leave some question or doubt about whether the misconception is present or to what degree it is present. Below are two examples of student responses with weak evidence of a misconception, using post-­‐assessment items. To see additional examples of student responses that illustrate these misconceptions, go to “Sample Student Responses” on page 36. Example A: Weak Evidence of M1 “cuz there is bigger number.” For item 4, this student selects “greater than,” which indicates the possibility of M1 thinking. However, the student’s explanation doesn’t make it clear which problem results in a bigger number and why. This makes it “Weak Evidence” of M1. 27 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Example B: Weak Evidence of M1 “It’s multiplication.” For item 5, this student selects “less than,” which indicates the possibility of M1 thinking. However, the student’s explanation doesn’t make it clear how the student is comparing division and multiplication. This makes it “Weak Evidence” of M1. What if the student selects one of the choices, but provides no explanation? If a student selects a Correct response choice or an M1 response choice but provides no explanation at all, then there is no evidence to support the choice, and should be scored as “Other” on the scoring template. What if the student’s choice matches a misconception and the student provides an explanation, but the explanation does not reflect the type of thinking typical of that misconception? If a student’s response choice suggests a possible misconception, but the student’s explanation does not support it, then the item is not considered to be indicative of the misconception, and again can be scored as “Other.” What other difficulties are likely to show up in students’ responses? While many possible difficulties related to operations with fractions may arise, there are several that appear frequently. For more information on these difficulties, go to the “Sample Student Responses” on page 36. 3. After you have analyzed each item for a student, use the guidelines below to determine whether the student has Misconception 1. 4. This diagnostic assessment has been validated to predict the possible presence of Misconception 1 for a student. That means that if a student shows evidence of the misconception on even just one item, the student is likely to have that misconception, regardless of whether the evidence is coded as “Strong” or “Weak.” The relative number of items with weak or strong evidence gives you information about how strongly the misconception may be present for the student. What if my student has only one item coded as M1 with “Weak Evidence,” and the rest are correct? Even if your student has only one item with “Weak Evidence” of a misconception, this diagnostic assessment is validated to predict that it is likely your student has that misconception. However, the presence of only one item with “Weak Evidence” of the misconception suggests that the misconception may not be very deeply rooted in this student’s thinking. You may want to keep an eye on this student during regular classwork to watch for other evidence of this misconception. 28 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring (OPTIONAL) SCORING PRACTICE ITEMS—POST-­‐ASSESSMENT The following sample student responses are provided as an optional practice set. If you would like to practice scoring several items to further clarify your understanding of the scoring process, you may try scoring the following 10 items. We recommend scoring one or two at a time, checking your scoring as you go against our key, found on page 32. Practice Example 1 “because same numbers but multipacation [multiplication] is gonna beat division everry time.” Practice Example 2 Practice Example 3 “When divide fraction, you flip the secend [second] one upside down and then multiple [multiply] then which is a lot bigger.” 29 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 4 “You would have to find a common denominator for 5 and 12 then add.” Practice Example 5 “Cuz it depends whats is the number.” Practice Example 6 “You will get the same answer, it will be equivalent, because you multiply on both.” Practice Example 7 “Scence your [Since you’re] multiplying one fraction and adding the other, the one you added would be less than the one you m ultiplied.” 30 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 8 “I think that number 1 is smaller because if you use the hold switch flip method, you will multiply giving a bigger answer.” Practice Example 9 Practice Example 10 “It’s a bigger number.” 31 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring SCORING PRACTICE ITEMS ANSWER KEY—PRE-­‐ASSESSMENT Practice Example 1 “because same numbers but multipacation is gonna beat division everry time.” This is an example of M1 with “Strong Evidence.” The student selects “Greater than,” indicating possible M1 thinking (see Table 2). The student’s explanation then clearly states that multiplication will “beat” division every time. Practice Example 2 This is an example of a response that would be coded as “Other.” The student selects “Less than,” which indicates possible M1 thinking (“division makes smaller”). However, the explanation shows that the student has a different misconception regarding equivalent fractions. Because the explanation does not support M1 thinking indicated by the response choice, it is considered “Other.” Practice Example 3 “When divide fraction, you flip the secend [second] one upside down and then multiple [multiply] then which is a lot bigger.” This is an example of a correct response with “Strong Evidence” (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment; it simply gives you more information about your student). The student selects “Less than,” the correct response, then describes an accurate procedure for finding the answer. 32 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 4 “You would have to find a common denominator for 5 and 12 then add.” This is an example of M1 with “Weak Evidence.” The student selects “Less than,” indicating possible M1 thinking, but the student’s explanation is vague and incomplete. The lack of clarity around whether the student is in fact using M1 reasoning makes it “Weak Evidence” of M1. Practice Example 5 “Cuz it depends whats is the number.” This is an example of M1 with “Weak Evidence.” The student selects “Greater than,” indicating possible M1 thinking, but the student’s explanation is vague and incomplete. The lack of clarity around whether the student is in fact using M1 reasoning makes it “Weak Evidence” of M1. Practice Example 6 “You will get the same answer, it will be equivalent, because you multiply on both.” This is an example of a response that would be coded as “Other.” The student selects “Equivalent,” which indicates one of the common difficulties (confusing multiplication and division of fractions). The explanation shows that the student believes that you multiply on both, giving you the same answer. Because the explanation does not support M1 thinking indicated by the response choice, it is considered “Other.” 33 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 7 “Scence your [Since you’re] multiplying one fraction and adding the other, the one you added would be less than the one you m ultiplied.” This is an example of M1 with “Strong Evidence.” The student selects “Less than,” indicating possible M1 thinking (see Table 2). The student’s explanation then clearly states that adding will be less than multiplying. Practice Example 8 “I think that number 1 is smaller because if you use the hold switch flip method, you will multiply giving a bigger answer.” This is an example of a correct response with “Weak Evidence” (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment; it simply gives you more information about your student). The student selects “Less than,” the correct response, but is a little bit vague about how the multiplying will give you a bigger answer. This makes it “Weak Evidence” of M1. Practice Example 9 This is an example of a response that would be coded as “Other.” The student selects “Greater than,” which indicates possible M1 thinking (“division makes smaller”). However, the explanation shows that the student has a different difficulty with subtraction of fractions. Because the explanation does not support M1 thinking indicated by the response choice, it is considered “Other.” 34 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Scoring Practice Example 10 “It’s a bigger number.” This is an example of M1 with “Weak Evidence.” The student selects “Greater than,” indicating possible M1 thinking. The student’s explanation shows an incorrect calculation for the multiplication problem, but a correct calculation for the subtraction problem. It also does not provide convincing evidence that the student is thinking that multiplication always makes bigger” (because of the incorrect calculation for the multiplication problem). This makes it “Weak Evidence” of M1. 35 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Sample Student Responses Review examples of student responses to assessment items. The Comparing Fraction Operations diagnostic assessment focuses on a particular misconception that students have, regarding comparing different operations with fractions. Sample student responses indicative of the misconception are provided separately below along with samples of correct student responses. In order to determine the degree of understanding and misunderstanding, it’s important to consider both the answer to the selected response as well as the explanation text and representations. Misconception 1: Multiplication Always Makes Bigger / Division Always Makes Smaller Students whose responses suggest misconception 1 overgeneralize from early work with whole numbers in which “multiplication always makes bigger” or “division always makes smaller.” They incorrectly assume this to be true in all cases of multiplication or division, regardless of the sizes of the fractions involved. The following student responses show examples of this misconception. Item Sample Student Responses with Evidence of Misconception 1
Notes Pre-­‐
Assmt #1 • The M1 selected response is chosen AND • The student’s explanation states that multiplication is greater than addition. “I gave this answer because I know that multiplication is greater than addition.”
Post-­‐
Assmt #3 “2
3 1
3 1
* is greater than 2 − cause 4 8
4 8
• The M1 selected response is chosen AND • The student’s explanation states that the multiplication problem is greater because it’s multiplication. 36 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Sample Student Responses your [you’re] multiplying 2
but 2
3 1
* 4 8
3 1
− your [you’re] subtracting.” 4 8
Post-­‐
Assmt #2 “Minesing [Minusing] will not give you such a low number.” Pre-­‐
Assmt #5 “When dividing the numbers get smaller and when your [you’re] multiplying the numbers get larger.” • The M1 selected response is chosen AND • The student’s explanation states that division will result in a lower number than subtracting (minusing). • The M1 selected response is chosen AND • The student explains that dividing makes number smaller and multiplication makes numbers larger. • The M1 selected response is chosen AND • The student explains that multiplication makes the answer larger Pre-­‐
Assmt #4 “I gave this answer because if you multiply the fractions the fractions will be greater than if you devided [divided] them.”
37 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Sample Student Responses Post-­‐
Assmt #2 • The M1 selected response is chosen AND • The student explains that multiplication makes the answer larger ”When you subtract and when divide, the subtraction answer is always larger than the division.”
Correct Reasoning Students with correct reasoning about representing fractions are often able to do one or more of the following: •
•
•
•
•
Understand that the numerator and denominator indicate the number of shaded parts and the total number of parts, respectively Recognize the need for equal-­‐size parts Mentally or physically partition parts of the figure into equal-­‐size parts Ignore lines that subdivide a fractional part in order to see equal-­‐size fractional parts in a figure Recognize equal-­‐size fractions that are not the same shape Item Sample Student Responses with Correct Reasoning Post-­‐
Assmt #4 “I drew lines and realised that it made 10 pieces”
Post-­‐
Assmt #1 Notes • Correct selected response is chosen AND • Student correctly partitions the pentagon into tenths • Correct selected response is chosen AND • Student redraws figure and labels each unshaded part with a 4, indicating 4 quarters 38 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Sample Student Responses Post-­‐
Assmt #2 “Because to [two] of the tringes [triangles] are filled in the shape.” Post-­‐
Assmt #4 “There is 3 saded in [shaded-­‐in] airas [areas]” Post-­‐
Assmt #3 “There are half a tryangle [triangle—] put them together” Post-­‐
Assmt #2 • Correct selected response is chosen AND • Student correctly partitions the heptagon into sevenths AND • Student explains that 2 of the 7 triangles are filled in • Correct selected response is chosen AND • Student correctly partitions one-­‐fifth of the pentagon into tenths AND • Student explains that there are 3 shaded-­‐in areas • Correct selected response is chosen AND • Student says “there are half a [triangle—] put them together,” indicating the student understands that the 2 eighths represent 1/4 • Correct selected response is chosen AND • Student correctly partitions the heptagon into sevenths AND • Student draws a new figure showing 7 equal-­‐size parts, with 2 shaded in 39 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Sample Student Responses Incorrect Reasoning That is Not One of These Misconceptions In addition to the misconception described above, students may also exhibit one or more of several common difficulties related to comparing fraction operations. These difficulties are described below. In some cases, students may select the response for M1, but will not provide convincing evidence that he or she has the misconception, and may instead show evidence of one of these difficulties (though this is not an exhaustive list). Therefore, it’s important to always look at the student explanation in conjunction with the selected response. NOTE: This diagnostic probe is not validated to test for this other common difficulty. However, it may still be helpful to be aware of it as you look through your students’ responses, since you may notice this type of reasoning in some of your students’ work. Difficulty #1: Calculating Instead of Estimating Some students do not use estimation strategies and instead use a variety of calculation approaches. They may not understand the meaning of estimating, lack estimation skills, or be uncomfortable with not getting an exact answer. No Estimation, with Correct Calculations When students use calculations to answer the question, their explanations often do not provide evidence of reasoning about the size of the fractions or about a generalized understanding of the effect of the operations on the results. No Estimation, with Incorrect Calculations When students’ calculate rather than estimate, they may be applying a variety of incorrect calculation strategies including but not limited to: • Confusing multiplication and division • Inverting 1st fraction when dividing • Adding/subtracting numerators and denominators as if they were just whole numbers; no common denominators • Comparing fractions based solely on the size of the denominator or numerator • Incorrect use of equivalent fractions when finding a common denominator Note: This diagnostic probe is not validated to test for this other common difficulty. However, it may still be helpful to be aware of it, since you may notice this type of reasoning as you look through your students’ work. Also note: for some problems, this incorrect line of thinking results in choosing the correct selected response! So be sure to look at a student’s explanation as well as their selected response. 40 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Sample Student Responses Correct Calculations Item Sample Student Responses Notes • The correct selected response is chosen HOWEVER The student arrived at the correct response simply by doing both problems. Pre-­‐
Assmt #2 “I know that the first answer would be 1/8. And on the next problem is 1 1/6 so the 1st one is less than.”
Pre-­‐
Assmt #3 • The correct selected response is chosen HOWEVER • The student arrived at the correct response simply by doing both problems. Incorrect Calculations Item Sample Student Responses Pre-­‐
Assmt #1 Notes • The incorrect selected response is chosen AND The student does not estimate, but completes both problems, incorrectly adding the fractions as if they were whole numbers. 41 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Pre-­‐
Assmt #1 Pre-­‐
Assmt #5 • The correct selected response is chosen HOWEVER • The student does not estimate, but completes both problems, incorrectly multiplying the fractions. The student flips the second fraction (as if dividing), then adds the results as whole numbers. • The incorrect selected response is chosen AND • The student does not estimate, but completes both problems, incorrectly dividing the fractions. The students finds common denominators, then tries to divide the numerators, and keeps the common denominator. Difficulty #2: Confusing Division of Fractions with Multiplication of Fractions Some students confuse the steps to divide fractions with the steps to multiply fractions. They will tend to see the multiplication and division of the same pair of fractions as equivalent, since, in their minds, both problems involve multiplying the fractions. Note: This diagnostic probe is not validated to test for this other common difficulty. However, it may still be helpful to be aware of it, since you may notice this type of reasoning as you look through your students’ work. The following student response shows an example of this: Item Sample Student Responses Notes • The student chooses “Equivalent…” AND …explains that you just switch the division sign to a multiplication sign. Pre-­‐
Assmt #4 “It is equivalent because all you have to do is switch the divid 42 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations sing [divide sign] to a x sing [sign].”
• The student chooses “Equivalent…” AND • …explains that they both “come out to 8/15.” Pre-­‐
Assmt #4 • The student chooses “Equivalent…” AND • …explains that you have to convert division into multiplication. Pre-­‐
Assmt #5 “You have to convert division into multiplication so if each equation has the same numbers, it will be the same anwser [answer].” Other Miscellaneous Difficulties You will also likely see a variety of other difficulties that may suggest some other areas in which to follow up with your students. Here are several such examples (though not an exhaustive list!): Item Sample Student Responses Notes This student may be confusing numbers less than one with negative numbers. Pre-­‐
Assmt #2 “I think it is less because when you divid [divide] it comes to be a negative number.”
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Comparing Fraction Operations Pre-­‐
Assmt #3 • This student may understand that multiplying by a fraction often makes the result smaller than the first number, as does subtraction, but doesn’t realize they won’t have the same result. “I[t]will be equivalent because they will both be less then [than] the first number.” • This student appears to be subtracting only the denominators as whole numbers, and comparing 4 to 2 to determine which problem is greater or less than the other. Pre-­‐
Assmt #2 Correct Reasoning: Students with correct reasoning: •
•
•
Understand that multiplying fractions can result in a smaller number in the product than one or both of the factors; Similarly, they understand that dividing fractions can result in a larger number in the quotient than one or both of the factors; Can reason that when comparing multiplication and division with the same fractions (and the fractions are less than 1), the division problem will produce the larger answer because you’re multiplying by a larger fraction (the reciprocal). Item Sample Student Responses Notes This student may be confusing numbers less than one with negative numbers. Pre-­‐
Assmt #2 “I think it is less because when you divid [divide] it comes to be a negative number.”
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Comparing Fraction Operations Pre-­‐
Assmt #1 “Because when you multiply fractions, the number gets smaller and when you add up fractions, the number gets bigger.” • The correct selected response is chosen AND • The student is thinking correctly about both addition and multiplication with fractions. • Pre-­‐
Assmt #2 “I know this because when you divide with fractions, it gets bigger but when you subtract them it gets smaller.” • The correct selected response is chosen AND • The student is thinking correctly about both subtraction and division with fractions. • Pre-­‐
Assmt #2 “Well I know dividing fractions less then one makes the answer biger [bigger] subtracting makes it smaller.” • The correct selected response is chosen AND • The student is thinking correctly about both subtraction and division with fractions. • Post-­‐
Assmt #4 • The correct selected response is chosen AND • The student is thinking correctly about both multiplication and division with fractions. “9/13 ÷ 2/3 is more because if you divide fractions using the same numbers as the multiplication, you will get more.” 45 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Administering the Post-­‐Assessment Learn how to introduce the post-­‐assessment to your students. If the Comparing Fraction Operations pre-­‐assessment shows that one or more of your students have the misconception or any of the related difficulties that are outlined in the Scoring Guide, plan and implement instructional activities designed to increase students’ understanding. The post-­‐assessment provided here can then be used to determine if the misconception has been addressed. Prior to Giving the Post-­‐Assessment •
Arrange for 15 minutes of class time to complete the administration process, including discussing instructions and student work time. Since the post-­‐assessment is designed to elicit a particular misconception after instruction, you should avoid using or reviewing items from the post-­‐assessment before administering it. Administering the Post-­‐Assessment •
Inform students about the assessment by reading the following: Today you will complete a short individual activity, which is designed to help me understand how you think about comparing the results of different fraction operation problems using estimation and/or mental math, a topic we have been working on in class. •
Distribute the assessment and read the following: This activity includes six problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about estimating and comparing fraction operations, it’s important for you to include some kind of explanation in the space provided. This can be a picture, or words, or a combination of pictures and words that shows how you chose your answer. You will have about 15 minutes to complete all the problems. When you are finished, please place the paper on your desk and quietly [read, work on ____] until everyone is finished. 46 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fraction Operations Administering the Post-­‐Assessment Monitor the students as they work on the assessment, making sure that they understand the directions. Although this is not a strictly timed assessment, it is designed to be completed within a 15-­‐minute timeframe. Students may have more time if needed. When a few minutes remain, say: •
You have a few minutes left to finish the activity. Please use this time to make sure that all of your answers are as complete as possible. When you are done, please place the paper face down on your desk. Thank you for working on this activity today. •
Collect the assessments. After Administering the Post-­‐Assessment Use the analysis process (found in the Scoring Guide PDF document under the Scoring Process section and found on page 8 of this document) to analyze whether your students have these misconceptions: Misconception 1 (M1): Multiplication Always Makes Bigger / Division Always Makes Smaller Some students who previously had the misconception will no longer have it—the ideal case. Consider your instructional next steps for those students who still show evidence of the misconception. 47 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Scoring Guide Template – Comparing Fraction Operations
Student:
Pre # 1
Cor
Str M1 Wk
Pre # 2
Other
N/A
Cor
Str Other
N/A
Cor
Str Other
N/A
Cor
Str Post # 1
Cor
Str Student:
M1 Wk
M1 Wk
Student:
M1 Wk
M1 Wk
Other
N/A
Cor
Str Student:
M1 Wk
Other
N/A
Cor
Str M1 Wk
Other
N/A
Cor
Str Student:
M1 Wk
Other
N/A
Cor
Str M1 Wk
Other
N/A
Cor
Str Other
N/A
Cor
Str Student:
M1 Wk
Other
N/A
Cor
Str M1 Wk
Other
N/A
Cor
Str M1 Wk
M1
Wk M1
Wk M1
Wk Other
N/A
Cor
Str M1
Wk M1
Wk M1
Wk M1
Wk Other
N/A Cor
Str M1
Wk Other
N/A Cor
Str M1
Wk Other
N/A Cor M1
Str Wk M1
Wk Other
N/A Cor
Str M1
Wk Cor
Str M1
Wk Cor
Str M1
Wk Other
N/A Cor
Str Cor
Str Other
N/A Cor
Str Other
N/A Cor
Str Other
N/A Cor
Str Other
N/A Cor
Str M1
Wk Other
N/A Other
N/A Cor
Str Other
N/A M1
Wk Cor M1
Str Wk Cor M1
Str Wk Other
N/A Cor
Str M1
Wk Other
N/A Cor M1
Str Wk Other
N/A Cor
Str Other
N/A Cor M1
Str Wk Other
N/A Cor
Str Other
N/A Cor
Str Other
N/A Cor
Str Other
N/A Cor M1
Str Wk M1
Wk Other
N/A Cor M1
Str Wk Cor
Str M1
Wk Other
N/A Cor
Str Cor M1
Str Wk Other
N/A Cor M1
Str Wk Cor
Str Cor
Str M1
Wk Other
N/A Cor M1
Str Wk Other
N/A Cor
Str Other
N/A Cor M1
Str Wk Other
N/A Cor M1
Str Wk Other
N/A Cor
Str Other
N/A Cor
Str Other
N/A Cor M1
Str Wk Other
N/A Cor
Str Other
N/A Cor M1
Str Wk Other
N/A Cor
Str M1
Wk Other
N/A M1
None
Likelihood?
Other
N/A M1
None
Likelihood?
Other
N/A M1
None
Likelihood?
Other
N/A M1
None
Likelihood?
Other
N/A Pre # 6
M1
None
Likelihood?
Other
N/A Post # 6
M1
Wk M1
None
Likelihood?
Post # 6
Post # 5
M1
Wk Other
N/A Pre # 6
Pre # 5
Post # 4
M1
Wk M1
None
Likelihood?
Post # 6
Post # 5
Pre # 4
Other
N/A Pre # 6
Pre # 5
M1
Wk M1
Wk M1
None
Likelihood?
Post # 6
Other
N/A M1
None
Likelihood?
Other
N/A Pre # 6
Post # 5
Other
N/A M1
Wk M1
None
Likelihood?
Other
N/A Post # 6
Pre # 5
Other
N/A M1
Wk Pre # 6
Post # 5
Other
N/A Likelihood?
Other
N/A Post # 6
Pre # 5
Post # 4
M1
Wk M1
Wk Post # 5
Pre # 4
M1
Wk Cor M1
Str Wk Pre # 5
Post # 4
Post # 3
M1
Wk Other
N/A Pre # 6
Other
N/A Post # 5
Pre # 4
Pre # 3
M1
Wk Cor
Str Post # 4
Post # 3
M1
Wk M1
Wk Cor M1
Str Wk Pre # 3
M1
Wk Other
N/A Pre # 4
Post # 3
Other
N/A M1
Wk Post # 4
Pre # 3
Other
N/A Cor M1
Str Wk Pre # 4
Post # 3
Other
N/A Pre # 5
Other
N/A Post # 4
Pre # 3
Post # 2
M1
Wk Cor M1
Str Wk Post # 3
Pre # 2
Post # 1
Cor
Str M1
Wk Pre # 4
Other
N/A Pre # 3
Post # 2
Pre # 1
Cor
Str Cor
Str Pre # 2
Post # 1
Cor
Str Other
N/A Post # 2
Pre # 1
Cor
Str M1
Wk M1
Wk Post # 3
Pre # 2
Post # 1
Cor
Str Cor
Str Post # 2
Pre # 1
Cor
Str Other
N/A Pre # 2
Post # 1
Cor
Str M1
Wk Post # 2
Pre # 1
Cor
Str Cor
Str Pre # 2
Post # 1
Cor
Str Pre # 3
Other
N/A Post # 2
Pre # 1
Cor
Str M1
Wk M1
None
Likelihood?
Other
N/A M1
None
48 Eliciting Mathematics Misconceptions | Assessment: Comparing Fraction Operations
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Comparing Fractions
Operations
Name: ___________________________________________
Without doing calculations, use your understanding of fractions to choose the best answer.
Greater than (>)
1.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
2.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
3.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Eliciting Mathematical Misconceptions
1 of 2 pages
Comparing Fractions
Operations
Without doing calculations, use your understanding of fractions to choose the best answer.
Greater than (>)
4.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
5.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
6.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
2 of 2 pages
Eliciting Mathematical Misconceptions
Fraction
Operations
Name: ___________________________________________
Without doing calculations, use your understanding of fractions to choose the best answer.
Greater than (>)
1.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
2.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
3.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Eliciting Mathematical Misconceptions
1 of 2 pages
Fraction
Operations
Without doing calculations, use your understanding of fractions to choose the best answer.
Greater than (>)
4.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
5.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Greater than (>)
6.
Less than (<)
Equivalent (=)
Explain or Show your reasoning.
Eliciting Mathematical Misconceptions
2 of 2 pages