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Math in FDK Classrooms
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TweetSam Hopkins
on 27 June 2014Transcript of Math in FDK Classrooms
Math in FDK Classrooms
Creating Options to Maximize Mathematical Learning in the Early Years Classroom
by extending large group explicit math instruction
into small group learning
into play
with intentional interaction
in response to childinitiated questions or invitations
in attempts to scaffold understanding of mathematical concepts
using meaningful mathematical knowledge
using observations to inform further teaching and interactions
What the Research Says...
The Importance of Math Language
The Early Years Learner
Prior to the beginning of the 21st century, mathematics education was devalued, with a greater focus in early childhood education on the development of social skills and early literacy. However, because we now know much more about the early years learner, there has been a shift towards more in depth and complex mathematics education in Kindergarten.
Children at Block Play
Building on Everyday Language
When children enter preschool, the have accumulated mathematical knowledge that is based on their experience of life thus far;
They express it with
everyday language
(e.g., “more than” or “bigger than”);
According to Diaz (2008), in each of Piaget’s constructivist theory, Vygotsky’s sociocultural theory, and Lakoff’s metaphor theory, there are elements of how
mathematical language can be
mediated
in everyday play.
Children construct meaning while using language in their interactions with their classmates and with their teachers.
Mediating
is purposeful interaction as the teacher uses "students' speech to bridge their everyday language to mathematical concepts." For example, a student during block play asks a teacher to help. The teacher says, "Instruct me." The student tells where to put the blocks, and the teacher asks for more information ("how?"), using the word "stack," which was understood and affirmed by the student. Another example of a student asking for help prompts the teacher to ask, "Where?" and then she rephrases the student's response with, "Oh, around the perimeter."
Types of Mathematical Language
Math concepts (including "big ideas") need to be presented in a context that is motivating and handson. Furthermore, teachers need to be actively engaged in observing children while they are playing and interacting socially, so that they can act on
teachable moments
and promote learning of mathematical concepts. Of interesting note is that a teacher’s attitude and professional development will have an effect on how much learning can be accomplished in these interactions.
Teachable Moments
Math Standards
References:
Samples of Learning Centres (Numeracy):
Practical Ideas for Creating a Rich Mathematical Learning Environment
The Early Years Teacher
A Successful Approach
Math Talk
Facilitating Experiences
As educators we need to facilitate experiences that allow '
mathematization
' to occur. Educators assist students in transforming their everyday mathematics into a more formalized understanding that can be transferred and applied to other situations.
The educator plays an integral role by making meaningful connections between math strands and real world experiences. These connections should link math concepts students have learned through their own experiences and the ones being taught in the classroom.
How important is it to use appropriate math language for children to develop an awareness and understanding of math concepts in the world around them? What should teachers be looking for when observing children in play, so that they can formulate guiding questions for the children that would increase their understanding of math concepts?
After recording the number of mathematical utterances, Diaz rated the interactions of the teacher, based on whether the teacher succeeded or failed to pick up on the child’s reference to mathematical concepts using everyday language. When these interactions were intentional on the teacher's part, and based on observations in real time, the enhanced math language led to further math concept development. Her published research gives examples of several interactions.
Diaz concludes from the collected data that some teacher interactions were successful in encouraging more student talk. Obviously, "ignoring" or missing the opportunity for interaction is not helpful, but it is surprising how often opportunities were missed, when the teachers knew the purpose for the testing (60%). Other interactions were: a) accept/repeat idea (18%); b) praise or encourage (6%); c) ask questions (6%); and d)
mediating
(10%).
Her study shows that when children interact with teachers while in play, their learning can be greatly enhanced.
Diaz (2008) refers to recommendations of the National Council of Teachers of Mathematics (NCTM). The NCTM is an excellent resource for math teachers of all grades, which recommends curriculum focal points (the content emphases), and connections (other meaningful and essential components for learning the content). The Ontario curriculum's math expectations are in line with their recommendations. To compare with the 5 strands in the Ontario Math curriculum (listed below), the first 3 are emphasized for content, while generally, the last 2 can also be learned when math is integrated with other subjects (science, music, etc.).
Consider the mathrich language which can result when these connections are made.
Number Sense and Numeration (Quantity Relationships; Counting; Operational Sense);
Measurement (Attributes,Units, and Measurement Sense; Measurement Relationships);
Geometry and Spatial Sense (Geometric Properties; Geometric Relationships; Location and Movement);
Patterning (Patterns and Relationships);
Data Management and Probability (Collection and Organization of Data;Data Relationships;Probability).
Sorting Out the Math Talk
Diaz (2008) conducted research of interactions between teachers and students of children in PreK classrooms. Children were observed in block play and their "utterances" (in everyday language) were recorded into the following categories (except for classification and dynamic words, all the rest of the utterances were being used by the children "less than expected by chance"):
Words for classification
 involving sorting into groups and categorizing;
Dynamic words
 involving motions or actions (verbs);
Words denoting a spatial relation
 involving distance, location, direction (e.g., here, there, in, inside, etc.);
Magnitude
 involving size (e.g., tall), comparisons between objects (e.g, same height), judgments (e.g., some, a lot);
Enumeration
 referred to quantity;
Pattern and Shape
 descriptive qualities that are given attention when replicating structures.
Math talk is simply bringing in mathematical terms into the normal classroom conversations. A simple example would be asking students if they have more or less of something.
There are 5 effective talk moves that create meaningful discussion within the FDK classroom.
In order to create a positive math environment, the teacher must lead by example. If the teacher wants students to use mathematical terms, then they must be used in the classroom. Mathematical terms cannot be just a 'one time vistor' in the classroom; talk is important. Students in an FDK classroom are encouraged to talk and communicate with each other. It is through communication and talk that students learn social skills and learn the task they are doing.
5 Productive Talk Moves
1. Revoicing
Repeat what the student has said in the form of a question.
"So you're saying your taller then the chair?"
2. Repeating
Have the student repeat what was said in their own words.
3. Reasoning
Ask students if they agree or disagree is what was said and why.
4. Adding on
prompt students to participate
5. Waiting
Wait for the student to answer, give them to think
1. When teaching math to students, use problems that have meaning for children. If Sarah has 2 cookies and Ben has 4 cookies, who has more?
How many are cookies are there all together?
2. No matter what the problem or question, students will have their own spin on things. They will invent, explain and come up with their own solution to problems.
3. Be creative! Let the students have fun and be creative. They will be more interested in what is being taught.
4. Encourage and support children  through scaffolded lessons and activities that the student can engage in.
5. Help students make connections. These connections will help a student learn the content.
Children's Math Books
Numbers & Counting
1. Double the Ducks
2. Every Buddy Counts
3. Henry the Fourth
4. Missing Mittens
5. Monster Musical Chairs
6. Animals on Board
7. Elevator Magic
8. More or Less
9. Spunky Monkeys on Parade
10. 100 Days of Cool
** All these books are by Stuart J. Murphy MathStart Books**
Measurement
1. The Best Bug Parade
2. A House for Birdie
3. Mighty Maddie
4. Bigger, Better, Best!
5. Racing Around
6. Super Sand Castle Saturday
** All these books are written by Stuart J. Murphy MathStart Books
Algebra
1. Beep Beep, Vroom, Vroom!
2. Seaweed Soup
3. Mall Mania
4. Probably Pistachio
5. 3 Little Firefighters
6. Same Old Horse
** All of these books are written by Stuart J. Murphy MathStart Books**
Being Knowledgeable & Attentive
In order for mathematization to occur educators should:
have a strong knowledge of the curriculum and mathematical concepts
be watching students attentively in order to make connections
For Example
As a child naturally creates and extends a pattern while making a necklace, the educator can ask questions that provoke the student to describe, make predictions and generalizations about the pattern.
Sam H., Kelly K., Veronica R., Rebecca O., Michelle V.
Model and Nurture Positive Attitudes, SelfEfficacy and Engagement
As educators gain confidence in teaching mathematical knowledge, and in helping students extend and formalize their understanding of mathematical concepts, students will also begin to develop more positive attitudes toward mathematics and an increase their sense of selfefficacy.
If both students and teachers are confident and enthusiatic about math, it will increase the amount of mathematical ideas and language used within the classroom.
It is incredibly important that students are exposed to rich mathematical instructional interactions in order to expand upon what they already know. If teachers feel confident in teaching the mathematical concepts they will push the limits of student knowledge in order to stretch their learning.
Confident, enthousiastic teachers, lead to confident and enthusiatic students.
Get Familiarized with Math Concepts and Vocabulary
Compiled by Betsy Reilly, Faculty of Education, Western University
Compiled by Dave Watson, Faculty of Education, Western University
(promoting the "Big Ideas")
When planning your daily routine, take into account that:
•Social interactions foster learning, therefore you need to ensure your day is planned with opportunities for discussion and conversation.
•Communication is key! Building communication through active engagement in small group discussions plays a significant role in enhancing and clarifying children’s mathematical understandings.
•Students need to be exposed to “cognitively demanding mathematical tasks”.
Two Common Mistakes
in Teaching Approaches
"I believe that play is the only important and developmentally appropriate approach for young children."
Situation 2:
Early 1900s
Young children were historically considered not cognitively capable of engaging in the thinking needed to understand mathematics

Thorndike
: believed that children were mathematically inept before second grade, therefore it was not worth even addressing mathematics with them
• Due to this way of thinking, there was virtually no mathematics programing during the early 1900s
Situation 1:
1960s
"I believe that math needs to be highly structured and scripted. Things need to be taught the way I was taught in Elementary school."
Leads children to rely on completely unstructured and incidental learning during play, with little interaction with mathematical concepts.
By the 1960s, there was the establishment of mathematics education beginning at the early elementary level
The change was because it was now thought children needed formal schooling in mathematics in order to think that way

Piaget:
his research showed young children as mathematically curious and as actively constructing mathematical knowledge by interacting with the physical and social world
• However, he also believed they were incapable of abstract and logical thinking until at least age 7
Leads children to a lower curiosity about math and affects the degree of higher level thinking as students believe all they need to know will be told to them by the teacher.
A Balanced Viewpoint
I want to make mathematics fun, engaging and playful! I can do this by incorporating various activities into my structured mathematics lessons. I will do this in ways that ensures activities and lessons are developmentally appropriate for my students.
This enables students to become active and responsible participants in classroom communication.
Small Group Lessons
Table Activities
Learning Centres
21st Century
Developmental psychologists transferred the focus from what young children could not do to what they
could
do
• Research has begun to highlight children’s competencies and education has followed this thinking
Young children are now seen as much more powerful mathematicians than past instructional practices reflect
• By participating in interactions with the social and physical world, young children engage in diverse types of mathematical thinking
• Specifically, this helps to develop intuitive foundational knowledge and skills in mathematics
• These include: onetoone correspondence, cardinality, and basic addition and subtraction
Present Day
Young children spontaneously invent mathematicsbased strategies during everyday play and often make predictions and successfully solve numberrelated reasoning problems
We now know that Kindergarten aged students engage in surprisingly complex intuitive mathematical thinking in:
• Numeracy
• Geometry
• Measurement
• Algebraic Thinking
• Data Analysis
From the current understanding of the early years learner, it is important to challenge our students and allow them to explore complex mathematical concepts
With a Focus on the "Big Ideas" in Math
Full transcriptCreating Options to Maximize Mathematical Learning in the Early Years Classroom
by extending large group explicit math instruction
into small group learning
into play
with intentional interaction
in response to childinitiated questions or invitations
in attempts to scaffold understanding of mathematical concepts
using meaningful mathematical knowledge
using observations to inform further teaching and interactions
What the Research Says...
The Importance of Math Language
The Early Years Learner
Prior to the beginning of the 21st century, mathematics education was devalued, with a greater focus in early childhood education on the development of social skills and early literacy. However, because we now know much more about the early years learner, there has been a shift towards more in depth and complex mathematics education in Kindergarten.
Children at Block Play
Building on Everyday Language
When children enter preschool, the have accumulated mathematical knowledge that is based on their experience of life thus far;
They express it with
everyday language
(e.g., “more than” or “bigger than”);
According to Diaz (2008), in each of Piaget’s constructivist theory, Vygotsky’s sociocultural theory, and Lakoff’s metaphor theory, there are elements of how
mathematical language can be
mediated
in everyday play.
Children construct meaning while using language in their interactions with their classmates and with their teachers.
Mediating
is purposeful interaction as the teacher uses "students' speech to bridge their everyday language to mathematical concepts." For example, a student during block play asks a teacher to help. The teacher says, "Instruct me." The student tells where to put the blocks, and the teacher asks for more information ("how?"), using the word "stack," which was understood and affirmed by the student. Another example of a student asking for help prompts the teacher to ask, "Where?" and then she rephrases the student's response with, "Oh, around the perimeter."
Types of Mathematical Language
Math concepts (including "big ideas") need to be presented in a context that is motivating and handson. Furthermore, teachers need to be actively engaged in observing children while they are playing and interacting socially, so that they can act on
teachable moments
and promote learning of mathematical concepts. Of interesting note is that a teacher’s attitude and professional development will have an effect on how much learning can be accomplished in these interactions.
Teachable Moments
Math Standards
References:
Samples of Learning Centres (Numeracy):
Practical Ideas for Creating a Rich Mathematical Learning Environment
The Early Years Teacher
A Successful Approach
Math Talk
Facilitating Experiences
As educators we need to facilitate experiences that allow '
mathematization
' to occur. Educators assist students in transforming their everyday mathematics into a more formalized understanding that can be transferred and applied to other situations.
The educator plays an integral role by making meaningful connections between math strands and real world experiences. These connections should link math concepts students have learned through their own experiences and the ones being taught in the classroom.
How important is it to use appropriate math language for children to develop an awareness and understanding of math concepts in the world around them? What should teachers be looking for when observing children in play, so that they can formulate guiding questions for the children that would increase their understanding of math concepts?
After recording the number of mathematical utterances, Diaz rated the interactions of the teacher, based on whether the teacher succeeded or failed to pick up on the child’s reference to mathematical concepts using everyday language. When these interactions were intentional on the teacher's part, and based on observations in real time, the enhanced math language led to further math concept development. Her published research gives examples of several interactions.
Diaz concludes from the collected data that some teacher interactions were successful in encouraging more student talk. Obviously, "ignoring" or missing the opportunity for interaction is not helpful, but it is surprising how often opportunities were missed, when the teachers knew the purpose for the testing (60%). Other interactions were: a) accept/repeat idea (18%); b) praise or encourage (6%); c) ask questions (6%); and d)
mediating
(10%).
Her study shows that when children interact with teachers while in play, their learning can be greatly enhanced.
Diaz (2008) refers to recommendations of the National Council of Teachers of Mathematics (NCTM). The NCTM is an excellent resource for math teachers of all grades, which recommends curriculum focal points (the content emphases), and connections (other meaningful and essential components for learning the content). The Ontario curriculum's math expectations are in line with their recommendations. To compare with the 5 strands in the Ontario Math curriculum (listed below), the first 3 are emphasized for content, while generally, the last 2 can also be learned when math is integrated with other subjects (science, music, etc.).
Consider the mathrich language which can result when these connections are made.
Number Sense and Numeration (Quantity Relationships; Counting; Operational Sense);
Measurement (Attributes,Units, and Measurement Sense; Measurement Relationships);
Geometry and Spatial Sense (Geometric Properties; Geometric Relationships; Location and Movement);
Patterning (Patterns and Relationships);
Data Management and Probability (Collection and Organization of Data;Data Relationships;Probability).
Sorting Out the Math Talk
Diaz (2008) conducted research of interactions between teachers and students of children in PreK classrooms. Children were observed in block play and their "utterances" (in everyday language) were recorded into the following categories (except for classification and dynamic words, all the rest of the utterances were being used by the children "less than expected by chance"):
Words for classification
 involving sorting into groups and categorizing;
Dynamic words
 involving motions or actions (verbs);
Words denoting a spatial relation
 involving distance, location, direction (e.g., here, there, in, inside, etc.);
Magnitude
 involving size (e.g., tall), comparisons between objects (e.g, same height), judgments (e.g., some, a lot);
Enumeration
 referred to quantity;
Pattern and Shape
 descriptive qualities that are given attention when replicating structures.
Math talk is simply bringing in mathematical terms into the normal classroom conversations. A simple example would be asking students if they have more or less of something.
There are 5 effective talk moves that create meaningful discussion within the FDK classroom.
In order to create a positive math environment, the teacher must lead by example. If the teacher wants students to use mathematical terms, then they must be used in the classroom. Mathematical terms cannot be just a 'one time vistor' in the classroom; talk is important. Students in an FDK classroom are encouraged to talk and communicate with each other. It is through communication and talk that students learn social skills and learn the task they are doing.
5 Productive Talk Moves
1. Revoicing
Repeat what the student has said in the form of a question.
"So you're saying your taller then the chair?"
2. Repeating
Have the student repeat what was said in their own words.
3. Reasoning
Ask students if they agree or disagree is what was said and why.
4. Adding on
prompt students to participate
5. Waiting
Wait for the student to answer, give them to think
1. When teaching math to students, use problems that have meaning for children. If Sarah has 2 cookies and Ben has 4 cookies, who has more?
How many are cookies are there all together?
2. No matter what the problem or question, students will have their own spin on things. They will invent, explain and come up with their own solution to problems.
3. Be creative! Let the students have fun and be creative. They will be more interested in what is being taught.
4. Encourage and support children  through scaffolded lessons and activities that the student can engage in.
5. Help students make connections. These connections will help a student learn the content.
Children's Math Books
Numbers & Counting
1. Double the Ducks
2. Every Buddy Counts
3. Henry the Fourth
4. Missing Mittens
5. Monster Musical Chairs
6. Animals on Board
7. Elevator Magic
8. More or Less
9. Spunky Monkeys on Parade
10. 100 Days of Cool
** All these books are by Stuart J. Murphy MathStart Books**
Measurement
1. The Best Bug Parade
2. A House for Birdie
3. Mighty Maddie
4. Bigger, Better, Best!
5. Racing Around
6. Super Sand Castle Saturday
** All these books are written by Stuart J. Murphy MathStart Books
Algebra
1. Beep Beep, Vroom, Vroom!
2. Seaweed Soup
3. Mall Mania
4. Probably Pistachio
5. 3 Little Firefighters
6. Same Old Horse
** All of these books are written by Stuart J. Murphy MathStart Books**
Being Knowledgeable & Attentive
In order for mathematization to occur educators should:
have a strong knowledge of the curriculum and mathematical concepts
be watching students attentively in order to make connections
For Example
As a child naturally creates and extends a pattern while making a necklace, the educator can ask questions that provoke the student to describe, make predictions and generalizations about the pattern.
Sam H., Kelly K., Veronica R., Rebecca O., Michelle V.
Model and Nurture Positive Attitudes, SelfEfficacy and Engagement
As educators gain confidence in teaching mathematical knowledge, and in helping students extend and formalize their understanding of mathematical concepts, students will also begin to develop more positive attitudes toward mathematics and an increase their sense of selfefficacy.
If both students and teachers are confident and enthusiatic about math, it will increase the amount of mathematical ideas and language used within the classroom.
It is incredibly important that students are exposed to rich mathematical instructional interactions in order to expand upon what they already know. If teachers feel confident in teaching the mathematical concepts they will push the limits of student knowledge in order to stretch their learning.
Confident, enthousiastic teachers, lead to confident and enthusiatic students.
Get Familiarized with Math Concepts and Vocabulary
Compiled by Betsy Reilly, Faculty of Education, Western University
Compiled by Dave Watson, Faculty of Education, Western University
(promoting the "Big Ideas")
When planning your daily routine, take into account that:
•Social interactions foster learning, therefore you need to ensure your day is planned with opportunities for discussion and conversation.
•Communication is key! Building communication through active engagement in small group discussions plays a significant role in enhancing and clarifying children’s mathematical understandings.
•Students need to be exposed to “cognitively demanding mathematical tasks”.
Two Common Mistakes
in Teaching Approaches
"I believe that play is the only important and developmentally appropriate approach for young children."
Situation 2:
Early 1900s
Young children were historically considered not cognitively capable of engaging in the thinking needed to understand mathematics

Thorndike
: believed that children were mathematically inept before second grade, therefore it was not worth even addressing mathematics with them
• Due to this way of thinking, there was virtually no mathematics programing during the early 1900s
Situation 1:
1960s
"I believe that math needs to be highly structured and scripted. Things need to be taught the way I was taught in Elementary school."
Leads children to rely on completely unstructured and incidental learning during play, with little interaction with mathematical concepts.
By the 1960s, there was the establishment of mathematics education beginning at the early elementary level
The change was because it was now thought children needed formal schooling in mathematics in order to think that way

Piaget:
his research showed young children as mathematically curious and as actively constructing mathematical knowledge by interacting with the physical and social world
• However, he also believed they were incapable of abstract and logical thinking until at least age 7
Leads children to a lower curiosity about math and affects the degree of higher level thinking as students believe all they need to know will be told to them by the teacher.
A Balanced Viewpoint
I want to make mathematics fun, engaging and playful! I can do this by incorporating various activities into my structured mathematics lessons. I will do this in ways that ensures activities and lessons are developmentally appropriate for my students.
This enables students to become active and responsible participants in classroom communication.
Small Group Lessons
Table Activities
Learning Centres
21st Century
Developmental psychologists transferred the focus from what young children could not do to what they
could
do
• Research has begun to highlight children’s competencies and education has followed this thinking
Young children are now seen as much more powerful mathematicians than past instructional practices reflect
• By participating in interactions with the social and physical world, young children engage in diverse types of mathematical thinking
• Specifically, this helps to develop intuitive foundational knowledge and skills in mathematics
• These include: onetoone correspondence, cardinality, and basic addition and subtraction
Present Day
Young children spontaneously invent mathematicsbased strategies during everyday play and often make predictions and successfully solve numberrelated reasoning problems
We now know that Kindergarten aged students engage in surprisingly complex intuitive mathematical thinking in:
• Numeracy
• Geometry
• Measurement
• Algebraic Thinking
• Data Analysis
From the current understanding of the early years learner, it is important to challenge our students and allow them to explore complex mathematical concepts
With a Focus on the "Big Ideas" in Math