Grade 3
Mean: 20.64
StdDev: 9.87
Total MARS Raw Scores
Third Grade – 2005
pg.
1
MARS Test Performance Level Frequency Distribution Table and Bar Graph
2005 – Number of Students tested in 3rd grade: 11972
Frequency Distribution of MARS Test Performance Levels, Grade 3
2000
Perf. Level
1
2
3
4
% at
19%
28%
35%
18%
Perf. Level
1
2
3
4
% at
9%
14%
33%
44%
Year of Testing
2001
% at least
100%
81%
53%
18%
% at
14%
20%
41%
25%
2003
% at least
100%
91%
77%
44%
% at least
100%
86%
66%
25%
2002
% at
8%
20%
31%
41%
2004
% at
14%
17%
32%
38%
% at least
100%
86%
69%
38%
% at least
100%
92%
72%
41%
2005
% at
19%
18%
36%
26%
% at least
100%
81%
63%
26%
Bar Graph of 2005 MARS Test Performance Levels, Grade 3
Total Student Count: 11,972
Third Grade – 2005
pg.
2
3rd grade
Task 1
Television Time
Student
Task
Read a graph and answer questions concerning the amount of time a child
spends watching television on a Saturday.
Core Idea
5
Data
Analysis
Collect, organize, display, and interpret data about themselves and
their surroundings
• Identify important features of a set of data
• Compare data using quantitative measures
• Communicate reasoning using numbers, pictures and/or words
Understand the meanings of operations and how they relate to each
other, make reasonable estimates, and compute fluently
• Understand different meanings of addition
• Communicate reasoning using numbers, pictures and/or words
Core Idea
2
Number
Operations
Third Grade – 2005
pg.
3
Third Grade – 2005
pg.
4
Television Time
Grade 3
Rubric
The core elements of performance required by this task are:
• to look at a graph and determine what it shows about the questions
Based on these, credit for specific aspects of performance should be assigned as follows
1.
points
Gives correct answer: cartoons
section
points
1
1
2.
Gives correct answer: 30 minutes
1
1
3.
Gives correct answer: 20 minutes
1
1
4.
Gives correct answer: 140 minutes (accept 2 hours 20 minutes)
1
Gives correct explanation such as:
20 + 30 + 50 + 40 =
or I added them all together (with the correct answer)
1
2
5
Total Points
Third Grade – 2005
pg.
5
Looking at Student Work on Television Times:
Third grade students did quite well on this task. There were a variety of strategies used
by students to show their work for thinking about addition and use of measurement.
Student A shows the comparison subtraction to find “how much longer” in part 3. The
student is able to read the number scale on the graph to write an equation or number
sentence to find the total time in part 4. The student also shows how the bars were
converted to the numbers in the addition problem.
Student A
Third Grade – 2005
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6
Only 1/4 of the students showed the subtraction in part 3. Students may have just used a
counting strategy. The numbers may have been so easy; they did the subtraction in their
head. Student B breaks the addition in part 4 into smaller, more manageable bits to help
solve the problem. Notice how the student makes use of labels in part three “more
minutes” and in part 4 to demonstrate understanding of the situation.
Student B
Third Grade – 2005
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7
While the task did not require students to convert from minutes to minutes and hours, it
gave students the opportunity to demonstrate this knowledge. Student C shows a student
who is comfortable with converting from minutes to hours and gives a glimpse into the
thinking behind the process.
Student C
While Student D does not show a number sentence to show what numbers were counted,
the student does reveal the thinking needed to properly convert the minutes to hours and
minutes.
Student D
Third Grade – 2005
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8
Student E shows a popular misconception when working with time conversions. The
student changes the 140 minutes into 1 hour and 40 minutes.
Student E
Third Grade – 2005
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9
Some students had difficulty working with the scale of the graph. Student F is able to use
the scale correctly to find “how much longer” Terrie spent watching comedies. However
the student does not label this answer or any of the answers with a measurement unit. In
part 2 and part 4 the student confuses the boxes with the number they represent. The
student appears to have some confusion on the idea of “unit” and working with scale.
Student F
Teacher Notes:
Third Grade – 2005
pg.
10
Frequency Distribution for Task 1 – Grade 3 - Television Time
Television Time
Mean: 3.68
StdDev: 1.33
MARS Task 1 Raw Scores
Score:
Student Count
%<=
%>=
0
177
1.5%
100.0%
1
719
7.5%
98.5%
2
1718
21.8%
92.5%
3
1932
38.0%
78.2%
4
3033
63.3%
62.0%
5
4393
100.0%
36.7%
The maximum score available on this task is 5 points.
The cut score for a level 3 response, meeting standard, is 3 points.
Most students, about 92%, were able to recognize the category for most tv viewing and
read a value from the graph for a particular category. Many students, 79%, were able to
read and interpret the graph and recognize the need for addition for finding the total hours
of tv viewing and write the appropriate number sentence using values from the graph.
37% of the students could meet all the demands of the task including calculating the total
time for viewing using an appropriate unit of scale and doing comparison subtraction and
labeling the answer with a measurement label. Less than 1% of the students scored no
points on this task.
Third Grade – 2005
pg.
11
Television Time
Points
Understandings
0
Less than 1% of the students
scored no points on this task.
1
Students could identify the
favorite category of tv programs,
by recognizing the bar with the
most.
Students who scored no points could not
identify the favorite t.v. category. They may
have put a general statement, tv program,
rather than choosing one category. Another
error was to pick game shows as the favorite,
because it was the first one listed.
Some students struggled with the scale of the
graph. They counted individual boxes instead
of groups of 10’s. 10% of the students could
not follow the bar across to the scale and put
20 minutes instead of 30 minutes for sports or
read the bar for game shows instead of the bar
for sports.
Many students could find the correct
number of minutes for the comparison
subtraction but did not label the answer
with minutes (20%) Only 1/3 of the
students showed the subtraction in part 3.
Some students had difficulty comparing
the bars with 8% getting a difference of 10
minutes. 8% just gave the value of the
larger bar rather than making a
comparison. 8% ignored the scale of the
graph and gave an answer of 2 minutes.
Students could read and
12% of the students are still giving
3
interpret the graph. Students
explanations, like “I counted”, “I looked at
understood that finding the total the graph”, or “skip count by 10’s”. They
hours of tv viewing required
are not documenting their thinking with
addition and could read the
the exact values. Less than 3% of the
values of each category to write students tried to add all the numbers on
an appropriate number
scale rather than the values of the
sentence.
individual categories.
2/3 of the students with this score forgot
4
the minutes in part 3. The other students
may have incorrectly converted 140
minutes to 1 hour and 40 minutes or did
not show their work in part 4.
Students could read and
5
interpret a graph, including
using a scale other than 1.
Students could do comparison
subtraction. Students could
work with units of time,
including making proper
conversions. Students could
show their work with a number
sentence.
12
Third Grade – 2005
pg.
2
Students could read and
interpret the graph, by
identifying the category with
the most and by finding the
value for one of the categories.
Misunderstandings
Based on teacher observations, this is what third graders knew and were able to do:
• Identify most on a bar graph
• Read the value of a bar when the unit is not going up by 1’s
• Recognize when to use the number operation of addition
• Add accurately
Areas of difficulty for third graders:
• Understanding a using a scale going up in units of 10’s
• Using appropriate measurement labels for their answers
• Converting between minutes and hours and minutes
Strategies used by successful students:
• Labeling the tops of each bar
• Showing the subtraction process for “how much more”
• Breaking up the addition problem into smaller parts
• Using appropriate labels for number calculations, like 20 minutes more
Questions for Reflection on Television Time:
When working with graphs do your students get opportunities to see and to
interpret scales that don’t count by one’s?
• Do students use the graph to make comparisons: “How much more?” or “How
many more times?”
Look at your student work in part 3, how many of your students put:
•
20
minutes
•
•
20
10
40
30
2
Why do you think students got answers of 10? How might this be a different type
of error than answers of 40? What might students have been thinking who got
answers of 2? Which students are having trouble tracking values from the bar to
the scale, understanding the scale, and understanding comparison subtraction?
In part 4, how many of your students put:
2hr.40min. 140 min. 1hr.40min 2hr.10min. 104 min.
Other
or
210 min.
What is the classroom norm for showing work? Are students expected to
document their thinking with number sentences? Are students expected to label
the results of their calculations?
o In part 3, how many of your students showed the comparison subtraction?
o In part 4, how many of your students showed their thinking by:
Using a number
“I looked at the
“I counted the
“I counted by 10’s”
sentence
graph”
numbers”
•
Third Grade – 2005
pg.
13
Implications for Instruction:
Students need practice making and interpreting graphs where the vertical scale is not one.
Students should have opportunities to gather and record their own data, so they can
experience a need for changing scale. While this graph uses a scale of 10, students should
be exposed to a variety of scales, such as 2’s, 5’s, 25’s, etc. Part of understanding graphs
is the ability to use the information to make decisions. How might this information be
useful? What is the impact of the visual representation? Students should also be asked to
write summary statements about the graphs. Students should be used to looking at graphs
as a way to make comparisons. Students at this grade level should be very comfortable
with comparison subtraction, how many more marbles does Angie have than Bill? How
much taller is Fred than Susan? How many more people like baseball than football?
Students at this grade should also be starting to think about using multiplication for
comparison. Example, Terrie spends twice as much time watching comedies as she does
watching game shows. Students should be able to recognize situations where using
addition is the correct number operation. They should be able to document this process
by using a number sentence instead of saying, “I counted” or “I looked at the graph”.
When doing problems with a context, students should be in the habit of labeling their
answers with the appropriate measurement unit, such as minutes, hours, and inches.
Students should be able to convert between simple units of time, such as changing
minutes to hours and minutes. They should be able to think about how many groups of
“sixty minutes” are in the total and see the difference between a large number of minutes
and its representation into hours and minutes. Students should have a variety of
strategies for making this conversion.
Teacher Notes:
Third Grade – 2005
pg.
14
3rd grade
Student
Task
Core Idea
4
Geometry and
Measurement
Task 2
Katie’s Kitchen
Name and create two-dimensional shapes that might appear in a
kitchen. Recognize a floor tile as it is flipped and turned as part of a
design.
Recognize and use characteristics, properties, and relationships of
two-dimensional geometric shapes and apply appropriate
techniques to determine measurements
• Recognize geometric ideas and relationships and apply them
to problems
• Identify and compare attributes of two-dimensional shapes
• Use visualization, spatial reasoning, and geometric modeling
to solve problems
Third Grade – 2005
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15
Third Grade – 2005
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16
Third Grade – 2005
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17
Katie’s Kitchen
Grade 3
Rubric
The core elements of performance required by this task are:
• name and create two dimensional shapes
• recognize flips and turns
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answer: triangle
2.
Gives correct answer: hexagon
3.
Gives correct answer: rhombus (accept quadrilateral or parallelogram)
4.
Gives correct answer: circle
5.
Gives correct answer: square (accept rectangle)
6.
Gives correct answer: rectangle
6 correct answers
section
points
3
Partial credit
5 or 4 correct answers
3 or 2 correct answers
(2)
(1)
7.
Draws a pentagon (a shape with 5 sides)
1
8.
Draws an octagon (a shape with 8 sides)
1
9.
Gives correct answer: 3 different tile patterns
1
3
1
1
Shades in one of each of the three squares
shown in diagram with no extras.
3
Partial credit
(2)
Shades in two correct squares with no extras.
(1)
Shades in one correct square with no extras.
4
9
Shades in correct squares with extras: minus one point for each extra
Total Points
Third Grade – 2005
pg.
18
Looking at Student Work on Katie’s Kitchen:
Some students had difficulty with the purpose of the numbers in the first part of the task.
Students gave a list of shape names in the diagram without thinking about matching
numbered blanks to the numbered shapes in the diagram. Student A not only knows the
correct mathematical terms for the shapes, but makes clear with diagrams exactly the
shape being addressed with each term. Notice in part 9, Student A also makes clear that
he is thinking about different tiles rather than just tiles.
Student A
Third Grade – 2005
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19
Student A, continued
Drawing the pentagon and octagon is quite challenging for many students. They struggle
with fine-motor skills or they make the lines so small, the lines start to blend together.
Student B does a nice job of indicating the number of intended sides.
Student B
Third Grade – 2005
pg.
20
Student C has tried to clarify the number of intended lines by clearing showing the
vertices of each shape and numbering the lines. Many students struggled with what was
meant by “tile” in part 9. 12% of the students put answers like 5, 6 or 8 different tiles in
part 9. 6% gave really large answers like 16 or 25. However in looking at their work,
most students don’t give clues about what they were thinking. Student C has numbered
the 6 pieces (large tile, small circle, ring around the small circle, large ring, triangle, and
center circle). Teachers might want to ask some probing questions of students with these
answers to find out more about their thinking.
Student C
Third Grade – 2005
pg.
21
For some students drawing the pentagon and the octagon is an exercise. They are
reproducing shapes from learned prototypes. For other students this is a problem-solving
design task. Students have information about the number of sides and then need to
experiment to make a shape with the correct number of sides. Student D manages to
come up with 8 sides for the octagon, but in a very unusual way.
Student D
Student E helps to clarify the intended sides by clearly marking the vertices.
Student E
Student F again shows that for many students drawing the shapes is a problem-solving
task rather than a reproduction of memorized knowledge.
Student F
Third Grade – 2005
pg.
22
Even having a clear idea of the general shape for a pentagon, Student G demonstrates the
challenge of motor skills needed to replicate the shape. Again, Student G must problem
solve to make the octagon.
Student G
Student H compensates for lack of motor skills by indicating the number of intended
sides for the shape. Because the sides are so small and the angles are quite large
distinguishing the differences between sides in the octagon is difficult.
Student H
Teacher Notes:
Third Grade – 2005
pg.
23
Some students had difficulty recognizing rotations in part 9. Almost 18% of the students
thought there were four different tiles. See the work of Student I.
Student I
Other students appeared to be looking at each little piece of the design. This is indicated
by the work of Student J, who shows some of the pieces below the answer of 24.
Student J
Third Grade – 2005
pg.
24
Reference Note – Developing Fine-Motor Skills
From the website: www.shrewsburyma.gov/schools/beal/readiness/finmotractivities
Activities:
Students need to develop hand strength and dexterity in order to grip a pencil correctly.
• Molding and rolling play dough into balls- using the palms of the hands facing
each other and with fingers curled slightly towards the palm.
• Rolling play dough into tiny balls (peas) using only the finger tips.
• Using pegs or toothpicks to make designs in play dough
• Tearing newspaper into strips and then crumpling them into balls. Use to stuff
scarecrow or other art creation.
• Scrunching up 1 sheet of newspaper in one hand. This is a super strength builder.
• Picking up objects using large tweezers such as those found in the “Bedbugs”
game. This can be adapted by picking up Cheerios, small cubes, small
marshmallows, pennies, etc. in counting games.
• Using small-sized screwdrivers like those found in an erector set.
• Lacing and sewing activities such as stringing beads, Cheerios, macaroni, etc.
• Using eye droppers to “pick up” colored water for color mixing or to make artistic
designs on paper.
When scissors are held correctly, and when they fit a child’s hand well, cutting activities
will exercise the very same muscles which are needed to manipulate a pencil.
• Cutting junk mail, particularly the king of paper used in magazine subscriptions
cards.
• Making fringe on the edge of a piece of construction paper.
• Cutting play dough with scissors.
• Cutting straws.
The joints of the body need to be stable before the hands can be free to focus on specific
skilled fine motor tasks.
• Wheelbarrow walking, crab walking, and wall push-ups.
• Toys: Orbiter, sill putty, and monkey bars on the playground.
Attach a large piece of drawing paper to the wall. Have the child use a large marker and
try the following exercises:
• Make an outline on the paper. Have the child trace over your line from left to
right or from top to bottom. Trace each figure 10 times. Then have the child
draw the figure next to yours.
• Play connect the dots. Make sure the child’s strokes connect dots from left to
right and from top to bottom.
• Trace around stencils- the non-dominant hand should hold the stencil flat and
stable against the paper, while the dominant hand pushes the pencil firmly against
the edge of the stencil.
Third Grade – 2005
pg.
25
Frequency Distribution for Task 2 – Grade 3 - Katie’s Kitchen
Katie’s Kitchen
Mean: 4.93
StdDev: 2.39
MARS Task 2 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
9
287
2.4%
100.0%
395
5.7%
97.6%
1503
18.3%
94.3%
1679
32.3%
81.7%
1694
46.4%
67.7%
1435
58.4%
53.6%
1347
69.7%
41.6%
1296
80.5%
30.3%
1526
93.2%
19.5%
810
100.0%
6.8%
The maximum score available for this task is 9 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, about 95%, could name 4 or 5 correct shapes. More than half the students,
67%, could name 4 or 5 shapes, draw an octagon, and recognize some of the different
tiles with maybe one extra. A few students, 29%, could name all of the shapes except for
the rhombus and the pentagon, find the number of different tiles and shade them with no
extras, and draw the octagon. 7% of the students could meet all the demands of the task,
including drawing a pentagon, and correctly identifying the mathematical term for a
rhombus shape. 2% of the students scored no points on this task. All the students with
this score in the sample attempted the task.
Third Grade – 2005
pg.
26
Katie’s Kitchen
Points
Understandings
All students in the sample
0
attempted the task.
2
Students could name 4 or 5 shapes
correctly.
4
Students could name 4 or 5 shapes
and draw either the pentagon or the
octagon, and fill in some of the
different tiles (usually with extras).
7
Students could name most of the
shapes, draw the octagon, find the
number of different tiles, and shade
in the 3 examples with no extras.
8
Students could meet all the
demands of the task except for
naming one shape.
Students could give the
mathematical terms for common
shapes, construct pentagons and
octagons, recognize differences in
tile patterns and recognize shapes
that had been rotated.
9
Third Grade – 2005
Misunderstandings
Some students did not think the
numbers on the answer lines
matched the numbers in the
diagrams. Example, 4% of the
students put circle for number 1
instead of triangle.
For the hexagon, more than 10%
thought it was an octagon and 20%
called it a pentagon. For the
rhombus, 68% used the common
name diamond instead of the
mathematical term.
For many of these students,
designing and drawing the shape
was not a rote exercise but a
problem-solving task. Some
students had difficulty with the fine
motor skills needed to complete the
drawings, but may have been able to
indicate the number of intended
sides. Almost 20% of the students
had difficulty recognizing shapes
that had been rotated and colored in
two of the middle border tiles in
part 9. 14% of the students did not
see the tiles as different and saw 9
tiles.
Students, at this level, still had
difficulty naming the hexagon and
the rhombus. They also had
difficulty drawing the pentagon.
24% of all students drew a 6-sided
figure for the pentagon. 11% of all
students were not willing to attempt
drawing a pentagon.
60% of the errors at this level, were
for using the word diamond instead
of rhombus.
pg.
27
Based on teacher observations, this is what third grade students knew and were able to
do:
• Name common shapes, such as triangle, circle, square, rectangle, hexagon
• Draw an octagon
• Notice some tiles that were different
Areas of difficulty for third grade students:
• Using the mathematical term rhombus, instead of the common word diamond
• Drawing a pentagon
• Fine motor-skills need for drawing
• Recognizing rotations of shapes
Strategies used by successful students:
• Marking the vertices on their drawings with dots to make them show up clearly
• Numbering the sides or putting the number of sides next to their drawings
• Using accurate labels, like putting “different” tiles in part 9
Questions for Reflection on Katie’s Kitchen:
What types of activities do you students have to learn shape names? Are the
words learned as vocabulary or in the context of needing to know in order to talk
about experiences and thinking about what they are doing?
• When looking at shapes, do students get opportunities to discuss the attributes or
focus on the important qualities of the shapes? How many sides does the shape
have? Are any of the sides equal? Are the angles the same size or different?
• Do your students get opportunities to sort and to classify shapes? Who makes up
the categories for sorting the shapes? Do students get to compare different ways
of sorting the shapes and talk about which is the most efficient or if the categories
are exclusive?
• How often do students in your class draw their own shapes instead of using
blocks, patterns or stencils? What are the advantages of having students draw
their own shapes?
• What special activities do you provide for students who have poor fine-motor
skills? What evidence do you see in the student work to indicate problems with
fine-motor skills?
Look at student work for part 9, how many of your students put:
•
1
4
5
6
8
9
15,16,17
24,25
What might the student be thinking about if he/she put 1 or 9?
What might the student be thinking about if they put 5,6,8?
What might the student be thinking about if they put one of the larger answers?
Which numbers might be caused by not recognizing rotations?
Third Grade – 2005
pg.
28
Implications for Instruction:
Students at this grade level need to have experiences with a variety of shapes and have
those shapes appear in many different orientations. It is important for them to develop
the visual discrimination to find shapes that are the same, but that have been turned or
rotated. Students should be able to recognize any three-sided figure as a triangle, rather
than have a set image of all triangles being equilateral. Students will remember the
names of shapes better when learned within the context of other activities, when knowing
the names is important for the purpose of discussing their ideas. Students should also
have the opportunity to make their own shapes, given certain properties like number of
sides, number of equal sides, parallel lines. Students should have access to tools, like
rulers, when drawing shapes. Students benefit from having opportunities to sort and
classify shapes, particularly if they invent their own categories. This allows them to
develop the logical reasoning involved in categorizing and helps them to focus on
specific attributes within the shapes rather than just the general shape. Some students at
this grade level have not developed fine-motor skills necessary for the tasks of drawing
complex shapes. Specific activities in art, P.E., or other times of the day should address
this issue. A list of interesting activities can also be given to parents to work on at home.
Teacher Notes:
Third Grade – 2005
pg.
29
3rd grade
Student
Task
Core Idea
1
Number
Properties
Core Idea
2
Number
Operations
Task 3
Number Cards
Given different sets of four number cards, find three- and four-digit
numbers given certain constraints and be able to explain how to find the
largest four-digit even number.
Understand numbers, ways of representing numbers, relationships
among numbers, and number systems
• Develop understanding of the relative magnitude of whole
numbers and the concepts of sequence, quantity, and the relative
positions of numbers
• Understand the place-value structure of the base-ten number
system including being able to represent and compare whole
numbers
Understand the meanings of operations and how they relate to each
other, make reasonable estimates, and computer fluently
• Understand the effects of adding and subtracting whole numbers
• Communicate reasoning using numbers, pictures and/or words
Third Grade – 2005
pg.
30
Third Grade – 2005
pg.
31
Third Grade – 2005
pg.
32
Number Cards
Grade 3
Rubric
The core elements of performance required by this task are:
• show understanding of whole numbers
Based on these, credit for specific aspects of performance should be assigned as follows
points
1. Gives correct answer: 8753
1
2.
1
a. Gives correct answer: 7538
b. Gives a correct explanation such as:
8 is the only even number, so write it at the end in the units column.
Write the biggest number (7) in the thousands position, the next biggest
(5) in the hundreds position and the smallest (3) in the tens position.
section
points
1
1
1
3
3.
a. Gives correct answer: 3875
1
b. Shows work such as:
5378 – 4000 = 1378 and 4000 – 3875 = 125
2
Partial credit
Considers only 3875 and gives reasons for arranging the digits in that
order.
4.
(1)
a. Gives correct answer: 262
1
b. Shows work such as:
6000 – 5738 = 262
1
2
9
Total Points
Third Grade – 2005
3
pg.
33
Looking at Student Work on Number Cards
Few students were able to meet all the demands for justification of this task. Student A is
able to explain the placement of all the digits for part 2 of the task. While the student
doesn’t think of the smallest number in the 5000’s, the student does consider numbers in
both the 3000’s and the 5000’s when trying to compare possibilities closest to 4000.
Student A was able to use addition successfully to find the missing addend in part 4.
Student A
Third Grade – 2005
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34
Student A, part 2
Third Grade – 2005
pg.
35
Thinking about how to measure distance from 4000 was a struggle for students. Student
B had a score of 8 points. The student knew to use subtraction to compare a number with
4000, but could not figure out how to structure the subtraction problem into a format that
she knew how to solve. Students need to have experiences comparing numbers above and
below a target amount. Student B understands the relationship between addition and
subtraction and is able to use that to solve for the missing addend in part 4.
Student B
Third Grade – 2005
pg.
36
Student C is able to use the number line to help determine which number is closer to
4000. The student only compares values below 4000. In part 4 the student is no longer
thinking about place value. The student just thinks, “ What can be added to each digit to
make a 10?” Almost 10% of the students made this error.
Student C
Third Grade – 2005
pg.
37
While the number line can be a powerful tool to help reason about the size of numbers
and how far apart they are, Student D does not try numbers below 4000. Does the
student have the habit of mind to consider more than 1 possibility or is the number line a
tool for justifying the first answer considered?
Student D
Student E has all answers correct and is able to show his strategy for solving for the
missing addend in part 4. The student must have some understanding of place value to
get so many answers correct. However the student does not appear to have much
experience with making a mathematical justification. The student reasons about the rules
of the card game, rather than any thinking about place value.
Student E
Third Grade – 2005
pg.
38
Student E, part 2
Student F also lacks language to describe place value. In part 2 the student just describes
how to position the cards, rather than offering a rationale for the placement of the cards.
In part 3, the student is willing to consider more than 1 possible value to get closest to
4000. Unfortunately that was not enough to reach the correct value. Reasoning about the
effect of the digits lower than the 1000’s place in getting close to 4000 was very difficult
for students. The student was not able to compute the subtraction correctly in part 4.
Student F
Third Grade – 2005
pg.
39
Student F, part 2
Some students do not have a complete understanding of even numbers. For students like
Student G, having an even digit somewhere seems to make it an even number. Student G
shows a clear gap in her understanding of place value for numbers in the 1000’s. The
student seems to think that the number after 3090 is 3400 or 4000. Students need many
counting experiences around transition points in large numbers. The student is able to
use addition to find the missing addend in part 4 and understands about the numbers
carrying over from each column. However, the process seems to indicate that the student
is doing this very mechanically, thinking about individual digits, rather than thinking
about size or place value.
Third Grade – 2005
pg.
40
Student G
Third Grade – 2005
pg.
41
Student H helps to highlight the difficulty students had considering the effect of all the
places in finding a number closest to 4000. The student could think about the leading
number and its value, 3000 is close to 4000. Then the student thinks to get a 0 in the ones
place will get her closest to 4000, so “8 is closed to 10”. The student does realize that the
hundreds place is the biggest determinate in getting closed to 4000.
Student H
Student I thinks the leading number makes it even or one even number any where is
enough to make it even.
Student I
Third Grade – 2005
pg.
42
Student J does not seem to understand place value or how to make a large number. The
student ignores the directions for using the cards, but still can’t make the largest number
using the digits chosen. In part 2, Student J thinks only even digits can be used in an
even number. Again the digits chosen are not arranged to make the largest possible
number. In par 3 the student makes the closest number to 4000 by using digits other than
those on the number cards. Work in part 4 suggests the student has worked with base-10
blocks, but can’t use them effectively to reason about the missing addend. The student
also attempts to use addition to find the missing addend, but doesn’t line up the digits in
the addition problem according to place value.
Student J
Third Grade – 2005
pg.
43
Student J, part 2
Third Grade – 2005
pg.
44
Student K thinks only about the leading digit in determining size of a number or deciding
whether it is odd or even. The student is not able to think of the role of all the digits in
determining the size the number.
Student K
Teacher Notes:
Third Grade – 2005
pg.
45
Number Cards
Mean: 3.89
StdDev: 2.53
MARS Task 3 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
9
1160
9.7%
100.0%
1495
22.2%
90.3%
1389
33.8%
77.8%
1569
46.9%
66.2%
1439
58.9%
53.1%
1372
70.4%
41.1%
1340
81.6%
29.6%
1051
90.3%
18.4%
847
97.4%
9.7%
310
100.0%
2.6%
The maximum score available for this task is 9 points.
The minimum score for a level 3 response, meeting standards, is 5 points.
Most students, about 89%, could arrange 4 digit-cards to make the largest possible
number. Many students, about 77%, could make the largest number and could make the
number closest to 4000 or the largest even number. More than half the students, 54%,
could make the largest number, make the largest even number, and find a missing addend
by using the inverse operation or adding up. 19% could start to make some justifications
for how the digits were placed to make the largest even number or the number closest to
4000. About 9% of the students could meet all the demands of the task including making
a justification of how to order the cards for the largest even number and make at least 1
comparison subtraction to find the number closest to 4000. 11% of the students scored
no points on this task. All of the students in the sample attempted the task.
Third Grade – 2005
pg.
46
Number Cards
Points
Understandings
Misunderstandings
All
the
students
in
the
sample
Students
did
not understand the constraints of
0
attempted the task.
using only the digits shown on the cards.
Almost 10% used other numbers for part one.
6% picked arranged the given digits, but could
not succeed in making the largest number. A
majority of those students made numbers in the
3000’s.
Students could arrange the Students had difficulty with the concept of
1
number cards to make the what makes an even number. Some students
largest 4-digit number.
used other digits, so that all 4 digits in the
number would be even. More than 10% of the
students thought the 8 should be placed in the
thousands place to make an even number.
Students could arrange a given Students had difficulty making a justification
2
set of 4 digits to make the for their answers. More than 10% just
largest number and either the described how they placed cards without
number closest to 4000 or the mentioning the place value of the cards or
largest even digit.
purpose to the ordering scheme.
Students could make the largest Almost 10% thought about adding each digit
3
4-digit number using the cards separately (2+8=10, 3+7=10, 7+3,=10) and so
and find a missing addend using arrived at an answer of 372 as the missing
subtraction or an “adding up” addend. More than 10% made renaming errors
strategy.
and got answers like 362,272,and 232.
Students
could
find
the
largest
44% of the students who could not identify the
4
number, largest even number, number closest to 4000, used digits not given
and find the missing addend.
in the number cards. 48% of the students, who
could not find the number closest to 4000,
picked an incorrect value in the 3000’s. This
indicates that students are not sure how to use
numbers other than the leading digit or leading
first 2 digits to determine size of numbers.
Students
could
find
largest
and
Students struggled with making justifications.
5
largest even number, number In part 3, 28% of the students did not attempt
closest to 4000, and find a to explain their answer choice. 17% only
missing addend.
compared a number in the 3000’s to a number
in the 4000’s. 6% talked about 3 being close
to 4, ignoring the idea of place-value.
Students were still struggling with making a
8
complete justification for finding a number
close to 4000.
Students could find largest and
9
largest even numbers, number
closed to 4000, find missing
addend, and explain the logic of
their choices using ideas about
even numbers and place value.
47
Third Grade – 2005
pg.
Based on teacher observations, this is what third graders knew and were able to do:
• Arrange a set of digits to make the largest 4-digit number
• Understand that the one’s digit determines whether a number is even or odd
• Understood that subtraction is the inverse of addition and could use this to find a
missing addend
• Use an “adding on” strategy to find the missing addend
Areas of difficulty for third graders:
• Consider possibilities above and below 4000 to find the closest number
• Make a mathematical justification to explain how to make the largest even
number
• Make a mathematical justification for finding how close a number is to a target
number
• Understanding which place value determines the “evenness” of a number
• Using an inverse operation to find a missing addend
Questions for Reflection on Number Cards:
What do you think your students know about odd and even numbers? What is your
evidence?
Look at student work for part 2. How many of your students put answers of:
7538
Numbers with 8
in the thousand’s
place
Using only 8,
like
8888, _ _ _ 8,
or 8 _ _ _
Using only even
digits, like
2468 or 8864
Using only even
numbers for the
cards, like
10203040 or
18162014
How sophisticated are your students in thinking about size of a number? Do they look
only at the leading digit or the first two digits? Are they able to reason about how all the
digits contribute to the size of the number? What is you evidence?
Look at student work for part 2 again. How many of your students, who made errors,
tried to make the largest even number, by:
Starting with
the 8
Starting with
the 7, but not
ending in 8
Starting with
the 5
Numbers
starting with
the 3
Inserting 2digit numbers
for the placevalue cards
What types of errors in answers 1 or 3 indicate this inability to reason about all the digits
to determine size? Incorrect choices in part 3 can be quite revealing.
Students struggled with making justifications. In part two, how many of your students
were able to reason correctly about 8 being the only even number and so 8 should be
placed in the units position?
How many students only talked about where to place the cards and ignored or did not
make reference to place value or size of the number being formed?
Third Grade – 2005
pg.
48
In part 3, students struggled with how to determine whether a number was close to 4000.
Some students were able to use a number line to help them think about this idea. Did any
of your students use a number line?
Of those, how many did not use the number
line effectively or showed lack of understanding about numbers of this size? (See work of
Student D and Student G)
Some students were able to use subtraction to find the distance between a given number
and 4000. How many of your students, tried:
Comparison sub.
Comparison sub. of Considered
Did not use
for at least 1
more than 1
numbers over 4000 subtraction to think
number to 4000
number to 4000
as well as under
about “close to”
4000
4000
Do your students know that subtraction is the inverse of addition? How many of your
students could use comparison subtraction to find the missing addend in part 4?
How many of your students made subtraction errors in part 4? Make a list of errors.
What do those errors reveal about misunderstandings students have around using the
subtraction algorithm?
While a very good strategy for solving subtraction problems mentally is to use an adding
on or adding up strategy, many students may have used addition because they didn’t
know the relationship between addition and subtraction. How many of your students in
part 4, used:
Addition with
correct answer
Addition with
incorrect
answer
Thinking about each digit
separately rather than with a
place value, such as 3 +7=10
An answer with no
relationship to
problem, like 100
or 936
How much time have students had this year working with place value? Do you ask a
variety of questions that help assess this understanding? Do students have opportunities
to compare numbers where they need to pay attention to more than one digit to determine
which is largest?
Teacher Notes:
Third Grade – 2005
pg.
49
Implications for Instruction:
Students need to develop a good understanding of place-value as it is the basis of the
base-ten number system. A good understanding of place value is necessary, as it is the
foundation for the concept of numbers, making comparisons, and the processes of the
four operations of number. Students should be exposed to a variety of representations of
place-value and base-ten system, including number lines, base-ten blocks, money, placevalues charts, and 100’s charts. Students need to be asked questions, like what is one
more, one less, ten more, ten less, 100 more, or 100 less than a given number. Students
need to count by 1’s, by 10’s, by 100’s from given numbers. Students need to make
comparisons of numbers where more than the leading digit or leading digits are needed to
make the decision about which is larger. Students should also be given tasks that show
that the same digit can have different values in different places. (For example, 2133 – the
value of the digit 3 in the tens place is 30, whereas the value of the digit 3 in the ones
place is 3.) Teachers should avoid confusing students with words like place value.
Instead teachers should consider words like “the digit is in the thousands place” or “the
value of the digit is . . . ” . Comparisons should involve not only identifying which
number is more, but how much more. Games where students try to get closed to a target
number help students learn to reason about “distance”, by finding the numbers higher or
lower can be equally close to the target. The games help students consider more than one
possibility for getting close to a number. When teachers are discussing comparisons, it is
important for them to talk about the value of the digits. For example: The first digit in
3452 represents 3 thousands. The first digit in 563 represents 5 hundreds. So 3452 is
greater than 563.
When learning addition and subtraction, students need to see the connection between the
two operations. Students need to be able to use inverse operations to find missing
numbers. At this grade level, students might talk about “doing” and “undoing” to find
unknowns. Adding up is a good strategy to help solve many subtraction problems,
especially for solving problems mentally. However, students need to have a firm grasp of
place value in order to use it efficiently and accurately. Trying to think about each
column separately allows for many computation errors.
Teacher Notes:
Third Grade – 2005
pg.
50
3rd grade
Student
Task
Core Idea
2
Number
Operations
Task 4
Sponsored Walk
Calculate the money that students will raise as they participate in a
sponsored walk. Find and explain how far a student must walk to earn at
least $20.
Understand the meanings of operations and how they relate to each
other, make reasonable estimates, and computer fluently
• Understand the meaning of multiplication as repeated addition,
make reasonable estimates and compute fluently
• Develop and use strategies to estimate and judge the
reasonableness of results
• Communicate reasoning using numbers, pictures and/or words
Third Grade – 2005
pg.
51
Third Grade – 2005
pg.
52
Third Grade – 2005
pg.
53
Sponsored Walk
Grade 3
Rubric
The core elements of performance required by this task are:
• choose and use number operations in a real context
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives correct answer: $50
Shows work such as:
6+4
or
= 10
10 x 5 =
points
1
5 x 6 = 30
5 x 4 = 20
30 + 20 =
1
1
1ft
Accept repeated addition
2.
section
points
4
Gives correct answer: $5
1
Shows work such as:
$30 ÷ 6 =
1
Accept repeated addition/subtraction
2
3.
Gives correct answer: 7
1
Gives a correct explanation such as:
If she walks 6 laps she will raise 6 x $3 = $18,
so she will have to walk another lap to raise at least $20.
2
Partial credit
See work such as: 20 ÷ 3 = 6
or 6 x 3 = 18
(1)
Total Points
Third Grade – 2005
3
9
pg.
54
Looking at Student Work on Sponsored Walk
Student A has a very clear strategy for finding the solutions to the problems. In part one
the student organizes a table to show how much money for Jack, for Bill, and altogether.
The student recognizes the situation as one using multiplication. (It would be interesting
to interview the student to find out the purpose of the tally marks. Is it one mark for each
lap?) In part 2 the student shows the $30 being separated into 6 equal parts. Modeling or
acting out an operation is a good process at the early learning stages of a new operation.
The student is then able to convert the “action” into a number sentence. In part 3 Student
A is able to make sense of the extra and arrive at the answer of 7 laps.
Student A
Third Grade – 2005
pg.
55
Student B makes a nice use of labels to make sense of the problem. In part 1 the use of
laps indicates the number of groups of $6 or groups of $4. In the later parts of the
problem, the student writes down the algorithm to help keep track of the steps for a
process that is probably not yet routine. Student B can use division in part 3 and can
interpret the remainder as needing one more lap.
Student B
Third Grade – 2005
pg.
56
When confronted with a new problem situation, a student may need to go through several
trial and error steps. In part 1 Student C appears to be cutting the amount of money in
half and notices that that adds to the $5. The student then tries to add the money together
to $10 as well as adding the 5 laps twice to get 10. The student tries multiplying some
other numbers together, which have somehow been derived from the original dollar
amounts (4 laps times 2 laps = 8 laps). While the student never reaches a clear picture of
part one, Student C is able to meet with good success after this struggle on other parts of
the task. In part 2 Student C uses a double number line to keep track of the groups of 6
needed to reach a total of $30. This strategy is the backbone for proportional reasoning
and understanding equivalent fractions or ratios. The student uses another form of the
double number line in part 3, but forgets that he is solving for number of laps and at least
$20.
Student C
Third Grade – 2005
pg.
57
Student C, continued
Third Grade – 2005
pg.
58
Student D has correct and efficient strategies for parts 1 and 2 of the problem. The
student shows the seeds of mathematical understanding around division, but doesn’t have
the formal language or notation for expressing those ideas. The student realizes that the
$20 should be divided by 3. The student can use multiplication to find out that 3 x 7 =
21, which would be $20 with $1 left over. The student knows something about
remainders or leftovers and tries to express that with the idea of 20r1, or the idea that
Sarah would walk 7 laps and have $1 left over. This is a nice shapshot of the beginning
understandings of learning a new concept.
Student D
Student E has strategies to help make sense of the task, but has not yet made the
connection between repeated addition and multiplication, or at least is more comfortable
with the repeated addition. In part 2 the student is clear that 6 groups of 5 equals 30, but
forgets that the 6 was the number of laps Maria finished and that the problem asked for a
solution to the amount of sponsorship for each lap. Understanding which part of the
computation is being asked for in the question is a struggle for many students. In part 3
Student E again loses track of the question. The student focuses on the constraint “at
least” and ignores the constraint of making $20.
Third Grade – 2005
pg.
59
Student E
Student F shows a similar problem to Student E’s thinking in part 2 in interpreting what
the question is asking. While Student F has a strategy that could have given her the
correct solution, the student does not count the number of groups or laps. Student does a
good job of comparing distance to 21 and showing why this is the closest possible choice.
Student F
Third Grade – 2005
pg.
60
Student G again shows the difficulty of determining what the question is asking. The
student knows that 5 times 6 equals 30 and talks about the 5, but then uses the $30 as the
answer because it’s the part that appears after the equal sign. Many students confuse the
equal sign with the idea that “the answer follows” rather than the idea of sameness or
equality for both sides.
Student G
A big part of third grade mathematics curriculum is to help students make sense of the
operation of multiplication, understanding that in situations with equal groups
multiplication is the best operation. Student H is not making sense of equal groups and
just uses addition for all the problems. Would labeling help the student see the
unreasonableness of adding dollars to laps? Has the student learned multiplication
without context so the connection to having equal groups is not clear in a problemsolving application?
Third Grade – 2005
pg.
61
Student H, continued
Student I shows difficulty with basic computation and a lack of understanding of
multiplication and forming equal groups. In part 3 Student I tries addition,
multiplication, and subtraction to try and make sense of the problem. The student is not
really clear about the meaning behind any of the operations being used.
Student I
Third Grade – 2005
pg.
62
Student I, continued
Student J again shows no experience with problems in context. In part one the student
picks up on the number 5 and does skip counting. It is unclear why he chose to count that
particular number of times. In part 2 the student skip counts by 3’s, which isn’t even a
number in the problem. In the final part Student J tries to just reason about the situation
from experience. The student does not see this as a mathematical question.
Third Grade – 2005
pg.
63
Student J
Teacher Notes:
Third Grade – 2005
pg.
64
Frequency Distribution for Task 4– Grade 3 – Sponsored Walk
Sponsored Walk
Mean: 4.49
StdDev: 3.24
MARS Task 4 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
9
2225
18.6%
100.0%
915
26.2%
81.4%
970
34.3%
73.8%
814
41.1%
65.7%
943
49.0%
58.9%
839
56.0%
51.0%
1200
66.0%
44.0%
1271
76.7%
34.0%
782
83.2%
23.3%
2013
100.0%
16.8%
The maximum score available on this task is 9 points.
The minimum score for a level 3 response, meeting standard, is 4 points.
Many students, about 80%, were able to find the total money earned by Jack and Bill
after completing 5 laps or they were able to find the number of laps needed to earn at
least $20. More than half the students, 54%, were able to find the total money earned by
Jack and Bill and show all the steps for figuring it out. Some students were able to find
the money earned by the two boys and work backwards to find the number of laps Maria
walked to earn $30. 16% of the students were able to meet all the demands of the task,
including reasoning about a remainder to find the number of laps Sarah needed to walk to
earn at least $20. 20% of the students scored no points on this task. 95% of the students
with a score of zero attempted the task.
Third Grade – 2005
pg.
65
Sponsored Walk
Points
Understandings
95%
of
the
students with this score
0
attempted the task.
2
4
6
Students with this score generally
knew to add the amount of money per
lap for Jack and Bill and could total
that to $10.
Students could use all the facts in part
one to find the total amount of money
earned by Jack and Bill and show all
their steps.
Students could find the total money
earned by Jack and Bill. They could
also work backwards from the total
amount of money to find the number
of laps Maria ran.
Students could understand the context
of part 3, but weren’t successful with
interpreting the constraint “at least”
or understanding how to interpret a
remainder. Many students used
multiplication to think about the
problem and arrive at a correct or
nearly correct solution.
Students could use multiplication and
9
addition to solve for the amount of
money earned by two boys in a walk.
Students could work backwards from
a given amount of money to the
number of laps run. Students could
interpret the remainder in a division
problem or use multiplication to find
the number of laps needed to earn a
given amount of money.
Third Grade – 2005
7
Misunderstandings
Students had difficulty choosing the
correct operation needed to solve the
problem. Many students chose
addition for all problems or
multiplication for all problems.
Example: 6 laps x $30 total = $180
per lap
The students did not know to take
this rate and multiply it by the
number of laps to find the amount of
money earned.
More than 15% of the students had a
final answer of $10 for part 1. 6% of
the students added all the numbers in
part 1 to get an answer of 15. Adding
dollars and laps together didn’t
bother students.
8% of the students multiplied instead
of dividing in part 2. 13% of the
students could write a correct number
sentence for part 2, but could not
decipher which part of the number
sentence represented the answer
(picking either the 30 or 6 instead of
the 5). 7% of the students just added
the numbers in the problem, not
recognizing the operation appropriate
for solving the problem.
Students were thinking about 6 x 3 =
18 and 7 x 3= 21. They picked 6 laps
because they didn’t want to go over
$20. They did not understand how
the remainder in 20 divided by 3
applied to the context of earning a
given amount of money.
pg.
66
Based on teacher observations, this is what third graders knew and were able to do:
• Show their calculations
• Skip count, multiply, divide, and add
• Work backwards
Areas of difficulty for third graders:
• Choosing the answer from a number sentence
• Understanding “at least” in the context of earning money
• Interpreting a remainder in a division problem or multiplication problem that is
close but not exactly matching the target number
Questions for Reflection on Sponsored Walk:
When learning multiplication in your classroom, what types of activities do you
do with students to help them focus on making “equal” groups?
• How frequently do students solve multiplication problems in context?
• What is the classroom norm for putting labels on quantities when solving
problems in context?
• Do students have discourse about labels in multiplication and addition problems?
What opportunities do students have to make sense of or see the nonsense of
adding money plus laps or multiplying money times money? Do they learn to use
label-meaning as a tool for checking the reasonableness of their process?
• How many of your students are still relying on repeated addition instead of
multiplication to solve problems?
• Are your students learning fact families as part of the strategies for multiplication
or will the division be a whole new concept for them? Can students work
backwards from a product to the factors?
• What kinds of modeling or acting out do students use in your classroom to help
them choose the appropriate number operation?
Look at student work in part 1. How many of your students gave an answer of :
•
$50
$20
$10
$24
$15
Other
What is the student who got an answer of $15 thinking about or not understanding?
How is this different from what the student who got an answer of $10 is thinking and
understanding? What kinds of experiences do these students need to understand the
operation of multiplication and its use in solving problems?
Look at student work in part 2. How many of your students gave an answer of:
$5
6
$30
$180
$36
$90
Other
What might have led a student to get answers of 6 or $30? What understanding are
these students demonstrating? What little hints might help these students solve the
problem?
Third Grade – 2005
pg.
67
What are the misunderstandings that led students to get answers of $180 or $36?
Why are these more serious errors that the first two? What substantial types of
instruction do these students need?
Teacher Notes:
Implications for Instruction:
Students need considerable time learning about number operations and when they are
used. Modeling or acting out situations helps students to make sense of new operations.
Using bar models or pictures to help represent equal groups is a good tool for building
an understanding of multiplication. Students need to work with multiplication in context
of problems to make sense of what is the group (laps, people, money) and identify the
number of groups. Labels are an important tool for identifying the group and number of
groups. Seeing how labels change during multiplication or stay the same during
addition is a mathematical process, later called dimensional analysis. But this tool can
be used now for sense-making around solving problems. Students should have frequent
opportunities to discuss labels. Does it make sense to add money and laps? people and
pounds? Can you have a multiplication problem of money times money? Why or why
not? When learning multiplication, students should be learning fact-families and starting
to make sense of the inverse operation. While they may not know the formal steps for
division, students should be comfortable working backwards from a product to find a
missing factor.
Third Grade – 2005
pg.
68
3rd grade
Task 5
Teddy Bears
Student
Task
Analyze and describe how Kate makes teddy bears using two eyes, a
nose, and three buttons.
Core Idea
3
Patterns,
Functions,
and Algebra
Understand patterns and use mathematical models to represent and
to understand qualitative and quantitative relationships
• Describe and extend numeric patterns
• Represent and analyze patterns using words and tables
• Solve simple problems involving a functional relationship
• Communicate reasoning using numbers, pictures and/or words
Third Grade – 2005
pg.
69
Third Grade – 2005
pg.
70
Teddy Bears
Grade 3
Rubric
The core elements of performance required by this task are:
• find and use number patterns
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
section
points
Gives correct answers:
All 6 values correct
6x1
Allow 1 point for each correct value.
6
2.
Gives correct answer: 8 teddy bears
1
Gives a correct explanation such as:
26 eyes make 13 bears, 11 noses make 11 bears, 24 buttons make 8 bears.
1
2
8
Total Points
Third Grade – 2005
pg.
71
Looking at Student Work on Teddy Bears:
Using a table with four columns was quite challenging for many students. Student A was
able to follow the logic of the table, make appropriate compensations for the breaks in the
table (the number of teddy bears was not continuous in three separate places), and
consider all three of the material constraints in part 2 to determine the number of teddy
bears that could be made. Student A systematically eliminates each material until all
categories are accounted for.
Student A
Third Grade – 2005
pg.
72
Student B is able to use division and pictures to make sense of the material constraints in
part 2.
Student B
Third Grade – 2005
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73
Student C is also able to meet all the demands of the task. The student is not yet
comfortable with division, but can use repeated subtraction to find the number of sets of
buttons. While the student uses the term “pairs” of buttons, in a problem-solving setting
precise use of vocabulary is not as important as communicating mathematical ideas.
Student C
Third Grade – 2005
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74
Student D can reason about buttons, noses, eyes and total number of bears and fill in most
of the table correctly. The student is able to calculate, using division with number
sentences to find the total number of bears that can be made with each material
individually. Student D is unable to identify what is significant about those calculations
and chooses the 11 noses as determining the number of teddy bears instead of the 8 sets
of buttons.
Student D
Third Grade – 2005
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75
Student E is confused by the demands of using the table. The student may have some
understanding of the relationship of each material to making a teddy bear as shown by the
13, 11, and 8 in part 2. However, like Student D, Student E is unable to use this
information correctly to determine which material will run out first. Student E seems to
think of each material making separate bears, which when put together will total 32 bears.
The student is not thinking of the constraints that each bear must have eyes, nose, and a
set of buttons.
Student E
Student F is able to follow the logic of the chart. While the student is still struggling with
the formal process of division the student shows an understanding of the logic of division
and is able to invent a strategy for finding the number of teddy bears that can be made
with 26 eyes. The student does not consider the effects of the other materials in making
complete teddy bears.
Student F
Third Grade – 2005
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76
A very common error was to equate the number of noses with the number of teddy bears.
See the work of Student G.
Student G
Many students had trouble with the final row of the table. They may have picked the
wrong number of teddy bears, but then used the rules to find other values in the row that
are consistent with the number of teddy bears picked. Student H does not think about
how the materials relate to making a possible numbers of bears, but uses experience to
think that you can’t make a bear without a body to put the materials on.
Student H
Third Grade – 2005
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77
Understanding a situation and using a table designed by someone else are not the same
thing. A good action research problem would be to give this problem to students without
the table and see if they do better or worse. Student I appears to understand the problem
situation, as demonstrated by the clear explanation and correct answer for part 2. The
student seemed stumped by the table.
Student I
Third Grade – 2005
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78
Student J does not appear to have any understanding of the table. The student also does
not seem to recognize what operation is needed to go from the total amount of a material
to the number of teddy bears that can be made with that material. The student seems to
think that when there are 3 numbers in a problem use addition.
Student J
Teacher Notes:
Third Grade – 2005
pg.
79
Frequency Distribution for Task 5 – Grade 3 - Teddy Bears
Teddy Bears
Mean: 3.65
StdDev: 2.77
MARS Task 5 Raw Scores
Score:
Student
Count
%<=
%>=
0
1
2
3
4
5
6
7
8
2123
17.7%
100.0%
1669
31.7%
82.3%
1166
41.4%
68.3%
1217
51.6%
58.6%
644
57.0%
48.4%
931
64.7%
43.0%
2178
82.9%
35.3%
563
87.6%
17.1%
1481
100.0%
12.4%
The maximum score available for this task is 8 points.
The minimum score for a level 3 response, meeting standards, is 3 points.
Most students, about 83%, could find the number of noses for 2 teddy bears. Many
students, 61%, could find the number of eyes for 5 bears, the number of noses for 2 bears,
and the number of buttons for 10 bears. Some students, 37%, could work with a table
using 4 columns of materials and discontinuous in three separate places to find the
number of bears, eyes, noses, or buttons. This may have included working backwards
from the amount of one quantity to find the number of bears. 14% of the students could
meet all the demands of the task, including determining which of the 3 materials limited
the number of complete bears that could be made and quantifying that number of bears.
17% of the students scored no points on this task. 70% of those students attempted the
task.
Third Grade – 2005
pg.
80
Teddy Bears
Points
Understandings
Only
70%
of the students with
0
this score attempted the task.
1
Students generally knew that 2
teddy bears had 2 noses.
3
Students could find the eyes for
5 bears, the number of noses for
2 bears, and the number of
buttons for 10 bears.
6
Students could work with a table
using 4 columns of materials and
discontinuous in three separate
places to find the number of
bears, eyes, noses, or buttons.
This included working
backwards from the amount of
one quantity to find the number
of bears.
8
Students could work with 3
material constraints to find out
how many of each were needed
for different numbers of bears.
They could follow the logic of a
table designed by someone else
and fill in values. They could
work backwards from number of
eyes or buttons to number of
bears. Students could determine
which material limited the
number of bears that could be
made.
Third Grade – 2005
Misunderstandings
Students are not familiar with a table
with this many columns. They had
difficulty tracking from the beginning
category, number of bears, to the later
columns, number of noses and number
of buttons. Students might have done
better without the table in making sense
of the problem.
Finding the number of noses for the
final row was dependent upon finding
the number of bears with 24 eyes.
Students had difficulty working
backwards from the number of eyes to
the number of bears. 10% of all
students thought there were 15 bears in
the final row. Almost 10% thought the
next number of bears after 10 bears
would be 11 bears. Often students’
answers for the rest of the row matched
the number of bears picked.
Students struggled with the logic of
which material would run out first.
Many students considered that the
number of noses is always equal to the
number of bears, which is true for the
table. Other students only considered
the number of bears that could be made
with the first quantity, which was eyes.
Using a variety of strategies, because
this was not a familiar fact, students
were able to determine there were
enough eyes for 13 bears.
pg.
81
Based on teacher observations, third grade students were able to:
• Fill out the beginning rows of a table
Areas of difficulty for third graders:
• Using a table set up by someone else
• Work backwards from a total number of eyes or buttons to find the number of
teddy bears.
• Determine which of 3 materials limited the number of teddy bears that could be
made and then use that information to quantify the number of possible bears.
Strategies of successful students:
• Drawing pictures
• Using repeated subtraction
• Decomposing a number into parts for which division facts were known
• Explaining why there were too many of some materials and how that effected the
answer
Questions for Reflection on Teddy Bears:
•
•
•
•
•
•
•
What opportunities have you students had working with tables? How many rows
or columns do they usually see?
Have they worked with tables for patterns? tables for data or frequency? tables for
quantities of materials?
When working with different types of tables, do students get opportunities to
discuss key features of the table and how they might differ from other types of
tables?
Do students get the opportunity to solve problems where making their own table
might be a useful strategy? Do they compare the differences between their tables
and the tables of their classmates?
Have students worked with tables where the data is not increasing for one base
unit at a time? Do you think students noted that the table was not going up by the
some number of bears each time? For example, how many of your students
thought there were 11 bears in the final row?
When looking at student work, do you think they might have made more sense of
the problem without the table?
What strategies did you students use to do the division parts of the problem?
o How many used repeated subtraction?
o How many used multiplication?
o How many used drawing and counting?
o How many decomposed the numbers into more manageable parts?
Look at student work in part 2, how many of your students:
8 and
discussed
all 3
materials
8 and
discussed
only
buttons
Third Grade – 2005
11
13
61- did not
considered considered understand
only noses only eyes operation
needed
12
Other
pg.
82
Teacher Notes:
Implications for Instruction:
Students at this grade level need many experiences with tables and need to have the use
of the tables made explicit. The teacher might ask questions about what each column
means or represents. The teacher might ask students, “If we know the number of bears,
how can we find . .. .?” The teacher might ask students to put a number sentence or rule
at the top of each column. The teacher might also ask questions about what they notice
about the column labeled number of bears. Will the students pick up on the idea that the
numbers are not consecutive? The students might then discuss ideas, like, “Will this
change our rule? Why or why not?” “Will this effect patterns within a row? Why or why
not?” In learning about tables and their uses, students should be asked how the table
might help to solve problems. Ask the students to think of questions the table might help
them solve.
Students also need opportunities to make their own tables and see the connection between
the table and the context of the problem. Students should work with tables that increase
in set increments and tables that are discontinuous. This will help them avoid blindly
following patterns within a table and think more about the problem context. Research
shows that students often start following a pattern when filling in values and disconnect
from the context of the problem. Students should have strategies to work from a total
number of parts to finding the number of objects that can be made. Students need to
think about many situations with multiple constraints and determining which constraint
limits the amount of teddy bears or cookies that can be made.
Teacher Notes:
Third Grade – 2005
pg.
83