part 1 - Noyce Foundation

Transcription

Core Idea
Task
Score
Votes
Number Operations
The task asks students to find and compare the total number of votes for two
candidates and then to use multiplication to find a weighted value for their votes.
Roger’s Rabbits
Algebra
The task asks students to identify and extend patterns and use a table. Successful
students could also give rules for extending both elements in the pattern, the number
of doors and the number of blocks needed to make a row of rabbit hutches.
Winning Lines
Number Operations
The task asks students to work with a “magic square” type number game to identify
numbers that add to a given total or to generate a series of number sets that add to a
given total. Successful students could reason about why some numbers were not
possible to use to make a given sum and meet the rules of the game.
Quilt Making
Geometry
The task asks students to work with 2-dimensional shapes and their properties, such as
symmetry and angles. Successful students knew the names for rhombus,
parallelogram, and right triangle.
Sum Bugs
Number Properties
The task asks students to solve problems using multiplication and division.
Successful students could generate numbers to fit 3 or more constraints, such as even
number divisible by 5 and 3 with 3 digits.
Grade 4 – 2008
Copyright ©2008 by Noyce Foundation
All rights reserved.
1
Grade 4 – 2008
2
Grade 4 – 2008
3
Votes
This problem gives you the chance to:
• work with a weighted point system
Some students vote for class president.
Each student can vote for a first choice and a second choice.
Amos, Brie and Carl received more votes than the other students.
Number of ‘first choice’ votes
Number of ‘second choice’ votes
Amos
8
12
Brie
6
15
Carl
9
7
1. Who got the most votes, Amos, Brie or Carl?
______________________
2. Who got the most ‘first choice’ votes?
_______________________
3. A ‘first choice’ vote gets 2 points and a ‘second choice’ vote gets 1 point.
Explain why Carl gets 25 points altogether. _________________________________
_____________________________________________________________________
_____________________________________________________________________
4. Figure out the points for:
a. Amos ______________________________________________________________
b. Brie _______________________________________________________________
5. Who should be class president?
_______________________
7
Grade 4 – 2008
Copyright ©2008 by Mathematics Assessment Resource Service
4
Votes
Rubric
The core elements of performance required by this task are:
• work with a weighted point system
Based on these, credit for specific aspects of performance should be
assigned as follows
section
points points
1.
Gives correct answer: Brie
1
1
2.
Gives correct answer: Carl
1
1
3.
Gives correct explanation such as: Carl gets 2 x 9 + 7 = 25votes
2
2
4.a. Gives correct answer: 28
1
b. Gives correct answer: 27
1
2
1ft
1
5.
Gives correct answer: Amos
Total Points
Grade 4 – 2008
7
5
Votes
Work the task. Look at the rubric. What are the key mathematical ideas in this task?
Look at student work for part 1 and 2. While most students got this part correct, the most
common error for both questions was Amos. Did any of you students have this error?
In each case, can you figure out what the student might have been thinking?
Now look at student work for part 3. Again, most students were very successful. For
students that made errors what made this confusing? Did they just restate the prompt in
words? Did they think the answer was 16? Did they misinterpret the rule? Did students
just write a number sentence equal to 25 without thinking about a rule, but just trying to
get the desired goal? Can you figure out what they were thinking?
Now look at work for part 4. How many of your students put:
28,27
20, 21
23,24
Other
What were the students thinking that caused these misconceptions?
Did you see students just making up number sentences to show work, but with numbers
that did not relate directly to the problem?
How many of your students showed their calculations on their paper? Why is this
important when you want to give students feedback on their thinking?
What work stood out for you as exemplary? What were the qualities that you valued in
the work?
Grade 4 – 2008
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Looking at Student Work on Votes
Student A shows all the calculations. Notice that the student puts all 3 answers for both
part 1 and part 2 to show what is being compared. The student again shows all three
choices for the comparison in part 5. This is an important mathematical idea, that
students in 6th and 7th grade struggled with.
Student A
Notice that the next student only considers two of the three choices for totals in part 1. So
the student only looks at those two scores to think about part 2.
Grade 4 – 2008
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Student B shows the totals underneath the diagram and has a clear explanation for finding
Carl’s points in part 3. The student does not understand that the rule for finding points in
part 3 also applies to part 4. The student just brings down the totals from part 1.
Student B
Grade 4 – 2008
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Student C has trouble tracking or organizing information. The student makes correct
calculations for part 1, by using the numbers lined up in the diagram. The student
understands the rule for finding points and applies it correctly to part 3. However in
calculating the total for Amos, the student uses the “1 point” value for Amos and the “2
point” value for Carl. In finding the points for Brie, the student uses the 2 point value for
Brie, but then adds the “1 point” value for Carl. Do students in your class have
opportunities to work problems with long reasoning chains? Are students given
opportunities to organize work for themselves without scaffolding, so that they learn
techniques for keeping track of information?
Student C
Grade 4 – 2008
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Student D is able to add the totals to answer question 1, although there is no work shown.
In part 3 the student adds the 2 points to the 9 to get 11 and adds the 1 point to the 7 to
get 8. The student does not understand the multiplicative nature of the rule. The student
then needs to “imagine” how to get the rest of the points needed to get the total of 25. In
part 4, the student merely adds the (2+1) to the total, but without the “mystery number of
6” that was added to Carl to get the desired total.
Student D
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Student E has a similar confusion about adding rather than subtraction. The student tries
to apply the “add 3” rule at the top of the page to find Carl’s points, but that does not
work out. Instead the student adds the first place votes together because it gets close to
25 and then the 2 points gives the needed results. The student then goes back and applies
his original interpretation of the rule to part 4.
Student E
Grade 4 – 2008
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Student F does not distinguish between the most in part 1 and the most in part 2.
However the student does understand how to find total votes and uses it in part 4 instead
of part 1. The rule for part 3 is not understood and the student is not confident enough to
attempt this part of the task.
Student F
Grade 4 – 2008
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Student G does not understand the rule in part 3. The student just implies that the total
votes should be added together. Notice that the student does not understand place value
when adding the totals in part 4. What classroom activities help students to develop a
number sense about place value? How does use of language in the classroom contribute
to understanding place value? What language in the classroom contributes to
misunderstanding place value?
Student G
Grade 4 – 2008
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Student H also does not understand the rule for part 3 and sticks with the idea of finding
totals. The student does want to show off computation skills and is even able to work
with division having a decimal solution. How can we use this eagerness to explore with
math and develop it?
Student H
Grade 4 – 2008
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4th Grade
Student Task
Core Idea 1
Number
Operations
Task 1
Votes
Work with a weighted point system.
Understand meanings of operations and how they relate to each
other, make reasonable estimates, and compute fluently.
• Develop fluency in multiplying whole numbers.
The mathematics of this task:
• Making mathematical comparisons between all the options
• Interpreting a multiplicative rule (seeing the difference between an additive rule
and a multiplicative rule)
• Multiplication and addition computation, place value
Based on teacher observations, this is what fourth graders knew and were able to do:
• Show thinking
• Add totals
• Compare numbers
• Basic multiplication and addition
• Read and interpret a chart
Areas of difficulty for fourth graders:
• Determining the difference between points and votes (finding totals in part 4
rather than using the rule from 3)
• Using additive rather than multiplicative relationships for the rule in 3
• Repeating the prompt rather than calculating in part 3
Strategies used by successful students:
• Showed their work and put totals under the appropriate name
Grade 4 – 2008
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The maximum score available on this task is 7 points.
The minimum score needed for a level 3 response, meeting standards, is 5 points.
Most students, 97%, could total the points and identify the largest total votes and largest
first place votes. Many, 84%, could also choose the person with the highest points in part
5. More than half the students, 64%, could recognize the most votes, most first place
votes, most points, and use the multiplicative rule to find Carl’s points. 43% of the
students met all the demands of the task including using the multiplicative rule on all
three candidates. Only 1% of the students scored no points on this task. All the students
in the sample with this score attempted the task.
Grade 4 – 2008
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Votes
Points
Understandings
All
the
students in the sample
0
with this score attempted the
task.
2
3
Students could identify the most
votes and most total votes.
for 3 different situations.
5
and use a multiplicative rule to
find Carl’s points.
7
and apply a multiplicative rule to
the votes of all the candidates.
Students were able to verify a
answer using words and
numbers.
Grade 4 – 2008
Misunderstandings
Students confused most total votes and
most first place votes. 6% of the students
chose Amos for part 1. 6% chose Amos for
part 2.
Students did not choose the highest number
for most points.
Students could not understand the
multiplicative rule in part 3. 4% of the
students just restated the prompt in words.
3% thought that Carl’s score was 16 not 25.
Students often did not use the rule from 3
in finding points in part 4. 14% used the
total points from part 1.5% added 3 to the
totals, using an additive rule instead of a
multiplicative rule.
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Implications for Instruction
Students at this grade level need to recognize multiplicative situations. Students might
need practice with models to help them identify operations. Students should be looking
for rules in patterns and use the verbal rules to write number sentences.
Some students need more practice working and solving problems with longer reasoning
chains. They had difficulty tracking what they had calculated. Students need to have
many experiences working problems without scaffolding, so that they can learn to
organize their own thinking.
Students also need to work problems where they need to write down the problems
themselves. A few students did not know how to line up their numbers when writing the
addition problems.
Ideas for Action Research – What’s My Rule?
Some students had difficulty interpreting the rule in part 3 of the task. Students at this
grade level should work lots of pattern problems with rich interesting rules, like Hexagon
Desks (2003), Piles of Oranges (2004), and Squares and Circles (2005). These problems
present a good context to press students to move from drawing and counting or repeated
addition to looking for verbal rules to solve for any number in the pattern. In focusing in
on finding and writing rules, students gain experience with how rules work and move to
more complex styles of thinking.
Students might also benefit from being given rule descriptions, such as 3 times a number
minus 4 and asked to calculate the solution for different starting values. This could be a
good class warm up activity, working on computational skills as well as algebraic
thinking.
A good game, also appropriate for a class warm up, is What’s My Rule? Put up a table on
the board:
_input _________________________
output______________________
You might start by filling in an input of 2 and an output of 5. Then maybe put in an input
of 7 and an output of 15. Then give a new input, see if any students can fill in the new
output. Put up several inputs and have students try to add new outputs. Then ask
students to think of the rule for the game. In this case the rule was 2 times the number
plus one. This also has students work with understanding how rules operate and also
provides computational practice.
Grade 4 – 2008
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Roger’s Rabbits
• identify and extend patterns
• work with tables
Roger keeps pet rabbits. He keeps them in a row of rabbit hutches.
The hutches are on blocks so that they don’t get damp.
This is hutch #1.
It is for one rabbit.
It has 2 doors and 2 blocks.
This is hutch #2.
It is for two rabbits.
This is hutch #3.
It is for three rabbits.
1. Describe hutch #4.
_______________________________________________________________
_______________________________________________________________
2. Fill in the empty spaces in the table below.
Hutch #
Number of doors
Number of blocks
1
2
2
Grade 4 – 2008
2
4
3
3
4
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3. How many doors will be needed for hutch # 8?
Explain how you figured this out.
__________
___________________________________________________________________
___________________________________________________________________
How many blocks will be needed for hutch #8?
____________
4. Roger says that for hutch # 12 he will need 11 blocks. Roger is wrong.
How many blocks will he need? ________________
Explain how you figured this out.
___________________________________________________________________
___________________________________________________________________
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Grade 4 – 2008
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Roger’s Rabbits
Rubric
• identify and extend patterns
• work with tables
points
section
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Gives a correct description: 4 rabbits, 8 doors and 5 blocks.
2.
Completes the table correctly.
2x1
2
2
Partial credit
2 correct numbers
3.
4.
(1)
Gives correct answer: 16 doors
1
Gives correct explanation such as: the number of doors is twice the hutch
number or draws diagram.
1
Gives correct answer: 9
1
Gives correct answer 13 blocks.
1
Gives correct explanation such as: He will need two blocks for hutch
number 1 and then one block for each of the next blocks.
1
Total Points
Grade 4 – 2008
2
3
2
9
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Roger’s Rabbits
Work the task. Look at the rubric? What are the big mathematical ideas in this task? What
strategies do you anticipate your students might use to solve this task?
Were your students able to give a description of the hutch #4?
Did there descriptions help them to identify the patterns in the task?
Look at their work for part 3. Were students able to identify the number of doors for hutch # 8 as
16?____________ 8?______________ other?_________________
• How many students drew a picture?
• Continued the table or did repeated addition by 2’s?
• How many gave a verbal rule, such as the number of doors is t times the pattern #?
• How many students multiplied by 2?
Now look at their work for the number of blocks. How many students put:
9
8
10
12
16
Other
How do you think the students were thinking about the blocks who made some of these errors?
What attributes of the pattern were they not seeing? Where were they making faulty
generalizations?
Now look at the work for part 4. While many students could find the number of blocks in part 3,
they often did not use that same strategy for part 4. When looking at errors check to see if they were
using the same or a different strategy. How many of your students put:
9
8
12
15
16
24
13
20
other
Why do you think finding the blocks was more difficult here than in part 3?
In your classroom, how do you help students move from drawing and counting strategies to thinking
about describing how the pattern grows and using multiplication?
What is the benefit for the student? How does the mathematics of finding a rule contribute to the
mathematics needed in later grade levels?
Grade 4 – 2008
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Looking at Student Work on Roger’s Rabbits
Student A searches for patterns and relationships in the numbers. Notice how the table is labeled to
show how the patterns grow in part 2. Then above part 3 the student verifies the relationship
between the hutch number and the number of doors and then writes a written rule to describe the
relationships. Notice the habits of mind to test conjectures, label work and organize work, and look
for relationships in the numbers.
Student A
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Student B is able to answer all the questions in the task, but the student is relying on strategies from
earlier grade levels. The student needs to draw and count every part of the pattern or use repeated
addition. Notice that the student doesn’t think to continue a drawing or continue a table, but starts
from scratch each time. In the work by question one the student seems to understand the idea of
multiplying by 2 doors for each hutch, but the student doesn’t seem to know how to make the next
developmental step to using the rule to replace tables and drawings. How do we help students leave
behind comfortable strategies and make the effort to learn new ones?
Student B
Grade 4 – 2008
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Student C is able to do parts 1, 2 correctly. The student is able to see the doubling pattern for the
doors. However the student doesn’t use the information in part 1 or 2 to help find the number of
blocks. Student does not observe that the first hutch has two blocks. What questions for self-talk
might have helped this student? How do we help students move from noticing how a pattern grows
to noticing the relationships between two variables (hutch # and # of blocks)? Do we give students
enough opportunities to explain how they know their answers are true? To listen to the arguments of
other students and hear other points of view? To debate different conjectures?
Student C
Grade 4 – 2008
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Student D has to draw pictures, but doesn’t have that diagram literacy to think about what are the
important or essential features. The student draws the screens on the door and even draws in the
rabbits. The student understands how the pattern grows, but in the table doesn’t use enough detail
(omits the hutch # row) to find the correct solution for the number of blocks in part 4. Notice the
student uses the table successfully for the number of blocks in part 3, but doesn’t notice the
relationship. The student is not thinking about patterns, but just number crunching.
Student D
Grade 4 – 2008
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Student E is able to see the pattern for the blocks, but the student looks at the picture and sees one
door instead of two doors per hutch. The student doesn’t use the verbal clues by the side of the
drawings.
Student E
Grade 4 – 2008
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Student F does not examine the physical model carefully. In part 1 the student may think the blocks
are 2 less than the hutch number. The student is able to find the number of blocks in part 3, but
there is no work to show where the number came from. Then in part 4 the student doubles to find
the number of blocks. The thinking is not consistent. What would be your next steps with this
student?
Student F
Grade 4 – 2008
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Like Student F, Student G does not give enough detail to understand what he is thinking. Questions
1 and 2 are correct, but what is added to get 12? The student can find the number of blocks in part
3, but uses a different rule to find the blocks in part 4.
Student F
Grade 4 – 2008
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Student G does not understand the basics of the pattern. The hutches are drawn disconnected, so
that even a counting strategy will not help the student solve the pattern. What type of help is
available for this student at your school? Where would you go with this student in terms of
providing fundamental help?
Student G
Grade 4 – 2008
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4th Grade
Student Task
Core Idea 3
Patterns,
Functions,
and Algebra
Task 2
Roger’s Rabbits
Identify and extend patterns. Work with tables.
Understand patterns and use mathematical models to represent and
to understand qualitative and quantitative relationships.
• Represent and analyze patterns and functions using words,
tables, and graphs.
• Find the results of a rule for a specific value.
• Use inverse operations to solve multi-step problems.
• Extend geometric patterns and be able to describe the growth
• Use a table to continue a growth pattern
• Extend the pattern using rules, look for relationships between two variables (hutch # and
number of doors & hutch # and number of blocks)
• Recognize doubling patterns
• Understand a pattern with a constant
Based on teacher observation, this is what fourth graders knew and were able to do:
• Extend the table
• Count by 2’s
• Recognize that there was a pattern and verbalize it
• Over-generalizing the rule for both parts, if they doubled for doors they doubled for blocks
• Understanding that the number of blocks was not the same as the hutch #
• Consistency in finding the number of blocks (using different rules to find blocks in part 3
and 4)
Grade 4 – 2008
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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, 93%, could explain the number of doors and blocks in hutch 4 and extend the pattern
using a table. Many students 87% could also find the number of doors for hutch #8. 81% could
explain how they found the number of doors. Almost half the students, 46%, could meet all the
demands of the task including seeing that the number of blocks was always one more than the hutch
number. Less than 2% of the students scored no points on the task. All the students in the sample
with this score attempted the task.
Grade 4 – 2008
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Roger’s Rabbits
Points
Understandings
All the students in the sample
0
4
5
6
7
9
task.
Students could describe the
number of doors and blocks
needed for hutch #4 and could
use the table to extend the
pattern.
Students could extend the pattern
using a table and using a
description. They could also
give the number of doors needed
for hutch #8.
using a table and using a
description. They could also
give the number of doors needed
for hutch #8 and explain their
thinking.
using words and a table. They
understood that the doors
doubled. Students could find the
number of blocks in part 3, but
not in part 4.
Students could extend a pattern
in words, tables, and using rules.
Good students looked for
connections between variables.
Grade 4 – 2008
Misunderstandings
Some students described hutch 3 in part 1.
Some students thought their were 6 blocks
for hutch #4.
Some students did not attempt to do the
table. Some thought both patterns were
growing by 2’s. Some students thought
both patterns were growing by 1.
They didn’t know how to explain their
thinking for finding the number of doors.
Students could not find the number of
blocks for hutch #8. 7% of the students
thought there would be 10 blocks. 5%
thought there would be 8 blocks. 3%
thought there would be 12 blocks.
8% of the students thought the number of
blocks for hutch #12 was 12. 5% thought
there would be 24 blocks. 3% thought
there would be 9 blocks.
Struggling students still relied on drawing
and counting or adding on to extend the
pattern. They weren’t asking themselves
questions to dig deeper into the patterns to
find the relationships, like the number of
doors is double the hutch number or the
number of blocks is one more than the
hutch number.
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Students need to look at visual patterns and be able to describe how they are growing. At second grade and
third grade students have been able to draw and extend patterns by counting and repeated addition. At this
grade level students should start to think more deeply about the patterns. Students should develop self-talk,
with questions such as, “what is growing?” and “what is staying the same?” They should be starting to
examine relationships between parts of the pattern. If I know the hutch number or pattern number, how can
I find the number of doors or the number of blocks? If I know the number of blocks how can I find the
hutch number? Students should start to make generalizations with words to find for any part of the patterns
without drawing and counting or using repeated addition. Part of pattern recognition is to look for rules.
Students should be able to fill in a simple table about the number of doors and blocks needed for different
numbers of rabbit hutches. Students should be comfortable recognizing patterns that grow by 2’s. Students
should be thinking about things that grow at a constant amount, like going up by 2’s, as a multiplication
pattern.
Ideas for Action Research – Searching for Deeper Patterns, Learning Self-talk
Students need to learn a process for thinking about patterns in deeper ways. Consider the patterns
below (from Teaching Student-Centered Mathematics, grades 3-5 by Van de Walle):
Students should be given materials and paper to build and draw what comes next in the pattern.
They should start to develop questions about how the pattern grows. What is new and different
about the next shape? What changes from one piece in the pattern to the next? Ask students to
explain why their extension follows the pattern. Students should also look for rules. For example,
students might not be able to find a mathematical rule to extend pattern b, but they should be able to
give a verbal rule for drawing the 10th term without needing to build every one in between. Students
at this grade level should also start to justify their rules by giving trying their rules on the numbers
from a table to see if it follows the pattern.
Grade 4 – 2008
34
Winning Lines
• work with a ‘magic square’ type number game
Gina and Sam are playing a card game.
They place number cards on a large game board.
A target number is written inside a circle at the top of each board.
To win a point they need to make a line of three numbers whose
sum is the target number.
The three numbers can be written in a column, a row or a diagonal.
14
7
6
1
5
3
9
2
8
4
In any game the same number cannot be used more than twice.
No zeros are allowed.
1. Gina and Sam have completed the game shown above. The target number is 14.
Draw lines through the five winning lines.
12
2. Here is a game board that has already been started.
4
One point has been won because 4 + 3 + 5 = 12.
Write numbers on the empty cards to win at least three more points.
3
5
Draw lines through your winning lines.
3. Here is a new game board.
9
Fill in the numbers to win at least four points
Fill
Draw in the winning lines.
Explain why the number 8 cannot be used in any winning line.
_____________________________________________________________________
_____________________________________________________________________
Grade 4 – 2008
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35
Winning Lines
Rubric
• work with a ‘magic square’ type number game
points
Based on these, credit for specific aspects of performance should be assigned as follows
1.
Draws a line on all 5 winning lines with no extras.
Partial credit.
Draws a line on 4 or 3 winning lines with no extras.
Draws a line on 2 or 1 winning lines with no extras.
section
points
3
14
7
6
1
5
3
9
2
8
4
(2)
(1)
3
2.
Creates at least 3 more winning lines and draws in the lines to indicate
where they are.
Partial credit.
Creates 2 or 1 winning lines and draws lines to indicate where they are.
or
Creates 3 correct winning lines and 1 error.
3.
Fills in the digits to win at least 4 points.
2
(1)
(1)
2
1
Gives correct explanation such as: If the number 8 was used it would mean
that only 1 could be used only once to reach the target number and it needs
to have 3 digits to be a winning line.
Total Points
1
2
7
Note: Where student uses a digit more than twice or uses the digit “0” treat as a misread and subtract
one point from the total.
Grade 4 – 2008
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Winning Lines
Work the task. Look at the rubric. What are the key mathematical ideas in this task?
Look at student work in part 1. How many of your students:
Found
Omitted Omitted 1 Miscalculated Forgot both
all 5
diagonal
row or 1
the center
horizontal
lines
column
column
rows
Found only
1 winning
line
Other
What do you think caused some of these errors? Are students given enough opportunities to look for
multiple solutions? Are students given enough opportunities to interpret and apply rules for
themselves?
Now look at work for part 2. How many of your students:
Found all
Made
Found only
Found only
winning
addition
2 winning
1 winning
lines
errors
lines
line
Didn’t draw in
the lines, but
made winning
lines
Other
What struck you as you looked at student work? What do you think was challenging for students?
In looking at how students filled in the numbers in part 2 and 3, what strategies did students use for
keeping track of their work:
• Cross out numbers from a list after use
• Make a list of combinations to reach the target number
• Show addition to the side
• Other strategies?
In part 3, make a list of the 3 best explanations for why 8 can’t be used:
Make a list of some of the worst explanations. What was needed to improve these explanations?
Grade 4 – 2008
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Looking at Student Work on Winning Lines
Student A circles the constraints. Notice the student lists numbers to be used in the solution and has
a method for knowing when that number has been used up. See how the explanation is complete.
Student A
Grade 4 – 2008
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Student B is missing one piece needed to complete his justification in 3. What is needed to round
off this argument?
Student B
Student C makes a list to keep track of the numbers used and writes down possible combinations. It
appears that the student thinks a number can be used only once and that only single digits are
allowed.
Student C
Grade 4 – 2008
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Student D also seems to think that a digit can only be used one. The student may also think that
only numbers can be added if it is possible to make a winning line. The student might have misread
the number of winning lines needed.
Student D
Grade 4 – 2008
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Student E makes lists of winning combinations, but does not find enough combinations for part 3.
Why might this part have caused difficulty for the student?
Student E
Grade 4 – 2008
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Student F does not find all the winning combinations in part 1. The student finds 3 winning
combinations in part 2 but does not draw the lines. Also the student uses too many 3’s. In part 3 the
student does not find enough winning combinations and does not draw lines. The student’s
explanation in part 3 does not seem to relate to the context of the problem or may be a description of
the 8 in relationship to the 9. There is not enough evidence to be a convincing argument. What
question might you pose to get further explanation from the student?
Student F
Grade 4 – 2008
42
Look at the work of Student G. What evidence do you see of understanding around the task? What
are some of the constraints that the student doesn’t understand? What are some misconceptions?
Student G
Grade 4 – 2008
43
What might be confusing for Student H? How do we help students who are struggling with this kind
of problem solving? What experiences does the student need?
Student H
Grade 4 – 2008
44
How many constraints does Student I ignore? How do we help students develop the ability to
identify and work with constraints?
Student I
Grade 4 – 2008
45
Fourth Grade
4th Grade
Student Task
Core Idea 2
Number
Operations
Task 3
Winning Lines
Work with a “magic square” type number game.
Understand meanings of operations and how they relate to each
other, make reasonable estimates, and compute fluently.
• Develop fluency in multiplying whole numbers.
• Identify constraints: total a given number, mark winning lines with a line, don’t use zero,
don’t use a number more than twice
• Addition, working backwards from a total to addends
• Explaining why some outcome is impossible in relationship to the given constraints, making
a justification
Based on teacher observations, this is what fourth graders know and are able to do:
• Add numbers
• Number sense, find winning combinations
• Skipping the first problem
• Identifying or interpreting the constraint “a number can’t be used more than twice”
• Adding in extra constraints: numbers must be 9 or smaller, a number can only be used once,
numbers can only be put in the diagram if the outcome will make a winning combination
Strategies used by successful students:
• Making a list of winning combinations
• Listing numbers and marking them off after being used twice
• Checking both columns and rows for winning combinations
Grade 4 – 2008
46
The maximum score available for this task is 7 points.
The minimum score for a level 3 response, meeting standards, is 4 points.
Most students, 88%, could either draw in 1 or 2 missing lines in part 1 (numbers given) or they
could explain why using 8 was not possible to make a winning line in part 3. Many students, 67%,
could put in numbers to make 3 winning lines in part 2 and draw the lines to show winners and
explain why 8 could not be used. More than half the students, 55%, could also make 4 winning
lines in part 3. Almost 13% could meet all the demands of the task including finding all winning
combinations in part 1. Almost 12% of the students scored no points on this task. All of the students
in the sample with this score attempted the task.
Grade 4 – 2008
47
Winning Lines
Points
Understandings
All the students in the sample
0
task.
1
3
4
Students could either draw in 1
or 2 missing lines in part 1
(numbers given) or they could
explain why using 8 was not
possible to make a winning line
in part 3.
Students could find and draw a
line through 3 winning rows in
part 2 and explain why 8 is not a
possible combination in part 3.
Students could find all solutions
for part 2 and 3 and explain why
the 8 can’t be used.
Students did not notice or understand the
directions for part 1 or had difficulty
explaining part 3. 6% said 8 is a really
high number. 5% said you would need to
add a zero to get 9. 3% said there are only
3 squares. 5% did not attempt to give an
explanation.
In part 1, 28% left it blank. 4% omitted the
left column and 4% only drew a line
through the left column. 3% omitted the
diagonal.
19% only put two combinations in part 2.
Perhaps they counted the sample row. 8%
only made one winning line. 7% made
addition errors in part 2. 5% left part 2
blank.
Students did not solve part 1 correctly.
Students did not find 4 solutions for part 3.
16% only found 3 solutions. 14% only
found 2 solutions. 9% only found 1
solution. 5% left the final square blank.
6
7
Misunderstandings
Students could add numbers to
give a target total. They could
find a variety of solutions to
make the same target number.
Students were able to work with
more than one constraint to find
their solutions. They could
justify why 8 could not be part
of a winning combination in 3
using the constraints as part of
their argument.
Grade 4 – 2008
48
Students need to have practice working with addition in problem-solving settings. Students should be able
to find rows or columns of numbers that add to a given total or find three numbers to make a given total. At
this grade level simple addition facts such as these should be automatic or easily recalled.
Students need opportunities to work problems with a variety of constraints and longer chains of
reason. Students too frequently are only given tasks where a solution is quick. They need
challenges that test their perseverance and build an attitude that effort is an important quality to
being successful in mathematics.
Ideas for Action Research – Reading Professional Literature and Experimenting
with the Ideas in the Classroom
Take some time to read professional literature on learning and motivation with your colleagues.
Then try to design an activity or series of activities to test these ideas in your classroom. In Lessons
Learned from Research published by NCTM, there is an article on motivation and the link to success
in mathematics. (“Finding from Research on Motivation in Mathematics Education: What Matters in
Coming to Value Mathematics”)
“Five factors influence motivation:
1. Motivations are learned.
2. Motivation hinges on students’ interpretations of their successes and failures
3. Intrinsic motivation is better than engagement for reward
4. Inequities are influenced by how different groups are taught
5. Teachers matter.”
“Because of cultural attitudes about what a good learner is, we educators begin to identify which
children are “quick” at mathematics and which are “slow” while the children are very young, as
early as kindergarten or first grade. Grouping practices, which exclusively segregate the faster
learners from the slower, reinforce these attitudes. By the time they reach the middle grades,
students begin to perceive mathematics as a special subject area in which fast students succeed.”
“Researchers’ best estimates are that a person needs to be successful about 70% of the time for
continued engagement to be considered both challenging enough to warrant effort and easy enough
for the student to believe that he or she will experience success.”
“Research shows that when students attribute their successes to ability mediated by effort, they tend
to succeed and that when they attribute their failures to lack of ability regardless of effort, they tend
to fail. Moreover when students conceive of ability as fluid, or subject to improvement through
effort, they tend to work even harder in mathematics and thus, are better achievers that students who
believe that ability is fixed.”
These excerpts give some incites into the importance of stressing effort in the classroom as a reason
for success. Researchers out of the Dana Center in Texas are using programs with middle schoolers
to show them brain research on how effort increases their ability. They are using this research in
lessons to help students prepare for success in algebra in their computer-based program Agile
Minds.
Grade 4 – 2008
49
Another important piece of research literature is the paper, “Inside the Black Box”, by Wilam and
Black. This article looks at the importance in the type of feedback that students receive. The article
talks about feedback being one of the key factors to students success. Students need to have specific
feedback about their work that allows them to see where they are on a learning continuum, where
they need to improve, and goals to work towards. This type of specific feedback regarding specific
aspects of the work is more helpful than grading systems, which in general are just a form of
ranking. Again, good feedback helps students to identify where to apply their effort.
After reading these articles, or similar articles, discuss with your colleagues specific activities to try
with students to help reward effort. Then compare results after everyone has had an opportunity to
try them in their own classrooms. Try to gather specific information about particular students and
how their work or effort improves over the course of the activities and the feedback design program.
During the year, continue to revisit this issue of motivation and ways to improve student feedback.
Grade 4 – 2008
50

part 1 - Noyce Foundation

Transcription

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