Students were asked to read Counting on Frank by Rod Clement
and to write a letter to the author commenting on at least one example
of the mathematical claims made.
This sample of student work
was produced under the following conditions: 
 alone 
 in a group 
 in class 
 as homework 
 with teacher feedback 
with peer feedback 
timed 
 opportunity for revision 


Four students discussed the mathematics in depth in Counting on Frank
before they wrote their analyses. The students spent class time as well
as many recesses figuring out how to test the claims in the book. For example,
the students discussed how to use the classroom sink to test the claim in
the book about the bathroom filling up with water. The students worked together
to develop strategies for testing the claims about the peas and the ballpoint
pen. The teacher encouraged the students’ discussions, and provided
time and materials in school for them to work out their reasoning. The students
met outside of class to work together on the writing. Students completed
the writing individually, at home.
The students completed this activity near the end of the school year. Earlier
in the year, the students had studied area and perimeter concepts in depth,
as well as applications of multiplication and division in problem solving
activities.
This work sample illustrates
a standardsetting performance
for the following parts of the standards:

a 
Arithmetic and Number Concepts:
Add, subtract, multiply, and divide whole numbers. 
b 
Arithmetic and Number Concepts:
Demonstrate understanding of the base ten place value system
and use this knowledge to solve arithmetic tasks. 
c 
Arithmetic and Number Concepts: Estimate,
approximate, round off, use landmark numbers, or use exact
numbers, as appropriate, in calculations.

d 
Arithmetic and Number Concepts:
Describe and compare quantities by using concrete and real world
models of simple fractions. 
e 
Arithmetic and Number Concepts:
Describe and compare quantities by using simple decimals. 
f 
Arithmetic and Number Concepts:
Describe and compare quantities by using whole numbers up to
10,000. 
g 
Geometry and Measurement
Concepts: Use basic ways of estimating and measuring the size
of figures and objects in the real world. 
i 
Geometry and Measurement
Concepts: Select and use units for estimating and measuring
quantities. 
j 
Geometry and Measurement
Concepts: Carry out simple unit conversions. 
a 
Problem Solving and Reasoning:
Formulation. 
b 
Problem Solving and Reasoning:
Implementation. 

c 
Problem Solving and Reasoning:
Conclusion. 
a 
Mathematical Skills and Tools:
Add, subtract,
multiply, and divide whole numbers correctly. 
b 
Mathematical Skills and Tools:
Estimate numerically and spatially. 
c 
Mathematical Skills and Tools:
Measure accurately. 
d 
Mathematical Skills and Tools:
Compute time. 
f 
Mathematical Skills and Tools:
Use +, , x, ÷, /,…and . (decimal point) correctly
in number
sentences and expressions. 
h 
Mathematical Skills and Tools:
Use recall, mental computations, pencil and paper, measuring
devices, manipulatives, calculators and advice from peers, as
appropriate, to achieve solutions. 
a 
Mathematical Communication:
Use appropriate mathematical terms, vocabulary, and language. 
b 
Mathematical Communication:
Show mathematical ideas in a variety of ways. 
c 
Mathematical Communication:
Explain solutions to problems clearly and logically. 
d 
Mathematical Communication:
Consider purpose and audience when communicating about mathematics. 
e 
Mathematical Communication:
Comprehend
mathematics from reading assignments and from other sources. 


a
Arithmetic and Number Concepts: The student
adds, subtracts, multiplies, and divides whole numbers, with and
without calculators; that is:…
• multiplies, i.e.,…uses simple rates.
The consistent
and effective use of multiplication throughout the student work
shows a deep understanding at the elementary level.
• divides, i.e., puts things into groups, shares equally; calculates
simple rates.
• analyzes problem situations and contexts in order to figure
out when to add, subtract, multiply, or divide….

The student
successfully figured out how and when to use arithmetic in these
multistep problems. The analysis is particularly strong because
the situations were general claims made by a character in a book,
i.e., there was no hint about how to proceed to test the claims.
• solves arithmetic problems by relating addition, subtraction,
multiplication, and division to one another.
The student
added to complete a multiplication problem. The teacher also verified
that the student divided 700 by 4 and multiplied the answer by 3
with a calculator to figure out that “three quarters of 700
equals 525.”
• computes answers mentally….
The student,
as verified by the teacher, computed some of the arithmetic mentally,
here and in other places in the work.
b
Arithmetic and Number Concepts: The student
demonstrates understanding of the base ten place value system and
uses this knowledge to solve arithmetic tasks; that is:…uses
knowledge about ones, tens, hundreds, and thousands to figure out
answers to multiplication and division tasks, e.g., 36 x 10, 18
x 100, 7 x 1,000, 4,000 ÷ 4.
c
Arithmetic and Number Concepts: The student
estimates, approximates, rounds off, uses landmark numbers, or uses
exact numbers, as appropriate, in calculations.
The student used a welldeveloped sense of when it is appropriate
to round off or estimate in these situations. Elsewhere, the student
successfully used exact numbers in calculations.
d
Arithmetic and Number Concepts: The student
describes and compares quantities by using concrete and real world
models of simple fractions; that is, finds simple parts of wholes….
The student
used coffee cups and measured out 4
cups. The computation (“three quarters of 700 is 525”)
is advanced for this level.
e
Arithmetic and Number Concepts: The student
describes and compares quantities by using simple decimals….
The student
took part of the decimal answer to a division problem (13.13) and
used the nearest whole number (13) to make sense of the given situation,
i.e., bags of peas.
f
Arithmetic and Number Concepts: The student
describes and compares quantities by using whole numbers up to 10,000;
that is, connects ideas of quantities to the real world….
The
student called the author’s bluff by actually connecting the
ideas in the book to the real world.
In each problem,
the student dealt successfully with quantities larger than 10,000
(10,240 seconds; 43,680 peas; 13,500 feet). The clarity and thoroughness
of communication provides evidence of understanding of these quantities
and represents a strong performance for this level.
g
Geometry and Measurement Concepts: The student
uses basic ways of estimating and measuring the size of figures
and objects in the real world, including length, width, perimeter,
and area.
The student
went beyond the standards by measuring the volume of the bathroom
in cubic feet. The teacher verified that the students had constructed
the formula for area themselves earlier in the year through conceptual
activities. The students’ motivation to test the claim about
the bathroom led them to a discussion of how to measure the size
of the sink and the bathroom, and they constructed the idea of “layers”
of length times width. The teacher then supplied them with the formula
L x W x H.
The student
demonstrated understanding of area by listing some of the “millions
of possibilities…that are close” for 264 square feet.
The student also demonstrated understanding by applying the concept
of area to the size of a bathroom floor, and noticing that an area
of 264 square feet would be “huge.”
The student measured the length of the pen and the length of the
line and used these measurements to solve the problem.
i
Geometry and Measurement Concepts: The student
selects and uses units, both formal and informal as appropriate,
for estimating and measuring quantities such as…length, area,
volume, and time.
The
student used centimeters to measure the pen, and feet to measure
the line.
The student
used a coffee cup to measure volume.
The student
used seconds to measure time.
j
Geometry and Measurement Concepts: The student
carries out simple unit conversions, such as between cm and m, and
between hours and minutes.
The student converted peas per day to peas per year, seconds to
minutes, minutes to hours, and feet to yards.
a
Problem Solving and Reasoning: Formulation.
Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials,
and strategies to use, i.e., does not merely fill in a given chart,
use a prespecified manipulative, or go through a predetermined
set of steps.
The student made the decisions to use the sink, the peas, the coffee
cup, the ballpoint pen, and the 100 feet long piece of paper.
• uses previously learned strategies, skills, knowledge, and
concepts to make decisions.
Throughout the three problems, the student drew upon knowledge and
skills related to multiplication, division, and measurement (i.e.,
length, area, and volume).
• uses strategies, such as using manipulatives or drawing sketches,
to model problems.
The student devised effective strategies, such as timing how long
the water took to fill the sink, using the coffee cup to measure
the peas, and measuring the decrease in length of the ink in the
pen.


b
Problem Solving and Reasoning: Implementation.
The student makes the basic choices involved in planning and carrying
out a solution; that is, the student:
• makes up and uses a variety of strategies and approaches
to solving problems….
• makes connections among concepts in order to solve problems.
There are many connections between arithmetic and measurement throughout
the work.
• solves problems in ways that make sense and explains why
these ways make sense, e.g., defends the reasoning, explains the
solution.
Statements
like, “His whole kitchen would be small like a grocery bag
or the peas are as big as softballs,” show that the student
made sense of the problem.
Throughout the work, the student impressively, consistently, and
accurately explicated the reasoning used.

c
Problem Solving and Reasoning: Conclusion.
The student moves beyond a particular problem by making connections,
extensions, and/or generalizations….
The assignment asked the student to focus on one claim made in the
book. The student chose to extend the assignment by analyzing and
writing about more than one claim. This is a reasonable extension
for elementary school level.
a
Mathematical Skills and Tools: The student
adds, subtracts, multiplies, and divides whole numbers correctly;
that is:
• knows single digit addition, subtraction, multiplication,
and division facts.
The student showed a command of basic facts, which was needed to
carry out these and other calculations.
• adds…numbers with several digits.
• multiplies and divides numbers with one or two digits.
• multiplies and divides three digit numbers by one digit numbers.
b
Mathematical Skills and Tools: The student
estimates numerically and spatially.
c
Mathematical Skills and Tools: The student
measures length, area,…height,…and volume accurately in
both the customary and metric systems.
The teacher verified the accuracy of the measurements.
d
Mathematical Skills and Tools: The student
computes time (in hours and minutes)….
f
Mathematical Skills and Tools: The student
uses +, , x, ÷, /,…and . (decimal point) correctly
in number sentences and expressions.
h
Mathematical Skills and Tools: The student
uses recall, mental computations, pencil and paper, measuring devices,…manipulatives,
calculators…and advice from peers, as appropriate, to achieve
solutions….
The student drew upon many resources to complete the assignment.
a
Mathematical Communication: The student uses
appropriate mathematical terms, vocabulary, and language, based
on prior conceptual work.
The student used many mathematical terms appropriately, including
“estimate,” “measure,” “average,”
“multiply,” “divide,” “seconds,” “minutes,”
“area,” “long,” “high,” “equal,”
“cm,” “ft.”
The use
of the term “cubic feet” and the formula for volume go
beyond what is required at this level.
The student
wrote “yds” after 13,500, but this mislabeling did not
interfere with the correct mathematics in the work.
b
Mathematical Communication: The student shows
mathematical ideas in a variety of ways, including words, numbers,
and symbols….
c
Mathematical Communication: The student explains
solutions to problems clearly and logically, and supports solutions
with evidence, in both oral and written work.
d
Mathematical Communication: The student
considers purpose and audience when communicating about mathematics.
The student
focused consistently on the purpose of disputing the claims in the
book and wrote directly to the author.


e
Mathematical Communication: The student comprehends
mathematics from reading assignments and from other sources.
The student
even comprehended mathematical information from a contractor about
the average size of a bathroom. 
The student misspelled a few words (e.g., “definetely”)
and made some grammatical mistakes (e.g., capitalizing “book”
in the middle of the sentence). The teacher did not require the students
to fix the errors. 
