Goal #2 Development
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Theories
During the mid-1900’s theorists like B.F. Skinner and Watson believed children learned by following the ideals of behaviorism. In other words, children’s behavior and learning was motived by external stimuli that elicited certain responses. Skinner expanded on this by developing operant conditioning. This led Skinner to conclude that “animals and humans would repeat acts that led to favorable outcomes, and suppress those that produced unfavorable results,” (Shaffer, 2000 as cited by Standridge, Melissa). Behaviorism was not adequate at explaining how human children learned to “use language, to have ideas, to be creative and to solve all kinds of problems,” (Anghileri, Julia 2005).
More recent theories of how children learn mathematics have come to be termed ‘constructivist’ and are based on the view that learning is not simply a case of ‘transmitting’ knowledge from one person to another but that children are actively engaged in constructing their own knowledge from their personal experiences. Mathematical knowledge is not something that is acquired by listening to teachers and reading textbooks but something that learners themselves construct by seeking out meanings and making mental connections in an active manner (3). Certainly teachers need to act as facilitators and provide students with opportunities to seek out and construct meaning for mathematics.
Kindergarten to 2nd grade
Children can be exposed to numbers from a very young age. As a teacher, I try to encourage children to develop an understanding of numbers simply by giving them time to play with numbers. Children also need to talk about numbers with each other and with their teacher and parents.
Students in kindergarten are able to recognize patterns and categorize objects. One way to practice this is through subitizing. Subitizing is “instantly seeing how many”. As children explore numbers they can begin to recognize a group of objects as a certain number of objects. One way to promote this skill is to do an activity where students race to see which team can tell how many circles are on a card, for instance. The top row in this image illustrates different cards students can use when they first begin learning how to subitize. As children "develop conceptual subitizing" they can begin to play with "more complex patterns" as seen in the second row.
(Image: Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics).
Additionally, students in the primary grades should begin with concrete tools such as place value blocks before they move to more abstract symbols such as = or +. When students start counting by 5’s and 10’s for instance, they should begin by doing this with actual objects such as beans or place value blocks. Children need to understand what “5” and “10” mean. So when they count 10, 20, 30… and so on, the students actually understand “how many” objects they have. During the course of my master’s program I had one of my second grade students count by tens and then draw stars to represent each ten. Much to my surprise she made 10 stars instead of 100 stars. Right away, I could see that she did not understand what was going on when she counted by tens. By working with place value blocks and other concrete objects such as a number line, she was able to understand what was actually happening when she counted by tens. Piaget would call this stage of thinking concrete operations. Generally, children ages 8-11 (2nd-5th grade) learn through concrete operations, (Clemson, David & Clemson, Wendy, 1994).
During the primary years children also spend a great deal of time learning about addition and subtraction. First, I provide concrete examples of these skills for my students by using either place value blocks, number lines, or even household objects. As the students become more comfortable with certain addition facts, like doubles or doubles +1 it is important to begin teaching mental math skills. Here is a statement taken from EDMA 614 Numeration and Operations: Mathematics for K-8 Teachers on how to include mental mathematics in the classroom:
"I think the biggest tool to use in “how” to teach mental math is practice. As mentioned previously my students are working on basic facts as well as mental math on a daily basis (for the most part). When we do our daily math warm-up and/or math readiness activity we generally do at least a few mental math problems focusing on one mental math strategy. We also spend time during our daily lesson explaining how we knew the answer to the math problem. During these discussions students present their strategy for solving the problem. Even the students who use the same strategy may have done the math differently. For instance, we were doing word problems yesterday using the total and parts diagram. One student added the two parts by using his knowledge of doubles. Then he added in the remainder to his answer from the doubles. Another student actually explained partial sums. I was really surprised too because she was one of my lower students. I always thought partial sums were difficult for my students; but, now I know it really does make sense to some students. Finally, when we use white boards in class I have started writing my thinking on the Smart Board. I put it in a little bubble and model for my students what is going on in my head. Even though we may have seen the strategy for mental math 3 or 4 times I will still take the time to model at least one before we practice. My students seem to be more successful when I illustrate what I am thinking."
3rd to 5th grade
Students in third through fifth grade, according to Piaget, still learn through concrete operations. So, students in my third grade classroom often spent time working with place value blocks, number lines, and different sets of objects. As students work with these concrete objects they begin to understand what is occurring when they add 4 blocks to 5 blocks, or why 5 blocks plus 6 blocks requires making a ten and a one.
Third grade students spend a great deal of time becoming comfortable with the four operations: addition, subtraction, multiplication, and division. Each math program approaches teaching these skills in different formats. I have always found it difficult to stick strictly to one math curriculum. I prefer to use the curriculum provided by the district and supplement when I see fit. Sometimes the units presented in the textbook are outstanding, while other units seem to offer strategies for students or different algorithms to solve problems, but don’t necessarily give them the time to work with the different strategies. In my classroom, I really try to help my students understand the different algorithms for solving problems involving the four operations. I also encourage my students to be able to construct meaning as they solve different problems. When my students work through word problems I will often have them try them alone first. Then I will give them time to work with their group members to solve the problem again. After 5-10 minutes (sometimes 15) one person from each group will explain how the problem was solved to the rest of the class. Then I award them points; I give a 1, 2, or 3. Things I consider are how well they explained their response, effort, and how they got to the answer. I am not always looking for a correct response, but more for perseverance and critical thinking.
Students in third through fifth grade also spend time working on estimation. Because students ages 8-11 think in concrete terms I like to begin estimation by using a number line. Once students see why they should round up or round down then I add in some fun things to get us moving like a “rounding song” or rounding bingo. Then I create things like this rounding chart to help support children who are still having difficulty. I think it is also important with each of the skills that are taught to connect them to real-life mathematics.
K-8 Development
Other skills that are important as children develop in grades K-8 included measurement, geometry, and developing skills needed to solve algebraic problems later in children's mathematics education. Prior to working on my master's degree I did not really understand the best ways to teach geometry, however after studying van Hiele's levels of geometric thought I now have a better understanding of how students learn geometry.
"Prior to taking this course I had never heard of the van Hiele levels of geometric thought. When I was preparing my first “community problem” I struggled with creating a problem that was appropriate for my third grade students. I discovered I did not truly understand what my students should know in terms of geometric thinking. So, I decided to study the van Hiele levels of geometric thinking.
In looking at these levels I found it interesting that students progress through the levels by experiencing geometry. In other words, students do not progress based on age or grade level, rather they move to the next level based on their experiences (Breyfogle & Lynch, 2010). So, now my goal is to help my students “experience” geometry."
Ultimately, teaching mathematics means presenting children with the information they need to be successful, and giving them the time and support they need to construct meaning for mathematics.
Artifacts:
EDMA 655: Concept Description and Reflection 2: The van Hiele Levels of Geometric Thinking
Lesson Plan for EDMA 654: (Students move from concrete to more abstract thinking)
Sources:
Anghileria, Julia. (2005). Children’s Mathematical Thinking in the Primary Years: Perspectives on
Children’s Learning. Continuum International Publishing; London, GBR. Retrieved online
June 5, from http://egandb.uas.alaska.edu:2081/lib/uasoutheast/docDetail.action?
docID=10224723&p00=anghileri%2C%20julia
Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics.
Standridge, Melissa. (2002). Behaviorism. In M. Orey (Ed.), Emerging perspectives on learning, teaching
and technology. Retrieved June 5, from http://projects.coe.uga.edu/epltt/
During the mid-1900’s theorists like B.F. Skinner and Watson believed children learned by following the ideals of behaviorism. In other words, children’s behavior and learning was motived by external stimuli that elicited certain responses. Skinner expanded on this by developing operant conditioning. This led Skinner to conclude that “animals and humans would repeat acts that led to favorable outcomes, and suppress those that produced unfavorable results,” (Shaffer, 2000 as cited by Standridge, Melissa). Behaviorism was not adequate at explaining how human children learned to “use language, to have ideas, to be creative and to solve all kinds of problems,” (Anghileri, Julia 2005).
More recent theories of how children learn mathematics have come to be termed ‘constructivist’ and are based on the view that learning is not simply a case of ‘transmitting’ knowledge from one person to another but that children are actively engaged in constructing their own knowledge from their personal experiences. Mathematical knowledge is not something that is acquired by listening to teachers and reading textbooks but something that learners themselves construct by seeking out meanings and making mental connections in an active manner (3). Certainly teachers need to act as facilitators and provide students with opportunities to seek out and construct meaning for mathematics.
Kindergarten to 2nd grade
Children can be exposed to numbers from a very young age. As a teacher, I try to encourage children to develop an understanding of numbers simply by giving them time to play with numbers. Children also need to talk about numbers with each other and with their teacher and parents.
Students in kindergarten are able to recognize patterns and categorize objects. One way to practice this is through subitizing. Subitizing is “instantly seeing how many”. As children explore numbers they can begin to recognize a group of objects as a certain number of objects. One way to promote this skill is to do an activity where students race to see which team can tell how many circles are on a card, for instance. The top row in this image illustrates different cards students can use when they first begin learning how to subitize. As children "develop conceptual subitizing" they can begin to play with "more complex patterns" as seen in the second row.
(Image: Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics).
Additionally, students in the primary grades should begin with concrete tools such as place value blocks before they move to more abstract symbols such as = or +. When students start counting by 5’s and 10’s for instance, they should begin by doing this with actual objects such as beans or place value blocks. Children need to understand what “5” and “10” mean. So when they count 10, 20, 30… and so on, the students actually understand “how many” objects they have. During the course of my master’s program I had one of my second grade students count by tens and then draw stars to represent each ten. Much to my surprise she made 10 stars instead of 100 stars. Right away, I could see that she did not understand what was going on when she counted by tens. By working with place value blocks and other concrete objects such as a number line, she was able to understand what was actually happening when she counted by tens. Piaget would call this stage of thinking concrete operations. Generally, children ages 8-11 (2nd-5th grade) learn through concrete operations, (Clemson, David & Clemson, Wendy, 1994).
During the primary years children also spend a great deal of time learning about addition and subtraction. First, I provide concrete examples of these skills for my students by using either place value blocks, number lines, or even household objects. As the students become more comfortable with certain addition facts, like doubles or doubles +1 it is important to begin teaching mental math skills. Here is a statement taken from EDMA 614 Numeration and Operations: Mathematics for K-8 Teachers on how to include mental mathematics in the classroom:
"I think the biggest tool to use in “how” to teach mental math is practice. As mentioned previously my students are working on basic facts as well as mental math on a daily basis (for the most part). When we do our daily math warm-up and/or math readiness activity we generally do at least a few mental math problems focusing on one mental math strategy. We also spend time during our daily lesson explaining how we knew the answer to the math problem. During these discussions students present their strategy for solving the problem. Even the students who use the same strategy may have done the math differently. For instance, we were doing word problems yesterday using the total and parts diagram. One student added the two parts by using his knowledge of doubles. Then he added in the remainder to his answer from the doubles. Another student actually explained partial sums. I was really surprised too because she was one of my lower students. I always thought partial sums were difficult for my students; but, now I know it really does make sense to some students. Finally, when we use white boards in class I have started writing my thinking on the Smart Board. I put it in a little bubble and model for my students what is going on in my head. Even though we may have seen the strategy for mental math 3 or 4 times I will still take the time to model at least one before we practice. My students seem to be more successful when I illustrate what I am thinking."
3rd to 5th grade
Students in third through fifth grade, according to Piaget, still learn through concrete operations. So, students in my third grade classroom often spent time working with place value blocks, number lines, and different sets of objects. As students work with these concrete objects they begin to understand what is occurring when they add 4 blocks to 5 blocks, or why 5 blocks plus 6 blocks requires making a ten and a one.
Third grade students spend a great deal of time becoming comfortable with the four operations: addition, subtraction, multiplication, and division. Each math program approaches teaching these skills in different formats. I have always found it difficult to stick strictly to one math curriculum. I prefer to use the curriculum provided by the district and supplement when I see fit. Sometimes the units presented in the textbook are outstanding, while other units seem to offer strategies for students or different algorithms to solve problems, but don’t necessarily give them the time to work with the different strategies. In my classroom, I really try to help my students understand the different algorithms for solving problems involving the four operations. I also encourage my students to be able to construct meaning as they solve different problems. When my students work through word problems I will often have them try them alone first. Then I will give them time to work with their group members to solve the problem again. After 5-10 minutes (sometimes 15) one person from each group will explain how the problem was solved to the rest of the class. Then I award them points; I give a 1, 2, or 3. Things I consider are how well they explained their response, effort, and how they got to the answer. I am not always looking for a correct response, but more for perseverance and critical thinking.
Students in third through fifth grade also spend time working on estimation. Because students ages 8-11 think in concrete terms I like to begin estimation by using a number line. Once students see why they should round up or round down then I add in some fun things to get us moving like a “rounding song” or rounding bingo. Then I create things like this rounding chart to help support children who are still having difficulty. I think it is also important with each of the skills that are taught to connect them to real-life mathematics.
K-8 Development
Other skills that are important as children develop in grades K-8 included measurement, geometry, and developing skills needed to solve algebraic problems later in children's mathematics education. Prior to working on my master's degree I did not really understand the best ways to teach geometry, however after studying van Hiele's levels of geometric thought I now have a better understanding of how students learn geometry.
"Prior to taking this course I had never heard of the van Hiele levels of geometric thought. When I was preparing my first “community problem” I struggled with creating a problem that was appropriate for my third grade students. I discovered I did not truly understand what my students should know in terms of geometric thinking. So, I decided to study the van Hiele levels of geometric thinking.
In looking at these levels I found it interesting that students progress through the levels by experiencing geometry. In other words, students do not progress based on age or grade level, rather they move to the next level based on their experiences (Breyfogle & Lynch, 2010). So, now my goal is to help my students “experience” geometry."
Ultimately, teaching mathematics means presenting children with the information they need to be successful, and giving them the time and support they need to construct meaning for mathematics.
Artifacts:
EDMA 655: Concept Description and Reflection 2: The van Hiele Levels of Geometric Thinking
Lesson Plan for EDMA 654: (Students move from concrete to more abstract thinking)
Sources:
Anghileria, Julia. (2005). Children’s Mathematical Thinking in the Primary Years: Perspectives on
Children’s Learning. Continuum International Publishing; London, GBR. Retrieved online
June 5, from http://egandb.uas.alaska.edu:2081/lib/uasoutheast/docDetail.action?
docID=10224723&p00=anghileri%2C%20julia
Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics.
Standridge, Melissa. (2002). Behaviorism. In M. Orey (Ed.), Emerging perspectives on learning, teaching
and technology. Retrieved June 5, from http://projects.coe.uga.edu/epltt/