|
These tasks all focus on the same arithmetic and number concepts and skills,
although they were drawn from four different classrooms.
In Sample 1, the teacher included the following instructions on a classroom
test:
Using what you know about multiplication, find the answer to the equation
below. Show all your work and clearly explain how you got your answer.
63 x 46 =
In Sample 2, the teacher gave the following written instructions as part
of a class exercise:
Solve each problem two ways.
522 - 367 =
87 x 9 =
In Sample 3, the teacher gave the following oral instructions at the
end of a long unit of work:
Make a catalogue of all the ways the class has come up with to multiply
large numbers.
Make up a two digit by two digit multiplication problem and use these
ways to find the answer.
In Samples 4 and 5, the teacher gave the following written instructions
to the students:
There were 3 bags of M & M candies. Four children decided to open
all three bags and share the M & M’s fairly. Each bag had 52
M & M’s in it. How many M & M’s did each child get?
In Sample 4, the teacher orally added the instruction, “Show all
your work and clearly explain how you got your answer.” In Sample
5, the teacher included this direction with the written instructions.
| These samples of student work
were produced under the following conditions: |
| - alone |
in a group |
| - in class |
as homework |
| with teacher feedback |
with peer feedback |
| timed |
opportunity for revision |
Sample 1 was part of a classroom test. For Samples 2 and 3, students
were permitted to talk as they worked. Samples 4 and 5 were produced
as class work.
Sample 1 was produced before the class had received any instruction
about two digit by two digit multiplication, although they had experience
in developing strategies for simpler computation. |
Sample 1
|
Sample 2 was produced in a classroom where
students’ computation strategies were regularly shared, and several
were posted on the wall.
Sample 3 was produced after a long unit of work in which students
created, developed, and learned a range of strategies for single and
double digit multiplication.
Samples 4 and 5 were produced after minimal instruction about division.
The students did have some experience with developing and explaining
various strategies for other computation problems.
These work samples illustrate standard-setting
performances 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
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
in calculations. |
| d |
Arithmetic
and Number Concepts: Describe and compare quantities by using
simple fractions. |
 a |
Mathematical
Skills and Tools: Add, subtract,
multiply, and divide whole numbers correctly. |
| 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, and pencil
and paper 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. |
|
a
Arithmetic and Number Concepts: The
student adds, subtracts, multiplies, and divides whole numbers,
with and without calculators; that is: |
•
|
adds, i.e., joins things together,
increases. |
 
 
All students added as they solved larger problems.
|
| • |
subtracts, i.e., takes away,
compares, finds the difference. |
 |
| • |
multiplies, i.e., uses repeated addition,
counts by multiples, combines things that come in groups, makes
arrays…. |
|
The student combined doubling with repeated addition.
|
|
The student used repeated addition of 85. |
The student showed how the problem could be solved by making
62 circles with 85 stars in each circle. |
The student drew a base ten block array. |
| • |
divides, i.e., puts things into groups, shares
equally…. |
The student “split” 80 into 40 and 40 in order to
multiply. |
The student divided 156 by 4 although the place value position
of the quotient is incorrect. |
The student divided 100 by 4, 50 by 4, and 6 by 4. |
| • |
analyzes problem situations and contexts
in order to figure out when to add, subtract, multiply, or divide. |
|
| 
Sample 3
|
• counts 1, 10, 100, or 1,000 more
than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000
more than 23,000, 100 less than 9,000.
The
student counted 3, 30, and 100 more.
• 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.
 The
students broke the numbers apart, e.g., 40 x 9, 8 x 9.
The student
broke 156 into 100, 50, and 6 and divided each part by 4.
c
Arithmetic and Number Concepts: The student
estimates, approximates, rounds off, uses landmark numbers, or uses
exact numbers, as appropriate, in calculations.
|
| • |
finds simple parts of wholes. |
|
The student applied concrete knowledge to split “80 in half.”
|
|
The student demonstrated knowledge of adding like fractions. |
|
The student recognized equivalent fractions and simple fractions as
parts of whole numbers. |
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. |
 b
Mathematical Communication: The student shows mathematical ideas in
a variety of ways, including words, numbers, symbols, pictures, charts,
graphs, tables, diagrams, and models. |
|
| c
Mathematical Communication: The student explains solutions to problems
clearly and logically, and supports solutions with evidence, in both
oral and written work. |
 
Each sample includes explanations for the solutions given. |
There are some errors of spelling, grammar, and usage in these samples
of student work. For example, in Sample 3, the student spelled a few
words inconsistently (e.g., “multiplied” is correct while
“multiplyed” is not) and misspelled others (e.g., “Catalouge”
instead of “Catalogue”). In Sample 4, there are some spelling
errors (e.g., “divied” instead of “divided”) and
some slips in tenses. All of the samples were taken from work done
in class which were not edited or revised. |
|
Here,
as in other places, the students’ knowledge of basic facts
is demonstrated in correct solutions to larger problems.
Here,
as in other places, the samples demonstrate correct work.
These parts
of the samples demonstrate correct use of symbols. This is also evident
throughout the samples.