Third Grade Mars 2006 Task Descriptions Overview of Exam Core Idea Task Score Data Analysis Flower Garden The task asks students to read, interpret, and draw conclusions from information on a bar graph representing data about flowers in a garden. Successful students can record data on a graph and compare values. They can use addition to find total values on a graph. Patterns, Functions Houses in a Row and Algebra The task asks students to find and extend a pattern about toothpicks are needed to make a geometric design. Successful students can recognize and extend a visual geometric pattern, reason about constant growth and extend the pattern forward. Successful students can also use inverse operations to think backwards from a total to where that total would fit in the pattern. Number Operations The Answer is 36 The task asks students to work with number calculations to get the answer 36. Successful students think about missing numbers in addition, subtraction, and multiplication situations. They can use regrouping for addition and subtraction with accuracy and reason about place value. Number Operations Pens and Pencils The task asks students to calculate buying groups of objects and to find change. Successful students can use addition or multiplication to find the cost of buying groups of an item. They can use decimals and convert easily from cents to dollar notation. They can use subtraction to find change. Geometry and Garden Design Measurement The task asks students to find the area of shapes on a grid, compare areas, and design a new shape with a given area. Successful students count to find area on a grid, know to quantify the value of all shapes before making a comparison, and design a shape to fit the given constraints about area. 1
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The Flower Garden This problem gives you the chance to: represent data using a bar graph draw conclusions from the data Cameron grows flowers in his garden. He grows sunflowers, daisies, roses, lilies and sweet peas. Cameron draws a bar graph to show how many flowers are in his garden today. 12 10 Number of flowers 8 6 4 2 0 Sunflowers Daisies Roses Lilies Sweet peas 1. How many sunflowers are there in Cameron s garden today? 2. How many more roses are there than sunflowers? Show your work. 3. On Cameron s bar graph, show that there are 8 sweet peas in his garden today. 4. Cameron waters all the daisies, roses and lilies. How many flowers does he water in all? Show how you figured this out. 5. Cameron picks the flowers he has most of to give to his granny. Which flowers does he pick? MAC Rubrics 2006 Test 3 8 4
The Flower Garden Rubric The core elements of performance required by this task are: represent data using a bar graph draw conclusions from the data Based on these, credit for specific aspects of performance should be assigned as follows points section points 1. Gives correct answer: 3 1 2. Gives correct answer: 7 Shows correct work such as: 10-3 3. Draws correct bar on graph for 8 sweet peas 1 4. Gives correct answer: 23 Shows correct work such as: 7 + 10 + 6 5. Gives correct answer: roses 1 1 Total Points 8 1ft 1 1 2 1 2 1 3 5
3 rd Grade-Task 1: The Flower Garden Work the task and examine the rubric. What do you think are the key mathematics the task is trying to assess? Look at student work in part 1, how many of your students put: 3 2 or 4 2 1/2 or other fractional amount 26 5 Other How were student errors related to issues of understanding scale? How did your students deal with the idea of a bar falling between two lines? Why might a student get an answer of 26? What is the student probably thinking? Does this answer show an understanding or lack of understanding about scale? Why might a student get an answer of 5? Look at student work in part 4, how many of your students put: 26 21/24 22 34 3 Other Which errors were caused by calculation errors? Is there evidence in some of the wrong answers to support the notion that the student understands scale? How do you know? Look at student work in part 2. Did your students recognize that this situation was a subtraction situation or did the word more trigger an addition response? Did your students pick the value for roses or sunflowers? Did your students try to compare two values next to each other on the graph, like roses and daisies or daisies and sunflowers? What might this indicate as an instructional implication? 6
Looking at student work on The Flower Garden: Student A shows a good understanding of scale. Notice how the student added the odd numbers on the left side of the graph. Student A understands comparison subtraction and can write a number sentence for the comparison in part 2. When computing the addition in part 4, Student A groups the numbers to make the addition easier and takes advantage of the easiness of adding ten. This may indicate a good understanding of place value. Student A Student B identifies and labels the given information to help make sense of the task. This allows the student to pick an operation and write an appropriate number sentence. Notice that Student B seems to understand equality in part 4, by using 2 separate equations to show the calculations. 7
Student B Student C is able to successfully deal with the issues of scale. The student also seems to understand when to use addition and subtraction in word problems. It is interesting to note that the student had the correct number sentence and calculation for part 4, but then erased the answer. What do you think confused the student? Student C 8
Student D is able to understand and think about scale. The student has difficulty thinking about comparison. The student may understand the concept of comparing using subtraction. The student may understand the inverse relationship between addition and subtraction. However, the student cannot pick out which part of the number sentence in part 2 represents the solution. If you were writing a brief note to the student, what question would you ask to probe her thinking? What else strikes you as you look at this piece of student work? Student D Student E is able to think correctly about scale for sunflowers and daisies. The student adds incorrectly in part 4. In part two, the student regroups to take a number away from 10. What might you be concerned about as a teacher? What other experiences might the student need? Student E Student F has difficulty understanding the scale. Can you find two pieces of evidence for this? The student has two different strategies for thinking about the value of a bar falling between the lines. Why do you think the strategies changed as the student progressed through the task? 9
Student F Student G shows a common misconception about graphing in part 1. Why do you think the student picked 12 as the number of sunflowers in part 1? Student G has trouble with the idea of comparison and chooses addition instead of subtraction for part 2. Why do you think the student put 5 in part 4? What might the student be thinking? Student G 10
Third Grade 3 rd Grade Task 1 The Flower Garden Student Task Core Idea 5 Data Analysis Core Idea 2 Number Operations Represent data using a bar graph and draw conclusions from data. Use comparison subtraction to solve problem. Collect, organize, display, and interpret data Describe important features of a set of data (...most, least and comparison.) Represent data using tables, line plots, bar graphs, and pictographs. Understand different meanings of addition and subtraction of whole numbers and the relationship between the two operations. Mathematics of the task: Ability to interpret a bar graph. Ability to draw conclusions from data. Ability to extend/represent data on a bar graph. Ability to recognize and use axis scale other than one. Ability to make a comparison. Based on teacher observation, this is what third graders knew and were able to do: Read information on a graph Adddatatoagraph Add and subtract with accuracy Show their work Areas of difficulty for third graders: Working with scale, interpreting a value that falls between the lines Comparing two elements, finding how many more Identifying most on a graph Trying to regroup to find 10-3= Strategies used by successful students: Putting odd numbers on the scale of the graph Listing the values of each item on the graph Labeling their work to help identify what is being calculated Underlining what is being asked, for example in part 4 underlining daisies, roses, and lilies 11
MARS Test Task 1 Frequency Distribution and Bar Graph, Grade 3 Task 1 Flower Garden Mean: 5.38 StdDev: 2.45 MARS Task 1 Raw Scores The maximum score available on this task is 8 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, about 91%, could interpret values on a scale, record data on a graph, and identify the most on a graph. Many students, about 74%, could also find the total for a group of items on a graph or compare the value of two items on a graph. 53% of the students could interpret scale, add accurately, identify the most from a graph, and record data on a graph. 29% met all the demands of the task including using subtraction accurately to make a comparison of two items on a graph. Approximately 2% of the students scored no points on this task. 75% of the students with this score attempted the task. 12
TheFlowerGarden Points Understandings Misunderstandings 0 75% of the students with this score attempted the task. Students often did not attempt to enter data on the graph. Many students struggled with most flowers in part 5. They may have put down more than 2 Students could generally record data on a bar graph and identify the most. 3 Students could record data on a graph, read a value falling between the bars, and identify the most. 4 Students could record data on a graph, read a value falling between the bars, and identify the most. one type of flower. Students had difficulty finding the value for sunflowers. About 12% just rounded to an adjacent bar (4 or 2). 8% found the total of all the flowers (instead of just sunflowers). A few recognized that it was more than 2, so they made the value 2 1/2. About 10% of all students did not attempt comparison subtraction in part 2. About 7% just put down the value of roses, effectively not making a comparison at all. Others common answers were 4, 8, and 3. Students struggled with comparison subtraction. They may have made calculation errors or forgot to show their work. 6 Students could complete all of the task, except finding the total for daisies, lilies, and roses. The most common error for part 4 was 22, incorrectly reading 6 as the value for the daisies. Another common response was 3, counting the types of flowers rather than the amount of the flowers. 7 Students missed part 3, adding data to the graph, or part 5, identifying the most. 8 Students could meet all the demands of the task: work with a scale of 2, identify the most, do comparison subtraction, and find the total of items on a graph. 13
Implications for Instruction Students at this grade level need to work with scales other than one unit, such as 2 s or 10 s. They need to be able to quantify values falling between the benchmark lines on the graph. Students need to struggle with issues, like, What does it mean to be in between? Does it make sense to read the value as the next bar up or the next bar down? Why or why not? Do all values falling between two lines on a bar graph represent the numerical value 1/2? These are important ideas that need to be discussed explicitly with students. Another critical issue of scale is the common misconception that the largest number on the scale is the same as the total of all the values represented by the bars on the graph. How could a lesson be designed to help students confront and overcome this misconception? Students at this grade level should be expected to identify information on a graph and use that information to make further calculations. Identifying most or least is important, but they should frequently be asked to make comparisons, find totals, or calculate subtotals. Graphs should be seen as a starting point for making decisions and the calculations are an important aspect of making sense of the graphs and the ideas being represented. Comparison is a big mathematical idea. Students first start making sense of comparison by finding the distance between two values. How much taller is Jane than Sam? How many more roses than sunflowers? While adults see this as subtraction, children have trouble learning to associate this with subtraction. This is not the action of taking away that they normally associate with subtraction. Students need to work subtraction problems with a variety of contexts and problem types to learn all the aspects of the operation. Looking for key words, such as more, may further add to their confusion. If children are limited to working with subtraction out of context, they will never grapple with the different ways that subtraction can be used. Ideas for Action Research: Student Use of Diagrams: As students progress through the grades, they move from acting out the ideas of a word problem to using numbers. Students should be able to move from drawing a picture to being able to use symbols, like tallies or circles to represent the object. A nice progression is to look at progression from pictures to models in the Japanese text books. See how the thinking progresses from discrete objects to objects within a bar, to a bar with lines, and finally to an empty bar or bar with units. Drawing pictures or using a diagram can be a powerful tool to help students make sense of operation and understand or visualize what the question is asking and what is known. 14
Discussion on progression of diagram use in Japanese texts Janna Look at the work of Janna. Do you think she understands the use of pictures to make sense of how many more? What might that look like? Do you think Janna is just drawing the picture to please the teacher (I need to show my work)? What kinds of questions would you like to ask Janna to really understand how she is using the diagram? If Janna is really using the diagram to make sense of the problem, what kinds of questions or discussions might help Janna to become more efficient in her use of diagrams? Action Idea 1: Try giving this task to your current students and see how many have work similar to Janna s. Interview them to explore their thinking. Action Idea 2: Make a chart or transparency of Janna s drawing without the subtraction. Ask students in your class how they could use the drawing to help solve the problem. How does it help us understand the idea of comparison, how many more? Do they have suggestions on ways to improve Janna s drawing to make it easier and more efficient? See what responses they come up with. 15
Investigating Comparison Subtraction Look at Vera s work. What do you think she is thinking? Is she counting the spaces between the 10 and the sunflowers, forgetting about the scale? (See marks on roses.) Is she having trouble moving visually from roses to sunflowers and finding the difference between roses and daisies? What does this student understand about scale? Now look at the work of Saul. (The 10 roses is scored incorrectly.) What do you think he is thinking about? Might his answer just as easily have been 3? Action Idea: Give this task to your students. Interview some of the ones with answers of 3, 4, and 10. What were they thinking? Write them down to discuss with your colleagues. Did all the students have the same misconceptions or were their misunderstandings different? What are the implications for instruction? What would be your next steps for instruction? Try an intervention and then re-interview them to see how their thinking has changed or where it has stayed the same. 16
Number Talks Look at the work of Kathy. When does an algorithm get in the way of understanding? Do you think Kathy is making sense of quantity or place value? What does Kathy know about numbers? What might Kathy be misunderstanding about numbers? Action Idea: Try some number talks in your classroom. Get students to talk about strategies besides the algorithm. See if Kathy s thinking changes over time. Does this make sense? Look at Ron s work. Where do you think these number sentences come from? Do students in you class get opportunities to explain how their number sentences relate to what is going on in the story? Did any of your students write number sentences that yield the correct answer, but seem to not match the ideas of the story? Interview them. Where do these come from? What surprised you about their responses? 17
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Houses in a Row Rubric The core elements of performance required by this task are: find a pattern in a sequence of diagrams usethepatterntomakeaprediction Based on these, credit for specific aspects of performance should be assigned as follows 1. Draws a correct diagram showing 4 houses in a row. 1 2. Gives correct answer: 21 1 3. Gives correct answer: 31 Draws a correct diagram or gives a correct explanation such as: The numbers go up by 5s, so I added 10 to 21. 4. Gives correct answer: 8 Gives a correct explanation such as: Counts on from 6 houses need 31 toothpicks, 7 houses need 36 toothpicks, 8 houses need 41 toothpicks. Draws a correct diagram or gives a correct explanation such as: The first house needs 6 toothpicks. Each extra house needs 5 toothpicks. 41 6=35, 35 5=7, 1+7=8 5. Gives a correct explanation such as: The first house needs 6 toothpicks. Each extra house needs 5 toothpicks. 6+5x10=56 or The number of toothpicks shows a repeating pattern of 1 and 6 on the units digits. Gives correct answer: 56 points 1 1 1 1 1 or 1 section points 1 2 Total Points 8 1 1 2 2 20
3 rd Grade Task 2: Houses in a Row Work the task and examine the rubric. What do you think are the key mathematics the task is trying to assess? Look at your student papers. Did your students have trouble with the visual discrimination and seeing the attributes of the houses? Did they draw discrete houses? Did they forget to include the vertical line separating the roof from the house? Did they add in extra details, like flowers or chimneys? What might these students not understand about the purpose of the task, diagram, or the mathematics? If you had time for an interview, what question would you want to ask the student? How many of your students thought the pattern increased by 6 s? What kinds of questions might push them to see the overlap as houses become connected? Look at work for part 3, how many of your students put: 31 21 32 or 33 36 Over 36 Other Now think about the strategies used by students who were successful, did they: Draw & Count Add 5 s Notice a pattern 1,6,1,6... Notice overlaps +12-2 Other The thinking for extending the pattern in part 4 can become more difficult. If students noticed that the pattern was going up by 5 s, did they understand the meaning of the extra one? Do you think they could explain where the one came from by using the diagram? 21
Look at work for part 4. How many of your students put: 8 6 7 9 Other What do you think they didn t understand about the pattern that led to the three common errors? What kinds of discussion could you facilitate with your students to get them to confront their misconceptions? Do you think your students could describe how the pattern grows in words? Give a rule for finding the toothpicks for any number of houses? How might describing the pattern growth help them to find a rule? How would different ways of seeing the pattern result in different rules? Would all the rules give the same answer? Understanding the Mathematics of Proportions and Functions with Constants Consider the cost of buying t-shirts at Nancy s, $6 each, or J-mart, $1 to enter the shirt sale and $5 for each shirt. Complete the tables. Nancy s Number of shirts 1 2 3 4 5 6 Total cost 6 12 J-mart Number of shirts 1 2 3 4 5 6 Total cost 6 11 Now use your tables to answer these questions. Could you take the cost of 3 shirts and double it to find the cost of 6 shirts? Are your answers the same as the values in your chart? Why or why not? To find the cost of 15 shirts, could you take the cost of 6 shirts plus 6 shirts plus three shirts? Why or why not? To find the cost of 15 shirts, could you take the cost of 5 shirts and multiply by 3? Why or why not? What is different about the two functions? Can you try to define why the addition and multiplication work for one store but not the other? How does this relate to the mathematics in Houses in a Row? 22
Looking at student work on Houses in a Row: A big piece of the mathematics of this task is seeing what happens when the houses connect, how groups of houses are different from the same number of individual houses. Student A thinks about each house having 6 toothpicks, but realizes that when the houses connect there are overlaps. Toothpicks need to be subtracted for each overlap. With questioning the student could probably explain that there are always one less overlaps than the number of houses. Algebraically, this pattern could be expressed as 6x (x - 1)= toothpicks. Student A 23
Another way of seeing the pattern is to think about the first house having six toothpicks and each additional toothpick house contains only 5 toothpicks because the left side is already made by the previous house. At a later grade level this might be expressed algebraically as toothpicks = 5(x-1) +6. See the work of Student B. Student B As students move through the grades, they should start to think in groups or units other than ones. Student C is able to think about the pattern growing by a unit of 5 and then reason about how many houses or units are being added. This allows the student to move beyond the cumbersome strategies of drawing and counting or doing multiple additions. 24
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The big mathematical idea is understanding that this function has a constant of 1, every house has 5 toothpicks + there is an additional 1 toothpick needed only by the first house (or there is a growth rate of 5 starting from 1 not zero). Student D understands this idea. Student E does not understand about the initial first toothpick. The student treats the pattern as the proportion 5x. How would describing the pattern or using colors to show the pattern help Student E see his mistake? Student D 26
Student E If the table were a proportion (5x): 1 2 3 4 5 6 5 10 15 20 25 30 Then it would be quite valid mathematically to take the second term (10) and multiply it by 3 to get 30 for the 6h term or add the 4 th term + the 4 th term + 3 rd term to equal the 11 th term (20+20+15 = 55 or 5 (11)= 55). However, this reasoning is not valid for functions with a constant. If numbers from a table with constants are added then the initial first toothpick is counted more than once. Students try to make generalizations such as these about patterns from work with proportional situations and don t realize that the procedures will not work for all patterns. See the work of Student F and G. 27
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Some students have difficulty thinking about the growth, what stays the same and what changes. Student H sees that the first house has 6 toothpicks and incorrectly assumes that all the houses will have 6 toothpicks. While in part 2 the student can fill out a table the student doesn t connect the growth in the table to the growth in the pattern. Student H We want all students to have access to the problem using a variety of strategies. The focus of third grade number strand is to start seeing and thinking in groups, being able to see equal size groups as representing a unit that can be used in multiplication. In algebraic thinking students should be able to distinguish attributes of a geometric pattern and replicate the pattern. While many student need to draw and count, students should start to transition into more efficient strategies, like adding on or continuing a table. A few students will start to work towards generalization and finding and describing rules using multiplication and constants. 29
Third Grade 3 rd Grade Task 2 Houses in a Row Student Task Core Idea 3 Patterns, Functions, and Algebra Find a pattern in a sequence of diagrams. Use the pattern to make prediction. Use inverse operations to solve a problem. Develop a mathematical justification about why something is or isn t true. Understand patterns and use mathematical models to represent and to understand qualitative and quantitative relationships. Describe and extend geometric and numeric patterns. Model problem situations with objects and use representations such as graphs and tables to draw conclusions. Describe quantitative change. Solve simple problems involving a functional relationship. Mathematics in the Task: Ability to see and extend an existing geometric pattern and visualize the attributes. Ability to extend a table of numeric values based upon a specific geometric pattern. Ability to explain and quantify the growth of a numeric pattern. Ability to work backwards. Ability to reason and give mathematical justification for why a given answer is incorrect mathematically and then correct the error. Based on teacher observation, this is what third graders knew and were able to do: Extend the pattern in pictures See the growth rate of 5 Extend the pattern numerically in a table Areas of difficulty for third graders: Understanding how the first term is different from the others Reasoning about the remainder of 1 when they divided in part 4 Using drawing and counting or adding on accurately Communicating their strategy, how they figured out their answer Strategies used by successful students: Drawing and counting Adding 5 s or groups of 5 Continuing the table Noticing that all odd houses end in 6, and all even houses end in 1 (while this was helpful in solving this specific problem, it would be difficult to use with large numbers) Multiplying by 6 and subtracting the overlaps Seeingthepattern6+5+5+5.. 30
The maximum score available on this task is 8 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, about 92%, could draw the fourth house. About 81% could also extend the values in the table. Many students, 61%, could draw the house, extend the table, find the value for the 6 th house, and find the number of houses made with 41 toothpicks. More than half the students, 59%, could also either explain how they found the 6 th house or how they found the number of houses for 41 toothpicks. Almost 29% could meet all the demands of the task including extending the pattern to 11 houses, and explaining mathematically how they found all their answers. 7% of the students scored no points on the task. All of the students in the sample with this score attempted the task. 31
Houses in a Row Points Understandings Misunderstandings 0 All the students in the sample with this score attempted the task. Students had difficulty with the geometric pattern. They often drew distinct, discrete houses rather than houses that were connected or they left out the toothpick separating the roof 1 Students could draw 4 toothpick houses. 2 Students could draw the 4 houses and find the total number of toothpicks. 4 Students could draw 4 houses, extend the pattern to 6 houses, and find the number of houses that could be made with 41 toothpicks. 5 Students could draw the houses, extend the table, and then do one of the other three parts of the task. The split was pretty even between missing part 3, 4 or 5. from the house. Students had trouble thinking about what happens when houses are connected. They added by 6 s instead of 5 s. They had difficulty extending the pattern to 6 houses. They may have made counting or computational errors. Many students could fill out the table correctly, but when asked about 6 houses in a row put the answer for 4 houses instead (9%). Students had difficulty explaining how they found their answers for 6 houses or 41 toothpicks. In working backwards, many students could not explain where the remainder one came from when they divided by 5 or they got the correct answer by dividing by 6 with computation errors. Students had difficulty explaining their thinking. They may have had trouble working backwards or extending the pattern to 11. Usually they tried to do one of the parts using groups of 6 or made mistakes in addition or subtraction. 7 Students had difficulty with the explanations in 3, 4, or 5. 8 Students could extend the pattern in words, tables, and by house number. Students could work backward from a number of toothpicks to find the number of houses. Students could express their thinking in words, calculations, or use pattern recognition. They were also able to make sense of the constant. 32
Implications for Instruction Students at this grade level need lots of activities to develop their visual discrimination. Activities with sorting help them to focus on attributes of different shapes. Having students pick their own categories for sorting adds the dimension of developing their own logic skills by forcing them to identify the visual attributes they are seeing. Students need opportunities to draw their own shapes and work with puzzles to develop a feel for how shapes are made and how angles fit together. Working with extending geometric patterns, sharpens the number of details that they pay attention to. Students should be asked to describe the pattern in words; telling what stays the same, what changes, and how the pieces fit together. This not only helps them with the visual discrimination; but also gives them important clues that leads to generalizations instead of recursive patterns. In developing algebraic thinking in students of this age, teachers need to move students gradually away from drawing and counting strategies, which are inefficient and cumbersome, to thinking about growth rates and change. This process will lead students to developing adding on strategies or continuing tables. As students develop an understanding of the meaning of multiplication, they will hopefully start to think in groups (of five, in this case, but dependent on the specific problem). Students should start to make the connection that if they are adding 4 more units to the pattern, then they can multiply 4 times the growth rate and add that on. While students at this grade level aren t expected to have a full understanding of a constant, they should notice where the pattern starts. It is going up by 4 s starting at 3 or going up by 5 s starting at 6. The geometric context should help students reason about why the first term is different. Ideas for Action Research: Creating an Investigative Classroom, Learning to share for sense-making- After giving students an opportunity to work this task alone, then they should compare their ideas with a group. Put up an example of a student dividing 41 by 6. Say you saw a student in another class solving the task this way. In pairs have students try to figure out what the student is thinking about. Then ask them if they think the student is right or wrong and why. Now show them work like that of Student C, but take away the extra explanations. So for part 3: I did 5 x2=10, then 10+21=31. For part 4, I did 2 x 5 = 10, then 10 plus 31 = 41. For part 5, I did 41 + 15 = 56. Ask them to figure out in their pairs what the student is doing. Why is the student multiplying? How does the student know how many to multiply by? Where does the 2 come from? Where does the 5 come from? What does the 15 mean? Are there other examples that you think would be useful to put before students? How does this process reengage students with the mathematics of the task? How does this promote useful discourse in the classroom? Why did I suggest taking away the labels or extra explanation off the work for student C? How does this push the thinking done by students? Understanding Growth A different approach to working the pattern problem would be to have students describe how the pattern grows: what stays the same, what changes, what happens when two pieces are combined? How is the first figure different than the next one that is added? If students can describe that the pattern is going up by 5 s, see if they can outline the 5-ness with a colored marker. See if students can develop a verbal rule to find for any house number. 33
Some students will see the pattern as growing by 6 every time, but with overlaps. Ask them if there is a way to predict the number of overlaps given any number of houses. Again, see if students can develop a verbal rule to find the number of toothpicks needed for any house number. While not all students are ready to make generalizations, these types of questions can develop their logical reasoning skills so that at some future time they can move away from drawing and counting. What other questions might help students think about the relationship between their number calculations and the geometric pattern? Why is this important? Both of these activities give students opportunities to talk meaningfully about mathematics and practice using mathematical vocabulary. It also emphasizes that mathematics is not just about getting an answer, but mathematics is also about sense-making and justification. 34
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The Answer is 36 The core elements of performance required by this task are: work with number calculations to get the answer 36 Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answers, start top right in a clockwise direction: 6 19 47 57 4 12 23 Rubric points 7x1 2. Gives correct answers: - and + 2 2 Total Points 9 section points 7 36
3 rd Grade Task 3: The Answer is 36 Work the task and examine the rubric. What do you think are the key mathematics the task is trying to assess? Look at the three addition problems. How do you think the order that the blank appeared (first or second) effected the strategies that students used to solve the problem? Do you think one blank position might be more difficult than the other? Why? Look at student work for the 3 addition problems. For 17 +, how many of your students put: 19 20 29 36 14 Other What might have caused their confusion? Now look at + 13, how many of your students put: 23 20 53 17 24 Other Now think about the 3 subtraction problems. What strategies might students use to solve these problems? Do you think the strategies would be different for solving the problems with the blanks first or second? What other factors might make some of these more difficult than others? Look at student work for the 3 subtraction problems. For 83 -, how many of your students put: 47 59 43 20 57 119 Other What misunderstanding might have led to some of the common errors? Look at - 21. How many of your students put: 57 15 52 36 47 Other Are the misunderstandings different from those in the previous problems? While the students were not required to show their work for this task, did your students show their strategies on the side? What evidence do you see in their work about misunderstanding place value? Misunderstanding order of operations? Misunderstanding equality? What is your evidence? 37
Looking at Student Work This task looks at student understanding of the relationship between addition and subtraction, place value, order of operations, and making sense of equality. While the task does not require students to show work, many students use the space to show their thinking or make calculations. Student A shows that inverse operations can be used to solve missing part problems for addition and subtraction. Student A 38
Student B tries to solve all the problems using subtraction. While this works for 86- or 48-, it is not useful for solving a problem in the format minus a quantity. The student doesn t seem to make sense of the idea that to get an answer of 36, the initial quantity, what you start with before the subtraction, must be larger than 36. The student is not thinking about how the process of subtraction effects the change in the initial quantity. What activities might help the student develop better number sense? Student B 39
Student C has a similar problem reasoning about the effects of operation on the solution. In thinking about -21, the student successfully uses addition to find the missing number. However this process does not work for 83-. Why doesn t the inverse operation apply to this problem as well? What would you want the student to be thinking about as she worked this task? Student C 40
Student D uses guess and check to try and find the solution to many parts of the task. What does the student need to understand about the operation of addition and subtraction to be able to use a more reliable method? Do you think this student would benefit from number talks? How might number talks contribute to the student s understanding of place value? Student D 41
Student E is also using a guess and check strategy to find 17 + =36. Notice that this student cannot reason about the distance between the original guess and the final answer. The student s guesses go up only one number at a time. What kinds of experiences or questions might help this student think more about the global picture? Student E 42
Some students have not moved to computation with subtraction. Student F is still relying on a counting up or counting down strategy to find the difference. While this is useful to learn about the process at lower grade level, third graders should be able to usemore efficient strategies. Number talks might help this student move from going backwards one step at a time to moving in groups of 10. Some sort of intervention is needed to help this student move beyond this strategy. Student F 43
Third Grade 3 rd Grade Task 3 The Answer is 36 Student Task Core Idea 2 Number Operations Work with number calculations to get the answer 36 using addition, subtraction and multiplication. Understand the meanings of operations and how they relate to each other, make reasonable estimates, and compute fluently. Understand multiplication as repeated addition, an area model, an array, and an operation on scale. Develop fluency in adding and subtracting whole numbers. The mathematics in the task: Ability to add double digit numbers, with or without carrying. Ability to subtract double digit numbers, with or without regrouping. Ability to multiply single digits. Ability to use inverse operations to solve problems with addition and subtraction and show understanding of how operations effect the size of the answer. Based on teacher observation, this is what their graders knew and were able to do: Solve for missing addends Add with accuracy Multiply by one digit numbers Reason about order of operations and fill in missing operation signs Areas of difficulty for third graders: Order of operations Place value and regrouping Finding missing information in subtraction problems Strategies used by successful students: Inverse operations Calculations Guess and check Count backwards or count forwards Show their work Checking their answers, e.g. 36 + 21 = 57 /57-21=36 44
The maximum score available on this task is 9 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, about 94%, could find the missing addends for 30 + and for + 23 and knew the fact that 4 x 9 =36. About 90% could also correctly fill in the missing operation signs in the bottom part of the task. More than half the students, 63%, could find all the missing addends; they may have struggled with one or more of the subtraction problems. 83- seemed to be the most difficult. 29% of the students understood the operations of addition and subtraction and could use inverse operations to find missing parts. Only 2% of the students scored no points on this task. Half of the students in the sample with this score attempted the task. 45
TheAnsweris36 Points Understandings Misunderstandings 0 Only 2% of the students scored zero. Half of them attempted the task. 4 Students could find the missing addends for 30 + and for + 23 and knew the fact that 4 x9=36. 6 Students could find all the missing addends and knew 9x 4. Students had difficulty with 17+ =36. 4% made a calculation error and wrote 20. 3% had difficulty with place value and put 29 instead of 19. Students had difficulty thinking about how to find the missing parts in subtraction problems. For -21, 10% of the students subtracted 21 from 36 to get 15 as the missing number. 2% thought the answer was 36 and 2% thought it was 47. 8 Students still struggled with subtraction. The most difficult part was 83- =36. 9% put 59. 4% put 43. 2% put 119. 9 students understood the operations of addition and subtraction and could use inverse operations to find missing parts. Implications for Instruction Students need help with subtraction. They should know a variety of strategies for thinking about addition and subtraction of 2-digit numbers. An understanding of part/part/whole relationships in addition and subtraction allows students to be more flexible in finding missing parts in addition and subtraction and to see the connection between the two. Work with the number line or empty number line help students make sense of size and relationship of the numbers. Thinking about the positions on the number lines might eliminate some of the mistakes involved in the traditional algorithms of carrying and regrouping. The bar model also helps develop these relationships and gives students a visual picture of the action of the operation. Learning to understand the operations is as important as the computational facts. Use of number talks as class warm-ups can familiarize students with how these models work. Liping Ma, in her book Knowing and Teaching Elementary Mathematics, talks about the importance of emphasizing composing and decomposing numbers using addition and subtraction on numbers between 0-20 or going around 10. Some students would benefit from some deep work on this area. Developing a clear understanding of how the operations work with these numbers, makes moving to larger numbers almost self-teaching. Work with the dot cards from Kathy Richardson, number talks, even games like dominos can help students start to visualize how the base-10 system works and make sense of addition and subtraction as they develop their own strategies for adding and subtraction based on the composition and quantity of the numbers. 46
Ideas for Action Research Cognitive Dissonance Many researchers think misconceptions are best dealt with by creating situations where students need to confront their misunderstandings in a context that helps them discover why those ideas are incorrect. Consider giving your students a puzzle like: Sally says that to find the missing part in a subtraction problem, you always use the opposite operation addition. She proves her point by showing that for: - 21= 36, so 21 +36 equals the missing part is 57. Sam says that to find the missing part in a subtraction problem, you always use subtraction. His example is: 83- =36, so 83-36 the missing part 47. Who is right? Why? How can you convince someone? Have students work in pairs to find a convincing argument. If students don t come up with the idea of using a model on their own, ask them if they could use a number line or bar model to help make their justification. Now give students some similar missing parts problems to solve. Do they have better strategies? Are they more accurate? Going Around Ten Read Liping Ma s Chapter about Subtraction with Regrouping: Approaches to Teaching the Topic. See if you can use the ideas sparked by the Chinese teachers to plan some number talks. Work with students in small groups for about 15 minutes a day for a week or two. Maybe try some groups with just lower achieving students and some groups with a mix. How do students think about composing or decomposing the numbers? Does this work help students develop a better understanding of arithmetic? What changes do you notice as students work with these ideas? Was there a difference in how students thought in the different groups? How did the lower achieving students in both groups progress in their understanding? Give an anecdote that shows why this was helpful. Use of models Pick a model, like the empty number line, and give students some opportunities to solve addition and subtraction problems using the number line. Have students explain their strategies? Notice which students are moving one step at a time and which students can move or think in larger groups, like fives, tens or larger. As you work with this idea for a couple of weeks, use your seating chart to keep track of strategies used by different students. Are more students thinking in groups or chunks over time? As they get better, see if this model can be expanded to thinking about multiplication or division. 47