NWSC Math Cohort Meeting

Noni J Bamberger
Christine Oberdorf
Karren Schultz-Ferrell
Publisher: Heinemann 2010
PreK and Elementary educators are by and
large nurturing and supportive and have
students interest in mind.
We want the students to enjoy learning and
end each year with all the skills and concepts
they should have.
But these positive characteristics sometimes
lead us to unwittingly encourage some
serious error patterns, misconceptions, and
*Number and Operations
Data Analysis and Probability
Assessing Children’s Mathematical Progress
* Counting with Number Words
* Thinking Addition Means “Join Together” and
Subtraction Means “Take Away”
* Renaming and Regrouping When Adding and
Subtracting Two-Digit Numbers
* Misapplying Addition and Subtraction Strategies to
Multiplication and Division
Multiplying Two-Digit Factors by Two-Digit Factors
Understanding the Division Algorithm
Understanding Fractions
Adding and Subtracting Fractions
Representing, Ordering, and Adding/Subtracting
What the Research Says
Five principles of Counting [Gelman and Gallistel (1986)]
1) One-one principle. Each item to be counted has a “name,” and we count each item only
once during the counting process
2) Stable order principle. Every time the number words are used to count a set of items, the
order of the number words does not change.
3) Cardinal principle. The last number counted represents the number of items in the set of
4) Abstraction principle. “Anything” can be counted and not all the “anythings” need to be
of the same type.
5) Order-irrelevance principle. We can start to count with any object in a set of objects; we
don’t have to count form left to right.
What to Do
* Ask students to skip-count from different numbers.
* Read counting books. (In Moira’s Birthday by Robert
Munsch for instance, students can count to 200 in a
variety of ways since 200 kids are invited to Moira’s
* Support students in applying critical-thinking skills to
the counting sequence by presenting number-logic
riddles. When students are familiar with format of
logic riddles, allow them to create riddles for
classmates to solve.
What to Do (continued)
* Encourage students to create number logic riddles for
three-digit through seven-digit numbers, depending
on students’ grade level.
* Provide small groups of students with a large box of
objects to count. Challenge students to determine a
way to count the objects in the box. Each group must
have a different way of counting. When the task is
complete, ask each group to present its counting
strategy and to justify why the method is efficient.
What to Do (continued)
* Present opportunities for students to count both
common and decimal fractions.
* Provide experiences for students in which they
place either common fractions or decimal
fractions on a number line. This exercise also
supports students’ understanding of relative
magnitude (the size relationship one number has
with another).
What to Look For
* Is the student able to count using a variety of
* Does the student use logical thinking skills
effectively to solve riddles about counting?
* Do students use reasoning to justify why their
method of counting is the most efficient for
counting a large amount of objects?
Questions to Ponder
1) What common path games can help students
develop their understanding of the one-one
counting principle?
2) What additional activities or strategies can you use
to help your students become successful
Take the next 5 (five)
minutes to mark the places
in your materials where
students work on
What the Research Says
Problem Types – Basis of Cognitively Guided Instruction
[Carpenter and Moser 1983; Carpenter, Carey, and Kouba 1990]
1) Join problems.
2) Separate problems.
3) Part-Part-Whole problems.
4) Compare problems.
What to Do
* Before using symbols, provide students with various
materials that can be used to create part-whole
representations of numbers (bi-colored counters,
connecting cubes, teddy bear counters, Cuisenaire
* Use correct terminology for the addition and subtraction
signs (+ plus) and (– minus).
* Provide students with opportunities to solve story
problems that include all four problem types: join,
separate, part-part-total.
What to Do (continued)
* Provide manipulatives to model story problems, along
with symbolically recording what they do.
* Have students share their strategies they used to get
their answers, reinforcing correct terminology.
* Use dominoes to have students model part-whole
addition number sentences.
* Use classroom routines to generate meaningful
comparative subtraction problems.
* Have students generate their own story problems.
What to Look For
* When students share their equations, listen for the
correct use of “minus” and “plus.”
* When students solve comparison subtraction story
problems, look to see if students create two sets.
Then look to see if they use an appropriate strategy to
determine how many more or fewer one set is
compared to the other.
* When students generate their own stories, look to see if
they are developing a variety of problems based on
the types and structures taught.
Questions to Ponder
1) What manipulatives do you currently have to
reinforce the idea of part-whole for addition and
2) How might you communicate to families the way
you’ll be teaching the concepts so that
misconceptions and overgeneralizations about
addition and subtraction do not occur?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on addition and
subtraction concepts.
What the Research Says
* “When children focus on following the steps taught
traditionally, they usually pay no attention to the
quantities and don’t even consider whether or not
their answers make sense.” [Richardson 1999, 100]
* Teaching so that students can understand the
traditional algorithm seems to be a real challenge.
What the Research Says (continued)
* For students to understand place value, they need to connect
the concept of grouping by tens with the procedure of how
to record numerals based on this system of counting.
* Counting is fundamental to constructing an understanding of
base-ten concepts and procedures.
* Models that are both proportional and groupable should be
used before models that are proportional but not
What the Research Says (continued)
* “Given the opportunity, children can and do invent increasingly
efficient mental-arithmetic procedures when they see a
connection between their existing, count-by-tens knowledge
and addition by ten.” (Baroody and Standifer 1993, 92)
* Many mathematics educators recommend spending a good deal
of time with manipulative models while simultaneously
practicing mental computation before putting pencil to paper
to solve expressions (Baroody and Standifer 1993, 92).
What to Do
* Have students use their understanding of counting by
ones to group larger quantities, making it easier to
count up or back to determine a sum or a difference.
* Have students use a five-hundreds chart to look for
patterns, and determine simple sums and differences
by moving around the chart.
* Use estimation activities so students get regular practice
estimating quantities and then determining the actual
amount by grouping by hundreds and tens to see how
What to Do (continued)
* Have students share their solutions and the
strategies that they used to get their answers.
* Play games that require students to bundle,
connect, or place objects together when there are
ten of the object. Have students record the
quantity of hundreds, tens and ones and the
number that this represents.
What to Do (continued)
* Give students a three-digit number and have them
represent, either through modeling, pictures, or
symbols, all of the ways to show this number
using hundreds, tens and ones only.
* Have students use whatever strategy is efficient
and effective in getting a sum or difference as
long as it makes sense to them
What to Look For
* Do students understand the value of each digit
rather than looking at the digit in isolation?
* Are students able to compose and decompose
* Do students see addition and subtraction as
inverse operations?
Questions to Ponder
1) What hundreds-chart activities help students
better understand two-digit numbers?
2) What manipulative materials help students add
and subtract? How can you use them?
3) How can you use estimation activities to reinforce
ideas of tens and ones?
4) What research supports your instruction of place
value, addition, and subtraction?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on two-digit place
value concepts.
What the Research Says
* In grades 3-5, multiplicative reasoning emerges and should
be discussed and developed through the study of many
different mathematical topics. Students’ understanding of
the base-ten number system is deepened as they come to
understand its multiplicative structure. That is, 484 is 4 x
100 plus 8 x 10 plus 4 x 1 as well as a collection of 484
individual objects (NCTM PSSM 2000, [144]).
What to Do
* Number Lines - Use number lines (rulers,
yardsticks, and meter sticks) to model
multiplication and division situations.
* Equal groupings – Making equal groups to model
multiplication allows students to create equal
sets and reinforces the notion that all sets are the
same size.
What to Do (continued)
* Partial Products and Partial Quotients – The partial
products strategy emphasizes the importance of
place value when multiplying whole numbers and
provides an alternative algorithm that
emphasizes the whole number rather than
isolated digits within a number.
* The same can be done by pulling out equal groups
for division and then finding the total quotient by
adding all of the partials.
What to Do (continued)
* Area Model of Multiplication – By using an area
model in conjunction with a rounding strategy,
students see the value of the rounded product
and how it compares to the actual product.
What to Look For
* All groups must be the same size when solving
multiplication and division problems.
* Adjustments made to any one group (for ease of
computation) must also be applied to all groups
when solving multiplication and division
Questions to Ponder
1) Which multiplication and division strategies
maintain the emphasis of place value?
Which do not?
2) When teaching multiplication and division, how do
you decide which models and representations to
use so that students understand the
multiplicative concept?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on multiplication and
division concepts.
What the Research Says
* Children spend a good deal of time learning and then
practicing multi-digit addition. Consequently, it’s not
uncommon that they combine algorithms when they
do not have a complete understanding of place value
(decomposing numbers) as well as what it means to
What the Research Says (continued)
* Ruth Stavy and Dina Tirosh, in How Students (Mis-)Understand
Science and Mathematics (2000), attribute some errors as
based on “intuitive rules.” Schemas about concepts and
procedures are formed by students. Without a firm
understanding of new content, students return to “relevant
intuitive rules” that they have come to rely on. They may not
make sense in the specific situation that they are now in, but
unless new knowledge makes sense these rules persist.
What the Research Says (continued)
* Jae-Meen Baek spent time with students in six classrooms in
grades Grades 3-5 to observe the different algorithms that
were invented, as well as to see whether students were
utilizing the traditional algorithm. Since none of the
teachers taught rules or formal algorithms to students,
many developed procedures that made sense to them.
What the Research Says (continued)
* “Many children in the study developed their invented
algorithms for multi-digit multiplication problems in a
sequence from direct modeling to complete number
to partitioning numbers into non-decade numbers to
partitioning numbers into decade numbers” (Baek
1998, 160).
What the Research Says (continued)
* Not only does this observation lead one to believe that
multi-digit multiplication algorithms can be invented
by students, but it also leads one to believe that when
this is done, students have a clearer understanding of
how to multiply.
What to Do
* Before doing any computation have students estimate
the product based on the numbers in the expression.
Any strategy (front-end, rounding, compatible
numbers) can be used to determine this estimate.
* Have students “expand” the factors of the multiplication
34 x 27 = (30 + 4) x (20 + 7) or
= (34 x 20) + (34 x 7) or
= (30 x 27) + (4 x 27)
What to Do (continued)
* Try three ways to see if different results will be
1) Build an array that matches the expression using either
base-ten blocks or centimeter grid paper.
2) Dissect the array by comparing it with the expanded
form of the expression so students can see all of the
different factors and partial products that will form from a
two-digit by two-digit expression.
What to Do (continued)
* Try three ways to see if different results will be
3) If it’s the first expanded form that’s used, have students
explain where the 30x20 part of the array can be found and
label this as 30x20=600. then have the students explain
where the 4x2=80 part of the array can be found and label
this. Do this for the 30x7 part and the 4x7 part.
* Try another expression and work on it as a whole
group before having students do this either
independently or with a partner.
What to Look For
* As students work through your chosen activities,
look to see if they are using a strategy that is
both efficient and effective. Also, be sure to ask
students to explain how they know that they have
used all of the digits in each of the factors as
they’ve multiplied.
Questions to Ponder
1) What are some other common errors students
make when multiplying multi-digit numbers?
2) What are some effective strategies that you’ve
used to help students understand why their
procedures aren’t yielding the correct answers?
Take the next 5-8 minutes to
mark the places in your
materials where students work
on two-digit by two-digit
What the Research Says
* “The traditional long-division algorithm is difficult for many
students. Many never master it in elementary school and
fewer develop meaning for the procedure or the answer”
(Silver, Shapiro, and Deutsch 1993).
* First reason – the procedure contains so many steps, and for
each step students need to get an exact answer in the
What the Research Says (continued)
* Second reason – the algorithm treats the dividend as a
set of digits rather than an entire numeral. Students
are taught to ignore place value as they routinely work
through a procedure they don’t necessarily
What to Do
* Encourage students to estimate the quotient using their
mental multiplication skills so they have a sense of
whether their answer is reasonable.
* Before using symbols, provide students with a story
problem that is meaningful to them.
* Provide students with manipulatives to model the story
that is given, but also have them symbolically record
what they’ve done.
* Ask students to share the strategies they used to get
their answers and discuss whether their answers make
What to Do (continued)
* “Try out” someone’s procedure that is both efficient
and effective to see if students are able to use
this same strategy.
* Use number-sense activities to foster mental
computation and an understanding of how to use
multiples of ten to arrive at answers.
* Look at ways to adjust numbers to make them
easier to use for computing.
What to Look For
* Check to see if students have a procedure that will
always work.
* As students use their own division algorithm, make
sure that they can articulate why it works and
how they know their answer makes sense.
Questions to Ponder
1) How do you help students move from representing
a division problem with manipulatives to using a
paper-and-pencil algorithm?
2) How do you get a student who has an effective but
inefficient algorithm to adopt one that is more
Take the next 5-8 minutes
to mark the places in your
materials where students
work on division
What the Research Says
Sherman, Richardson, and Yard (2005) suggest several reasons for
student difficulties in learning about fractional concepts and
* They memorize procedures and rules before they have developed
a conceptual understanding of the related concepts.
* Early instruction in mathematics focuses on whole numbers so
children over-generalize what they know about whole-number
computation and apply this knowledge to fractions.
What the Research Says
Sherman, Richardson, and Yard (2005) suggest several reasons for
student difficulties in learning about fractional concepts and
* Estimating rational numbers is more difficult than estimating whole
* Recording fractional notation is difficult and confusing for students if
they do not yet understand what the top and bottom numbers
represent. Knowing which is the numerator and which is the
denominator and what those numbers mean is critical.
What the Research Says
Chapin and Johnson (2000) list four critical interpretations
of fractions necessary for computing successfully:
1) Part of a whole or parts of a set.
2) Fractions as a result of dividing two numbers.
3) Fractions as the ratio of two quantities.
4) Fractions as operators.
What to Do
* Provide students multiple opportunities to share various
objects that support them in thinking more flexibly
about fractions. Present nontraditional shapes (such
as triangles) for students to divide.
* Involve students in discussions following their work with
fractions. Introduce fraction vocabulary and talk about
fractional parts rather than fractional symbolism.
Then expect students to explain what the symbolic
representation means.
* Encourage students to “fair share” a grid outline in
different ways.
What to Do (continued)
* Provide opportunities for students to solve word
problems involving both area and set models of
* Provide students with opportunities to develop
understandings about fractional concepts in a variety
of real-life connections. Reach beyond the “pizza”
* Present sharing problems that include the “set” model of
fractions to help students establish important
connections with many real-world uses of fractions.
What to Do (continued)
* Provide opportunities for students to explore
fractions such as sixths and eighths. Their
understandings about “halving” will help them as
they work with a variety of fractions.
* Count fractional parts with students so they see
how multiple parts compare to the whole.
What to Look For
* Are students able to fairly share an area in different
* Are students able to divide a variety of shapes or
objects accurately?
* Are students able to represent fractional parts in a
variety of nontraditional ways?
Questions to Ponder
1) What specific difficulty have your students had or
what overgeneralization have they made about
fractions? Discuss this misconception with
members of your teaching team or with an
instructional support teacher.
2) How will you plan instruction so that students can
develop a better understanding of this fractional
Take the next 5-8 minutes
to mark the places in your
materials where students
work on fraction concepts.
What the Research Says
* “In terms of instructional approaches, lessons are too often
focused on procedures and memorizing rules rather than
on developing conceptual foundations prior to skill
building” (Sherman, Richardson, and Yard 2005, 139).
* Many researchers have concluded that the complex topic of
fractions is more challenging for elementary students than
any other area of mathematics (Bezuk and Bieck 1993).
What the Research Says (continued)
* Before students study how to add and subtract fractions, they
need to understand the meaning of fractions through
various models, as well as how to use the language of
* Watanabe (2002, 457) delineates three models frequently
used in elementary materials – the linear model, the area
model, and the discrete (set) model.
What the Research Says (continued)
Van de Walle (2007) provides some important “big ideas” that
students must understand for computational understanding:
* Fractional parts are equal-size portions or equal shares of a
whole or unit. They don’t necessarily look alike.
* The special names for the numbers that make up a fraction tell
how many equal-size parts make up the whole (the
denominator) and how many of the fractional parts are being
considered (the numerator).
What the Research Says (continued)
* The National Mathematics Advisory Panel (2008) notes
that one key instructional strategy to link conceptual
and procedural knowledge of fractions is the ability to
represent fractions on a number line.
What to Do
* Allow students to create their own materials or draw
their own representations when adding and
subtracting fractions.
* Introduce activities in which children count by fractions.
Begin by using manipulatives with which children are
already familiar—pattern blocks, for example.
* Have students use fraction pieces to count by halves,
thirds, fourths, sixths, eighths, and even twelfths. If
they get good at this, have them combine fraction
What to Do (continued)
* Introduce story problems that reinforce what it means to
add and subtract fractions. Don’t have students
record the equation until they have shared their
strategies for getting their answers. Ask “What do you
notice about how fractions are added or subtracted
when the denominators are the same?”
* Give students opportunities to compare fractions. This
opportunity to visualize the value of a fraction will
help in making sense of the computation when
finding sums and differences.
What to Do (continued)
* Use the number line to represent fractions,
compare the magnitude of fractions, and to add
or subtract fractions with like denominators.
* Reveal patterns on the multiplication chart as an
example of equivalent fractions.
What to Look For
* The denominator names the total number of pieces
needed to form the whole.
* The numerator indicates a specific number of
pieces of the unit.
* While the numerator changes when adding and
subtracting fractions with like denominators, the
denominator remains the same in the sum or
Questions to Ponder
1) What big ideas about fractions must students
understand before they are able to build a
conceptual knowledge of equivalence?
2) What opportunities or supports can you use to
empower students to manipulate values by using
their own number sense rather than simply
relying on procedures?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on adding and
subtracting fractions.
What the Research Says
* A simple yet powerful introduction to decimals is to ask students to
represent two related decimal numbers using several
representative models (Van de Walle and Lovin 2006).
* Chapin and Johnson (2000) state, “Finding examples of decimals,
explaining what the decimal numeral means in the context of its
use, indicating the general value of the decimal numeral, and then
stating what two whole numbers the decimal is between helps
students recognize that the decimal amount is the sum of a whole
number and a number less than one.”
What the Research Says (continued)
* The Chapin and Johnson (2000) approach reaffirms the
importance of avoiding “naked” mathematics and instead
teaching mathematics skills and concepts with a context.
* Decimal number sense should be a focus during
instruction so that students recognize an unreasonable
answer (Sherman, Richardson, and Yard 2005).
What to Do
* Correctly name the decimal fraction. When a
decimal fraction is read correctly, the name
reinforces the place value of each digit. Prevent
students from getting into the habit of saying
“six point three” rather than “six and threetenths” when reading a decimal fraction.
* Use a variety of concrete models to represent
decimal fractions. Students need multiple
representations for decimal fractions.
What to Do (continued)
* Provide opportunities to reinforce place value. With
experience, students will recognize the
relationship among adjacent values and see that
moving to the left (by one digit) means ten times
larger and moving to the right denotes one-tenth
of the value. Additionally, students must have
opportunities to recognize that a value can be
named using different units.
What to Look For
* Students verbalize the decimal fraction
* Students are able to state the value of a
specific digit within a decimal fraction.
* Students can construct more than one
visual representation for a decimal
Questions to Ponder
1) How can you reinforce the notion that the quantity
represented by a digit is the product of its face
value and its place value?
2) What comparisons would you expect students to
identify between operations with whole numbers
and operations with decimals numbers?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on representing
Working in Groups of 3,
take the packet of Number and
Operations Activities. Go through the
Activities, talking, taking notes,
planning how activities of this sort
might be used in conjunction with your
curriculum materials.
Discuss how these activities can
Undo Misconceptions
Understanding Patterns
Meaning of Equals
Identifying Functional Relationships
Interpreting Variables
Algebraic Relationships
What the Research Says
* “When students identify patterns furnished by the teacher,
books, or the classroom environment or when they
memorize—store various patterns and recall them—they
internalize the concept of pattern and realize that it is the
same irrespective of the changes in the periodic themes
that create different patterns.”
(Hershkowitz and Markovits 2000, 169)
What to Do
* As the study of patterns begins, be sure to
make students aware that there are patterns
that repeat as well as patterns that grow.
* Use cubes, links, square tiles, and other
manipulatives to show challenging repeating
patterns that students can identify, extend,
and then create their own.
What to Do (continued)
* Present students with materials they see
every day and ask them to look for
patterns within these things.
* Expose students to patterns that appear in
nature and within their environment.
* Introduce games and activities in which
students need to use patterns in order to
complete a task or win a game.
What to Do (continued)
* Examine each multiplication sequence for patterns
that repeat (both in the ones place and in the
tens place). For example: 3, 6, 9, 12, 15, 18, 21,
24, 27, 30, 33, 36, 39, 42, 45, 48, and so on. The
“pattern unit” in ones place is: 3, 6, 9, 2, 5, 8, 1,
4, 7, 0. The “pattern unit” in the tens place is: 0,
0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, and so on.
Students can make predictions about what will
come next in the ones and tens place and then
extend this to include the hundreds place.
What to Do (continued)
* Students should use a 1-1000 chart to
extend the idea of noting patterns that
they’ve begun looking at in the early
primary grades with a 1-100 chart.
* Introduce games and activities that
require students to use patterns in order
to complete the task and win the game.
What to Look For
* Students are able to describe or name the core
unit or pattern core of a repeating pattern.
* Students are able to extend a repeating pattern
that has a somewhat simple core pattern (AB,
ABC, AAB, ABB, …).
* Students are able to create a repeating pattern
and extend it.
Questions to Ponder
1) Where in your academic curriculum
do you introduce and reinforce
2) How do you reinforce repeating and
growing patterns with your students?
Take the next 5-8 minutes to
mark the places in your
materials where students
work on ideas about
repeating and growing
What the Research Says
* Teachers and curriculum materials view arithmetic and
algebra as distinct and different. This impedes student
understanding of critical ideas such as equality, and they
encounter difficulties in later grades.
* A nonmathematical sense of equals sign is “one of the major
stumbling blocks for students when they move from
arithmetic to algebra” (Falkner, Levi, and Carpenter 1999).
What the Research Says (continued)
* We must be sure that students understand that a
balance must exist on either side of the equals
sign—that it represents the relationship of equality.
* “A concerted effort over an extended period of time is
required to establish appropriate notions of
equality” (Falkner, Levi, and Carpenter 1999, 233).
What to Do
* Provide multiple part-whole experiences to
strengthen number sense.
* Allow students to represent two-digit numbers in
a variety of ways using connecting cubes.
* Provide pairs of students with a two-pan balance
and weighted teddy bear counters to explore
equality. Encourage students to create multiple
equations using the relationship among the
What to Do (Continued)
* Provide opportunities for students to
explore with a number balance. This
manipulative helps students develop an
understanding of equality and inequality,
number comparisons, addition, and
subtraction. Symbolically representing the
number balance equation connects the
concrete to the more abstract.
What to Do (Continued)
* Provide student experiences in which they
create equivalent representations using
Cuisenaire Rods. For example, assign a
value of 10 to the orange rod. Then ask
students to find different ways to
represent that value with different rods,
such as 10 white rods are as long as one
orange rod.
What to Do (Continued)
* Let students explore unknowns in equations by
placing number squares to make an equation
* Ask “Is this true?” regularly and present
equations that are recorded in nontraditional
ways (7=2+5 or 11+3=20-6). Expect
students to support their answer with an
What to Look For
* Are students able to represent numbers
in different ways?
* Are students able to demonstrate a
variety of representations of different
* Are students able to represent equations
in a variety of nontraditional ways?
Questions to Ponder
1) “Is it true?” is one strategy you can use to help
your students understand the meaning of the
equals sign. Create several examples of
number sentences your students could
discuss. How well can they discuss and
demonstrate true or untrue?
2) What other instructional experiences can you
introduce that will help your students better
understand the equals sign represents the
relationship of equality?
Take the next 5-8 minutes to mark
the places in your materials where
students work on ideas about the
equals sign and equality.
Also use this time to share what you
remember about the September
2010 Cohort Meeting and the
presentation on Equality.
What the Research Says
* NCTM (2000) defines two specific expectations for
grades Grades 3-5 students with regard to
understanding patterns, relations, and functions:
Describe, extend, and make generalizations about geometric
and numeric patterns.
Represent and analyze patterns and functions using words,
tables, and graphs.
What the Research Says (continued)
* The inclusion of “words, tables, and graphs” emphasizes
the notion that students need experiences with
multiple representations of functional patterns.
* “It is important to see that each representation is a way
of looking at the function, yet each provides a
different way of looking at or thinking about the
function” (Van de Walle 2007).
What the Research Says (continued)
* Teachers must expose students to a variety of methods
of communicating functional relationships, including
physical models, pictorial models, symbolic models,
and verbal models.
* Providing varied representations gives students a
comprehensive look at this component of algebraic
What to Do
* Provide opportunities for students to explore
growing patterns. Growing patterns are
precursors to functional relationships (in a
functional relationship, any step can be
determined by a step number, without
calculating all the steps in between). Students
observe the step-by-step progression of a
recursive pattern and continue the sequence.
What to Do (continued)
* Choose a meaningful context for functional
-Money spent on candy
-Ingredients needed for a recipe
-Time required to finish a race
-Fuel needed for a vacation
What to Do (continued)
* Allow students to construct physical models of
functional relationships using tiles,
toothpicks, connecting cubes, or other
hands-on materials. The act of placing
toothpicks in a specified pattern or
connecting cubes in a sequence can provide
insight into the relationship between the two
What to Do (continued)
* Compare physical models with pictorial or
symbolic representations.
* Model the language of the dependent
relationship and encourage students to
describe the relationship.
-The amount of money I spend depends on how much
candy I buy.
-The amount of fuel we use is directly related to the miles
What to Do (continued)
* Reinforce number sense through estimation. When
students are able to articulate the intuitive
understanding of the relationship, they may
estimate and solve the function simultaneously.
Estimation may also help a student more readily
recognize an error.
* Have students graph the relationships revealed in a
function as a visual picture while learning about
rates of change.
What to Look For
* Students test their rule for the pattern among many
terms to confirm their rule is correct.
* Students are able to describe a rule verbally as well
as pictorially.
* Students can extend increasing and decreasing
Questions to Ponder
1) What varied representations can be used to
illustrate a functional relationship?
2) Using the sequence of triangles or other
combinations of geometric figures, what
questions can you pose to students to assess
their current level of understanding?
Take the next 5 minutes to
mark the places in your
materials where students
work on patterns and
functional relationships.
What the Research Says
* “Students may have difficulty if they view algebra as
generalized arithmetic. Arithmetic and algebra use the
same symbols and signs but interpret them
differently” (Billstein, Libeskind, and Lott 2007, 40).
* This can be very confusing to students, particularly if
their arithmetic concepts and skills are weak.
What the Research Says (continued)
* “Many students think that all variables are letters that
stand for numbers. Yet the values a variable takes are
not always numbers, even in high school
mathematics” (Usiskin 1988, 10).
* In middle school a variable can be used to represent
identifying points on polygons.
What the Research Says (continued)
* In high school logic, p and q are used to stand for
* The idea that a letter can replace a number only is a
misconception many students have—one that is
supported by educators who view problems like
5+x=12 as algebra, but 5+ =12 as arithmetic.
What the Research Says (continued)
* In elementary grades letters appear as abbreviations.
The letter m is used to represent the word meter.
* And even when students realize that a letter is being
used to replace a numerical value, many still assume
this is a unique value rather than a general number
(Kuchemann 1981).
What to Do
* Have students identify the elements in a repeating
pattern with letters of the alphabet as well as
other descriptors. Students will probably already
know how to do this, but reinforce that a circle,
square, triangle core unit may also be called an
ABC pattern.
What to Do (continued)
* Label polygons with letters identifying each vertex
of the shape. Middle school students aren’t
confused when they see this and neither should
third- through fifth-grade students. When
learning about angles in geometry class it seems
perfectly sensible to have fifth graders identify
these angles with letters that correspond.
What to Do (continued)
* Have students look for places where a letter is used
to represent some word. “Mathematical
Equations” can be created in which students
replace the letter with a word that makes the
equation true.
* Begin replacing the “box” in an arithmetic equation
with a letter.
What to Do (continued)
* Point out how letters are used in formulas that are
being learned. Before students learn formulas to
determine the area and perimeter of polygons or
the volume of solid figures, they should know the
words that the letters represent.
What to Look For
* Students aren’t just looking at the digits and the
sign and then following the sign. Be sure they are
trying to make sense of the open expression.
* Students are using a letter to label things that
might have been assigned a numeral (length of a
* Students can use a number balance to determine
the missing addend or sum.
Questions to Ponder
1) What manipulatives could you use to help young
students understand how to solve for a missing
2) Once formulas have been introduced into the
curriculum, how can you help students see the
different uses for variables?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on variables.
What the Research Says
* NCTM (2000) says “instructional programs from
prekindergarten through grade 12 should enable all
students to:
create and use representations to organize, record, and communicate
mathematical ideas
select, apply, and translate [from] among mathematical representations
to solve problems
use representations to model and interpret physical, social, and
mathematical phenomena”
What the Research Says (continued)
* It is critical that students have numerous opportunities
to represent problem solving using concrete and
pictorial representations before using abstract
* Ennis and Witeck (2007) propose that using abstract
representations such as numbers and equations
requires a deep understanding of a topic.
What the Research Says (continued)
* Moving students too quickly to abstract representations
encourages them to perform certain procedures by rote
without understanding why these procedures work or what
they mean.
* “However, we would be negligent as well if we did not help
students make the connection between ideas and equations
and see how equations can help us solve problems and
visualize ideas” (Ennis and Witeck 2007).
What the Research Says (continued)
* Choosing from a variety of representations (concrete,
pictorial, equations) helps students understand that
some representations are more useful than others
when solving a particular problem.
* “Little understanding is being developed when a
representation is used in a procedural way (Van de
Walle and Lovin 2006).
What the Research Says (continued)
* By encouraging students to use a representation in a
way that makes sense to them, we allow them to think
and reflect about the mathematical idea involved in
solving the problem.
What to Do
* Provide opportunities for students to explore, and
then talk about, a variety of manipulatives. Model
the correct vocabulary that is specific to each
manipulative. Plan lessons in which students use
these manipulatives.
* Encourage students’ use of multiple
representations. Create and model an
environment in which all explanations and
representations are honored and respected.
What to Do (continued)
* Allow students to freely select from different
representations to use in solving any problem.
Initially, model conventional ways of representing
mathematical situations, but eventually allow
students opportunities to choose representations
that they are comfortable using. Knowing which
type of representation is useful in which situation
is an important milestone in mathematical
understanding and reasoning for students.
What to Do (continued)
* Ask students to explain and show how they are
thinking about a problem during and following a
problem-solving task. When students hear how
others use representations to show how they are
thinking about a mathematical idea, it helps them
to consider other perspectives. Communicating
their thinking requires students to reflect on their
problem solving and reasoning; listening to
students’ explanations enables teachers to
determine what students know and can do at any
point in time.
What to Do (continued)
* Provide opportunities for students to solve many
open-ended tasks with a representation they
have chosen. Follow these problem-solving tasks
with classroom discussions.
* Model for students how to record their way of
solving a problem using numbers. Be sure to
record the students’ methods both horizontally
and vertically.
What to Do (continued)
* Observe when students use a representation to
determine if they understand representation and
how to use it effectively to solve the problem.
* Use literature to engage students in problem
solving and ask them to represent their solutions
in a representation that makes sense to them.
What to Look For
* Students’ flexibility in their use of representations
to show their thinking.
* Student use of appropriate vocabulary in describing
a strategy or representation.
Questions to Ponder
1) Why is it important to model many ways to use
representations? Why is it important for students to
explain and show how they used a representation to
communicate their thinking about how to solve a
2) Think about the types of representations your students
are now using. Do they use the same representation
for every problem? Do they choose one representation
over another one because it more efficient in helping
them think about the task?
Take the next 5-8 minutes to
mark the places in your
materials where students
work on algebraic
representations for their
problem solving.
Working in Groups of 3,
take the packet of Algebra, Patterns,
and Functions Activities. Go through
the Activities, talking, taking notes,
planning how activities of this sort
might be used in conjunction with your
curriculum materials.
Discuss how these activities can
Undo Misconceptions
Categorizing Two-Dimensional Shapes
Naming Three-Dimensional Figures
Navigating Coordinate Geometry
Applying Reflection
Solving Spatial Problems
What the Research Says
* Although NCTM recommends that elementary curricula ask
students to use” concrete models, drawings, and dynamic
geometry software so that they can engage with geometric
ideas” (NCTM 2000, 41), “research continues to indicate
that, regrettably, little geometry is taught in the elementary
grades, and that what is taught is often feeble in content
and quality” (Fuys and Liebov 1993).
What the Research Says (continued)
* The works of Pierre and Dina van Hiele (Van de Walle 2007,
400-404) has led the way for other mathematics
researchers to better understand the different levels of
geometric understanding and the variety of experiences
that students need in order to move comfortably to the
next level. We know that levels have nothing to do with the
grade students are in or with their age, but rather relate to
experiences to which students are exposed and in which
they participate.
What to Do
* Go on a shape hunt and have students identify
shapes in their classroom, school, and home
* Combine geometry with number concepts by
having students find different shapes on activity
* Select math-related literature that shows children
accurate plane figures.
* Develop “concept cards” of examples and
What to Do (continued)
* Develop some “best examples” (“clear cases
demonstrating the variation of the concept’s
attributes” [Tennyson, Youngers, and Suebsonthi
1983, 282]) for each of the two- and threedimensional shapes included in your curriculum.
* Ask students questions about these examples to
determine whether they recognize the important
properties of each.
What to Do (continued)
* Encourage students to describe, draw, model, identify,
and classify shapes as well as predict what the results
would be for combining and decomposing these.
* Take care in selecting posters, math-related literature,
and other commercial displays. Often these items
include inaccurate examples of shapes (show
rectangles with only two long and two short sides)
and incorrect shape names (ellipses are labeled
“ovals” and rhombuses are labeled “diamonds”).
What to Do (continued)
* Allow students to create shapes from a variety of
materials so they see regular as well as irregular
*Have students use Venn diagrams to list common
attributes and to classify figures.
* Play games like “Guess My Shape,” where clues are
given, students draw a shape after each clue, and
then determine the shape being described after
all the clues have been read.
What to Do (continued)
* Incorporate other areas of geometry into activities
with shapes (for example, creating tessellations
and transforming shapes through rotations,
translations, and reflections, as well as
combining shapes) to give students opportunities
to spend more time manipulating and exploring
with plane figures.
What to Look For
* Listen to see if students are able to classify shapes
in a variety of ways.
* Look to see if students can name shapes regardless
of their position.
* Watch to see if students can create both regular
and irregular polygons.
Questions to Ponder
1) How can you use geometry activities in other
content areas to help students see the purpose of
understanding shapes and concepts surrounding
2) How does your knowledge about shapes impact
your comfort level in teaching geometric ideas?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on two-dimensional
What the Research Says
* Fuys and Liebov (1993) identify various misconceptions
children have about geometric shapes. When they
undergeneralize, students include irrelevant characteristics.
When they overgeneralize, they omit key properties.
* Language-related misconceptions occur when they create
their own inaccurate definitions (for example, diagonal
means slanty).
What the Research Says
* Clements and Battista (1992) found that children often form a
geometric concept by noticing characteristics and
developing an “average representation” for any new
* Research on visual discrimination by Hoffer (1977) indicates
that young children sometimes lack the ability to
distinguish similarities and differences between objects.
What to Do
* Look at various objects in the classroom
and name these figures correctly. The
geometry words should be placed on a
mathematics word wall, along with various
pictures or other real objects and wooden
or foam objects that are made from these
solid figures.
What to Do (continued)
* Offer students a range of activities in which they find,
color, name, and discuss the solid figures they need
to learn. This enables them to generalize the
characteristics of a solid figure so that size, color, and
other unimportant attributes aren’t confusing.
* Ask questions such as, “Is it OK for this to be a cone
even though it’s smaller than this cone?” “Can this still
be a cylinder even though it’s not red?” “Would this
still be a called a sphere even it was made out of
What to Do (continued)
* Ask students to determine the number of vertices,
faces, and edges each figure has, and be sure to have
them name the plane figures making up these
faces/surfaces. Unfolding and then refolding “nets,”
made of cardstock, allows students to decompose
these figures to better see what each is made out of.
* Play “Guess the Shape”, which provides students with a
logical thinking activity while it reinforces the name of
the solid figure and the plane figures as well as other
What to Do (continued)
* Provide students with geometric analogies so
that they begin to look at the attributes of
specific solid figures and see the difference
between these and plan figures..
* Teach students how to draw various threedimensional figures. Be sure to discuss the
attributes of these figures, naming the
surfaces as plane figures.
What to Do (continued)
* Have students listen to The Important Book, by
Margaret Wise Brown. Give every four students
a different labeled picture of a solid figure. Use
a sphere, cube, cone, cylinder, pyramid, and
rectangular prism. Have students brainstorm
things that the shape reminds them of and
attributes of the shape. Reread The Important
Book and ask students what style the author is
using on each page. Have them then work
together to create their own page in a book
called The Important Thing About Solid Figures.
What to Look For
* Are students able to describe and classify
geometric solids in a variety of ways?
* Can students name geometric solids
regardless of their position?
Questions to Ponder
1) How might you reinforce the names and
characteristics of solid figures while still
introducing the names of plane figures?
2) How can you use what you know about teaching
letter and word recognition to support teaching
shape recognition?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on three-dimensional
What the Research Says
* NCTM (2000) expects students to be able to specify location
and describe spatial relationships using coordinate
geometry. Students in grades Grades 3-5 should be able to:
Describe location and movement using common language and
geometric vocabulary.
Make and use coordinate systems to specify locations and describe
Find the distance between points along horizontal and vertical lines of a
coordinate system.
What the Research Says
* “Coordinate systems are an extremely important form of
representation” (Van de Walle 2007, 437). This
important idea translates into students’ ability to
analyze other geometric ideas such as
transformations. Later, coordinate geometry plays a
crucial role in the representation of algebraic
What to Do
* Encourage students to articulate descriptions of
location, direction, distance related to current of
future positions.
* Generate a list of vocabulary words for location,
direction and distance.
Location: over, under, behind, between, above, below
Direction: left, right, up, down, north, south, east, west,
diagonal, clockwise
Distance: near, far, long, short, inches, miles
What to Do (continued)
* Create a series of steps using the location,
direction, and distance words, and allow
students to act them out in order to reach a
specific destination. When students generate
directions for others to follow, or attempt to
follow the directions of their peers, they
become aware of the importance of direction
and distance when seeking location.
What to Do (continued)
* Have students work in small groups with a large
piece of graph paper. Let each student select a
game piece and place it on the grid. Students
should name the location and then respond to
questions such as:
How would you describe your location compared to mine?
How far are you from the origin?
Which is the shortest path for you to reach another
Do you share a coordinate with anyone else?
What to Do (continued)
* Provide activities for students to plot points and
additional activities that require students to name
* Make connections to real-world applications using
stories and maps. For example: Plan field trips,
plan scavenger hunts, use city street maps to
explore multiple ways to get to the same
* Expand the coordinate grid to include negative
values when students are ready.
What to Look For
* Are students able to correctly identify the
coordinates for a specific space/point?.
* Can students navigate the coordinate grid
by describing location, distance, and
Questions to Ponder
1) How can you provide meaningful opportunities for
students to navigate a coordinate grid and
describe location, direction, and distance? How
can they be involved in the construction of such
2) Think about the errors your students make when
they are naming or plotting points? What
questions can you ask to diagnose the
Take the next 5 minutes to
mark the places in your
materials where students
work on foundational ideas
about coordinate
What the Research Says
* NCTM (2000) says that in prekindergarten through
grade 2 all students should be able to:
Recognize and apply slides, flips, and turns.
Recognize and create shapes that have symmetry.
What the Research Says (continued)
* NCTM (2000) says that all students in grades Grades 3-5
should be able to:
Predict and describe the results of sliding, flipping, and turning
two-dimensional shapes.
Describe a motion or a series of motions that will show that two
shapes are congruent.
Identify and describe line and rotational symmetry in two- and
three-dimensional shapes and designs.
What the Research Says (continued)
* “Younger students generally ‘prove’ (convince
themselves) that two shapes are congruent by
physically fitting one on top of the other. But students
in grades Grades 3-5 can develop greater precision as
they describe the motions needed to show
congruence” (NCTM 2000, 167).
What the Research Says (continued)
* Young children also create pictures with rotational symmetry
using, for example, pattern blocks. But they will have
difficulty explaining what they did and recognizing that the
figure shows rotational symmetry. These informal
explorations are important because they prepare students
to be able to understand and describe rotational symmetry
in later grades.
What the Research Says (continued)
* Providing concrete materials and introducing paperfolding activities are important; however,
technological experiences enhance students’
understanding of transformations, symmetry, and
What to Do
* Allow students to role-play flips (reflections),
slides(translations), and turns(rotations) with their
* Provide students with pattern blocks, attribute blocks,
or tangram puzzles. They will naturally use
transformations to create designs. Ask them to
explain how a design was made to reinforce the
vocabulary of flip, slide, and turn.
* Find all possible different arrangements for five
connected squares.
What to Do (continued)
* Let students use materials to model vertical,
horizontal, and diagonal reflections across a line
of reflection.
* Make rotational tools. Students rotate a figure
around a point to view how its position looks at
different points (quarter- or half-turns).
What to Do (continued)
* Allow students to investigate transformations on
the computer.
* Show examples and nonexamples of symmetrical
designs and pictures.
* Model how to place a mirror and/or a GeoReflector
perpendicular to a design or picture to show a
symmetrical reflection. Let students explore
symmetry with these tools.
What to Do (continued)
* Provide paper-folding experiences.
* Provide geoboards for students to create
symmetrical designs or pictures. Geoboards
allow concrete experience with rotational
symmetry. Students make a design on the
geoboard and predict how it will look when
turned or rotated. They record predictions on dot
geoboard paper and check their predictions by
actually turning the geoboard.
What to Look For
* Are students able to describe and model
transformations accurately?
* Are students able to recognize, model, and
describe symmetry and congruence?
Questions to Ponder
1) How will you change the way you are currently
teaching transformations? If you feel a change is
not necessary, explain what you are currently
doing to help students understand this concept.
2) What is one task that has helped your students
understand the idea of symmetry? Is there a way
you will change the task to provide students with
a richer understanding of symmetry? Describe
your enhancement.
Take the next 5 minutes to
mark the places in your
materials where students
work on transformations,
symmetry, and
What the Research Says
* The van Hieles propose instruction rather than
maturation as the most significant factor contributing
to the development of geometric thought (Burger and
Shaughnessy 1986).
* Geometric thinking can be enhance through meaningful
What the Research Says (continued)
* “Any activity that requires students to think about a
shape mentally, to manipulate or transform a
shape mentally, or to represent a shape as it is
seen visually will contribute to the development
of students’ visualization skills” (Van de Walle
2007, 443).
What to Do
* Provide visualization opportunities for students to
develop their “mind’s eye.”
* Allow students to manipulate and build using a
variety of materials, such as multilink cubes,
wooden blocks, and connecting cubes. These
tactile experiences provide opportunities to view
different transformations of figures.
What to Do (continued)
* Present small groups of students with several nets
and geometric solids. Challenge the students to
match each net to its corresponding solid. Allow
students to confirm their predictions by cutting
and folding the nets to form the solids.
Additionally, instruct students to trace each face
of a solid to form their own nets. They may then
cut out their nets and fold to form solids.
What to Do (continued)
* Let students make two-dimensional representations of
three-dimensional figures using Cartesian graph
paper or isometric graph paper.
* Ask students to draw various polygons using a ruler.
Students may then cut out the polygons and draw a
line segment connecting any two points on the
polygon. Encourage students to name the original
polygon and the two new polygons created with the
line (slice).
What to Do (continued)
* Challenge students to:
- Slice a triangle to make a trapezoid and a
- Slice a pentagon to make two quadrilaterals.
- Slice a hexagon to make two pentagons.
* Have students fold a piece of paper and make a
single cut through both layers. Have them predict
what will result when they unfold what they have.
Then have them discuss what happened and how
it compared to their prediction.
What to Do (continued)
* Provide puzzles for students to complete using
tangrams, pattern blocks, pentominoes, and
GeoReflectors. (For example: make specific
shapes out of tangrams and pentominoes; use
GeoReflectors for symmetry and reflections.)
*Expose students to real-world applications of twodimensional drawings such as blueprints, house
plans, or aerial-view photographs.
What to Do (continued)
* Present students with sets of cards showing the
top, front, and side view of a figure made with
connecting cubes. Students then construct the
figures using the visual clues.
* Look for geometric shapes and figures in works of
What to Look For
* Can students describe shapes and figures and
relate them to real-world objects.
* Are students able to match two-dimensional
representations with the corresponding threedimensional objects?
* Are students able to describe mental images of
Questions to Ponder
1) What tools and materials are available at your
school to enhance your instruction of geometry
concepts? What may be needed to enhance your
2) It is important for students to develop a strong
vocabulary when studying geometry. How can
you effectively incorporate vocabulary
development into your instruction of spatial
problem solving?
Take the next 5 minutes to
mark the places in your
materials where students
work on spatial problem
Working in Groups of 3,
take the packet of Geometry Activities.
Go through the Activities, talking,
taking notes, planning how activities of
this sort might be used in conjunction
with your curriculum materials.
Discuss how these activities can
Undo Misconceptions.
Reading an Analog Clock
Determining the Value of Coins
Units Versus Numbers
Distinguishing Between Area and
* Overgeneralizing Base-Ten Renaming
What the Research Says
* Fully understanding how to read the hour and minute hands
of an analog clock demands a “conscious switching for
quarter and half turns in relation to either the hour past of
the hour approaching” (Ryan and Williams 2007, 99).
* Later, these clockwise fractions of a turn need to be converted
into fifteen-, thirty-, and forty-five minute intervals.
What the Research Says (continued)
* To complicate matters, students need to learn to read
an analog clock both clockwise and counterclockwise.
* These skills, are hard to master, especially given the
non-decimal nature of time.
* “Measuring time causes problems for children right
through the primary school” (Doig et al. 2006).
What the Research Says (continued)
* Confusion over the hour and minute hands, the
language associated with time (quarter past, quarter
of, half past), and the fractions associated with
periods of time create all sorts of problems.
* By the intermediate grades elapsed time can be more
than a challenge for many students.
What the Research Says (continued)
* In Connect to Standards 2000: Making the Standards
Work at Grade 2 (Fennell et al. 2000), the authors
suggest that when first introducing how to tell time
on an analog clock, teachers use only the hour hand,
so that students learn its relative position as time
passes. They also suggest placing the numbers on the
clock on a number line to demonstrate the divisions
of a clock.
What the Research Says (continued)
* In Understanding Mathematics in the Lower Primary Years (1997),
Haycock and Cockburn indicate that part of the problem students
have learning to tell time and grasping the passage of time is “the
multitude of words relating to time” : “how long, second, minute,
hour, day, week, fortnight, month, quarter, year, leap year,
decade, century, season, spring, summer, autumn, winter,
weekend, term, lifetime, sunrise, sunset, past, present, future,
evening, midnight, noon, earlier, prior, following, never, always,
once, eventually, instantly, in a jiffy, meanwhile, sometime,
sooner, during” (1997, 103-104).
What to Do
* Using an hour-hand-only clock, position the hand
directly on a numeral and have students say the
‘o’clock time. Then position the halfway between
two numerals and have students say the halfhour time. Do the same for quarter-past, quarter
of, and three-quarters past.
* Have students match clock faces with phrase cards
to connect the time vocabulary with the face on
an analog clock.
What to Do (continued)
* Play “How Many Minutes After?” which has students
making a connection between the numeral that
the minute hand is pointing to and the number of
minutes after the hour this represents.
* Provide students with story problems that give
them practice drawing the hands on an analog
clock face, writing the digital time, and working
on elapsed time problems where they write to
explain how they got their answer.
What to Look For
* Can students correctly match the phrase, digital time,
and analog clock face?
* Do students have a strategy for determining the number
of minutes after the hour each of the numerals on a
clock face represents?
* Are students able to articulate different ways to say the
time at quarter-hour intervals?
* Can students solve elapsed-time problems and write to
explain how their answer was obtained?
Questions to Ponder
1) What are some other activities that you’ve done in
your classroom that help students learn how to
tell time (on the hour, half-hour, quarter-hour,
five-minute intervals, or to the minute?
2) How can you use the technology that is already in
your classroom to help students learn to tell
Take the next 5-8 minutes
to mark the places in your
materials where students
work on telling time on an
analog clock.
What the Research Says
* Randall Drum and Wesley Petty (1999) find that although coins
are a concrete model, because their value is nonproportional
to their size, they become abstract when those values are
* In their research, Douglas Clements and Julie Sarama (2004)
found that it takes a long time to master money skills because
children have to both count on and skip count in different
increments. In traditional instruction, young students are often
expected to use mental computation before they completely
understand the concept of addition.
What the Research Says
* Van de Walle and Lovin’s research (2006) reveals
that in order for coin values—5, 10, 25—to
make sense, students need to understand what
they mean. Children look at a dime without
thinking about the countable pennies it
What to Do
*Use children’s literature to pose problem-solving
questions that support students’ understanding
of counting larger amounts.
* Skip-count by 5s, 10s, 25s, and 50s. Use coins to
help facilitate the skip-counting.
* Play “earn $1.00” with pairs of students throwing
two dice and gradually earning their dollar as
they make exchanges from pennies to nickels to
quarters to fifty cent pieces and/or to dollars.
What to Do (continued)
* Play “shift count” having students count by dimes,
holding up your hand to shift to nickels then
quarter then back to dimes, etc..
* Provide opportunities for students to place coin
amounts in order according to values.
* Provide sets of coins for students to count that are
only a nickel and some pennies or a dime and
some pennies. Later, include a quarter and some
pennies, etc.
What to Do (continued)
* Provide hundreds charts for students to visually and
concretely see the amount of pennies in nickels,
dimes, and quarters (Drum and Petty 1999, 264-68).
* Encourage students to mentally add numbers that
represent the values of different coins.
* Target an amount and challenge students to find all the
different ways to make that amount using specific
* Provide opportunities for students to solve logic riddles
about coins. Expect students to justify their answers.
What to Look For
* Are students able to organize coin sets before
determining the value?
* Can students find multiple ways to make a given
Questions to Ponder
1) What difficulties do your students encounter when
they are counting coins?
2) What new strategy can you implement to help
students count coins or make change?
Take the next 5 minutes to
mark the places in your
materials where students
work on the value of coins.
What the Research Says
* Constance Kamii (2006) says that teachers almost always ask
students to produce a number about a single object rather
than asking them to compare two or more objects (156).
* The purpose for measurement is not immediately obvious
when children measure numerous isolated pictures of
objects. It’s not surprising they view these as procedural
tasks only.
What the Research Says (continued)
* In Engaging Young Children in Mathematics (2004), Clements and
Sarama describe foundational concepts related to length
measurement (301-304).
Partitioning. Dividing an object into same-size units.
Unit iteration. Iterating a unit repeatedly along the length of an
Transitivity. The understanding that if the length of one object is
equal to the length of a second object, which is equal to the length
of a third object that cannot be directly compared to the first
object, the first and third are also the same length.
What the Research Says (continued)
Conservation. The understanding that as an object moves, its
length does not change.
Accumulation of distance. When you iterate a unit along an object’s
length and count the iterations, the number works convey the space
covered by all units counted up to that point.
Relation between number and measurement. Many children fall
back on their earlier counting experiences to interpret measuring
tasks. Students who simply read a ruler procedurally have not
related the meaning of the number to its measurement.
What the Research Says (continued)
* “Although researchers debate the order of the
development of these concepts and the ages at which
they are developed, they agree that these ideas form
the foundation for measurement and should be
considered during any measurement instruction”
(Clements and Sarama 2004, 304).
What to Do
* Ask students to estimate the size of an object first
before measuring it. Expect students to explain
why they think their estimate is reasonable.
* Allow students to measure real-world objects,
rather than only pictures on paper. Environmental
objects force children to approximate measure, a
more realistic application of measurement.
What to Do (continued)
* Encourage students to measure the same object
with a variety of nonstandard units and standard
units. This reinforces the importance of a unit’s
size, and that some units are more efficient for
measuring an object.
* Provide students who are having difficulty, with
rulers that have fewer markings .
What to Do (continued)
* Bridge nonstandard units to standard units by
providing manipulatives that are “standard” size.
* Allow students to make their own rulers to help
them understand that it’s not just a tool used to
complete a procedural task. Engage students in
comparing student-made rulers to standard
What to Do (continued)
* Use rulers with the “0” mark a short distance from
the edge. Students will be engaged in thinking
about both endpoints when measuring with this
type of ruler. They will also be focusing on units
(and not markings) when they measure.
* Ask students to develop and explain strategies for
measuring curved and crooked lines or other
hard-to-measure objects.
What to Look For
* Do students understand that when units are small,
more are needed to measure, and when units are
larger, fewer units are needed.
* Can students explain why they think their
estimates are reasonable?
* Can students identify the starting point on each
Questions to Ponder
1) What difficulties have your students had in
measuring length? How will you change your
instruction to avoid future misconceptions?
2) What activities or strategies can you use to help
your students develop the foundational concepts
described in this section?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on understanding
measurement units versus
What the Research Says
* Big ideas are that area is a measure of covering and that
perimeter (length around) is a measure of distance.
* Area and perimeter are often taught together or in
immediate succession. This combination or sequence
may cause confusion, because both area and
perimeter require students to consider the boundaries
of shape.
What the Research Says (continued)
* “It is common fallacy to suppose that the area of a region is
related to its perimeter” (Leibeck 1984).
* Haycock (2001) notes that this relationship proves a
contradiction to ideas of conservation. He stresses that
rearranging a constant perimeter to form a new shape
conserves the perimeter, but not the area. It is therefore
important to provide meaningful problems that allow
students to experience this phenomenon.
What the Research Says (continued)
* “Through problem-solving tasks, students develop an
understanding of math content and ultimately use
that content understanding to find solutions to
problems. Problem solving is both the process by
which students explore mathematics and the goal of
learning mathematics” (O’Connell 2007).
What the Research Says (continued)
* It is important for students to find their own
strategies and algorithms to measure specific
attributes rather than simply plug numbers into
formulas presented to them without context or
What to Do
* Allow students to explore area by covering the
surface of a variety of objects with nonstandard
units. Then move on to covering surfaces using
congruent units such as index cards or multilink
cubes. Using a consistent unit allows students to
compare the areas of different shapes.
* Connect the concepts of area and perimeter to
meaningful scenarios like those in children’s
What to Do (continued)
* Give students a set amount of squares and
triangles (pattern blocks work well) to use to
make a design. Compare the varying designs
created. Compare the area of each. Discuss the
notion of conservation of area. Get students to
recognize that although the designs vary in
appearance, the area is consistent for all.
What to Do (continued)
* Have students use pattern blocks or square tiles to
create a variety of designs with a constant perimeter.
Instruct students to compare the areas of the shapes
made with like materials. Discuss what trends they
notice about the area of shapes that all have the same
* Use a multiplication chart to illustrate the connection
between the area of a rectangle and multiplicative
arrays. Such connections help students construct their
own formulas based on their conceptual
understanding rather than mimicking a formula
without context
What to Do (continued)
* Provide cutouts of rectangles, triangles, trapezoids, and
circles and have students develop strategies to
compare the area of each.
*Have students shade, fold, or even cut square-grid paper
to explore the relationship between area and the
dimensions of the length and width:
What happens to the area of a rectangle if the length is
How is the area affected if both the length and width are
doubled? Why?
If the length and width were cut in half, what would happen
to the area?
What to Look For
* Do students understand that the distance around
the perimeter of a figure is different from the
amount of space covered by a figure?
* Can students cover a figure with units and count
the number of units used?
* Are students able to see that the size of the units
affects the number of units needed to measure
the area or perimeter of a figure?
Questions to Ponder
1) How can you help students distinguish
between area and perimeter in a
meaningful context?
2) What materials are available in your school
to make the concepts of measurement a
hands-on experience?
Take the next 5-8 minutes
to mark the places in your
materials where students
work on area and
What the Research Says
* In a 1984 Arithmetic Teacher article, Jim Hiebert writes,
“Effective instruction should take advantage of what
children already know or are able to learn and then relate
this knowledge to new concepts that might be more
difficult to learn” (22-23).
* The best way to help students see the relationship between
units of measure is by having them use measuring devices
during mathematics class.
What to Do
* Give students time to explore with whatever
units of measure are being used, prior to
giving students problems to solve
(whether they be story problems or
numerical expressions).
* Have students record different ways to
represent the same unit of measure.
What to Do (continued)
* Have students discuss what they might have to
do if they want to know what the difference
would be between two units of measure when
subtracting involves renaming. Ask students
to share their ideas for solving a story
problem and then discuss whether the
answers make sense. Try them out with units
of linear measure, liquid measure, and time.
What to Look For
* Look to see that students represent addition and
subtraction of time, length, or fractions with the
appropriate conversions.
* Have students use illustrations, whenever possible,
to represent different ways to name the same
* Provide students with manipulative materials to
represent fractions in multiple ways.
Questions to Ponder
1) What other instructional experiences will help
students better understand the base-ten
relationship and when (and when not) to use it
for renaming?
2) What other measurement tools or materials ought
to be introduced to students in third through fifth
grades so they can use them to solve problems?
Take the next 5 minutes to
mark the places in your
materials where students
work on conversions and
Working in Groups of 3,
take the packet of
Measurement Activities. Go through the
Activities, talking, taking notes,
planning how activities of this sort
might be used in conjunction with your
curriculum materials.
Discuss how these activities can
Undo Misconceptions.
* Sorting and Classifying
* Choosing an Appropriate Display
* Understanding Terms for Measures of
Central Tendency
* Analyzing Data
* Probability
Final Thoughts

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