MTTC Lower Elementary Test
Questions and Answers
A. Having one student count their own set of counters aloud does not address the
misconception that the number of counters in a set depends on how they are arranged.
B. This manner of addressing the misconception does not necessarily build upon the
students' understanding because the teacher did not check whether they grasped the
explanation or provide them with opportunities to explain in their own words why the
total number of objects in each group is the same.
C. CORRECT. By asking probing questions about the total number of counters each
student has, the teacher can help students move beyond a naïve conception that bigger
equals more and deepen their conceptual understanding of counting and cardinality.
D. The affordances created by combining the counters into a large group do not offer
these students more substantive insights into understanding counting and cardinality
concepts than the affordances created by - ANS-1. A kindergarten teacher observes as
a small group of students practice comparing numbers and quantities using
manipulatives. Each student has four counters. One student's counters are spaced
farther apart than the other students' counters, and several members of the group claim
that student has more counters than everyone else. The teacher can build on the
students' understanding of counting and cardinality by:
A. encouraging the one student to count their counters for the group.
B. identifying the error and moving the one student's counters closer together.
C. asking probing questions about the total number of counters each student has.
D. prompting the group to combine their counters and count how many they have in all.
A. This activity does not require students to compare numbers.
B. The skills required to measure and sort rational numbers—the numbers that would
be used to describe shoe lengths—are too advanced to be included in a first-grade
lesson activity about comparing number values.
C. This activity is not developmentally appropriate because the concepts of number
comparison should be introduced to students without requiring them to also apply
,additional mathematical knowledge that does not directly support their understanding of
these concepts.
D. CORRECT. The alignment and rigor of the activity is developmentally appropriate for
introducing first-grade students to the concept of comparing number values and the
kinesthetic activity promotes their engagement. - ANS-2. A first-grade teacher plans
initial lessons on comparing number values. Which of the following activities would be
developmentally appropriate and engaging when introducing this concept?
A. Students form multiple-digit numbers using index cards labeled with the digits 1, 2, 3,
and 4.
B. Students measure the lengths of classmates' shoes and then sort the shoes from
smallest to largest.
C. Students discuss the values of different piles of coins, such as a pile of 5 quarters
and a pile of 5 pennies.
D. Students stand between two different quantities and arrange their arms into a
greater-than or less-than symbol.
A. CORRECT. The teacher should review the meaning and function of the equal sign
because students who agree that only the first two equations are correct may be
interpreting the equal sign to be a symbol that indicates the result of the last operation
(i.e., they would likely believe that the third equation should be written as 10 − 3 = 7 + 2
or 10 − 7 = 5 + 2).
B. None of the equations shown makes use of addition and subtraction as inverse
operations.
C. A review of benchmark equations (e.g., sums and differences involving 5 and 10) is
not necessary because students have previously agreed that 7 = 10 − 3 and 7 = 5 + 2.
D. There is evidence that the students are interpreting "=" to mean "the result of the last
operation," and reviewing the concepts of "greater than" and "less than" would not
address this misconception directly or efficiently. - ANS-3. First-grade students consider
the following equations.
7 = 10 − 37 = 5 + 210 − 3 = 5 + 2
Most students state that the last equation is incorrect. In order to address the students'
misconception, the teacher should plan a review of which of the following concepts?
A. meaning and function of the equal sign
B. how addition and subtraction are related
,C. the use of benchmark equations to find the answer
D. the concepts of "greater than," "less than," and "equal to"
A. Students should explore the relationship between counting and addition before they
learn to add without a counting strategy.
B. Describing games in which dice may be used does not direct students' attention to
attributes of dice that makes them useful manipulatives for learning addition: the neat
arrangement of dots on the faces, the unique numbers of dots on each side, and the
many different outcomes that can be formed by rolling two dice.
C. CORRECT. Providing students with opportunities to explore manipulatives on their
own stimulates their curiosity, interest, and comfort with them and prepares students to
explore how they may be used as mathematical tools.
D. Posting addition tables does not support students' ability to understand how the
mathematical concepts used to count a single collection of objects can be extended to
count two combined collections of objects through addition. - ANS-4. A first-grade
teacher uses an activity involving dice to help students make the jump from counting to
addition. Students roll two dice, then determine the sum of the dots that are face up. On
a piece of paper, students draw their dice as an addition problem and write the problem
using numbers. One student's work is shown.
Two dice are shown above an equation. The left die shows 3 pips, the right die shows 4
pips, and the equation reads 3 plus 4 equals 7.
The teacher can increase students' success by taking which of the following actions
before explaining the activity?
A. teaching students how to add without using a counting strategy
B. providing context by describing games in which dice may be used
C. giving students the opportunity to become familiar with dice and their dots
D. posting addition tables at the front of the room and on each student's desk
A. The color yellow and the object name "cup" are not used to describe attributes of
three-dimensional shapes.
B. This question does not direct students to focus on the three-dimensional attributes of
different plates.
C. The name of an object is not a three-dimensional attribute.
, D. CORRECT. Students answer this question by describing the similarities and
differences of the three-dimensional attributes of the plate and the cup (e.g., "The plate
is short and flat, and the cup is tall", "The plate and the cup both have round edges"). -
ANS-5. The dramatic play area of a prekindergarten classroom is set up a like a kitchen,
with three-dimensional blocks standing in for plates and cups. While children work in
stations pretending to make and serve food to each other, the teacher can help the
children extend their understanding of three-dimensional attributes by asking them:
A. to find the yellow cup.
B. which plate is their favorite.
C. to bring a certain object to the table.
D. how the plate and the cup are the same and different.
A. The teacher conducts the conversation about the pattern in a way that solicits a wide
range of input from students without commenting about whether one description is more
efficient than another.
B. All students—not only those who misinterpreted the pattern—may wish to revise their
explanations when they consider the other students' responses.
C. CORRECT. By recording a wide range of student responses, the teacher exposes
students to a rich variety of word choices and mathematical details that students can
draw upon to refine their own explanation of the pattern.
D. Dismissing students' responses creates a counterproductive learning dynamic, and
while students may judge details in the responses to be accurate or inaccurate, the
teacher is not likely recording the responses with the intention of having the students
identify which should be removed. - ANS-6. A teacher shares this geometric pattern with
the class.
A pattern formed from geometric shapes arranged in a row is shown. The shapes, from
left to right, are a triangle, a square, a pentagon, a hexagon, a triangle, and a square.
The teacher asks students to explain the order of the shapes in the pattern. As students
share their explanations, the teacher writes these student explanations on the board.
"The shapes go from smallest to biggest. Then they start over."
"They are in order by the number of corners."
"The pattern is the least number of sides to the greatest number of sides, then it starts
again."
By writing these explanations on the board, the teacher:
Questions and Answers
A. Having one student count their own set of counters aloud does not address the
misconception that the number of counters in a set depends on how they are arranged.
B. This manner of addressing the misconception does not necessarily build upon the
students' understanding because the teacher did not check whether they grasped the
explanation or provide them with opportunities to explain in their own words why the
total number of objects in each group is the same.
C. CORRECT. By asking probing questions about the total number of counters each
student has, the teacher can help students move beyond a naïve conception that bigger
equals more and deepen their conceptual understanding of counting and cardinality.
D. The affordances created by combining the counters into a large group do not offer
these students more substantive insights into understanding counting and cardinality
concepts than the affordances created by - ANS-1. A kindergarten teacher observes as
a small group of students practice comparing numbers and quantities using
manipulatives. Each student has four counters. One student's counters are spaced
farther apart than the other students' counters, and several members of the group claim
that student has more counters than everyone else. The teacher can build on the
students' understanding of counting and cardinality by:
A. encouraging the one student to count their counters for the group.
B. identifying the error and moving the one student's counters closer together.
C. asking probing questions about the total number of counters each student has.
D. prompting the group to combine their counters and count how many they have in all.
A. This activity does not require students to compare numbers.
B. The skills required to measure and sort rational numbers—the numbers that would
be used to describe shoe lengths—are too advanced to be included in a first-grade
lesson activity about comparing number values.
C. This activity is not developmentally appropriate because the concepts of number
comparison should be introduced to students without requiring them to also apply
,additional mathematical knowledge that does not directly support their understanding of
these concepts.
D. CORRECT. The alignment and rigor of the activity is developmentally appropriate for
introducing first-grade students to the concept of comparing number values and the
kinesthetic activity promotes their engagement. - ANS-2. A first-grade teacher plans
initial lessons on comparing number values. Which of the following activities would be
developmentally appropriate and engaging when introducing this concept?
A. Students form multiple-digit numbers using index cards labeled with the digits 1, 2, 3,
and 4.
B. Students measure the lengths of classmates' shoes and then sort the shoes from
smallest to largest.
C. Students discuss the values of different piles of coins, such as a pile of 5 quarters
and a pile of 5 pennies.
D. Students stand between two different quantities and arrange their arms into a
greater-than or less-than symbol.
A. CORRECT. The teacher should review the meaning and function of the equal sign
because students who agree that only the first two equations are correct may be
interpreting the equal sign to be a symbol that indicates the result of the last operation
(i.e., they would likely believe that the third equation should be written as 10 − 3 = 7 + 2
or 10 − 7 = 5 + 2).
B. None of the equations shown makes use of addition and subtraction as inverse
operations.
C. A review of benchmark equations (e.g., sums and differences involving 5 and 10) is
not necessary because students have previously agreed that 7 = 10 − 3 and 7 = 5 + 2.
D. There is evidence that the students are interpreting "=" to mean "the result of the last
operation," and reviewing the concepts of "greater than" and "less than" would not
address this misconception directly or efficiently. - ANS-3. First-grade students consider
the following equations.
7 = 10 − 37 = 5 + 210 − 3 = 5 + 2
Most students state that the last equation is incorrect. In order to address the students'
misconception, the teacher should plan a review of which of the following concepts?
A. meaning and function of the equal sign
B. how addition and subtraction are related
,C. the use of benchmark equations to find the answer
D. the concepts of "greater than," "less than," and "equal to"
A. Students should explore the relationship between counting and addition before they
learn to add without a counting strategy.
B. Describing games in which dice may be used does not direct students' attention to
attributes of dice that makes them useful manipulatives for learning addition: the neat
arrangement of dots on the faces, the unique numbers of dots on each side, and the
many different outcomes that can be formed by rolling two dice.
C. CORRECT. Providing students with opportunities to explore manipulatives on their
own stimulates their curiosity, interest, and comfort with them and prepares students to
explore how they may be used as mathematical tools.
D. Posting addition tables does not support students' ability to understand how the
mathematical concepts used to count a single collection of objects can be extended to
count two combined collections of objects through addition. - ANS-4. A first-grade
teacher uses an activity involving dice to help students make the jump from counting to
addition. Students roll two dice, then determine the sum of the dots that are face up. On
a piece of paper, students draw their dice as an addition problem and write the problem
using numbers. One student's work is shown.
Two dice are shown above an equation. The left die shows 3 pips, the right die shows 4
pips, and the equation reads 3 plus 4 equals 7.
The teacher can increase students' success by taking which of the following actions
before explaining the activity?
A. teaching students how to add without using a counting strategy
B. providing context by describing games in which dice may be used
C. giving students the opportunity to become familiar with dice and their dots
D. posting addition tables at the front of the room and on each student's desk
A. The color yellow and the object name "cup" are not used to describe attributes of
three-dimensional shapes.
B. This question does not direct students to focus on the three-dimensional attributes of
different plates.
C. The name of an object is not a three-dimensional attribute.
, D. CORRECT. Students answer this question by describing the similarities and
differences of the three-dimensional attributes of the plate and the cup (e.g., "The plate
is short and flat, and the cup is tall", "The plate and the cup both have round edges"). -
ANS-5. The dramatic play area of a prekindergarten classroom is set up a like a kitchen,
with three-dimensional blocks standing in for plates and cups. While children work in
stations pretending to make and serve food to each other, the teacher can help the
children extend their understanding of three-dimensional attributes by asking them:
A. to find the yellow cup.
B. which plate is their favorite.
C. to bring a certain object to the table.
D. how the plate and the cup are the same and different.
A. The teacher conducts the conversation about the pattern in a way that solicits a wide
range of input from students without commenting about whether one description is more
efficient than another.
B. All students—not only those who misinterpreted the pattern—may wish to revise their
explanations when they consider the other students' responses.
C. CORRECT. By recording a wide range of student responses, the teacher exposes
students to a rich variety of word choices and mathematical details that students can
draw upon to refine their own explanation of the pattern.
D. Dismissing students' responses creates a counterproductive learning dynamic, and
while students may judge details in the responses to be accurate or inaccurate, the
teacher is not likely recording the responses with the intention of having the students
identify which should be removed. - ANS-6. A teacher shares this geometric pattern with
the class.
A pattern formed from geometric shapes arranged in a row is shown. The shapes, from
left to right, are a triangle, a square, a pentagon, a hexagon, a triangle, and a square.
The teacher asks students to explain the order of the shapes in the pattern. As students
share their explanations, the teacher writes these student explanations on the board.
"The shapes go from smallest to biggest. Then they start over."
"They are in order by the number of corners."
"The pattern is the least number of sides to the greatest number of sides, then it starts
again."
By writing these explanations on the board, the teacher:
