The Decimal Operations: Division assessment is designed to elicit information about two common misconceptions that students have when dividing decimals: Misconception 1 (M1): Overgeneralizing from Multiplication Counting All the Places Misconception 2 (M2): Counting Only Places in the Dividend Although you can access the assessment here at any time, we strongly recommend you reference the information below to learn more about those misconceptions, including how they appear in student work, and how to score pre- and post- assessments once you have given them to students. Contents Topic Background: Learn about dividing decimals.... 2 Student Misconceptions: Learn about student misconceptions related to the topic.... 3 Administering the Pre- Assessment: Learn how to introduce the pre- assessment to your students... 5 : Learn about the scoring process by reviewing the Guide....7 Sample Student Responses: Review examples of student responses to assessment items... 35 Administering the Post- Assessment: Learn how to introduce the post- assessment to your students... 43
Topic Background Learn about comparing decimals with embedded or trailing zeros Decimal division poses a particular challenge for students because of their prior experience with whole numbers. Students initial experience with division with whole numbers teaches them that division makes smaller, by partitioning some quantity into smaller equal- size portions. As they start to work with fractions and decimals, however, division by a number less than 1 makes bigger, a counter- intuitive idea for many students. As students work with division of decimals, they need to develop a deeper understanding of how division works as an operation, including the ability to reason about the size of quotient and estimating the approximate range of an answer to decimal division problems. If this understanding is not developed in students, then the procedural skills of learning how to move the decimal points in a decimal division problem, and place the decimal point correctly in the answer, can prove problematic for students. 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: Understand the place value system. 5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole- number exponents to denote powers of 10. 5.NBT.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. At grade 6, students should be able to: Compute fluently with multi- digit numbers and find common factors and multiples. 6.NS.3. Fluently add, subtract, multiply, and divide multi- digit decimals using the standard algorithm for each operation. 2 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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 how to divide decimals. The following common misunderstandings and misconceptions when representing fractions on a number line are targeted in the Decimal Operations: Division assessments: Misconception 1: Overgeneralizing From Multiplication Counting All the Places Some students consistently over- generalize from multiplication of decimals by simply counting the total number of digits to the right of the decimal point in both the dividend and divisor, and using that total number to place the decimal point in the quotient. Misconception 2: Counting Only Places in the Dividend Students with this misconception count the number of places to the right of the decimal in the dividend only, and use that number to place the decimal point in the quotient. When doing the problem using long division, this is equivalent to bringing the decimal point straight up into the answer, lining it up with where it sits in the dividend. Students may also turn the decimals into whole numbers, divide them as whole numbers, then replace the decimal in the result, according to where the decimal point is in the dividend. Note that this strategy results in the correct placement of the decimal point when the divisor is a whole number. 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 35 of this document. 3 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Student Misconceptions Resources Bergeson, T. (2000). Teaching and Learning Mathematics: Using Research to Shift From the Yesterday Mind to the Tomorrow Mind. Office of Superintendent of Public Instruction. Hanson, S. A., & Hogan, T. P. (2000). Computational Estimation Skill of College Students. Journal For Research In Mathematics Education, 31(4), 483. Tirosh, D., & Graeber, A. O. (1991). The Effect of Problem Type and Common Misconceptions on Preservice Elementary Teachers' Thinking about Division. School Science And Mathematics, 91(4), 157-63. 4 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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 dividing decimals. Each item within any EM2 assessment includes a selected response (multiple choice) and an explanation component. The decimal numbers used in this assessment include numbers in tenths and hundredths. The learning target for the Decimal Operations: Division assessment is as follows: The learner will use information about a whole number division problem to determine the correct placement of the decimal in a corresponding problem with decimal division 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 about dividing decimals using the information provided rather than the doing the actual calculations 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 dividing decimals, 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. 5 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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 Guide PDF document under the Process section and found on page 7 of this document) to analyze whether your students have these misconceptions: Misconception 1 (M1): Overgeneralizing from Multiplication Counting All the Places Misconception 2 (M2): Counting Only Places in the Dividend 6 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Learn about the scoring process by reviewing the Guide. The Decimal Operations: Division assessment is composed of six items with specific attributes associated with different misconceptions that are directly related to dividing decimals. We encourage you to carefully read the Guide to understand these specific attributes and to find information about analyzing your students responses. The scoring guide is intended for use with both the pre- assessment and the post- assessment for Decimal Operations: Division. To use this guide, we recommend following these steps: Read the Misconception Description above, and be sure you understand what the misconceptions are. You may want to view the video description of the misconceptions found in the included files. Numerous examples of student work illustrating this misconception are included in this guide. Familiarize yourself with the six assessment items and what they are assessing; Consider completing the optional scoring practice items and checking yourself with the answer key; Score your students work using the Analysis Process described below; Refer to the various examples found here and in the Sample Student Responses for guidance when you are unsure about the scoring. 7 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
PRE- ASSESSMENT The assessment is composed of six items with specific attributes associated with understandings and misunderstandings related to dividing decimals. Item 1 Item 2 Item Correct Response: 3 Understandings and Misconceptions Students with misconception 1 will reason that there are 2 places after the decimal point in 0.33, and 2 places in 0.11, so that the quotient must have 4 total places. They will tend to choose 0.0003. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.03, so that their answer has 2 places after the decimal, matching the dividend. Students with misconception 1 will reason that there are 2 places after the decimal point in 1.25, and 1 place in 0.5, so that the quotient must have 3 total places. They will tend to choose 0.025. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.25. Correct Response: 2.5 Item 3 Item 4 Correct Response: 0.171 Students with misconception 1 will reason that there are 2 places after the decimal point in 3.42, and none in 20, so that the quotient must have 2 total places. They will tend to choose 1.71. (Note that this is the same as the M2 response, so it s important to check the explanation to determine whether the student has misconception 1.) Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will also tend to choose 1.71, so that their answer matches the dividend. Students with misconception 1 will reason that there is 1 place after the decimal point in 4.8, and 1 place in 1.2, so that the quotient must have 2 total places. They will tend to choose 0.04. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.4, so that their answer has 1 place after the decimal, matching the dividend. Correct Response: 4 8 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Item 5 Students with misconception 1 will reason that there are 2 places after the decimal point in 1.35, and 1 place in 0.9, so that the quotient must have 3 total places. They will tend to choose 0.015. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.15, so that their answer has 2 places after the decimal, matching the dividend. Item 6 Correct Response: 1.5 Correct Response: 1.43 Students with misconception 1 will reason that there is 1 place after the decimal point in 42.9, and no places in 30, so that the quotient must have 1 total place. They will tend to choose 14.3. (Note that this is the same as the M2 response, so it s important to check the explanation to determine whether the student has misconception 1.) Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 14.3 so that their answer has 1 place after the decimal, matching the dividend. If students choose an incorrect response that does not indicate M1 or M2 thinking, review their explanations to determine what difficulty they are having. 9 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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, Misconception 2, or both. You can weigh the relative likelihood that your student has a 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 the Student Misconceptions on page 3 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 EITHER MISCONCEPTION 1. For each item, look at Table 1 to determine what the selected response might indicate. Say that a student selects choice B for item 3. Looking at Table 1 below, we see that choice B might indicate the presence of M1 or of M2. Note that some responses for a particular item may apply to more than one misconception. Therefore, it is particularly important to also consider the student s explanation in order to determine whether a misconception is present and, if so, which one. 10 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Table 1. Response Patterns for the Pre- Assessment Ite m # Correct M1 Likely Responses M2 Likely Responses 1 C A B 2 C A B 3 A B B 4 C A B 5 C A B 6 B C C What if there s no multiple- choice response selected? In that case, carefully consider the explanation the student gives. If the explanation leaves no doubt which of the fractions the student would have selected and no 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 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, carefully consider the student s explanation to determine what the response indicates, and note whether the evidence from the explanation is strong or weak. If the student provides a response on any item that aligns with a misconception, look next at the student s explanation to determine whether it supports M1 or M2. A Caution! Table 1 shows that some responses indicate only one possibility; for example, a response of choice C for item 1 indicates only the possibility of correct understanding. However, it is still necessary to check the student s explanation to confirm evidence that the student has correct understanding. It is not unusual for a student to choose a response that appears to indicate one thing but then to provide an explanation that appears to be contradictory. 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. 11 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Below are two examples of student responses with strong evidence of a misconception, using pre- assessment items. To see additional examples of student responses that illustrate these misconceptions, go to the Sample Student Responses on page 35. Example A: Strong Evidence of M1 For students with M1, the explanation will include clear evidence that the student is counting the total number of places in the dividend and the divisor, and matching the same number of decimal places in the quotient. (For a more detailed description of this misconception, see the video found in Student Misconceptions on page 3.) Since 33 11 = 3, I just moved the decimal 4 hops to the left. For item 1, this student chooses 0.0003, which indicates the possibility of M1 thinking (see Table 1). The student s explanation shows the student counting the total number of hops and using that to determine the placement of the decimal point in the quotient. This is strong evidence of M1. Example B: Strong Evidence of M2 For students with M2, the explanation will include clear evidence that the student is paying attention to the number of decimal places in the dividend only. (For a more detailed description of this misconception, see the video.) For item 2, this student chooses 0.4, which indicates the possibility of M2 thinking (see Table 1). The explanation states that the student converted to whole numbers, divided, then reinserted the decimal point. This shows the line- up- the- decimals reasoning that is indicative of M2. 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? I multiplied 4.8 and 1.2 by 10 to get whole numbers. Then I divied [divided] the whole numbers and I Got 4 so then I put in the decimal in the right spot. 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 12 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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 a misconception, using pre- assessment items. To see additional examples of student responses that illustrate these misconceptions, go to the Sample Student Responses on page 3. Example A: Weak Evidence of M1 For item 5, this student selects.015, which indicates the possibility of M1 thinking. However, the student s explanation provides no clear evidence of why the student chose to put the decimal point in the quotient with decimal three places. This makes it Weak Evidence of M1. Example B: Weak Evidence of M2 For item 1, this student chooses 0.03, which indicates the possibility of M2 thinking. However, it is unclear why the student is moving the decimal point two places, without having to make an inference, so it is considered Weak Evidence of M2. What if the student selects one of the choices, but provides no explanation? If a student selects an M1 or M2 response choice but provides no explanation at all, this is not considered convincing evidence of the misconception, and can be scored as Other on the scoring template. What if the student s choice matches a misconception and there is 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. 13 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
3. After you have analyzed each item for a student, use the guidelines below to determine whether the student has either misconception. This diagnostic assessment has been validated to predict the possible presence of M1 or M2 for a student. If a student is determined to show 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 or M2 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. 14 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
(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 18. Practice Example 1 When you divide 135 and 9 you get 15. When you dive [divide] 1.35 and 0.9 you have to move the decimal, and when you get the quotient, that s how many spaces you move it. Practice Example 2 You have two wole [whole] numbers that are very close but one s a little higher. I would think if you divided, it would be around one. Practice Example 3 One has a decimal, the other dosen t [doesn t]. 15 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 4 It would be.4 because the decimal moved left once on the dividend and divisor so when it moves over, 4 becomes.4. Practice Example 5 48 12 = 4 4.8 1.2 = not.40 and/or 4 Makes sense = 0.04 Practice Example 6 If 33 11 = 3, then if you put a decimal in front of it, it would.33.11 =.03. It can t be.3 or 3 or.0003 because they are either to [too] big or to [too] small. Practice Example 7 I know that both numbers are below zero so it can not be 3. Also, there are 4 numbers after the decimal point so it is not 0.03 leaving 0.0003. 16 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 8 Practice Example 9 I estimated [estimated] that 40 30 = 1.3 (repeating), so 1.43 was closest to 1.3 (repeating). Practice Example 10 I moved the decimal to the left twice. (shows arrow pointing two places to the left in 0.03) 17 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
SCORING PRACTICE ITEMS ANSWER KEY PRE- ASSESSMENT Practice Example 1 When you divide 135 and 9 you get 15. When you dive [divide] 1.35 and 0.9 you have to move the decimal, and when you get the quotient, that s how many spaces you move it. This is an example of M1 with Strong Evidence. The student selects.015, indicating possible M1 thinking. The student s explanation then clearly describes paying attention to the total number of decimal places in the dividend and divisor in order to place the decimal point in the quotient. Practice Example 2 You have two wole [whole] numbers that are very close but one s a little higher. I would think if you divided, it would be around one. This is an example of a Correct response with Strong Evidence. The student selected 1.43, then describes accurately estimating the answer. Practice Example 3 One has a decimal, the other dosen t [doesn t]. 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 the correct multiple- choice response, 1.43. The student s explanation, however, leaves it unclear how the student arrived at this answer from seeing that one has a decimal, the other doesn t. Therefore, it is considered Weak Evidence of a correct response. 18 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 4 It would be.4 because the decimal moved left once on the dividend and divisor so when it moves over, 4 becomes.4. This is an example of M2 with Strong Evidence. The student selects 0.4, indicating possible M2 thinking. The student s explanation makes it clear the student is lining up the decimal point with where it is in the dividend and divisor. Practice Example 5 48 12 = 4 4.8 1.2 = not.40 and/or 4 Makes sense = 0.04 This is an example of a response that is neither M1 nor M2, and would be coded as Other. The student selects 0.04, which indicates possible M1 thinking. However, the student s explanation shows that the student is not thinking about the total number of decimal places, but is instead using the process of elimination to rule out 4 and 0.4. It is unclear from the explanation why the student is ruling out these choices. Practice Example 6 If 33 11 = 3, then if you put a decimal in front of it, it would.33.11 =.03. It can t be.3 or 3 or.0003 because they are either to [too] big or to [too] small. This is an example of M2 with Weak Evidence. The student selects 0.03, indicating possible M2 thinking. However, the student s explanation seems to use a mix of process- of- elimination and wanting to line up the decimal point with the dividend and divisor. The lack of clarity in how the student is coming up with 0.03 as the answer makes it Weak Evidence of M2. 19 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 7 I know that both numbers are below zero so it can not be 3. Also, there are 4 numbers after the decimal point so it is not 0.03 leaving 0.0003. This is an example of M1 with Strong Evidence. The student selects 0.0003, indicating possible M1 thinking. The student s explanation then clearly mentions paying attention to 4 numbers after the decimal point. Practice Example 8 This is an example of M1 with Weak Evidence. The student selects 0.04, indicating possible M1 thinking. However, even though the explanation shows moving the decimal point two places, there is no explanation for why the student thinks this is the correct thing to do. This ambiguity makes it Weak Evidence of M1. Practice Example 9 I estimated [estimated] that 40 30 = 1.3 (repeating), so 1.43 was closest to 1.3 (repeating). This is an example of a Correct response with Strong Evidence. The student selects 1.43, the correct response. The explanation then describes a correct way to estimate the answer. 20 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 10 I moved the decimal to the left twice. (shows arrow pointing two places to the left in 0.03) This is an example of M2 with Weak Evidence. The student selects 0.03, indicating possible M2 thinking. The explanation describes moving the decimal point to the left twice, matching what s in both the dividend and divisor. There is no explanation for why the student thinks this is the correct thing to do. This ambiguity makes it Weak Evidence of M2. 21 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
POST-ASSESSMENT ITEMS The post- assessment is structured exactly the same as the pre- assessment, comprising six items with specific attributes associated with dividing decimals. Note Students who have either or both misconceptions may show evidence of one or the other misconception on different items in the assessment. For instance, a student may show evidence of M1 thinking on several items and M2 thinking on several other items. Refer to the Post- Assessment Analysis Process below for guidance on how to determine whether a student has a particular misconception. Item Understandings and Misconceptions Item 1 Students with misconception 1 will reason that there are 2 places after the decimal point in 0.48, and 2 places in 0.12, so that the quotient must have 4 total places. They will tend to choose 0.0004. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.04, so that their answer has 2 places after the decimal, matching the dividend. Correct Response: 4 Item 2 Students with misconception 1 will reason that there are 2 places after the decimal point in 1.75, and 1 place in 0.5, so that the quotient must have 3 total places. They will tend to choose 0.035. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.35. Correct Response: 3.5 Item 3 Correct Response: 0.183 Students with misconception 1 will reason that there are 2 places after the decimal point in 5.49, and none in 30, so that the quotient must have 2 total places. They will tend to choose 1.83. (Note that this is the same as the M2 response, so it s important to check the explanation to determine whether the student has misconception 1.) Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will also tend to choose 1.83, so that their answer matches the dividend. 22 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Item 4 Students with misconception 1 will reason that there is 1 place after the decimal point in 6.6, and 1 place in 1.1, so that the quotient must have 2 total places. They will tend to choose 0.06. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.6, so that their answer has 1 place after the decimal, matching the dividend. Correct Response: 6 Item 5 Students with misconception 1 will reason that there are 2 places after the decimal point in 1.53, and 1 place in 0.9, so that the quotient must have 3 total places. They will tend to choose 0.017. Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 0.17, so that their answer has 2 places after the decimal, matching the dividend. Correct Response: 1.7 Item 6 Correct Response: 1.26 Students with misconception 1 will reason that there is 1 place after the decimal point in 25.2, and no places in 20, so that the quotient must have 1 total place. They will tend to choose 12.6. (Note that this is the same as the M2 response, so it s important to check the explanation to determine whether the student has misconception 1.) Students with misconception 2 will pay attention primarily to the number of decimal places in the dividend. They will tend to choose 12.6 so that their answer has 1 place after the decimal, matching the dividend. 23 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
POST-ASSESSMENT ANALYSIS PROCESS You may want to review the bulleted items listed under Pre- Assessment Analysis Process. HOW TO DETERMINE IF A STUDENT HAS A 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 7. 1. For each item, look at Table 2 to determine what the selected response might indicate. Table 2. Response Patterns for the Post- Assessment Ite m # Correct M1 Likely Responses M2 Likely Responses 1 C A B 2 C A B 3 A B B 4 C A B 5 C A B 6 B C C What if there s no multiple- choice response selected? In that case, carefully consider the student s explanation. If the explanation leaves no doubt that the student would have chosen the misconception response and about how the student is reasoning, you can code it as Strong Evidence of that misconception. However, if the explanation leaves some question about what the student was thinking, code it as Weak Evidence of that misconception. For additional guidance on determining the strength of the evidence, see the What counts... information in step 2 below. 2. For each item, carefully consider the student s explanation to determine what the response indicates, and note whether the evidence from the explanation is strong or weak. If the student provides a response on any item that aligns with a misconception, look at the student s explanation to determine whether it supports M1 or M2. A Caution! Table 1 shows that some responses indicate only one possibility; for example, a response of choice C for item 1 indicates only the possibility of correct understanding. However, it is still necessary to check the student s explanation to confirm evidence that the student has correct understanding. It is not unusual for a student to choose a response that appears to indicate one thing but then to provide an explanation that appears to be contradictory. 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. 24 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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; it is quite clear from the combination of the selected response and the explanation. Below are two examples of student responses with strong 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 35. Example A: Strong Evidence of M1 For students with M1, the explanation will include clear evidence that the student is counting the total number of places in the dividend and the divisor, and matching the same number of decimal places in the quotient. (For a more detailed description of this misconception, see the video found in Student Misconceptions on page 3.) I took the information 175 5 = 35 and then I did three hops because that's where the decimal points are in the problem. For item 2, this student chooses 0.035, which indicates the possibility of M1 thinking. The student s explanation describes counting the total numbers of hops and using that to place the decimal point in the quotient. Example B: Strong Evidence of M2 For students with M2, the explanation will include clear evidence that the student is paying attention to the number of decimal places in the dividend only. I thought that if 6.6 1.1 = 0.6 you could do 66 11 and get 6 and just add in the decimal point. For item 4, this student chooses 0.6, which indicates the possibility of M2 thinking. The student s explanation describes the strategy of ignoring the decimal points, completing the calculation with whole numbers, then reinserting the decimal point to match the dividend and divisor. 25 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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 the Sample Student Responses on page 35. Example A: Weak Evidence of M1 I thought that it was 0.035 because it was 1.75 0.5 so I thought it would be 0.035. For item 2, this student chooses 0.035, which indicates the possibility of M1 thinking (see Table 2). However, it is unclear exactly why the student is counting three decimal places, without having to make conjectures about the student s thinking. This makes it Weak Evidence of M1. Example B: Weak Evidence of M2 I knew it was 0.6 because I knew that 6 1 = 6 and that's how I got my answer. For this item, this student chooses 0.6, which indicates the possibility of M2 thinking. However, the explanation is vague, and provides no convincing evidence that the student was focusing only on the dividend, making it Weak Evidence of M2. What if the student selects one of the choices, but provides no explanation? If a student selects an M1 or M2 response choice but provides no explanation at all, this is not considered convincing evidence of the misconception, and can be scored as Other on the scoring template 26 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
What if the student s choice matches a misconception and there is 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. 3. After you have analyzed each item for a student, use the guidelines below to determine whether the student has either misconception. This diagnostic assessment has been validated to predict the possible presence of M1 or M2 for a student. If a student is determined to show evidence of either 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 or M2 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. 27 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
(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 31. Practice Example 1 I chose my answer of 0.04 because there is two decimals in the sentence so you move the decimal twice. Practice Example 2 You would add a decimal in between the 1 and the 8. Practice Example 3 28 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 4 I knew it was 0.183 because I knew it had to be really small of a quotient because your [you re] splitting about 5 30 times. Practice Example 5 Since three digits are decimals, the answer also has three decimals. Practice Example 6 Practice Example 7 I saw one zero in front of 48 and one in front of 12 soo [so] I put them in front of 4. 29 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 8 I added zeros before 4 in the decimal spot where there is a single digit number in the equation. Practice Example 9 I knew because 20 and 25.2 are close and when you divide a number by itself, you get one. Practice Example 10 The quotient [quotient] asked me to divide.48.12 and I knew that it equaled 4 so I put my answer as 0.04. 30 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
SCORING PRACTICE ITEMS ANSWER KEY POST- ASSESSMENT Practice Example 1 I chose my answer of 0.04 because there is two decimals in the sentence so you move the decimal twice. This is an example of M2 with Strong Evidence. The student selects 0.4, indicating possible M2 thinking. The student s explanation makes it clear the student is matching the placement of the decimal point with where it sits in the dividend and divisor. Practice Example 2 You would add a decimal in between the 1 and the 8. 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 the correct multiple- choice response, 0.183. The student s explanation, however, leaves it unclear why the student arrived at this answer. Therefore, it is considered Weak Evidence of a correct response. Practice Example 3 This is an example of M1 with Strong Evidence. The student selects.0.06, indicating possible M1 thinking. The student s explanation then shows counting the total number of places. (The student then creatively shows moving two places over in Answer, moving the decimal from after the r to in between the w and e. ) J 31 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 4 I knew it was 0.183 because I knew it had to be really small of a quotient because your [you re] splitting about 5 30 times. This is an example of a Correct response with Strong Evidence. The student selected 0.183, then describes accurately estimating the answer. Practice Example 5 Since three digits are decimals, the answer also has three decimals. This is an example of M1 with Strong Evidence. The student selects 0.183, indicating possible M1 thinking. The student s explanation then clearly describes paying attention to the total number of decimal places in the dividend and divisor in order to place the decimal point in the quotient. Practice Example 6 This is an example of M2 with Strong Evidence. The student selects 0.35, indicating possible M2 thinking. The student s explanation makes it clear that the student is ignoring the decimals, completing the calculation with whole numbers, then reinserting the decimal to match the dividend. 32 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 7 I saw one zero in front of 48 and one in front of 12 soo [so] I put them in front of 4. This is an example of a response that is neither M1 nor M2, and would be coded as Other. The student selects 0.04, which indicates possible M2 thinking. However, the student s explanation shows that the student is not thinking about the total number of decimal places, but is instead focusing on the number of leading zeroes to determine the placement of the decimal point in the quotient. Practice Example 8 I added zeros before 4 in the decimal spot where there is a single digit number in the equation. This is an example of M1 with Weak Evidence. The student selects 0.0004, indicating possible M1 thinking. The student s explanation is unclear, and though we might be able to conjecture what the student is thinking, the ambiguity of the explanation makes it Weak Evidence of M1. Practice Example 9 I knew because 20 and 25.2 are close and when you divide a number by itself, you get one. This is an example of a Correct response with Strong Evidence. The student selects 1.43, the correct response. The explanation then describes a correct way to estimate the answer. 33 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Practice Example 10 The quotient [quotient] asked me to divide.48.12 and I knew that it equaled 4 so I put my answer as 0.04. This is an example of M2 with Weak Evidence. The student selects 0.04, indicating possible M2 thinking. However, the student s explanation leaves it unclear why they ended up with 0.04. The lack of clarity in how the student is coming up with 0.04 as the answer makes it Weak Evidence of M2. 34 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Review examples of student responses to assessment items. The Decimal Operations II (Division) diagnostic assessment focuses on two particular misconceptions that students have, regarding division of decimals. Sample student responses indicative of the misconceptions 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: Overgeneralizing From Multiplication Counting All the Hops Some students consistently over- generalize from multiplication of decimals by simply counting the total number of digits to the right of the decimal point in both the dividend and divisor, and using that total number to place the decimal point in the quotient. The following student responses show examples of this misconception. Item Sample Student Responses with Evidence of Misconception 2 Notes Post- Assmt #3 The M1 selected response is chosen AND Explanation shows student marking 2 hops in 5.49, 0 hops in 30.0, and noting a total of 2 hops. The student then moves the decimal point 2 places from 18.3 to 0.183. Post- Assmt #1 The M1 selected response is chosen AND The explanation uses four numbers to describe four places, clearly showing the student adding the total number of places. Since there are four numbers that are decimals, the answer has four numbers that are decimals. 35 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Post- Assmt #2 Pre- Assmt #4 Pre- Assmt #1 I took the information 175 5 = 35 and then I did three hops because that s where the decimal points are in the problem. I knew to move the decimal point 2 spaces to the left. Since 33 11 = 3, I just moved the decimal 4 hops to the left. The misconception selected response is chosen AND The explanation shows finding the total number of hops in the problem and using that to place the decimal in the quotient. The misconception selected response is chosen AND The explanation shows the student using the total number of places to determine the location of the decimal point in the quotient. The misconception selected response is chosen AND The explanation shows finding the total number of hops in the problem and using that to place the decimal in the quotient. 36 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Misconception 2: Counting Only Places in the Dividend Students with this misconception count the number of places to the right of the decimal in the dividend only, and use that number to place the decimal point in the quotient. When doing the problem using long division, this is equivalent to bringing the decimal point straight up into the answer, lining it up with where it sits in the dividend. Students may also turn the decimals into whole numbers, divide them as whole numbers, then replace the decimal in the result, according to where the decimal point is in the dividend. Item Sample Student Responses with Evidence of Misconception 3 Notes Pre- Assmt #1 Pre- Assmt #1 Post- Assmt #5 Well, sence [since] its.48.12 I try to think their [there] are no decimals so I divided 48 12 and got my answer than [then] I added a decimal. I chose my answer of 0.17 because there is two decimals in the sentence so you move the decimal twice. The M2 selected response is chosen AND The student describes dividing the decimals as whole numbers and replacing the decimal in the quotient. The M2 selected response is chosen AND The explanation shows the student counting 2 places in both the dividend and divisor, then matching that in the quotient. The M2 selected response is chosen AND The student is paying attention to the number of places in the dividend to determine the placement of the decimal point in the quotient. 37 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Post- Assmt #1 The M2 selected response is chosen AND The student describes moving up the decimal point. I moved up the decimal. Post- Assmt #1 Post- Assmt #4 Post- Assm t #4 The M2 selected response is chosen AND The student writes the problem to line up the decimal points and bring down the decimal point in the quotient. The M2 selected response is chosen AND The student shows all the correct steps of long division, but brings up the decimal point to line it up with where it sits in the dividend. The M2 selected response is chosen AND The student is paying attention only to the dividend to determine the placement of the decimal point in the quotient. Incorrect Reasoning That is Not An Example of Misconception 1 or 2 There may be other cases in which the student selects the response for a particular misconception, but does not provide convincing evidence that he or she has that misconception. Therefore, it s important to always look at the student explanation in conjunction with the selected response. In some cases, the student has a different set of difficulties than the specific misconceptions targeted by this probe. 38 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Difficulty #1: The Goldilocks Approach (Not Too Big, Not Too Small, But Just Right) Some students will look at the three response choices, and rule out the one that they believe is too big or too small, accepting the remaining one as the only reasonable answer, whether or not is actually mathematically reasonable. Often, this approach appears to be linked to students belief that dividing always makes the quotient smaller than either the dividend or the divisor (or sometimes both!). For example, in the student s thinking, 0.4 0.2 cannot be 2, because 2 is larger than both 0.4 or 0.2. In other cases, students will explain in some manner that the quotient of a decimal problem must also have a decimal in it (see the last two examples below). 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 responses show examples of this: Item Sample Student Responses with other Incorrect Reasoning Notes Post- Assmt #1 Post- Assmt #2 Pre- Assmt #1 I chose 0.04 because 4 was not a decimal, and 0.0004 was far too low so I felt 0.04 was in this range. I chose 0.035 because I felt like the answer couldn t be greater than the dividend. 0.33 0.11 =? I believe this answer [0.03] is correct because 30 10 = 3, and the closest answer to that with a decimal is 0.03. The M2 selected response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the dividend. Instead, the student is ruling out answers that he or she thinks are too big or too small. The M1 selected response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the total number of places. Instead, the student is ruling out answers that he or she thinks are too big or too small. The M2 selected response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the dividend. Instead, the student is estimating, and assuming the answer must have a decimal. 39 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Post- Assmt #2 I knew it was 0.35 because I knew 3.5 was too big and 0.035 was too small. The M2 selected response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the dividend. Instead, the student is ruling out answers that he or she thinks are too big or too small. Difficulty #2: The Number of Zeros is Somehow Related to the Placement of the Decimal Point Some students pay attention to the number of zeros that they perceive as significant in the problem, and use that in some way to determine the placement of the decimal point in the quotient. 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 responses show examples of this: Item Sample Student Responses with other Incorrect Reasoning Notes Post- Assmt #1 I did this because there were 2 zeroes in the equation 0.48 [writes 1 above the 0] 0.12 [writes 2 above the 0]. There was one zero in front of 5 so I put it in front of 35. The M2 response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the dividend. Instead, the student counts two zeros, and includes two zeroes in the answer. The M2 response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the dividend. Instead, the student pays attention to the number of zeroes in the equation and includes them in the answer. 40 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses I chose 0.06 because if 66 11 = 6 and there is decimals in the middle of the equation that you would have to put a decimal point after the 1 st 0 and before the 2 nd 0. The M1 selected response is chosen HOWEVER The student does not provide any evidence that he or she is paying attention to the total number of places. Instead, the student is paying attention to zeroes, and using them to determine the placement of the decimal point in the answer. Correct Reasoning: Students with correct reasoning: Understand that dividing decimals can result in a larger number in the quotient than in either the dividend or the divisor; Can reason accurately about the approximate size of the quotient; Can reason about division as a whole (the dividend) partitioned into some number of groups or copies of the divisor, or as the inverse of multiplication. Item Sample Student Responses with Correct Reasoning Notes Pre- Assmt #3 Pre- Assmt #4 I know that it is going to be below zero cause 20 is greater than 3.42 so the answer is 0.171. The correct selected response is chosen AND The explanation shows the student correctly estimating the answer (the students says below zero but likely means between 0 and 1 ). The correct selected response is chosen AND The student is correctly thinking conceptually about division. I know that 1.2 goes into 4.8 4 times so 4 is the correct answer. 41 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Sample Student Responses Pre- Assm #4 The correct selected response is chosen AND The student is correctly using estimation to determine the answer. I estimated that 5 1 = 5, so 4 is closest to 5. Post- Assmt #3 The correct selected response is chosen AND The student describes a correct procedure for dividing the numbers. You don t have to move the decimal point but 5 isn t divisable by 30. So you ll divide 5.4 by 30 then take what s left ant [and] add the 9 to it and divide that. You ll get adout [about] 0.183. Post- Assmt #1 The correct selected response is chosen AND The student uses multiplication to verify the division. I think the 4 because 0.12 x 4 would equal 0.48. 42 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Decimal Operations: Division Administering the Post- Assessment Learn how to introduce the post- assessment to your students. If the Decimal Operations: Division pre- assessment shows that one or more of your students have either or both of the misconceptions outlined in the 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 dividing decimals using the information provided rather than the doing the actual calculations, 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 dividing decimals, 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. 43 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
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 Guide PDF document under the Process section and found on page 7 of this document) to analyze whether your students have these misconceptions: Misconception 1 (M1): Overgeneralizing from Multiplication Counting All the Places Misconception 2 (M2): Counting Only Places in the Dividend 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. 44 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved
Guide Template Decimal Division Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Pre # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Post # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Pre # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Post # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Pre # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Post # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Pre # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Post # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Pre # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Post # 6 Likelihood? Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other Cor M1 M2 Other M1 M2 Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A Str Wk N/A None 45 Eliciting Mathematics Misconceptions Assessment: Decimal Operations: Division EDC 2015 All Rights Reserved