COMPARING DECIMALS 1 SCORING GUIDE HOW TO USE THIS GUIDE This Scoring Guide is intended for use with both the pre- assessment and the post- assessment for Comparing Decimals 1. To use this guide, we recommend following these steps: Read the Misconception Description below, and be sure you understand what the misconception is. You may want to view the video found under the Research- Based Misconceptions tab. Numerous examples of student work illustrating this misconception are included in this guide, but you may also want to refer to the additional examples of student work found under the Sample Student Responses tab. Familiarize yourself with the five 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 under the Sample Student Responses tab for guidance when you are unsure about the scoring. TABLE OF CONTENTS Misconception Descriptions p. 1 PRE- ASSESSMENT Pre- Assessment Items p. 3 Pre- Assessment Analysis Process p. 4 (Optional) Scoring Practice Items Pre- Assessment p. 8 Scoring Practice Items Answer Key Pre- Assessment p. 11 POST- ASSESSMENT Post- Assessment Items p. 14 Post- Assessment Analysis Process p. 15 (Optional) Scoring Practice Items Post- Assessment p. 19 Scoring Practice Items Answer Key Post- Assessment p. 21 (Optional) Scoring Template p. 24 MISCONCEPTION DESCRIPTION There are multiple research- based misconceptions related to comparing decimals, but this set of diagnostic assessments focuses on one in particular: overgeneralizing from experiences with whole- number comparisons when comparing the digits to the right of the decimal point. Because students are accustomed to thinking of a number with more digits as the larger number, they extend this rule to 1
decimals; they compare the decimal numbers according to how many digits appear to the right of the decimal point and assume that longer is larger. Students typically do not apply this thinking when given numbers with different digits in the ones place, such as comparing 2.36 and 5.1. Instead, they tend to appropriately compare the values of the digits in the ones place, in this case reasoning that since 5 is greater than 2, 5.1 is greater than 2.36. The EM2 Comparing Decimals 1 assessments have designated this misconception in the following way: Misconception 1 (M1): Using Whole- Number Thinking / A Focus on Longer Is Larger Students with this misconception consistently compare decimals by comparing the numbers to the right of the decimal point as if they were comparing whole numbers (e.g., they consider 0.34 to be greater than 0.8 because 34 is greater than 8). Because they are accustomed to thinking of numbers with more digits as larger numbers, they over- generalize from their experiences with whole- number comparisons and extend this rule to decimals. Resources The 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 The Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics (draft): K 5, Number and Operations in Base Ten. Retrieved from http://ime.math.arizona.edu/progressions/#products Irwin, K. (1996). Making sense of decimals. In J. Mulligan & M. Mitchelmore (Eds.), Children s Number Learning (pp. 243 257). Adelaide, Australia: MERGA & AAMT. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing Effective Fractions Instruction for Kindergarten Through 8th Grade: A Practice Guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Steinle, V., & Stacey, K. (2004). Persistence of Decimal Misconceptions and Readiness to Move to Expertise. In M. Johnsen Hoines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education PME 28, 4(1), 225 232. Bergen, Norway: Bergen University College. 2
PRE-ASSESSMENT PRE-ASSESSMENT ITEMS The pre- assessment is composed of five items with specific attributes associated with comparing decimals. Each item may elicit information about students understanding of place value when comparing decimals. Item Correct response: Less than (<) Correct response: Greater than (>) Correct response: Less than (<) Correct response: Less than (<) Correct response:greater than (>) Understandings and Misconceptions NOTE: This is considered a baseline item; it is included in the assessment to confirm that students understand the basic concept of reading decimal numbers. A lack of understanding of this concept would invalidate the rest of the diagnostic assessment, so this item is a double check that students have the basic understanding that forms the basis of this diagnostic assessment. However, in most cases, students for whom this assessment is appropriate will have this understanding. This item is not included in the determination of whether students have M1. Students with Misconception 1 will reason that 86 < 659, so 12.86 < 12.659. Students who only consider the numbers to the left of the decimal will select Equivalent. Students with Misconception 1 will reason that 788 > 88, so 0.788 > 0.88. Students who only consider the numbers to the left of the decimal will select Equivalent. Students with Misconception 1 will reason that 65 > 9, so 3.65 > 3.9. Students who only consider the numbers to the left of the decimal will select Equivalent. Students with Misconception 1 will reason that 3 < 189, so 0.3 < 0.189. Students who only consider the numbers to the left of the decimal will select Equivalent. If students choose an incorrect response that does not indicate M1 thinking, review their explanations to determine what difficulty they are having. 3
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 responses provide Strong Evidence or Weak Evidence of Misconception 1. If a student is determined to show evidence of Misconception 1 on even just one of items 2, 3, 4, or 5, the student is likely to have this misconception. (Item 1 is a baseline item and is not intended to provide information on the presence of 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 M1 response choice and have an explanation that supports Misconception 1. To learn more about how to tell whether an explanation supports Misconception 1, go to the Research- Based Misconceptions tab and watch the video 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 a student s name, then circle the appropriate information for each item on the pre- assessment (shaded) and, later, the post- assessment (in white). If a student s response does not fit Correct or M1 but is Other, draw a strike- through line. HOW TO DETERMINE IF A STUDENT HAS THE MISCONCEPTION 1. For each item, look at Table 1 to determine what the selected response might indicate. Item # Table 1. Response Patterns for the Pre- Assessment Decimal Numbers Being Compared Correct Response M1 Likely Response 1 0.352 0.476 Less than (<) n/a 2 12.86 12.659 Greater than (>) Less than (<) 3 0.788 0.88 Less than (<) Greater than (>) 4 3.65 3.9 Less than (<) Greater than (>) 5 0.3 0.189 Greater than (>) Less than (<) 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 that the student would have selected the M1 response choice and about how the student is reasoning, you can code it as Strong Evidence of M1. However, if the explanation leaves some question about what the student was thinking, code it as Weak Evidence of M1. For additional guidance on determining the strength of the evidence, see the What counts... information in step 2 below. 4
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 selects the M1 response choice, look next at the student s explanation to determine whether it also supports Misconception 1. An explanation can be categorized as Strong Evidence of M1, Weak Evidence of M1, or No Supporting Evidence of M1. What counts as Strong Evidence of M1 in the pre- assessment? In general, responses with strong evidence of M1 include a clear indication that the student is focusing on the number of digits to the right of the decimal point and is comparing those digits as whole numbers. Below are three examples of student responses with strong evidence of M1, using pre- assessment items. To see additional examples of student responses that illustrate this misconception, go to the Sample Student Responses tab and click on the button to download the PDF. Example A: Strong Evidence of M1 This student chooses the M1 response ( Less than ) and specifically refers to the number of digits in each decimal number. Example B: Strong Evidence of M1 12.86 is smaller than 12.659. Because 12.659 has one more number than the other This student chooses the M1 response ( Greater than ) and is clearly paying attention to the number of digits after the decimal point. Example C: Strong Evidence of M1 This one is greater because it has more digits after the decimal This student chooses the M1 response ( Less than ) and clearly indicates that he or she is paying attention to the digits to the right of decimal point and comparing them as whole numbers. What counts as Weak Evidence of M1 in the pre- assessment? Responses with weak evidence of M1 include some indication that the student is ignoring place value and is viewing the digits to the right of the decimal point as a whole number. However, these responses also generally require making more inferences about what the student was 5
thinking, or they leave some question or doubt as to whether the misconception is present or to what degree it is present. Below are three examples of student responses with weak evidence of M1, using pre- assessment items. To see additional examples of student responses that illustrate this misconception, go to the Sample Student Responses tab and click on the button to download the PDF. Example A: Weak Evidence of M1 They would not be equile [equal] because the first is bigger This student chooses the M1 response ( Greater than ). However, the explanation ( the first is bigger ) leaves some doubt as to what the student is thinking without having to make inferences. This makes it Weak Evidence of M1. Example B: Weak Evidence of M1 0.3 is smaller than 0.189 because its just a 0.3. This student chooses the M1 response ( Less than ) and explains that this decimal is smaller because it s just a 0.3. However, it s not clear why the student sees it as smaller without having to make inferences about what the student is thinking. This makes it Weak Evidence of M1. Example C: Weak Evidence of M1 Because there only is a 0.3 and on the other on [one] is 0.189 This student chooses the M1 response ( Less than ) and explains that there only is a 0.3. While the student clearly sees this decimal number as smaller, there is no information about why the student sees it as smaller, which makes it Weak Evidence of M1. What counts as No Supporting Evidence in the pre- assessment? If a student selects the M1 response choice but provides no explanation at all, this counts as No Supporting Evidence of M1. If a student s response choice suggests the possibility of M1 but the explanation does not support it, the item is not considered to be indicative of the misconception and can also be scored as No Supporting Evidence. 6
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. If a student is determined to show evidence of the misconception on even just one of items 2, 3, 4, or 5, the student is likely to have Misconception 1, 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 M1, this diagnostic assessment is validated to predict that it is likely your student has this misconception. However, the presence of only one item with Weak Evidence of M1 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. What if the student s explanation is contradictory to the multiple- choice response chosen? If you come across a response in which the explanation seems to contradict the response choice, it is considered a possible indication of M1. Look for additional evidence, either on these assessments or from the student s comments in class. 7
(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 and checking your scoring as you go against our key, found on p. 10. Practice Example 1 Becuase [Because] 65 is big [bigger] than 9. Practice Example 2 12.86 is smaller beause [because] 86 is smaller then [than] 659 and it won t work. Practice Example 3 There s an 8 in the tenths place other has a 6 in the tenths place Practice Example 4 0.189 is bigger because 0.3 is smaller then [than] 189 8
Practice Example 5 Because 189 is way bigger than 3 and it has a 9 in the thousands place Practice Example 6 I looked at the place value. Practice Example 7 30 > 18.9 Practice Example 8 07.88 is more than 0.88 because in 07.88 it has a 7 in the thousand place Practice Example 9 If you compare the 6 and 8 the larger number is 8 so 12.86 is grater [greater] than 12.659. 9
Practice Example 10 3 is smaller than 189 so 0.189 is big. 10
SCORING PRACTICE ITEMS ANSWER KEY PRE-ASSESSMENT Practice Example 1 Becuase [Because] 65 is big [bigger] than 9. This is an example of M1 with Strong Evidence. The student is clearly focusing on the digits to the right of the decimal point and comparing them as whole numbers. Practice Example 2 12.86 is smaller beause [because] 86 is smaller then [than] 659 and it won t work. This is an example of M1 with Strong Evidence. The student is clearly focusing on the digits to the right of the decimal point and comparing them as whole numbers. Practice Example 3 There s an 8 in the tenths place other has a 6 in the tenths place 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). This student is clearly paying attention to place value in order to compare the decimal numbers. Practice Example 4 0.189 is bigger because 0.3 is smaller then [than] 189 This is an example of M1 with Weak Evidence. Although the student selects the correct response for this item ( Greater than ), the explanation states the opposite: that 0.189 is bigger and 0.3 is smaller. It is difficult to know how this student is reasoning without making inferences, so this is considered Weak Evidence of M1. 11
Practice Example 5 Because 189 is way bigger than 3 and it has a 9 in the thousands place This is an example of M1 with Strong Evidence. The student s explanation specifically focuses on the digits to the right of the decimal and compares them as whole numbers. However, there is also an interesting reference to the 9 in the thousands place, suggesting that this student has some partial or flawed understanding of place value. Practice Example 6 I looked at the place value. 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 response for this item ( Greater than ); however, while place value is mentioned, the student does not explain what it is about the place value that helped the student decide. It is unclear from the student s explanation how the student is thinking about place value, making it Weak Evidence that the student is thinking correctly. Practice Example 7 30 > 18.9 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 response for this item and moves the decimal point two places to the right for each number to compare them; the student offers no other explanation. 12
Practice Example 8 07.88 is more than 0.88 because in 07.88 it has a 7 in the thousand place This is an example of M1 with Weak Evidence. The student selects the M1 response for this item ( Greater than ), and the explanation says that 07.88 has a 7 in the thousand place, which suggests that the student is thinking of whole numbers. However, this statement is both unclear and incorrect (i.e., in the decimal numbers presented, 7 is in the tenths place), leaving it unclear how the student is thinking about the comparison. This makes it Weak Evidence of M1. Practice Example 9 If you compare the 6 and 8 the larger number is 8 so 12.86 is grater [greater] than 12.659. This is a Correct example 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 is clearly paying attention to place value in order to compare the decimal numbers. The student underlines the numbers in the tenths place and explains how to compare them. Practice Example 10 3 is smaller than 189 so 0.189 is big. This is an example of M1 with Strong Evidence. The student s explanation specifically focuses on the digits to the right of the decimal and compares them as whole numbers. 13
POST-ASSESSMENT POST-ASSESSMENT ITEMS The post- assessment is structured exactly the same as the pre- assessment, comprising five items with specific attributes associated with comparing decimals. Each item may elicit information about the students understanding of place value when comparing decimals. Item Correct response: Less than (<) Understandings and Misconceptions NOTE: This item is considered a baseline item; it is included in the assessment to confirm that students actually understand the basic concept of reading decimal numbers. A lack of understanding of this concept would invalidate the rest of the diagnostic assessment, so this item is a double check that students have the basic understanding that forms the basis of this diagnostic assessment. However, in most cases, students for whom this assessment is appropriate will have this understanding. This item is not included in the determination of whether students have M1. Students with Misconception 1 will reason that since 41 < 187, 2.41 < 2.187. Students who only consider the numbers to the left of the decimal will select Equivalent. Correct response: Greater than (>) Students with Misconception 1 will reason that since 899 > 99, 0.899 > 0.99. Students who only consider the numbers to the left of the decimal will select Equivalent. Correct response: Less than (<) Students with Misconception 1 will reason that since 34 > 8, 6.34 > 6.8. Students who only consider the numbers to the left of the decimal will select Equivalent. Correct response: Less than (<) 14
Item Understandings and Misconceptions Students with Misconception 1 will reason that since 2 < 175, 0.2 < 0.175. Students who only consider the numbers to the left of the decimal will select Equivalent. Correct response: Greater than (>) If students choose an incorrect response that does not indicate M1 thinking, review their explanations to determine what difficulty they are having. POST-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 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 Misconception 1 on even just one of items 2, 3, 4, or 5, the students is likely to have this misconception. 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 M1 response choice and have an explanation that supports Misconception 1. To learn more about how to tell whether an explanation supports Misconception 1, go to the Research- Based Misconceptions tab and watch the video 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 p. 3. 1. For each item, look at Table 2 to determine what the selected response might indicate. Item # Table 2. Response Patterns for the Post- Assessment Decimal Numbers Being Compared Correct Response M1 Likely Response 1 0.279 0.345 Less than (<) n/a 2 2.41 2.187 Greater than (>) Less than (<) 3 0.899 0.99 Less than (<) Greater than (>) 4 6.34 6.8 Less than (<) Greater than (>) 5 0.2 0.175 Greater than (>) Less than (<) What if there s no multiple- choice response selected? In that case, carefully consider the student s explanation. If the explanation leaves no 15
doubt that the student would have chosen the M1 response and about how the student is reasoning, you can code it as Strong Evidence of M1. However, if the explanation leaves some question about what the student was thinking, code it as Weak Evidence of M1. For additional guidance on determining the strength of the evidence, see the What counts... information in step 2 below. 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 selects the M1 response choice, look at the student s explanation to determine whether it also supports Misconception 1. An explanation can be categorized as Strong Evidence of M1, Weak Evidence of M1, or No Supporting Evidence of M1. What counts as Strong Evidence of M1 in the post- assessment? Responses with strong evidence of M1 include a clear indication that the student is focusing on the number of digits to the right of the decimal point and is viewing those digits as whole numbers. Below are three examples of student responses with strong evidence of M1, using post- assessment items. To see additional examples of student responses that illustrate this misconception, go to the Sample Student Responses tab and click on the button to download the PDF. Example A: Strong Evidence of M1 It is greater because it s in the hundreds and the other one has 1 diget [digit] This student chooses the M1 response for this item ( Greater than ) and says that 0.175 is greater because it s in the hundreds and that 0.2 only has one digit, clearly viewing them as whole numbers. Example B: Strong Evidence of M1 It s greater than 0.2 because it hase [has] more numbers. This student chooses the M1 response for this item ( Greater than ) and is clearly using the number of digits in each as a way to compare the two decimals. 16
Example C: Strong Evidence of M1 This student chooses the M1 response for this item ( Less than ) and is clearly paying attention to the digits to the right of the decimal and comparing them as whole numbers. What counts as Weak Evidence of M1 in the post- assessment? Responses with weak evidence of M1 include some indication that the student is ignoring place value and is viewing the digits to the right of the decimal point as a whole number. 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 three examples of student responses with weak evidence of M1, using post- assessment items. To see additional examples of student responses that illustrate this misconception, go to the Sample Student Responses tab and click on the button to download the PDF. Example A: Weak Evidence of M1 This student chooses the M1 response for this item ( Less than ). The explanation, however, basically restates the student s selected response and leaves some question as to why the student selected Less than. This lack of clarity makes it Weak Evidence of M1. Example B: Weak Evidence of M1 I chose this because 2.41 is less than 2.187 by [because?] 41 is much smaller than 187 I got less than because the number on the left is less than the number on the right I think less than because (2.187) looks bigger than (2.41). This student chooses the M1 response ( Less than ). However, it is unclear why the student thinks 2.187 looks bigger than 2.41 without having to make inferences about the student s thinking. Therefore, it is considered Weak Evidence of M1. 17
Example C: Weak Evidence of M1 This student chooses the M1 response for this item ( Greater than ), but it is unclear what the student means by has bigger numbers without having to make inferences about the student s thinking. Therefore, it is considered Weak Evidence of M1. What counts as No Supporting Evidence in the post- assessment? I think 0.175 is greater than 0.2 because it has bigger numbers If a student selects the M1 response choice but provides no explanation at all, this counts as No Supporting Evidence of M1. If a student s response choice suggests the possibility of M1 but the explanation does not support it, the item is not considered to be indicative of the misconception and can also be scored as No Supporting Evidence. 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. If a student is determined to show evidence of the misconception on even just one of items 2, 3, 4, or 5, the student is likely to have Misconception 1, 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. 18
(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 and checking your scoring as you go against our key, found on p. 20. Practice Example 1 9 in 0.99 is bigger than 8 in 0.899 Practice Example 2 34 is grater [greater] than 8. Practice Example 3 I picked less than becaus [because] the aligater [alligator] whants [wants] to eat the bigest [biggest] number. Practice Example 4 6/8 > 6/34 Fractions Practice Example 5 0.175 is greater than 0.2 beacause [because] on the right its only 0.2 19
Practice Example 6 Because the bigger number eats the smaller number Practice Example 7 Even though theres the number zero befor [before] it the one on the left is still bigger. Practice Example 8 8 is bigger than 3 Practice Example 9 34 is a hieghr [higher] number than 8. Practice Example 10 189 is more than 3. Because three is more than 1. 20
SCORING PRACTICE ITEMS ANSWER KEY POST-ASSESSMENT Practice Example 1 9 in 0.99 is bigger than 8 in 0.899 This is an example of a Correct example 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 is clearly paying attention to place value in order to compare the decimal numbers. Practice Example 2 34 is grater [greater] than 8. This is an example of M1 with Strong Evidence. The student is clearly paying attention to the digits to the right of the decimal point and is comparing them as whole numbers. Practice Example 3 This is an example of M1 with Weak Evidence. This student chose the M1 response for this item ( Less than ). This student has clearly heard the mnemonic about alligators and inequality signs, but it is unclear how the student is deciding which is the bigger number without having to make inferences about the student s thinking. Therefore, it is considered Weak Evidence of M1. Practice Example 4 I picked less than because [because] the aligater [alligator] wants to eat the bigest [biggest] number. 6/8 > 6/34 Fractions This is an example of a response that is neither correct nor an indicator of M1. The student selects the correct response for this item ( Greater than ) but then changes the decimals into fractions incorrectly 21
to compare them. While 6/8 is indeed greater than 6/34, it is unclear how or what the student is thinking. Practice Example 5 This is an example of M1 with Weak Evidence. This student chooses the M1 response for this item and in the explanation says it s only 0.2, suggesting possible whole- number reasoning. However, there is not enough information in the explanation to determine this without making inferences about the student s thinking. Therefore, it is considered Weak Evidence of M1. Practice Example 6 0.175 is greater than 0.2 beacause [because] on the right its only 0.2 Because the bigger number eats the smaller 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 chooses the correct response for this item but does not provide sufficient evidence that he or she understands why 2.41 is greater than 2.187. Practice Example 7 Even though theres the number zero befor [before] it the one on the left is still bigger. This is an example of M1 with Weak Evidence. This student chooses the M1 response for this item ( Greater than ) and says the one on the left is still bigger. While this suggests possible whole- number reasoning, there is not enough information in the explanation to determine this without making inferences about the student s thinking. Therefore, it is considered Weak Evidence of M1. Practice Example 8 8 is bigger than 3 22
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 is clearly paying attention to place value in order to compare the decimal numbers. Practice Example 9 34 is a hieghr [higher] number than 8. This is an example of M1 with Strong Evidence. The student is clearly paying attention to the digits to the right of the decimal point and is comparing them as whole numbers. Practice Example 10 189 is more than 3. Because three is more than 1. This is an example of M1 with Weak Evidence. The student selects the M1 response ( Less than ). However, the explanation seems contradictory: The student is clearly comparing 3 and 189 as whole numbers, but then says, Because three is more than 1, suggesting the use of place value. Because it s unclear how the student is thinking about this comparison, it is considered Weak Evidence of M1. 23
Scoring Guide Template Comparing Decimals 1 Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None Student: Pre # 1 Pre # 2 Pre # 3 Pre # 4 Pre # 5 Likelihood? Post # 1 Post # 2 Post # 3 Post # 4 Post # 5 Likelihood? Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Cor M1 Cor M1 Cor M1 Cor M1 Cor M1 M1 Str Wk Str Wk Str Wk Str Wk Str Wk None Str Wk Str Wk Str Wk Str Wk Str Wk None 24