Balanced Assessment Test Second Grade Core Idea Task Score. Number Operations Pocket Money

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1 Balanced Assessment Test Second Grade 2008 Core Idea Task Score Number Operations Pocket Money SC This task asks students to demonstrate an understanding of whole numbers and how to represent and use them in flexible ways, including relating, composing, and decomposing numbers. Successful students demonstrate fluency in adding and subtracting whole numbers in context, and they are able to communicate reasoning using words, numbers, or pictures. Data Analysis Pets SC This task asks students to represent the same data in more than one way. Students who are successful are able to compare and combine values on the graph, and use words, numbers, or pictures to explain their reasoning. Students should demonstrate that they understand that the data represented in the tally table and in the bar graph is the same information, and should be consistent. Successful students will also provide a statement that shows they can differentiate between the types of information provided about the data set based on the type of graph used to represent the data. Geometry Don s Shapes SC This task asks students to describe and classify two-dimensional shapes. Successful students are able to identify the two shapes that make up a larger shape, using triangles, rectangles, and squares. They can sort a set of two-dimensional shapes based on the attribute of having straight sides or not. Successful students demonstrate that they can identify and compare shapes based on geometric attributes such as number of sides and angles, rather than size, appearance, or orientation. Algebra Building Walls SC This task asks students to identify and extend a growing pattern of blocks in a wall, using drawings and numbers to describe the growth. Successful students identify and use the growth element to determine the number of blocks used to create walls at various stages of the pattern. They can also communicate reasoning using words, numbers, or pictures to translate between representations of shape and number in the pattern. Number Operations Carol s Numbers SC This task asks students to use their knowledge of place value to arrange three digits to make the largest and smallest combination. Students are also asked to place numbers on a number line. Successful students were able to think about scale and order is arranging the numbers on the number line. 2 nd Grade

2 2 nd Grade

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4 Pocket Money Margie says, I have 10 in my pocket but I do not have a dime. 1. What coins could Margie have in her pocket? Jeff says, I can make 17 with the coins in my pocket. 2. Show two different ways to make 17 with coins. First way Second way 2 nd Grade

5 Tim says, I have one half-dollar, one quarter, one dime, one nickel, and one penny in my pocket. 3. If Tim pulls three coins from his pocket, what is the most money he would have in his hand? Show how you know your answer is correct. Anna bought some gum for 18. She gave 50 to pay for the gum. 4. How much change did Anna get? What coins did Anna get back? 8 2 nd Grade

6 Pocket Money Mathematics Assessment Collaborative Performance Assessment Rubric Grade 2 Pocket Money: Grade 2 Points Section Points The core elements of the performance required by this task are: Understand whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers Demonstrate fluency in adding and subtracting whole numbers Communicate reasoning using words, numbers or pictures Based on these credit for specific aspects of performance should be assigned as follow: 1 Gives correct answer such as: 2 nickels or 1 nickel 5 pennies or 10 pennies 2 Shows one correct way to have 17 Shows a second correct way to have 17 Partial credit: Shows 2 ways to get to a number other than 17 3 Gives correct answer: 85 Shows work such as: = 85 4 Gives correct answer: (1) Total Shows coins such as: 3 dimes, 2 pennies Or 1 quarter, 1 nickel, 2 pennies 1 ft nd Grade

7 2nd grade Task 1: Pocket Money Work the task and examine the rubric. What do you think are the key mathematics the task is trying to assess? In Parts 1 and 2, many students had difficulty understanding the constraints of the problems. In Part 1, students needed to show 10 without using a dime. Looking at their work, how many students disregarded or misunderstood the constraint and drew a dime or wrote 10? In Part 2, how many students could successfully show one way of making 17 but not two ways? Did they make 17 again? Did they make a completely different value? How many students made two values, but did not use the constraint of 17 and used two unrelated amounts? What the implications for future instruction? How can successful students make their strategies explicit? Looking at the student work in part 3, how many students put: Other What do these incorrect answers tell us about how students understood the problem? Which incorrect answers were attributable to errors of understanding the constraints? Which incorrect answers were attributable to errors of calculation? When you look at errors of calculation, what kinds of mistakes are students making? What evidence is there of understanding within the mistakes they ve made? When you look at papers from students who successfully solved this part of the task, which strategies did your students use to show evidence that they understood the constraints of the question? Drew pictures Listed all coins and indicated the three with the largest value Listed all coins and crossed out the two with the lowest value Other How might successful strategies be shared with students who were not able to determine the relevant information to solve the problem? 2 nd Grade

8 Look at student work in Part 4. We can learn both from errors and from successful strategies. How many students used the following strategies to successfully find the change due: Count up from 18 to 50 Draw Coins Standard Algorithm Count back from 50 to 18 Other Can you sort these strategies from most to least accurate? From most to least efficient? Are there tools, such as open number lines or bar models, that might help move some of the strategies to more efficient and more accurate? Look at the errors in Part 4. What types of errors are being made? Can you sort the errors into different categories? Which errors seem to be errors of understanding, such as a misconception of what change means? Which errors seem to be errors of calculations? Which strategies are students using when they make errors in calculations? What are the implications for future instruction? 2 nd Grade

9 Student A s work is typically representative of the students working at the Cut Score. This student is able to answer Parts 1 and 2 with drawings and labels, and attends to the constraints of both problems by making 10 without using a dime and showing two different ways of making 17. In Parts 3, the student ignores the constraint to sum the three coins with the highest value, and instead adds them all. Then he or she makes an error in place value alignment in the placement the vertical values, where the penny is being added in with the dimes. This student does not recognize a situation where making change requires them to find a difference between to values of coins. However, they can accurately model with drawings and labels their incorrect answer of 68. Student A Student B did not score any points on this task. Looking at their work in Parts 1 and 2, what evidence of understanding do you see in their thinking? What would they tell you if you asked them what those circles mean? How could they be more explicit in their recording? Student B 2 nd Grade

10 Student C did not attend to the constraints in either Part 1 or Part 2. Does this student think they have shown two different ways make 17? If they do, what is their understanding of different? Student C What system is Student D using to label the coins in Part 2? How can this student use what they already understand to learn how to be more explicit in their recording? Student D Student E is using a system that many successful students used in Part 3. What evidence is there that this student understands the constraints? What does the labeling demonstrate that student knows about the coins listed in the problem? Student E 2 nd Grade

11 Students F and G are trying to attend to the constraints of the problem. Student F s strategy for keeping track of what won t be used is similar to how Student E keeps track of the coins that will be used. How does Student F s written explanation help us understand what Student G s number sentence? Student F Student G 2 nd Grade

12 50 was a common incorrect answer in Part 3. What does Student H s explanation let us know about students who have made this mistake? Student H Most students who didn t understand the constraints in Part 3 did what Student I did. What misunderstanding do they have about the problem? What do they demonstrate that they understand about the coins listed in the problem? Student I 2 nd Grade

13 Student J is using a successful strategy to identify the three coins that meet the constraints, but the explanation reveals that the reasoning is based on the size of the coins, rather than the value. Student J Like Students K and L, many students were unsure what change meant in this problem. Students who couldn t determine the meeting likely gave one of these two answers. What does Student L s use of the the word got for gave tell us about their understanding of the problem? Student K Student L 2 nd Grade

14 Student M and Student N are using a similar strategy. What can Student M learn from the way Student N is using the strategy successfully? Would it be helpful for this student to use actual or plastic coins to create the set of coins created by both students? What kinds of counting or exchanges can the student make to connect the two sets? What would happen if the student tried to physically take away 18 from the set they created? Studet M Studet N Students who successfully found the difference between 18 and 50 in Part 4 used a variety of strategies. Student O does not show the number line, but the written description is clear and quantitative. Student P correctly sets up and uses the standard subtraction algorithm, and Student Q draws and labels a model to show their strategy of counting up from 18 to 50 using coin values. Student O Student Q Student P 2 nd Grade

15 Students Q T These students all attempted to find the difference in Part 4 by using the standard subtraction algorithm. What evidence is there the students are trying to remember and make sense of set of procedures? Do you believe that these students are making sense of the quantities? What evidence supports your belief? In what ways are these students misusing concepts of subtraction and place value? Student Q Student R Students Q and R got the same difference. Although Student R has labeled the ones place with an O and the tens place with a T, they are also demonstrating that they are not thinking about the quantities inherent in those labels. They see each set of digits as a separate problem, to be solved using subtract from the larger digit regardless of the value or placement of the digit. Why aren t these students pondering the reasonableness of this answer? If these students were asked to directly model how they would subtract these values, they would have to confront the quantities behind these numbers. Student S What evidence is there that Student S is thinking about place value? Where did the 10 come from above the 0? What experiences has this student had in thinking the quantity of 50? Do they recognize this number as 5 tens + 0 ones? Do they realize that the 10 in the ones place changes the value of the number to 5 tens + 10 ones? Is that still 60? In what ways is it hindering these students understandings to insist on using the standard subtraction algorithm? How can a teacher use what they are showing they understand as a way to focus on number sense, rather than rule following? 2 nd Grade

16 Student T Although Student T is still confused about the underlying place value of our number system, they are using a strategy in the second part of Question 4 that might be helpful for determining the accurate answer for the first part. Using what they know about pennies, they break the 2 into 1 and 1 in order to subtract it from 50. In what ways can 18 also be broken apart and then subtracted? How might this student benefit from the strategy used by Student U? Student U Student U is using a combination of strategies to find the difference. In addition to a calculation error, the student makes a tactical error when they compensate for subtracting 8 in the first line (18-8) and then adding 8 back in in the last line (40 + 8). Using an open number line, determine what the student was attempting do by making jumps that correspond to each number sentence. Do you see what the error is, using this model? Would this model be helpful for the student to also begin to make sense of compensation in subtraction? 2 nd Grade

17 Students V X used a counting up strategy to find the change in Part 4. Looking closely at Students V and W, what can be determined about their understanding of how counting works? What strategies does Student X use to eliminate the inefficient, errorprone counting by ones that the other two students are using? How might this student s work be used to connect with the work done by the other students? Student V is counting the numbers between 18 and 50. In the semi-erased work of Student W, you make out the numbers 18 to 49 with tally marks representing their counting. Number line work can help these students understand that we are counting the spaces between numbers, and not the numbers themselves. Student V Student W Student X 2 nd Grade

18 2nd Grade Task 1 Pocket Money Student Task Know the value of coins. Use coins to represent the same amount in more than way. Demonstrate fluency in adding and subtracting whole numbers in context. Follow one or more constraints in a problem. Communicate reasoning using words, numbers, or pictures. Core Idea 1 Number Operations Core Idea 2 Number Operations Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Understand whole numbers and represent and use them in flexibly ways, including relating, composing, and decomposing numbers. Demonstrate fluency in adding and subtracting whole numbers. Mathematics in this task: Knowledge of the value of coins Ability to use coins to show an amount of money in more than one way. The ability to attend to more than one constraint for a problem. Understand that change in this context means to find the difference between two coin amounts. Accurately add and/or subtract coin amounts. Explain mathematical reasoning using words, drawings, and symbols. Based on teacher observations, this is what second graders know and are able to do: They could show 10 cents multiple ways. They could show two different ways to make 17 cents. They were able to draw the correct coins to match an amount. Some could use the regrouping strategy correctly while subtracting Many drew coins as a strategy to explain their thinking or solve a calculation. Some increased their efficiency by counting by 5 s and 1 s to get the correct answer. Areas of difficulty for second graders: Understanding the concept of change. Confusing the size of a coin with the value of a coin. Following the constraints of a problem. Inability to use the standard subtraction algorithm to correctly find a difference. Strategies used by successful students: Successful students used what they knew about the values of coins to make amounts in more than one way. Successful students used drawings and lists to make sense of the problem and explain their thinking. Successful students understood that change in this context means that they should find the difference between two amounts of money. Successful students could find the difference between two amounts of money, using a variety of strategies including counting up and accurately regrouping in the standard algorithm. 2 nd Grade

19 2 nd Grade

20 Pocket Money Points Understandings 0 All the students in the sample with this score attempted the task. They could use drawings and symbols to represent an amount of money. 1-2 Students could make a total of 10 cents without using a dime. Many got both points from making 17 cents in two different ways. They could show the value using drawings and symbols. 3 4 Eighty-five percent of these students were able to answer parts 1 and 2 of the task. Most students who could find one way of showing 17 cents could also show a second way. These students were able to draw and label coins. 5 Students were able to demonstrate an understanding of the value of coins and the ability to represent given values in multiple ways. Nearly one-third of these students correctly identified half-dollar, quarter, and dime as the three coins with the most value in Part 3 and correctly summed the values. Fifty percent of the students were able to correctly make change from 50 cents for an 18 cent purchase. 6 7 Three-quarters of these students were able to identify the three coins with the highest value in Part 3, and were also able to accurately add them together for a correct answer. Eighty percent of these students could accurately find change in Part 4. They used a variety of strategies including drawings, counting back, counting up, and the standard algorithm for subtraction. 8 Students identified the coins with the three highest values and summed them correctly. They used strategies to keep track of the constraints, by drawing coins, listing all coins and identifying the relevant ones for the problem, etc. They understood change as finding the difference and accurately used the standard subtraction algorithm, counting up, and drawing coins to solve the problem. Misunderstandings Students couldn t hold both constraints of 10 cents and not a dime. They answered with one or the other by either showing a dime, or showing, for example, a quarter. Students continued to struggle with constraints, in Part 3. Common mistakes include using any three coins, or applying the wrong value to a chosen coin. Students who were able to accurately identify the three coins with the greatest values still struggled with inaccurate addition and subtraction. Students continued to struggle with Part 3. They also often misunderstood what it means to make change in Part 4. Twenty percent of the students added the two values instead of finding the difference between them. Others assumed that 18 cents would be the change, or claimed that 50 cents should be the change. Most students who set up the expression proceeded to solve it incorrectly. Half of the students continued to struggle with the constraints in Part 3, and either added the two most valuable coins or added all the coins. One-quarter of the students chose either 50 cents or 68 cents when asked to make change for an 18 cent purchase from 50 cents. About half the students who could not find the correct change for Part 4 solved the problem by stacking and then subtracting the ones as 8 0 and subtracting the tens as 5 1 resulting in the answer 48. The other half solved the problem by stacking and then regrouping a ten but failing to remove the regrouped ten from the tens place. They went on to subtract the ones as 10 8 and the tens as 5 1 resulting in the answer nd Grade

21 Implications for Instruction Working with money is an application of number and operations. There are certain things that students need to understand about the notations and conventions of coins and money in order to work successfully, but the basic strategies of operations and the basic ideas such as decomposition and multiple representations still apply. Students need many experiences to be able to name and identify coins, and also to understand the value of each coin. This can be particularly tricky for young children because the size of each coin is not necessarily commiserate with its value. Further, they need many experiences counting and spending money (including contextual play such as restaurants or stores) in order to make sense of which strategies are efficient and accurate for counting coins and making change, and which are not. Once students understand the specifics of working with coins and money, we turn once again to ways in which students can use tools and strategies to accurately find the difference between two numbers. Making change is a wonderful application of the inverse operations of addition and subtraction, and being able to count up to make change will help students develop a more profound notion of subtraction as think addition in other number situations, also. In addition to the operations and concepts of money, we must also think about the problemsolving nature of this task, and what it means for our classrooms. In what ways are we providing students with experiences in determining which information is relevant when deciding how to solve a problem? How are successful strategies shared by students who are able to identify the constraints of a problem? What opportunities do students have to explain their thinking using models or drawings? What feedback do they experience to help them know if their explanation was explicit and accurate enough? Ideas for Action Research The following activities and ideas are from the article, Teaching the Values of Coins by Randall Drum and Wesley Petty in the January 1, 1999 of the journal Teaching Children Mathematics. Plastic or actual coins are often used as manipulatives in the elementary classroom. However, the coins themselves are nonproportional to the values they represent. This means that rather than being concrete models that help develop understanding, they are actually abstract models when used to teach their values. In this article, the authors suggest creating proportionate models to represent the values of the coins. (These can be made in any size, but the authors recommend 20cm x 20cm for each 1 box.) Take a look at the models on the following page and reflect on how using them for the following activities could help students make sense of the values of coins and how to operate on money amounts. Looking at the model, how is the true relationship between the value of a dime and the value of a penny visually represented by this model? Teaching Values Place the cutouts representing the proportionate value of the coins over the top of the cent model. Relative Values of Coins Determine how many of one coin are needed to have the same value as a different coin by placing cutouts directly on top of other cutouts. Example: 5 nickel cutouts will be needed to completely cover 1 quarter cutout. Value of a Set of Coins Use plastic or actual coins to introduce the set of coins to be worked with. Cutouts of the proportional values should be placed on the cent model, largest value piece to smallest value piece. o In what ways does placing largest to smallest mimic the most efficient way to count a set of coins? How can the counting be linked to the open number line? Model Apparent Contradictions Use the proportional model to show that a set with fewer coins can have more value than a set with more coins. 2 nd Grade

22 Find the Difference Between Two Values How can you use the proportionate cutout models to model the concept of making change as finding the difference between two values? Create a Set of Coins When Given a Value Use the cent model to shade the value you will be working with. Completely cover the shaded area with the proportional coin models. o Multiple possibilities exist (except for the value of 4 ). How might students share their ideas to encourage multiple representations? What questions or prompts could the classroom teacher use to encourage them to make more than one representation? o Good way to practice using constraints ~ can they make the value using the fewest coins possible? Using exactly 6 coins? Etc. As you go through the activities, reflect on ways to connect the ideas they are developing about the value of, and about the addition and subtraction of, coins. Would it help to have them also create the sets they are modeling with actual or plastic coins after they have built their proportionate models? What value is there in recording their movements with the model pieces on an open number line? (Use a 1-jump for penny, a 5-jump for nickel, a 10-jump for dime, etc.) nd Grade

23 OUR PETS Sam asked his classmates: What kind of pets do you have? He made a tally table to show the answers. BIRDS CATS DOGS FISH NO PETS OUR PETS 1. Use the information in the tally table to fill in the bar graph below. OUR PETS N u m b e r s o f P e t s BIRDS CATS DOGS FISH NO PETS Kinds of Pets 2. Which kind of pet do the fewest children have? 2 nd Grade

24 3. How many dogs and cats do these children have in all? Show how you know that your answer is correct. 4. How many more children have dogs than fish? Show how you know that your answer is correct. 5. Using the same data, Sam found out how many of the children in his class have pets. He wasn t able to finish the table. Finish it for him. Have pets Do not have a pet 6. What is one thing you can learn from the bar graph that you can t learn from the table above? 10 2 nd Grade

25 Pocket Money Mathematics Assessment Collaborative Performance Assessment Rubric Grade 2 Pocket Money: Grade 2 Points Section Points The core elements of the performance required by this task are: Understand whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers Demonstrate fluency in adding and subtracting whole numbers Communicate reasoning using words, numbers or pictures Based on these credit for specific aspects of performance should be assigned as follow: 1 Gives correct answer such as: 2 nickels or 1 nickel 5 pennies or 10 pennies 2 Shows one correct way to have 17 Shows a second correct way to have 17 Partial credit: Shows 2 ways to get to a number other than 17 3 Gives correct answer: 85 Shows work such as: = 85 4 Gives correct answer: (1) Total Shows coins such as: 3 dimes, 2 pennies Or 1 quarter, 1 nickel, 2 pennies 1 ft nd Grade

26 2nd grade Task 2: Pets Work the task and examine the rubric. What do you think are the key mathematics the task is trying to assess? Looking at the student work, determine which students were able to accurately transfer the data from the tally table to the bar graph in Part 1. As you look at the student work in other parts of the task, think about how the accuracy of this bar graph affects the accuracy of all other calculations and statements students are required to complete to be successful at this task. For students who made mistakes in creating the bar graph, were these caused by confusion on how to read tallies? Inaccuracies in visual discrimination, such as moving data over one bar for each pet when transferring? Misunderstandings in how to make a bar graph, including starting from the top to shade, leaving the first bar blank, or using tallies in the bar graph? What learning experiences would benefit students who are struggling with this transfer of information? Look at the student work for Part 3. How many students put: Other For students who accurately summed to 22, which strategies did they use? How could those successful strategies be shared with other students? How might students arrive at the other answers of 32, 37, and 6? When you look at the student work for students who got an other answer, how many of them chose a correct process and strategy, but were using incorrect data because of an inaccurately constructed bar graph? Look at the student work for Part 4. How many students put: Other For students who accurately found a difference of 8, which strategies did they use? How could those successful strategies be shared with students? How might students arrive at the other answers of 20, 9, and 6? What does this tell us about student understanding? When you look at the student work for students who got an other answer, how many of them made an error in calculating? Which were based on an error in the data? What strategies did students use to find the difference between 14 and 6? Were some more accurate than others? More efficient than others? When analyzing Parts 3 and 4, think about examples in the student work where students used particularly effective tools, strategies, or explanations of their thinking. In what ways can these successes be shared with other students? Are there models that you don t see evidence of, such as the use of open number lines or bar models or drawings, that you think would have been helpful in clarifying student thinking, improving their explanations, and increasing their efficiency and accuracy? How might these tools and strategies also be introduced in meaningful ways? 2 nd Grade

27 Looking at the statements in Part 6, can you identify students who: Offer judgments on usability of a graph, such as one being harder or prettier? Make a statement that could easily be applicable to both graphs, such as I can count or There are 32 pets? Make statements about an incomplete tally count in the table from Part 5, such as stating that there are 22 or 3? Make unclear or incomplete statements about one or both graphs? What do these different answers tell you about the students understanding of the relationship between the three representations of the data? What evidence is there that students understand that each of these is a representation of the same data? Or is there evidence that they see these as unrelated, separate tasks? 2 nd Grade

28 In order to be successful at this task, students needed to move accurately and flexibly between three representations of one data set, and perform qualitative and quantitative analysis on the information. Student A represents the 23.9% of students who performed at the highest level on this task. The student accurately transferred the data from the tally chart to the bar graph, was able to use that data to compare and combine quantities, and could explain at least one difference in the kinds of information available on two representations of the same data. A small error in representing the total number of pets with tally marks on the last chart did not diminish the student s understanding of the data. When performing calculations, the student could justify the work using symbols, drawings, and numbers. Student A 2 nd Grade

29 Student B represents the 7.8% of students who performed at the lowest end on this task. The student loses points because they were not able to accurately transfer the data between the three representations, and moreover, they did not seem to understand that the charts and graph are all based on the same information. Although Student B had some good strategies for adding and subtracting, including using the graph to count both sets and to do a comparison by direct matching, all calculations were based on incorrect values in the bar graph, and the explanations provided were accurate, but incomplete. Student B The next few pieces of student work for Students C H demonstrate the variety of strategies students use to make sense of the data, and provide evidence of the ways that students see (or don t see) the connection between the different representations. Are your students using any of these counting and organizing strategies? Are your students unable to find correct solutions to qualitative and quantitative questions about the data because of errors in how they set up the data? 2 nd Grade

30 Student C struggled with the logistics of completing the bar graph, but seems to understand that the data should come from the tally chart. Student C In this work, Student D uses a strategy that helps him or her to answer Part 6. In totaling the number of pets, this student uses the extra tally marks to complete groups of 5. They mark one tally off of the Cats and add it to Birds. They mark one tally off of fish and add it to Dogs. Now this student can count by fives, with one group of two extra tally marks with the Cats. How many of your students attempted to complete the tally chart in Part 6 but made errors in the total number of tallies because of miscounting? Student D 2 nd Grade

31 Student E s work shows evidence of how they are managing the data. The student marked each box, verifying that they had shaded the number of boxes that corresponded to the number of tally marks. In order to find the difference between the number of dogs and the number of fish, this student counts the blank boxes that match the remainder of the shaded boxes in the dog category. Does this student even see the number labels on the vertical axis? If they were aware of this graph feature, would they need to count every box? Student E In the margin of Student F s paper, there is evidence of how they determined the total number of pets for the answer to Part 6. The numbers they are working with come from the tally chart in Part 1. Student F 2 nd Grade

32 Students G J solved Part 3 using a variety of strategies. Look at the strategies they ve used. Identify the evidence of student thinking. Are any of these strategies more efficient than others? Are any more accurate? How can student work be used to re-engage with the problem and begin to make connections between different strategies to deepen understanding? Student G Student H Student I Student J 2 nd Grade

33 Students K N solved Part 4. Look at their work. What strategies can you identify? What do they understand about comparing quantities? Student K Student L Student M Student N 2 nd Grade

34 Re-Engagement Student O and Student P both solved Part 4 using the graph to count the physical differences in the number of shaded boxes between the categories of dogs and fish. They both came to the correct answer. Look at the number sentences these two students wrote to go with their work. How does thinking addition and then writing subtraction seem to cause confusion for Student P? Would it help Student P to verbally describe their process, as Student O did? Does connecting to the action help choose an operation? Student O Student P I counted all the gray spots that are higher than the grey spots for fish. 2 nd Grade

35 Re-Engagement Students Q and R both used counting to solve the quantitative questions for this task. Only one of them, though, got credit for their explanations. What can you identify in Student Q s explanation that is not part of Student R s explanation? Tell the class that they have to solve the problem based only on the instructions you read them. Read Student R s explanation and ask, could they solve this problem? Would they be able to identify what information was missing? Can they ask a question or two to get clarification? Do the same with Student Q s work. Can they clearly articulate what makes Student Q s explanation more mathematically explicit than Student R s? Student Q Student R 2 nd Grade

36 2nd Grade Task 2 Pets Complete a bar graph to represent data given in a tally table. Find numerical and Student categorical information and compare values. Explain their thinking in finding quantitative information about the graphs. Complete a tally table using summary Task information from the data. Make a statement that compares the kinds of information found on two different representations of the same data. Core Idea 5: Data Analysis Core Idea 2: Number Operations Students organize, display, and interpret data about themselves and their surroundings. Describe and compare data using qualitative and quantitative measures. Represent the same data in more than one way. Represent and interpret data using bar graphs, tally charts, and other representations. Understand the meanings of operations and how they relate to each other, make reasonable estimates, and compute fluently. Demonstrate fluency in adding and subtracting whole numbers. Mathematics in this task: Complete a bar graph based on the information in a tally chart. Represent strategies of addition and subtraction for values based on the graphs. Use the same information to create a different interpretation of the data and record the information by completing a second tally table. Compare the type of information available on two representations of the same data. Represent and explain thinking and strategies using words, pictures, models, and/or symbols. Based on teacher observations, this is what second graders know and are able to do: Use the tally marks in the first chart to make a bar graph. Identify and name the least value in the graph. Use a variety of strategies to add and subtract values from the data, including finding landmark numbers, algorithms, using missing addend addition to find a difference, using the bar graph to count the number of extra or missing bars when comparing. Areas of difficulty for second graders: Adding the two values instead of finding the difference when comparing. Using the standard algorithm to subtract accurately. Most students only completed the tallies for No Pets on the second tally table. Making a clear statement that compares two representations of the same data. Strategies used by successful students: Use tallies, drawings, or other models to represent values and strategies when adding and subtracting. Use a variety of strategies to find the difference between two numbers. Identify the relevant information needed to choose the values and operations. Move between representations to make sure the values are the same in each representation. 2 nd Grade

37 2 nd Grade

38 Pets Points Understandings Misunderstandings 0-2 All the students in the sample with this score attempted the task. Some students could identify the pet with the least tallies. Nearly half the students described an accurate process to answer the quantitative questions in Parts 3 and 4, but based their calculations on incorrect data from the bar graphs they had created. 85% of the students at this level were unable to accurately transfer data from the tally table to the bar graph. Some students confused the meaning of least to be none in Part 2, as evidenced by answering with No Pets. Half the students made calculation errors in their addition and subtraction. 75% of the students made no attempt to complete the second tally table representation of the 3 More than half of these students received full credit on creating the bar graph, and the rest received partial credit. They could correctly identify the pet with the least tallies. 55% also described an accurate process to answer the quantitative questions in Parts 3 and 4. About a third attempted to complete the tally table in Part 5 for the category No Pets. 4-5 Two-thirds of these students got full credit for the bar graph they created. 40% of the students answered Part 3 correctly, and another 40% chose a correct process. Students used a variety of strategies to find the difference between the two values in Part 4, including direct matching, counting up, and subtraction or counting back. 60% of these students attempted to complete the tally table in Part 5 for the category No Pets and another third of them attempted to complete it for the category of Have Pets. 6 Students were able to transfer the data from the tally table to the bar graph. In Part 3, 80% of the students were able to identify the relevant information and accurately sum two numbers. Nearly 50% were able to accurately find the difference between two values in Part 4, and used a variety of strategies including direct matching and counting up. More than 60% of these students attempted to complete the tally graph in Part 5, and nearly half were able to make an accurate statement in Part Students working at this level were able to accurately complete the bar graph. More than 80% were able to correctly write and solve addition and subtraction equations to solve Parts 3 and 4. These students all attempted to answer Part 5, and almost 40% were able to accurately complete the tallies in both Have Pets and No Pets. An additional 42% were able to create the No Pets tallies. When using a data. Students at this level were working with better data, but continued to be plagued by errors in calculations. Many of these students incorrectly used the standard algorithm for subtraction. When using a matching strategy, some students used a tally mark model, which proved difficult for making a direct match for comparison purposes. Nearly half of these students were also using an inaccurate bar graph to do their calculations and comparisons. 60% of these students were not able to accurately answer Part 3. When answering Part 4, 80% gave an incorrect response, and a third of them added instead of finding the difference. These students were not able to make a clear or accurate statement comparing the information on the different representations of the graph. Attempts to complete the tally table in Part 5 indicate that they may not have seen the connection between the representations. Students continued to make calculation errors when subtracting, particularly with the standard algorithm. About 10% of these students incorrectly added the two values they were supposed to be comparing in Part 4. At this level, students who were unable to complete the second tally table were also unable to write an accurate statement in Part 6, and nearly 40 % of them didn t even attempt to complete the first section of Part 5. Some students continued to make small or isolated errors when creating the bar graph. This small percentage of graphing errors led to some incorrect calculations in Parts 3 and 4, but students still got process points for their explanations. Around 60% of these students were not able to complete the Have Pets part of the tally table, though nearly all attempted to. Common errors included 2 nd Grade

39 more accurate tally chart, two-thirds were able to make a clear and accurate statement in Part Students were able to accurately complete the bar graph. They could identify the relevant values in Parts 3 and 4 and use addition and subtraction to answer quantitative questions about the data. These students demonstrated an understanding of the relationship between the three representations of the data, and could create the second tally table and make a clear comparative statement about the type of data found in each representation. counting No Pets in for a total of 37 tally marks. 2 nd Grade

40 Teaching Implications In this task, we saw that students working at all but the highest levels struggled with answering the question How many more? when comparing two values on the graph. Direct Modeling In their book, Children s Mathematics (Heinemann, 1999), Thomas Carpenter and his colleagues reflect on their research with young children who have been given the opportunity to direct model their thinking and actions when operating on two values. These students were given many opportunities to use cubes to directly model various actions they interpreted from math stories involving combining, separating, or comparing values. Looking at the curriculum and materials for your classroom, where can you provide experiences for them experience direct modeling? Counting Strategies From these experiences, they invented counting strategies that were direct reflections of the actions they took with the cubes. From these counting strategies, they began to construct increasingly efficient counting strategies that involved groupings, doublings, etc. At each step along this progression, the students were engaged in constructing their own understandings and strategies. As you observe your own students, where can you guide them and use questioning techniques to help them identify and formalize the strategies they are using? Comparisons In many elementary math classrooms, young children who are asked to solve a task that involves a comparison to find out How many more between two values are often instructed to look for those key words which indicate that they should then subtract to find the answer. Interestingly, when children construct their own meaning, they very often don t think of comparisons as subtractive in nature. In fact, when these children had stacked two sets of cubes (for example a tower of 5 and a tower of 7) to represent to values, they tended to solve the comparison by directly matching the cubes that were the same, and then counting on to the smaller value to find out the difference between the two. In our example, they would start with 5, and count up two cubes until they reached 7. Those two cubes represent the difference between the numbers. Reflection If we insist that they use the definition finding the difference to mean subtract, are we overwriting their intuition about how this relationship works? How could using the techniques of Cognitively Guided Instruction described in Carpenter s book help us to build on the intuition they bring to the mathematical situation? Making a connection between the two equations 5 + = 7 and 7 5 = deepens the children s understanding of addition, subtraction, comparison, and inverse operations. It helps them understand the relationship between both operations, and between both values. When they understand this relationship, picking an operation becomes a trivial part of the mathematics, because they know the situation can be resolved using either addition or subtraction. In what ways could this deeper understanding support students who continue to struggle with the standard algorithm and counting backward models of subtraction? 2 nd Grade

41 Ideas for Action Research In this task, we also saw that students working at all but the highest levels struggled with understanding that the three representations in the task were all generated from the same data. Re-Engagement Have your students complete the task, Pets, and/or use this activity any time your students are working with data. Use the student work to re-engage with your students afterward. Have them look at the three graphs that are part of the task. When students understand that these represent the same data, that they are in fact related, they can begin to reflect on the idea that different representations might be helpful to answer different questions. They can begin to imagine other ways they might represent the data. In order to help students make connections between various representations of the same data, use a questioning strategy such as: Can students identify the question asked to make the original tally table in the task? Can they make the connection that the same question is answered with the bar graph? Can they identify the question asked to make the second tally table? Can they make a connection between that question and the data pool that answered that question? Where did this data come from? Who answered the question how many students did not have pets? Once students have connected the data in the different representations, are they more successful in thinking about and explaining differences in information reflected about that data? Does thinking about the data this way help them to accurately make the second tally table? If not, what other experiences would help students explore these two questions about data sets and multiple representations of them? 2 nd Grade

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