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Grade Two – 2006 pg.(c) Noyce Foundation 2006. To reproduce this document, permission must be granted by the Noyce Foundation:

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Second Grade Mars 2006 Task Descriptions

Overview of Exam

Core Idea Task

Data Analysis Our Class Graphs

The task asks students to read, interpret and answer a prediction question related to

two graphs. Successful students can find numerical data and compare values. They

can make a prediction as to which of two data sets might change if collected the

following day and defend their choice.

Number Operations In One Minute

The task asks students to find values on a chart and compare to find their differences.

Successful students consider the relationships between the quantities and may use

addition, subtraction, counting on or counting back to find the answers. They can use

regrouping for addition or subtraction with accuracy. Answers are proven using

words, numbers or pictures.

Geometry Describing Shapes

The task asks students to identify the attributes of two-dimensional shapes. Students

who are successful can identify the correct number of sides and corners of a given

shape as well as compare the lengths of the sides on a shape. They can name common

geometric shapes.

Number Operations Pam’s Shopping Trip

The task asks students to calculate buying caps and balls for a baseball team.

Successful students use repeated subtraction, equal sharing and forming equal groups

to find the price for one unit. They can use repeated addition or counting by multiples

to find the price for several items. Student use previously calculated information to

find the total cost of equipment for four players.

Measurement The Track Team

The task asks students to find the number of jumps two students will make from a

start to a finish line. Successful students find the total number of jumps by iterating

each jumping unit. Students compare units and predict whether the measures will be

greater or smaller when a different unit is used. Successful students can identify the

person to finish first and explain their thinking.


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Overall Results for Second Grade

Total MARS raw scores is the summation of Tasks 1 through 5 on the MARS test.


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MARS Test Performance Level Frequency Distribution Table and DescriptiveStatistics

2006 – Number of Students tested in 2nd

grade: 5982


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Our Class Graphs

On Monday, the students in Mrs. Gong’s class answered twoquestions and displayed the information in these two graphs.

1. How many children have black hair? ____________

2. Which color of hair do the fewest number of students have?

________________


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3. How many more students come by bus than by walking? _____.Show how you know your answer is correct.

On Tuesday, Mrs. Gong asked the same two questions.

4. If they made new graphs of the information on Tuesday, whichgraph might look different?

Circle one: What color is your hair?

How do you come to school?

Explain how you know your answer is correct.

__________________________________________________________

__________________________________________________________

__________________________________________________________

__________________________________________________________


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Mathematics Assessment CollaborativePerformance Assessment Rubric Grade 2

Our Class Graphs: Grade 2: Points Section

Points

The core elements of the performance required by this task

are:

• Represent and interpret data using graphs or other

representation

• Describe and compare data using qualitative and

quantitative measures.

• Communicate reasoning using words, numbers or

pictures.

Based on these credit for specific aspects of performance

should be assigned as follows

1 5 1

1

2 Red, accept 2 1

1

3 5

shows work such as 7 - 2

1

1 2

4 Circles How do you come to school?

Explains answer such as: a person’s hair color won’t change

but they may come by a different way to school tomorrow

Alternative Solution:

Circles What color is your hair?

Explains answer such as: a student might have changed the

color of his/her hair

1

2

or

1

2 3

Total Points 7 7


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2nd Grade – Task 1: Our Class Graphs

Work the task and examine the rubric.

What do you think are the key mathematics the task is trying to assess?_____________________

______________________________________________________________________________

______________________________________________________________________________

_____________________________________________________________________________

Look at student work for the first graph, parts 1 and 2:

Can students find information in the graph?

• Can students read and answer the questions in the first graph?

Sort out student work that contains errors.

• Are there similar types of errors?

• What might have led to the errors?

• What types of clarifications around the conventions of graphing might lead to more

success in reading displays and in recognizing graphical conventions?

Look at student work for the 2nd

graph, part 3:

Can students find information between the data? Can they compare quantities? Do they use

operations to combine? Do they use other methods? What strategies did successful students

use?

Subtraction

7 – 2 = 5 or

7 – 5 = 2

Addition

2 + ? = 7 or

2 + 5 = 7

Direct

comparison

of columns

Counting up

or

counting back

Other

Sort through student work that contains errors:

• Are their similar types of errors?

• What do they understand about comparing two values?

• What questions or clarifications might facilitate a more successful answer?

Look at student work for the prediction question, part 4:

Can students make predictions from the data? Can students reason through their prediction?

• Is the student prediction supported by at least one reason why one of the two graphs

might be different on Tuesday?

• Is the reasoning incomplete or idiosyncratic?

• What types of discussions, during similar in-class graphing experiences, might facilitate

students’ ability to make and support predictions?


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Looking at student work on Our Class Graphs:Second grade students did well on this task. Students had little or no trouble pulling information from the

two graphs provided. Occasionally, students erred when reading the names of the categories but were

able to find the values and provide them. Students had a variety of strategies for answering the

comparison question and unique explanations as to why one graph might be different on the following

day. Student A shows the comparison by subtracting to find the difference between the two values. In

predicting which graph might be different the following day, Student A shares a valid reason why a

student might change the way he comes to school.

Student A


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Student B uses a known addition fact to find the difference between those riding the bus and those

walking to school. In providing a rational why the How do you come to school graph might be different

the next day, Student B knows that lateness can be a factor in choosing a mode of transportation.

Student B

Student C shows two strategies for solving to find a difference thoroughly proving that the

subtraction problem is correct. The practical explanations given by Students C and D reflect

realities we all face in getting to work or to school each day.

Student C


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Student D

The work of Student E not only provides an explanation for why the school graph might be

different on Tuesday, it also addresses a reason for why the hair color graph would be more

likely to remain the same.

Student E

It is, of course, possible to provide reasoning in support of the hair color graph changing the

following day. Student F gives a logical reason as to why this might happen. Full credit was

given to logical reasoning as to why this graph might change.

Student F


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Students G and H compared two values from the chart below. Student G’s work is

representative of the work that was not given credit because it did not answer what was counted

to get a correct answer of 5. Look at the work of Student H. What might have precipitated using

the values of 8 and -1? What question(s) might you ask to clarify the thinking?

Student G

Student H

For many students, approximately 46%, finding a reason why one of the graphs might be

different was very difficult. Students I though L are examples of some of the incomplete

thoughts that second graders had around this prediction. It is important to expose our students to

questions such as these in order to show usefulness in data collection and analysis short of just

pulling data off of a graph. Many students described the graphs or wrote more about what they

saw in the graphs. Some students gave personal information rather than predicting. Student I

feels that the ways we come to school graph might be different because it is more important than

your hair color.

Student I


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Student J would make the How do you come to school graph different by rewording the title on

Tuesday. Student K knows that the same questions asked on Tuesday but put in a new graph

will just look different.

Student J

Student K


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The work of Student L is an example of the kinds of personal statements used to answer part 4.

With good questioning and prediction opportunities, students can begin to think about issues that

influence data rather than just participating in building graphs. As students share predictions and

conjectures with each other, students can have access to all parts of data analysis.

Student L


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Second Grade

2nd grade Task 1 Our Class Graphs

Student

Task

Read, interpret and answer questions around two class graphs. Answer a

prediction question related to the two graphs. Make a prediction as to

which of two data sets might change if collected the following day and

defend the choice.

Core Idea

5

Data

Analysis

Students collect, organize, display, and interpret data about

themselves and their surroundings

•• Interpret, describe and compare data using quantitative and

qualitative measures

•• Communicate reasoning using words, numbers, or pictures

Core Idea

2

Number

Operations

Understand the meanings of operations and how they relate to each

other, make reasonable estimates, and compute fluently

•• Demonstrate fluency in subtracting whole numbers

•• Communicate reasoning using pictures, numbers and/or words.

Mathematics of the task:

•• Ability to interpret bar graphs

•• Ability to make a comparison

•• Ability to make predictions from data

Based on teacher observation, this is what second graders knew and were able to do:

Find a difference by counting the lines, by subtraction or by addition

Read and interpret the bar graph

Write answers numerically

Areas of difficulty for second graders:

Explaining why the graph would change on Tuesday (very difficult concept for 2nd

grade)

Comparison means finding the difference

Expressing their reasoning in words

Didn’t understand that they were being asked to remake the same graph on a different

day

Strategies used by successful students:

•• Using subtraction, addition, matching or counting on to find the difference between two

values on the graph

•• Drawing a line and counting the number of values to find the difference

•• Reasoning that a different day might include a different way to school and that might

change the values in the graph

•• Reasoning that a child could dye or color hair and have the values of the graph change


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Frequency Distribution for Task 1 – Grade 2 – Our Class Graphs

Our Class GraphsMean: 4.60 StdDev: 1.89

Score: 0 1 2 3 4 5 6 7

Student Count 80 203 649 890 1063 1061 481 1555

% < = 1.3% 4.7% 15.6% 30.5% 48.2% 66.0% 74.0% 100.0%

% > = 100.0% 98.7% 95.3% 84.4% 69.5% 51.8% 34.0% 26.0%

There is a maximum of 7 points for this task.

The cut score for a level 3 response, meeting standards, is 3 points.

Most of the students, a little over 84%, were able to read the data and pull information from the

graphical displays. Approximately 70% of the students had successful methods of comparing

quantities in the data. 26% of all students could meet the challenging demands of this task

including predicting future similar events and giving a valid reason for their prediction. All

students attempted to solve this problem.


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Our Class Graphs

Points Understandings Misunderstandings

0 All students attempted to solve this

problem.

2% of the students scored a zero on

this task. Students were unable to

find the value for the number of

students with black hair, to name

the category with the fewest data

points or to show a comparison

between any two values on the

ways we get to school graph.

1 - 2 Most students could find the value

for the number of students with

black hair as well as find the

category with the fewest data points.

Students struggled to find the size

difference between two values on a

graph. Most named 7 as “how

many more”. Students scoring a

one or a two could not predict and

defend what graph might be

different if the data was collected

the next day.

3 - 4 Some students were able to compare

the two values in the “How we get to

school” graph by using subtraction,

addition, matching or by describing

what they did, i.e. “I counted the

lines between 7 and 2”.

Some of these students struggled

with the comparison question.

Frequent incorrect answers were 7,

9, and 72. Although some of the

students could predict which graph

might be different on Tuesday, they

were unclear or incomplete in their

evidence.

5 - 6 Most students compared the two

values using addition, subtraction or

by counting up or down between

values. About one- half of all

students could make and defend a

prediction for which graph might be

different if data were collected on

the next day.

Several students gave “9” as the

answer to the number of students

with black hair. In explaining how

they got an answer of 5 for the

comparison question, a few students

said that they counted or looked but

did not fully explain what they

looked at or how they counted.

7 Predicting and defending a choice to

which graph might be different was

difficult for all students. 26% of the

students were able to choose one or

the other graph and provide a valid

reason for their decision. In some

instances (about 10% of these

students), they explained both sides

of the prediction – why one might

change and why the other would not.


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Implications for Instruction:

Mathematicians and educators stress the importance of incorporating data analysis and statistics

into the elementary curriculum to prepare our students to be involve and informed participants in

society. Data and statistics is a real application of mathematics. As such, it affords many

opportunities to use the strategies, reasoning, and skills they are learning around counting,

measurement, number patterns and other school subjects.

Primary students can describe, organize, represent and analyze data. Although their work may

look different than that of upper grade students and adults, many of the processes are similar.

Research has found that the majority of students in elementary schools can read displays

accurately and know graphical conventions. This is also true of the students in our collaborative.

It is often in reading “beyond the data” (analyzing the data) that students in the early grades

struggle.

A study, “Young Students’ Informal Statistical Knowledge” presented in Teaching Statistics:

1999 Putt, etal, notes that 80% of 1st

and 2nd

graders gave idiosyncratic or incomplete responses

when attempting to analyze data from a line plot or a bar graph. This particular task, Our Class

Graphs, was written to look deeply at how students reason when analyzing data. Can they

recognize patterns and trends in the data? Can they make inferences and predictions from the

data when it is not explicitly implied in the representation? Can they support their predictions

with reasoning?

Central to all data and statistics activities should be dialogue and discussion. Discussions might

need to differentiate between the processes involved in collecting, organizing, and representing

data or to highlight the more efficient or readable displays. Graphical conventions need to be

made explicit and accessible to all students. Collecting and organizing is fun and exciting but a

significant amount of time should be devoted to reflection around the meaning of the data.

Students must have the opportunity to grapple with interpreting the results and thinking about

whether or not the results would remain the same over time or over different samples or different

populations.

Action Research Idea

Describing, Analyzing and Predicting Data:

Using a data question that you have used in the past such as: How do you get to school? What is

your favorite ice cream flavor? How many teeth have you lost?

Have students describe the data:

•• What information does this graph give us?

•• Give the students a false statement about the graph. Ask: Does this statement reflect the

information in our graph? Why or why not?

•• What could a visitor to our room learn from this graph?


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Have students predict and give reasons for predictions:

•• What if we asked each of you the same question tomorrow (or next month)? Would it be

the same or different? Why?

•• What if we asked the same question in another 2nd grade classroom? Would the results

be the same or different? Why?

•• What if we asked the same question of third graders? Fifth graders? Parents?

Continue to pose these kinds of questions as you graph throughout the year. Note how

students’ ability to evaluate and predict changes and grows with time.


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In One Minute

Lydia, Andres and Jim were playing “In One Minute”. First theyfound out how many numbers they could write in one minute.

Here’s what they could do.

Numbers

Lydia 67

Andres 49

Jim 51

1. Who wrote the most numbers? _______________

2. How many more numbers did Lydia write than Andres? _____

Show how you know your answer is correct.


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Next, they found out how many letters they could write in oneminute. Here’s what Lydia and Andres could do.

Letters

Lydia 46

Andres 84

Jim

3. Who wrote the most letters? _________________

4. Jim said to Andres, “I wrote 8 fewer letters than you did. How

many letters did Jim write? __________


5. How many more letters did Andres write than Lydia? ______



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Mathematics Assessment Collaborative

Performance Assessment Rubric Grade 2

In One Minute: Grade 2: Points Section

Points

The core elements of the performance required by this task are:

• Demonstrate fluency in adding and subtracting whole

numbers.

• Understand whole numbers and use them in flexible ways

such as relating, composing, and decomposing numbers.

• Understand the relative magnitude of whole numbers and

the concepts of quantity and the relative position of

numbers.

• Demonstrate an understanding of the base-ten number

system and place value concepts.

• Communicate reasoning using pictures, numbers and/or

words.Based on these credit for specific aspects of performance should be assigned as follows

1 Lydia 1

1

2 18

Shows work such as: 67 - 49

2

1

3

3 Andres 1

1

4 76

Shows work such as 84 – 8

1

1

2

5 38

Shows work such as 84 – 46

2

1 3

Total Points 10


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2nd

grade – Task 2: In One Minute


What do you think are the key mathematics the task is trying to assess? ______________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

Look at the student work for part 2, part 4, and part 5 and chart student answers below:

Part 2

18 * 17 or 19 49 or 67 22 116 Other

Part 4

76 * 75 or 77 84 or 8 92 0 Other

Part 5

38 * 37 or 39 46 or 84 42 130 Other

• What methods do they use to solve these problems?

o Do they count every number?

o Do they count by twos or fives? By tens?

o Do they count on or count back?

o Do they take numbers apart and recombine them with other parts of numbers?

Are they efficient? Are they successful?

o Do they try to make tens?

o What other strategies do they use to determine the answers?

• If there were errors in executing a strategy, where did the error occur?

o What are they able to do?

o How can you build success with their current strategy?

• Do they use the same method for all three problems?


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Looking at student work on In One Minute:

Student A

Students A, B, and C show strong understanding of place value and have fluency when adding

and subtracting two-digit numbers. Student A has taken apart and recombined the numbers to

make subtraction easier. In part 2, she kept the 67 whole but took the 49 apart into 40 and 9.

She then subtracted the 40 from the 67. Why was subtracting 2 from the remaining 9 such an

efficient strategy? A similar process was used to solve part 5. Why does she subtract 2 from the

6? In part 4, why partition 8 into 4 and 4? What about this strategy makes the subtraction

easier?


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Student B

Student B finds how much more by adding and subtracting the numbers in between. In part 2,

Student B subtracts 10, 7 and then 1. Why did the student choose those particular numbers? In

part 5, Student B uses a similar strategy with addition. What about those numbers facilitates a

correct answer? Both Student A and Student B use methods of adding and subtracting around

10s and do so with flexibility. What does Student B understand about place value and the base-

ten structure of our number system?

Student C

Student C shows us how he found the missing part between 84 and 46. Here is the solution

strategy for part 5. He used a similar method for parts 2 and 5. Notice how this student uses

what they know about ones addition facts to work with tens. The student knows that the 4 is a -4

and uses it to get from 50 to 46.


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Students D, E, and F count on to find “how many more” by finding the distance between the two

numbers. Student D is adding on from 46 to 86 using tens. 84 is where she wants to go and so

she subtracts 2 back from the 86. She errs in counting the number of tens between 46 and 86.

Student E counts on by ones from the smallest to the largest number. He successfully counts the

number of jumps from one number to the next, even though his numbering system may not

appear as such.

Student D

Student E

Student F attempts to use an open number line to quantify the distance between to find how

many more. The student is successful in finding friendly jumping spaces from 46 to 50, 50 to

70, 70 to 80 and 80 to 84. What error caused the incorrect answer of 28?

Student F


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Student G

Student G uses the standard subtraction algorithm to find “how many more”. Renaming the 67

into 50 and 17, the student subtracts nine circles from the 17 circles drawn.

Student H

In calculating 67 – 49 using the standard subtraction algorithm, most students with incorrect

answers gave a response of 22: 60 – 40 = 20 and 7 – 9 is 2. In using an algorithm, Student H

can see nothing inherently wrong with the incorrect answer as she is following steps and

procedures. Little sense-making is happening while finding the difference.

Student I

Students I and J used words, rather than numbers or pictures, to tell how they found “how many

more”. They had varying degrees of success. Student I explained how he counted to 84 and then

subtracted 46 to get the answer. Following this procedure correctly would result in an answer of

38. Student J not only gave an incorrect response but it isn’t clear where the “50” came from.

All students benefit from being asked to explain their reasoning and to prove how they know the

answer is correct.

Student I


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Student J

Student K

Many students used a direct modeling strategy to access this problem. Student K draws all 84

circles and crosses out 8 to find the number of letters written by Jim. This student has

successfully drawn and counted to find the correct answer. What similar strategy might be more

efficient?

Student L

Student L uses tally marks, in groups of tens, to compare Lydia’s score to that of Andres. Each

score is represented correctly, 67 and 49. What error caused the incorrect answer of 27? What

question might help this student reflect on this error?


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Student M

Whether counting on or counting back, this student has found a way to find the difference

between 67 and 49 using tallies. Could a system such as this give Student K a less cumbersome

way to find differences?

Student N

In approximately 5% of the student work, the differences were calculated correctly as Student N

has done below, however, the scores compared were not for those asked.

Student O

Student O’s paper is an example of 1% of the answers for number 5. Did the word “more”

trigger a response to add both scores together? Fortunately, only a very small percentage of all

student work reflected this misunderstanding.

Student P

Student P adds all the digits together and gets an answer without making sense of the question or

the answer.


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2nd grade Task 2 In One Minute

Student

Task

Find values on a chart and compare to find their differences. Provide

evidence of the reasoning behind the answers using words, numbers or

pictures.

Core Idea

2

Number

Operations

Understand the meanings of operations and how they relate to each

other, make reasonable estimates, and compute fluently

•• Demonstrate fluency in adding and subtracting whole numbers


Core Idea

1

Number

Properties

Understand numbers, ways of representing numbers, relationships

among numbers, and number systems

•• Understand whole numbers and use them in flexible ways such as

relating, composing, and decomposing numbers



•• Ability to find a difference between two one- or two-digit numbers

•• Ability to add or subtract double digit numbers

•• Ability to communicate reasoning around answers


Read the table, find the most number of letters or numbers & record

Had many strategies for finding differences: counting by tally marks, counting

up/down from the higher/lower number, adding or subtracting, using number lines

Choose the biggest number out of a group and complete a double digit subtraction

problem to find the difference between the two numbers


Knowing that “How many more than” means to subtract or add up to find the difference

in some way

Always subtracting the smaller number from the larger number when using the standard

subtraction algorithm

Some students couldn’t determine which numbers to subtract or to use when finding the

difference

The blank area of the second table was confusing for them. They thought it meant Jim

hadn’t really written any letters.


•• Finding the difference by subtraction, addition, finding the missing addend, counting up

or back

•• Using a number line, open number line, 100s charts or friendly numbers

•• Checking answers by using the inverse operation

•• Showing two ways to prove an answer, i.e. subtraction and tallies of difference


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Frequency Distribution for Task 2 – Grade 2 – In One Minute

In One MinuteMean: 6.32 StdDev: 3.08

Score: 0 1 2 3 4 5 6 7 8 9 10

StudentCount 81 147 834 387 518 479 455 512 634 361 1574

% < = 1.4% 3.8% 17.8% 24.2% 32.9% 40.9% 48.5% 57.1% 67.7% 73.7% 100.0%

% > = 100.0% 98.6% 96.2% 82.2% 75.8% 67.1% 59.1% 51.5% 42.9% 32.3% 26.3%



67% of the students met the essential demands of the task: finding which child had the “most”,

showing “how many more than” and providing proof of the answer. 97% of the students could

identify the person who had written the most letters and numbers. 65% of the students had a

successful strategy for finding one of the three subtraction problems. Less than 2% of the

students scored no points on this task.


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In One MinutePoints Understandings Misunderstandings

0 Less than 2% of all 2nd

graders had little or

no access to this task.

1 – 2 Students were able to identify the

largest value in both charts.

In finding the difference between two

values, students added the two together and

gave the sum or named one or the other

value as the amount that was “more than”

the other amount.

3 – 4 Students could identify the largest value

in both charts and subtract to find Jim’s

value or find one of the two differences.

Most successful students used counting

up or back by 10s and extras, writing

tally marks for each number between

two values or by finding the missing

addend. A few of these students (15%)

were successful with the standard

algorithm.

In attempting to find the difference

between two values, students had difficulty

in correctly counting the numbers between

two numbers (counted one more or one

less than the answer), in correctly

subtracting 67 – 49 (answer of 22), in

choosing a correct method of comparison

(67 + 49), or in subtracting using a

standard algorithm:

84 84 84

– 46 -46 -46

40 42 43

5 Most students successful in finding the

difference between two values used

counting up or back strategies, used

tallies to show the difference between

the two values, or counted to friendly

numbers (those with zeros) and then

counted the extras.

In finding the difference between two

numbers, some students erred by

incorrectly writing the size of their jumps,

i.e. 49 to 55 = 5, 55 to 65 = 5, 65 to 67 = 2.

Frequently students set up the standard

algorithm incorrectly or subtracted

incorrectly:

46 84 67

– 84 -46 -49

42 42 22

6 – 7 Successful strategies for finding the

differences continued as stated before.

Some students used circles or Xes

instead of tally marks. Other students

used addition but used partial sums that

were familiar such as: 49 + 1 = 50,

50 + 17 = 67.

Mistakes in finding differences surfaced

due to incorrectly subtracting using the

standard algorithm, in choosing the wrong

two values to compare and in adding both

values together to find the difference.

8 - 9 Students had many strategies for

successfully finding the difference

between 67 and 49. Students were able

to subtract 8 from 84 and obtain the

answer of 76. Many of these students

set up the 84 to 46 comparison problem

correctly.

Most students struggled with finding the

difference between 84 and 46. In doing so,

most erred in following the steps of the

traditional subtraction algorithm similar to

the ways listed earlier.

10 Students had a variety of flexible

strategies for successfully finding

differences between values.


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Implications for Instruction:How do students develop efficient strategies for adding and subtracting?

Elementary students naturally determine “how many” by counting – counting all, counting on, or

counting back. The problem is not that they count but that they don’t move beyond this strategy.

Our job is to provide opportunities for student to see and use number relationships to solve

problems. Students won’t move to more sophisticated ways of solving problems until they have

a strong foundation in the relationships within the numbers to 10 - doubles, near doubles, and

combinations to 10. These number relationships can then be applied to larger numbers (see

Student C). Our job is not to teach strategies but to provide situations in which students have

opportunities see various ways of solving problems and to recognize a value of a strategy they

choose for themselves.

How can we present opportunities for students to construct understanding of place value?

A very typical place value activity in primary curricula is to present a number, say 96, or a

number of base-ten blocks, say 9 ten sticks and 6 loose ones, and ask: How many tens? How

many ones? To answer this requires no real understanding of place value and provides little

understanding that might be useful in problem solving. Real knowledge of place value develops

slowly over time and with varied experiences.

Place value involves building connections between key ideas of place value – quantifying sets of

objects by grouping tens and treating the groups as units and using the written notation to capture

this information. Building connections between these representations and quantities yields an

understanding of place value.

Developing procedures for the addition and subtraction of multi-digit numbers should evolve

from the students understanding of place value. When given a problem such as 92 - 37, students

may wish to add to find the difference rather than subtract – “How can I find the difference

between 37 and 92? Well, 37 + 3 = 40, 40 + 50 = 90 and two more is 92. 3 + 50 + 2 = 55.”

Although this method is different from the standard addition or subtraction algorithm, it is no

less efficient, gives the correct answer and shows a great deal of number sense built on place

value. The beauty and simplicity of the base-ten number system is that it can lead to real

mathematical power as well as their ability to see mathematics a sense-making activity.

What would our instruction look like if we were diligent in helping all students make sense of

mathematics?

Classroom interactions should focus on the student methods for solving problems. The focus of

these interactions should be to improve the methods – make them more powerful and more

efficient. This places the attention on student understanding.

Our goal is to help students recognize the need to understand the methods they use as well as to

present as well as explain their method so that others will understand. “I can claim to understand

a method if I can help others understand it.”(Heibert, et all 1997 pg. 47)

Working to improve methods for solving problems means moving from methods that do not

work as well, and that may even be flawed, toward ones that work better. Mistakes are simply

methods that need to be improved. Offering up student methods for class discussion focuses


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the correctness of the methods and solutions to the logic of the mathematics rather than to the

teacher or to another student.

Ideas for Action Research

Addition and Subtraction Solution Strategies:

Most children pass through three stages in acquiring addition and subtraction skills: interpreting

problems with manipulative materials, using counting strategies, and finally relying on number

facts. These transitions do not happen all at once, and for a time children may move back and

forth depending on the difficulty of the problem and may sometimes use more than one strategy

at once. In order to determine what further work the children need with addition and subtraction,

we must observe them while they solve problems and listen to what they know and view their

written work to determine where their thinking might work better.

Action Research:

Use these two problems taken from Young Mathematcians At Work: Constructing Number

Sense, Addition and Subtraction, Fosnet and Dolk, Heinemann 2001.

Ms. DiBrienza is reading a book with Ginny that has 107 pages. She is on page 64. How many

more pages does she need to read?

Ginny is reading the same book but she is only on page 43. How many more pages does Ginny

need to read to finish the book?

Observe students while they work: How are they making sense of the problems? Do they use

manipulatives? Do they use a drawing? Do they use numbers alone? Do they work alone or in

pairs? Do they think about whether the answers make sense? What questions can you ask to

help them become more efficient in their solution methods?

Observe students’ work on paper: What successful strategies do students use? Did they use a

drawing? Are they using a counting strategy? Does their strategy reflect knowledge of number

facts and/or place value? What questions will you ask students to clarify their work or encourage

them to reflect on their method of solution?

Observe the students as they share their solutions: Can they explain their strategies so that others

can understand? How are the class strategies similar? How are they different? What questions

would you ask the student sharing or the listening students to build understanding of the

mathematics?


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Describing Shapes

1. Look at the shape on the geoboard below. Read the sentences andcircle whether they are true or false.

a. This shape is called a trapezoid. true false

b. This shape has 6 sides. true false

c. This shape has 4 corners. true false

d. Two sides of this shape are the same length. true false

2. Look at the shape on the geoboard below. Circle whether thestatements are true or false

a. This shape has 6 sides. true false

b. You could draw a line of symmetry. true false

c. All sides of this shape are the same length. true false

d. This shape is called a ____________ .


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3. Look at the shape on the geoboard below. Write three truesentences that describe this shape.

true

true

true


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Mathematics Assessment Collaborative

Performance Assessment Rubric Grade 2

Describing Shapes: Grade 2: Points Section

Points

The core elements of the performance required by this task are:

• Describe and classify two-dimensional shapes according

to their attributes and/or parts of their shapes.

• Develop an understanding of how shapes can be put

together or taken apart to form other shapes

• Communicate reasoning using pictures, numbers and/or

words.Based on these credit for specific aspects of performance should be assigned as follows

1 a) true

b) false

c) true

d) true

all four correct

2 -3 correct

2

(1) 2

2 a) true

b) true

c) false

d) hexagon

all four correct

2-3 correct

2

(1) 2

3 Gives three correct statements such as:

Four sides, four corners, can draw a line of symmetry, rectangle,

two sides match in length

3 x 1

3

Total Points 7


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2nd

Grade – Task 3 – Describing Shapes


What do you think are the key mathematics the task is trying to assess?______________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________

Look at student work for part 1 and part 2:

1: Were students able to correctly identify the attributes of the trapezoid (b,c,d)?

Attribute Understood Misunderstood

Number of sides

Number of corners

Side length

2: Were students able to correctly identify the attributes of the hexagon (a,b,c)?

Attribute Understood Misunderstood

Number of sides

Line of symmetry

Side length

• In each case, what might have led to the misunderstanding?

• What might the student have been thinking?

• What discussions might facilitate getting them to confront these misunderstandings?

• What other examples and non-examples might clarify the attributes?


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Look at the student work for part 3:

How many of your students were able to give true statements about the attributes of a rectangle?

Gave statements about the attributes of a

rectangle

Named the shape as a rectangle

1.

2.

Used statements about appearance such as:

1. “looks like a ______”

2. “fat”, “thin”, “wide”, “long”

3. “rectangle only sideways 3.

Other

Surprises?


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Looking at student work on Describing Shapes:

Student A was able to identify three attributes for each of the three shapes and correctly identify

the trapezoid and name the hexagon. This work reflects understanding in what each shape looks

like, the features that distinguish it and ways to describe shapes. It was a welcome surprise to

have this second grader correctly identified the rectangle as a parallelogram.

Student A

Student B identifies and describes the attributes of the three shapes actually noting four correct

attributes of the rectangle including one that relies on visualization (“looks like a table”).


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Student B

While naming shapes is important, even more important is observing the attributes or

characteristics of shapes such as the number of sides or number of corners, how they can be put

together or taken apart or other examples or non-examples of a particular shape. Student C

didn’t come up with the name for the hexagon but was able to re-describe the shape using one or

its attributes – the number of sides. Student D showed one way that the hexagon could be

subdivided into other shapes.


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Student C

Student D

Many students were still struggling with determining the length of the sides of each shape.

Student E could not find the sides with the same length in the trapezoid and incorrectly

identified all the hexagons side lengths as the same. If you were talking with this student about

these answers, what would you ask them to re-focus their thinking? What kinds of experiences

might this student need?

Student E


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Each shape name defines the figure. It carries the attributes with it. When students learn the

names of shapes without the attributes they end up with misconceptions. They are able to list

names but may over-generalize their identity to other shapes. Approximately half of those

students who were unable to name the hexagon named it by another shape’s name. Most

frequently named were diamond (see Student F), octagon, and trapezoid. Each of these shapes is

in some way similar to the appearance to this hexagon.

Student F

The appearance of the shape is a strong factor in students’ choices for description. Although

each of the following pieces of student work have used attributes to describe the rectangle, the

rotation of the shape is a focal point. Student G names 4 sides and 4 corners to the rectangle but

must say something about is being sideways. Student H also focuses on the sideway-ness of the

rectangle. Student H’s first statement is interesting. What about the picture might lead to this

misconception? Did you have students who gave attention or misunderstood the dots? What

about the presentation might have lead to this misconception? Student I is clear that this

rectangle is just “not the right way”. What kinds of experiences might you provide for this

student to open up the definition of rectangle?

Student G


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Student H

Student I

Over 85% of our students were able to meet the essential demands of this task: identify at least 2

attributes of each of the three shapes. Student J might have been able to due this as well but for a

fragile understanding around the attributes of these shapes. In describing the rectangle, Student J

names one characteristic of a rectangle – 4 sides, tells us what it is not – not a square, and relies

on descriptive language – “long” to identify this shape. Similar descriptive vocabulary was used

by other students in describing this rectangle – “big”, “wide”, “flat”.


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Student J

Less than 2% of our students scored a zero on this task. All of them at least attempted to do the

task. Many of them, like Student K, tried their best to make some sense of what they knew and

what they were being asked to do. Clearly Student K struggled to identify any of the attributes

of the three shapes. Where might the “six-ness” or “eight-ness” in part 3 have come from? How

would you open up a discussion with this student around his/her thinking? What experiences

might help the student make sense of the number of sides and the number of corners? The name

suggested by this student in part 2 could have been the name parallel(ogram) and incorrect and

the one suggested in part 3 could have been rectangle. In both instances, it makes one think

about how important it is to be able to describe and identify the attributes of a shape before

committing its name to memory.


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Student K


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2nd grade Task 3 Describing Shapes

Student

Task

Identify the attributes of two-dimensional shapes. Name common

geometric shapes. Give three attributes of a rectangle.

Core Idea 4

Geometry

and

Measurement

Students will recognize and use characteristics properties, and

relationships of two-and three-dimensional geometric shapes and

apply appropriate techniques to determine measurements

•• Identify and visualize two-dimensional shapes according to

their attributes and/or parts of their shapes


•• Ability to identify the attributes of common two-dimensional shapes.

•• Ability to recognize and name common two-dimensional shapes.


Most student could identify attributes of the shapes

Identify attributes of a trapezoid and a hexagon

Describe attributes of a rectangle

Could identify sides, corners, edges, and line of symmetry


Struggled with the names of the trapezoid and the hexagon

Identifying a rectangle as a square

Most students could give one attribute and many could give two attributes of a rectangle

Students struggled to identify the correct number of sides, corners, lengths of sides and

names of the trapezoid, hexagon, and the rectangle


•• Identified the correct number of sides, corners, lengths of sides and names of the

trapezoid and the hexagon

•• Listed at least two attributes of a rectangle

•• Did not rely on describing what the rectangle “looks like” but rather listed characteristics


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Frequency Distribution for Task 3 – Grade 2 – Describing Shapes

Describing ShapesMean: 5.20 StdDev: 1.54

Score: 0 1 2 3 4 5 6 7

Student Count 78 95 207 403 818 1392 1751 1238

% < = 1.3% 2.9% 6.4% 13.1% 26.8% 50.0% 79.3% 100.0%

% > = 100.0% 98.7% 97.1% 93.6% 86.9% 73.2% 50.0% 20.7%



Most students, 86.9% could identify at least 2 attributes of each of the three shapes. One half of

the students were able to correctly identify three attributes for each of the shapes and name two

of the three shapes. 20.7% of the students met all the demands of this task. 98% of the students

attempted the problem.


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Describing ShapesPoints Understandings Misunderstandings

0 - 1 These students could correctly identify

either the number of corners or the

number of sides on the trapezoid or the

hexagon.

These students were unable to identify

or name more than one attribute of a

trapezoid, hexagon or rectangle.

Approximately 2% of students did not

attempt this problem.

2 - 3 In addition to the understandings listed

above, students were able to identify the

number of sides and the number of

corners of two of the three shapes.

Most students could not identify the

trapezoid or the hexagon. Most of

these students called the hexagon by

another shape name. Some students

counted the number of dots on each of

two sides of the rectangle (6) as the

number of sides. Many students knew

that there was a “fourness” to the

attributes of a rectangle but were

unclear as to how to assign it. In

assigning attributes to the rectangle,

less than 5% of the students used

statements like “looks like a…” or “it

is the shape of wood”.

4 Students could identify the correct

number of sides, corners of all the

shapes and name the rectangle. The

majority of these students (83%) could

give two or three correct attributes of a

rectangle.

Students had difficulty in identifying

shapes with sides of equal length from

those with unequal length. 12% of the

students scoring a “4” could not name

the trapezoid. 23% could not name the

hexagon. In giving a name to the

hexagon, most students called it by

another shape name. Only 2% of these

students assigned a “looks like a…”

statement to the rectangle. 4% used

words like “bigger”, “flat”, “long” and

“wide”.

5 - 6 Most students were able to identify the

trapezoid and name the hexagon as well

as finding the number of sides, corners

and symmetry for both shapes. 98% of

the students scoring a 5 or a 6 could

name two or three attributes of a

rectangle.

The most prevalent challenge for

students was in identifying the length

of the sides of these shapes. If the

student was unable to identify the

hexagon by name, they were, again,

most likely to call it by another shapes

name with only 1% of those students

naming something that the hexagon

looked like.

7 83% of students met all the demands of

this task. These students correctly

identified the trapezoid, named the

hexagon and rectangle, and gave correct

answers with regard to the number of

sides, corners, lengths of sides,

symmetry or the shapes within the

shapes presented.


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Implications for Instruction:

Geometry is an important tool for understanding the world around us. It is also an essential part

of the study of mathematics – even in the primary grades. Geometry helps us navigate through

number, algebra and measurement as well as through other school subjects and in real world

applications.

It is noted that 4 year olds can identify circles accurately and squares fairly well. They are less

accurate at recognizing triangles – 60% and recognizing rectangles – 50% of the time. One goal

of the primary grades mathematics schooling is to move the K-2nd

grade students from their

current knowledge toward informal analysis of the component parts and the attributes of basic

shapes. Students can learn to distinguish shapes by moving from

Our job is to make it possible for students to learn which features will define a given shape.

Children’s awareness of shape and form will evolve over time when they have many informal

but important experiences in sorting and classifying, taking shapes apart into other shapes,

assembling shapes from other shapes and exploring shapes in a myriad of different locations in

space.

The material in many current textbooks gives examples that are only the “best” examples, i.e.

regular shapes. Non-examples of a given shape are seldom presented or discussed. Most

opportunities in these textbooks call for an identification of a shape from a picture but do not ask

why. Naming shapes is important but just as critical is knowing the properties of various shapes

and being able to distinguish from among the properties. Rather than simply learning the names

of basic shapes, children should discover:

•• How are shapes alike? How are they different?

•• Which shapes fit together? Which shapes leave spaces between?

•• What can we build with these shapes?

•• What other shapes can we make using these shapes?

What

they look

like

Features

that

distinguish

them

Ways to

describe

them

toward toward


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Idea for Action Research: Using Concept Cards

Develop a “concept card” (see example below) for the two- and three-dimensional shapes in your

curriculum. Be sure to include examples, non-examples, and undesignated figures that are not in

standard positions. To assess their understanding, duplicate the cards and ask each student to

describe the shape in writing on the duplicated card. Collect the work. Use their written work to

identify the shapes that each student recognizes and the shapes that each student can accurately

describe. Occasionally, provide student work for the class to discuss.

Engage the students in written activities and in discussion of the shapes at various times

throughout the year. You may want to use Guess My Rule activities as a variation. Keep a

record of each student’s progress.

Is it a Mer?Yes No

Where will these go?

What’s the rule?

Adapted from: “Concept Learning In Geometry”. Fuys and Liebov. Teaching Children

Mathematics. January 1997

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