Goal 4: Math Content Knowledge includes several different content areas. I understand the value of each strand and have spent time reflecting on each content strand. On this page I have reflected on many of the different strands under content knowledge; however, I did not include every strand.
Problem Solving
Statement:
Reflects how to develop and assess student problem solving knowledge and skills, accommodating learning styles and abilities, and helps students connect problem solving to the local community and the real world.
Through the course of my master’s program I gained valuable knowledge about problem solving. One of the standards for mathematical practice listed in the Alaska Common Core Standards, states that student should be able to "apply the mathematics they know to solve problems arising in everyday life, society and the workplace," (Alaska Common Core Mathematics Standards, 2012). Furthermore, NCTM states that students will solve problems by "applying and adapting a variety of appropriate strategies to solve problems," (NCTM Process Standards, 2014). During EDMA 608 I was able to review different problem solving strategies such as: drawing a diagram, guess and check, working backwards, acting it out, or by creating a systematic list. Reviewing these different strategies gave me the tools necessary to help my students use a number of different strategies to solve a variety of problems. During class my students would work on solving 1-2 word problems each day. First students would work on solving these problems independently; then after working on the problems for a bit the students would be given time to work in their teams to solve the problem. Then the teams would determine the best solution to the problem and would choose one team member to explain or justify the team’s solution. The team earned points based on how well they justified their solution; in other words, I did not simply award points based on the “correct” answer. By allowing students to make mistakes I was able to teach them the value of thinking critically and persevering in problem solving.
Besides reviewing different strategies for solving problems I also completed a case study during EDMA 608. Through the course of this study I learned the value of asking good questions rather than telling students how to solve problems. I was also reminded of how different each child is; what make sense to one student may make little sense to another student. I try to model different methods for solving problems so I can encourage my students to construct their own meaning as they work through problems during class. I also want my students to be able to apply their math skills to real world problems so I tried to include community problems in my classroom.
Mathematical Communication
Statement:
Accommodates learning styles and abilities to help them express themselves mathematically.
In my classroom I try to set up a good atmosphere and a strong relationship with my students from day one. I greet them each day as they enter my classroom and manage my classroom in such a way that students feel safe and comfortable explaining their thinking in mathematics. Some students always take a little longer to open up than other students but most students seem to feel comfortable in my classroom after they see other students share their thinking.
NCTM states that all students should be able to "communicate their mathematical thinking coherently and clearly to peers, teachers, and others," (NCTM Process Standards, 2014). Students are encouraged to share their thinking first with their peers. I show students how to explain their thinking out loud as I solve math problems for them by modeling my thinking out loud. Then I have a pair of students model how to solve a math problem out loud before students work in pairs. While students work in pairs they do Rally Coach, a Kagan Cooperative Learning Strategy that requires one student to explain their thinking while the other student coaches him/her. Once students feel comfortable sharing their thinking with a peer I start having the students share their thinking for a group and eventually the class. When students share their thinking for the whole class I always start with students who volunteer to share, eventually all the students want to share their thinking. One great tool I learned from EDMA 614 was to name strategies after students. The students are so excited when you say comments like, “Let’s use the Jenny Method.”
“I also like that mental math requires students to become aware of the idea that the same answer can be reached in many ways. We discussed recently how we knew two parts equaled a certain sum, and in this discussion we used your (Jennifer’s) idea of naming a strategy after a student. The kids loved this and are now excited to provide a written justification for their solutions. YAH! Finally, I like that my students are learning how to manipulate numbers and become comfortable doing so,” (taken from Mental Math Chapter 2 Reflection EDMA 614).
Another important component of mathematical communication is understanding and being able to assess how my students solve mathematical problems. During EDMA 614 I created a CGI or Cognitively Guided Instruction grid for problem solving. By using this grid I was able to see the different methods my students were able to use well for solving problems and determine the different methods I needed to devote more time to during class.
Mathematical Connections:
Statement:
Incorporates learning styles and abilities while fostering the application of mathematics beyond the classroom.
As an educator and someone who is passionate about math, two of the most common questions students ask me are: “Why do you love math so much?” and “When will I need math outside of the classroom?” Responding to the question: “Why do I love math so much?” is rather simple to answer because using numbers and thinking mathematically is fun! Explaining when students will need math outside of the classroom is more challenging to answer because I want to take the time to make sure students understand when and how they will use math in the real world. Working on my master’s degree has helped me develop community based problems that require my students to use their mathematical skills and thinking on real world problems. For example, many of my students had no idea they were using their geometry skills when they help mom put away groceries in the cabinet. I also encourage parents to set up problems for their children such as putting away toilet paper after shopping or using money to buy tickets at Fur Rondy. By incorporating community problems into my math instruction, my students are able to apply their mathematical skills to real world situations.
Mathematical Representation
Statement:
Incorporates students' learning styles while fostering their ability to reason mathematically.
Mathematical representation, according to the Annenberg Foundation, “can include written work, oral explanations, models with manipulative materials, and even the mental processes one uses to do mathematics,” (2013). Students in grades K-2 need to have many experiences representing mathematics with concrete materials. As students begin to learn the different concepts with concrete objects they can learn to grasp more abstract mathematical concepts.
“In 1969 Zoltan Diene’s work convinced researchers that the use of various representations of a concept, or ‘multiple embodiments,’ were needed to support students’ understandings (as cited in Moyer, 2002). Many years earlier Piaget (1952) suggested that children do not have the mental maturity to grasp abstract mathematical concepts presented in words or symbols alone and need many experiences with concrete materials…for learning to occur (as cited in Moyer, 2002),” (Taken from Teaching Algebra with the help of Manipulative Concept Description for EDMA 664).
When I begin new concepts my students generally use manipulatives to help them represent their mathematical thinking. Once my students are comfortable using concrete materials I encourage them to explain their reasoning verbally. For many students this seems to be easier than writing down their thoughts. For students who are in late 2nd grade and up my goal is to have them be able to represent their math in ‘multiple embodiments’ as Zoltan suggested.
Once students use concrete materials to represent math, they can then begin to represent mathematics with the use of symbols. Other ways students might represent mathematics might be creating different types of graphs such as pictographs or bar graphs. Or, students may choose to draw pictures. Students may also decide to represent their math problems by acting them out. Regardless of how students choose to represent mathematics, teachers should introduce students to a variety of mathematical representations and encourage students to represent math in a way that is clear to them.
Numbers and Operations
Statement:
Fosters students understanding of numbers and operations.
In kindergarten students are introduced to place value. According the Alaska Common Core Standards kindergartens should "work with numbers 11 to 19 to gain foundations for place value," (Alaska Common Core Standards, 2012). Then as students continue through elementary school they extend their thinking and continue to explore place value as they learn addition and subtraction, and later multiplication and division. Understanding place value is a key piece for students as they progress through elementary school.
When children begin to learn mathematics they learn about numbers first and then begin to learn the operations that can be performed with numbers. Young children learn 4 mathematical aspects of the number core; they are cardinality (how many are in a set), number word lists, 1-to-1 correspondence, and written number symbols (Cross, Christopher T. & Woods, Taniesha A., 2009). In other words, students first practice recognizing the amount of dots on dice or the number of crackers on a plate. Teachers can encourage students to recognize how many are in a set by doing subitizing activities like the one below. The top row depicts the type of cards teachers should use with students when they are beginning to subitize. As children "develop conceptual subitizing" they can begin to play with "more complex patterns" as seen in the second row. During EDMA 614 we discussed using the game “Set” to help our students learn how to subitize.
Statement:
Reflects how to develop and assess student problem solving knowledge and skills, accommodating learning styles and abilities, and helps students connect problem solving to the local community and the real world.
Through the course of my master’s program I gained valuable knowledge about problem solving. One of the standards for mathematical practice listed in the Alaska Common Core Standards, states that student should be able to "apply the mathematics they know to solve problems arising in everyday life, society and the workplace," (Alaska Common Core Mathematics Standards, 2012). Furthermore, NCTM states that students will solve problems by "applying and adapting a variety of appropriate strategies to solve problems," (NCTM Process Standards, 2014). During EDMA 608 I was able to review different problem solving strategies such as: drawing a diagram, guess and check, working backwards, acting it out, or by creating a systematic list. Reviewing these different strategies gave me the tools necessary to help my students use a number of different strategies to solve a variety of problems. During class my students would work on solving 1-2 word problems each day. First students would work on solving these problems independently; then after working on the problems for a bit the students would be given time to work in their teams to solve the problem. Then the teams would determine the best solution to the problem and would choose one team member to explain or justify the team’s solution. The team earned points based on how well they justified their solution; in other words, I did not simply award points based on the “correct” answer. By allowing students to make mistakes I was able to teach them the value of thinking critically and persevering in problem solving.
Besides reviewing different strategies for solving problems I also completed a case study during EDMA 608. Through the course of this study I learned the value of asking good questions rather than telling students how to solve problems. I was also reminded of how different each child is; what make sense to one student may make little sense to another student. I try to model different methods for solving problems so I can encourage my students to construct their own meaning as they work through problems during class. I also want my students to be able to apply their math skills to real world problems so I tried to include community problems in my classroom.
Mathematical Communication
Statement:
Accommodates learning styles and abilities to help them express themselves mathematically.
In my classroom I try to set up a good atmosphere and a strong relationship with my students from day one. I greet them each day as they enter my classroom and manage my classroom in such a way that students feel safe and comfortable explaining their thinking in mathematics. Some students always take a little longer to open up than other students but most students seem to feel comfortable in my classroom after they see other students share their thinking.
NCTM states that all students should be able to "communicate their mathematical thinking coherently and clearly to peers, teachers, and others," (NCTM Process Standards, 2014). Students are encouraged to share their thinking first with their peers. I show students how to explain their thinking out loud as I solve math problems for them by modeling my thinking out loud. Then I have a pair of students model how to solve a math problem out loud before students work in pairs. While students work in pairs they do Rally Coach, a Kagan Cooperative Learning Strategy that requires one student to explain their thinking while the other student coaches him/her. Once students feel comfortable sharing their thinking with a peer I start having the students share their thinking for a group and eventually the class. When students share their thinking for the whole class I always start with students who volunteer to share, eventually all the students want to share their thinking. One great tool I learned from EDMA 614 was to name strategies after students. The students are so excited when you say comments like, “Let’s use the Jenny Method.”
“I also like that mental math requires students to become aware of the idea that the same answer can be reached in many ways. We discussed recently how we knew two parts equaled a certain sum, and in this discussion we used your (Jennifer’s) idea of naming a strategy after a student. The kids loved this and are now excited to provide a written justification for their solutions. YAH! Finally, I like that my students are learning how to manipulate numbers and become comfortable doing so,” (taken from Mental Math Chapter 2 Reflection EDMA 614).
Another important component of mathematical communication is understanding and being able to assess how my students solve mathematical problems. During EDMA 614 I created a CGI or Cognitively Guided Instruction grid for problem solving. By using this grid I was able to see the different methods my students were able to use well for solving problems and determine the different methods I needed to devote more time to during class.
Mathematical Connections:
Statement:
Incorporates learning styles and abilities while fostering the application of mathematics beyond the classroom.
As an educator and someone who is passionate about math, two of the most common questions students ask me are: “Why do you love math so much?” and “When will I need math outside of the classroom?” Responding to the question: “Why do I love math so much?” is rather simple to answer because using numbers and thinking mathematically is fun! Explaining when students will need math outside of the classroom is more challenging to answer because I want to take the time to make sure students understand when and how they will use math in the real world. Working on my master’s degree has helped me develop community based problems that require my students to use their mathematical skills and thinking on real world problems. For example, many of my students had no idea they were using their geometry skills when they help mom put away groceries in the cabinet. I also encourage parents to set up problems for their children such as putting away toilet paper after shopping or using money to buy tickets at Fur Rondy. By incorporating community problems into my math instruction, my students are able to apply their mathematical skills to real world situations.
Mathematical Representation
Statement:
Incorporates students' learning styles while fostering their ability to reason mathematically.
Mathematical representation, according to the Annenberg Foundation, “can include written work, oral explanations, models with manipulative materials, and even the mental processes one uses to do mathematics,” (2013). Students in grades K-2 need to have many experiences representing mathematics with concrete materials. As students begin to learn the different concepts with concrete objects they can learn to grasp more abstract mathematical concepts.
“In 1969 Zoltan Diene’s work convinced researchers that the use of various representations of a concept, or ‘multiple embodiments,’ were needed to support students’ understandings (as cited in Moyer, 2002). Many years earlier Piaget (1952) suggested that children do not have the mental maturity to grasp abstract mathematical concepts presented in words or symbols alone and need many experiences with concrete materials…for learning to occur (as cited in Moyer, 2002),” (Taken from Teaching Algebra with the help of Manipulative Concept Description for EDMA 664).
When I begin new concepts my students generally use manipulatives to help them represent their mathematical thinking. Once my students are comfortable using concrete materials I encourage them to explain their reasoning verbally. For many students this seems to be easier than writing down their thoughts. For students who are in late 2nd grade and up my goal is to have them be able to represent their math in ‘multiple embodiments’ as Zoltan suggested.
Once students use concrete materials to represent math, they can then begin to represent mathematics with the use of symbols. Other ways students might represent mathematics might be creating different types of graphs such as pictographs or bar graphs. Or, students may choose to draw pictures. Students may also decide to represent their math problems by acting them out. Regardless of how students choose to represent mathematics, teachers should introduce students to a variety of mathematical representations and encourage students to represent math in a way that is clear to them.
Numbers and Operations
Statement:
Fosters students understanding of numbers and operations.
In kindergarten students are introduced to place value. According the Alaska Common Core Standards kindergartens should "work with numbers 11 to 19 to gain foundations for place value," (Alaska Common Core Standards, 2012). Then as students continue through elementary school they extend their thinking and continue to explore place value as they learn addition and subtraction, and later multiplication and division. Understanding place value is a key piece for students as they progress through elementary school.
When children begin to learn mathematics they learn about numbers first and then begin to learn the operations that can be performed with numbers. Young children learn 4 mathematical aspects of the number core; they are cardinality (how many are in a set), number word lists, 1-to-1 correspondence, and written number symbols (Cross, Christopher T. & Woods, Taniesha A., 2009). In other words, students first practice recognizing the amount of dots on dice or the number of crackers on a plate. Teachers can encourage students to recognize how many are in a set by doing subitizing activities like the one below. The top row depicts the type of cards teachers should use with students when they are beginning to subitize. As children "develop conceptual subitizing" they can begin to play with "more complex patterns" as seen in the second row. During EDMA 614 we discussed using the game “Set” to help our students learn how to subitize.
(Image: Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics).
As children move into 1st and 2nd grade they begin to start working more with the different operations. 1st and second grade students spend a great deal of time learning how to add or subtract numbers. One thing I learned in EDMA 614 is that, as educators, we have to be very aware of the language that we use when we teach mathematics. Saying “borrowing” for instance gives students the idea that at some point during the subtraction problem we are going to “give it back”, which is certainly not the case. I learned that a better term would be “unbundling” or “decomposing”.
Towards the end of 2nd grade students begin to work with multiplication and division. Then as students enter third grade they begin to work with fractions. During EMDA 614 I learned a great deal about fractions, multiplication, and division. As a class we discussed some common errors students make when performing different operations. One “golden nugget” that stuck with me is: I think it is important to understand the different errors students make because “bad habits are hard to break,” (p.68 Parker, Thomas H. & Bladridge, Scott J., 2003). –Taken from a Reflection for EDMA 614
During the course of my master’s program I was able to look more closely at how numbers and operations are taught throughout elementary school. Students need a firm understanding of numbers before they learn how to use numbers to add, subtract, multiply, or divide.
Different Perspectives on Algebra
Statement:
Incorporates learning styles and ability to foster perspectives on algebra.
When most people think of algebra they think of letters being used for numbers, which for many people makes little sense if any. During EDMA 664 I spent some time researching the history of algebra and discovered that algebra is more than simply using letters to represent an unknown. For instance, in 2006 Cathy Seely stated, “algebraic thinking includes recognizing and analyzing patterns, studying and representing relationships, making generalizations and analyzing how things change,” (as cited in Checkley, 2006, p 17). Furthermore algebra today includes a wide variety of topics. “These include the arithmetic of signed numbers, solutions of linear equations, quadratic equations, and systems of linear and/or quadratic equations, and the manipulation of polynomials, including factoring and rules of exponents” (Barton and Katz, 2007 p 186). Also included might be “matrices, functions and graphs, conic sections, and other topics” (186). As one can see, algebra today “covers a lot of ground” (186).
Educators, yes even kindergarten teachers, can begin to build the skills necessary for children to understand algebra. As Seely stated, “algebraic thinking includes recognizing and analyzing patterns” and “studying and representing relationships” (as cited in Checkley, 2006, p. 17). According to the Alaska Common Core Standards, students in 3rd grade should be able to, "solve problems involving the four operations, and identify and explain patterns in arithmetic," (2012). As a second/third grade teacher I spend time working with my students on recognizing patterns and solving problems related to these patterns. I also devote time to understand different symbols; for instance, my students last year spent a great deal of time actually understanding what the equal sign means. Rather than always saying, “is equal to” our class began saying “is the same as”. Students began to understand that the equal sign meant that each side of the equal side had to balance.
Working on building skills as simple as understanding patterns and symbols like the equal sign can serve as building blocks for the study of algebra. As I stated in one of my Concept Description and Reflection papers,
“Furthermore, it is important for us to help children have a deeper understanding of mathematics by incorporating algebraic thinking early in children’s education. This will help them be successful in later algebra courses and in solving problems in the “real world” involving algebraic ideas.”
“Teachers need to thoughtfully consider how and when they incorporate algebraic thinking into their daily curriculum.” (Taken from EDMA 664 Concept Description and Reflection).
Geometries
Statement:
Reflects how to develop and assess student knowledge and skills of geometric modeling, structures and shapes and demonstrates some ability to foster exploration and analysis of geometry. Incorporates learning styles and abilities to foster spatial visualization and geometric relationships.
Prior to taking EDMA 655 I did not take a course focused solely on the content and pedagogy for geometry and measurement. During EDMA 655 I researched the van Hiele theory of geometry which is a “five-level hierarchy of ways of understanding spatial ideas,” (Van de Walle, 2007, p. 346). Before I had completed research on the van Hiele theory of geometry I often found it difficult to create community problems using geometry that were appropriate for my students. By studying the van Hiele theory of geometry I learned how to appropriately assess where my students are at and how to provide instruction based on my students’ level of understanding. Here is my teacher reflection section from my Concept Description and Reflection on the van Hiele theory of geometry.
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.
I currently teach in a third grade classroom; I have 23 students. 2 of my students are new within the last week, since I teach at a school on base this is a common occurrence. One point mentioned by Van de Walle, states that vocabulary instruction needs to be taught at a student’s current geometric level. When we introduce vocabulary that is too difficult for students they are not able to construct meaning for that term or idea. I have recently had difficulty with the copious amounts of vocabulary provided when one studies geometry. Even though we do complete math journals as a class, one of my new students is having difficulty because he is behind the other students. Today I paired him up with a peer, they worked cooperatively on their math journals and the boy did a better job. Sometimes students seem to do better when their peers present vocabulary or concepts previously introduced by the teacher.
Many of my third grade students are in level 1 in the van Hiele levels of geometric thought. This past week we worked on symmetry. The following is an example of providing students with an appropriate geometric experience for level 1: Analysis. The students watched a short clip from Illuminations describing symmetry. Then we wrote in our math journals; we included a definition and a few examples and non-examples of symmetry. Finally, each child was given a paper with different shapes on it. Then the students had to cut the shapes out and sort them based on their properties (symmetrical & non-symmetrical). The children folded each shape to see if it was or wasn’t symmetrical. The children did this, and they did a good job.
In addition to providing experience with the appropriate level of geometric thought, I also work hard to incorporate the Common Core Standards of Mathematics into my classroom. Alaska recently adopted the Common Core Math Standards in June of 2012, which require teachers to make certain that students experience geometry so they have a true “understanding” of the geometric object of thought being assessed. For instance, standard 3.G1 states the student will,
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. (Alaska Mathematics Standards, 2012)
By using van Hiele’s levels of geometric thought as a guide I will be able to create meaningful experiences for my students. My students will use the knowledge they have gained from the Visualization level, taught in previous grades, to help them notice and sort shapes based on properties and similarities between shapes, rather than just visual similarities.
Data Analysis, Statistics and Probability
Statement:
Incorporates learning styles and abilities to help students investigate and apply statistical methods.
Most of my teaching experience has been in 2nd and 3rd grade so prior to working on my master’s degree I had never really looked at the content strands in the Common Core for Statistics and Probability. As a result of EDMA 656 I have a deeper understanding of the standards taught in 6th-8th grade (and even into high school); additionally, I understand how I can foster statistical thinking in my elementary students. For instance, my students can participate in lessons that require them to collect and analyze data which is a precursor for statistics content my students may encounter later in their mathematics education. Here is an example of a community problem I designed: EDMA 656 Shared Problem.
Measurement:
Statement:
Incorporates learning styles and abilities, and helps students apply measurement concepts and tools to the local community and the real world.
Measurement experiences for students should include hands-on measurement activities. In other words, students should not be simply measuring a line on a piece of paper or estimating which weight is correct for an object on a multiple choice test. Learning measurement should be concrete especially for primary children. In my classroom I prefer to begin by using non-standard units of measure such as measuring the length of something using paperclips, graham crackers, or perhaps a student’s foot. Using a student’s foot is also a good lead into why standard units of measure are necessary if we are to have a consistent unit to compare measurements. My students always seem to have a better grasp on standard measurement when they understand why a standard unit is important.
Including measurement in the curriculum is fun for students and I can easily suggest ways for parents to incorporate measurement at home as well. For example, pbskids.org offers many activities for elementary age students to do with their parents.
Resources:
Alaska Mathematics Standards. (2012, June). Alaska Department of Education and Early Development.
Retrieved February 5, 2013, from http://www.eed.alaska.gov/tls/Assessment/standards/
AKStandards_ELAandMath_080812.pdf
Barton, Bill and Katz, J.Victor (2007). Educational Studies in Mathematics, Vol. 66, Issue 2, pp. 185-201. [On-line].
Retrieved March 14, 2013 from http://egandb .uas.alaska.edu:2048/login?
url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=26618755&site=ehost-live
Breyfogle M. & Lynch C. (2010) “van Hiele Revisited,” Mathematics Teaching in the Middle School. [On-line], Vol.
16, No. 4, pp. 232-238. Retrieved February 4, 2013, from http://lesage.blogs.uoit.ca/wp-
uploads/2010/09/van-Hiele-Levels- Assessment_2010-NCTM.pdf
Checkly, Kathy (2006). The Essential of Mathematics, K-6: Effective Curriculum, Instruction, and Assessment [On-
line]. Retrieved January 26, 2013, from http://egandb.uas.alaska.edu:2081/lib/uasoutheast/doc
Detail.action?docID=10120221
Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics.
Cross, Christopher T. & Woods, Taniesha A. (2009). Mathematics Learning in Early Childhood: Paths Toward
Excellence and Equity. [On-line]. Retrieved June 21, 2014 from
http://egandb.uas.alaska.edu:2081/lib/uasoutheast/docDetail.actiondocID=10355555 &p00=cross%2C%20christopher%20t.%20woods%2C%20taniesha
Moyer, P. S. (2001). Are We Having Fun Yet? How Teachers Use Manipulatives to Teach Mathematics.
Educational Studies In Mathematics, 47(2), 175-197. [On-line]. Retrieved February 8, 2013 from
http://egandb.uas.alaska.edu:2303/ehost/pdfviewer/pdfviewer?sid=301653c5-3f17-4af1-9ea3
5ce25e8a0160%40sessionmgr104&vid=1&hid=104
NCTM Math Standards and Expectations. (2014). Retrieved July 25, 2014
from http://www.nctm.org/standards/content.aspx?id=4294967312
Parker, Thomas H. & Baldridge, Scott J. (2003). Elementary Mathematics for Teachers. Sefton-Ash Publishers.
Teaching Math Grades K-2. (2013). Annenberg Foundation, Inc. Retrieved online
from http://www.learner.org/courses/teachingmath/gradesk_2/session_05/index.html
Van de Walle, John A. (1998). Elementary and Middle School Mathematics Teaching Developmentally. (3rd ed., p.
11). New York: Addison Wesley Longman, Inc.
As children move into 1st and 2nd grade they begin to start working more with the different operations. 1st and second grade students spend a great deal of time learning how to add or subtract numbers. One thing I learned in EDMA 614 is that, as educators, we have to be very aware of the language that we use when we teach mathematics. Saying “borrowing” for instance gives students the idea that at some point during the subtraction problem we are going to “give it back”, which is certainly not the case. I learned that a better term would be “unbundling” or “decomposing”.
Towards the end of 2nd grade students begin to work with multiplication and division. Then as students enter third grade they begin to work with fractions. During EMDA 614 I learned a great deal about fractions, multiplication, and division. As a class we discussed some common errors students make when performing different operations. One “golden nugget” that stuck with me is: I think it is important to understand the different errors students make because “bad habits are hard to break,” (p.68 Parker, Thomas H. & Bladridge, Scott J., 2003). –Taken from a Reflection for EDMA 614
During the course of my master’s program I was able to look more closely at how numbers and operations are taught throughout elementary school. Students need a firm understanding of numbers before they learn how to use numbers to add, subtract, multiply, or divide.
Different Perspectives on Algebra
Statement:
Incorporates learning styles and ability to foster perspectives on algebra.
When most people think of algebra they think of letters being used for numbers, which for many people makes little sense if any. During EDMA 664 I spent some time researching the history of algebra and discovered that algebra is more than simply using letters to represent an unknown. For instance, in 2006 Cathy Seely stated, “algebraic thinking includes recognizing and analyzing patterns, studying and representing relationships, making generalizations and analyzing how things change,” (as cited in Checkley, 2006, p 17). Furthermore algebra today includes a wide variety of topics. “These include the arithmetic of signed numbers, solutions of linear equations, quadratic equations, and systems of linear and/or quadratic equations, and the manipulation of polynomials, including factoring and rules of exponents” (Barton and Katz, 2007 p 186). Also included might be “matrices, functions and graphs, conic sections, and other topics” (186). As one can see, algebra today “covers a lot of ground” (186).
Educators, yes even kindergarten teachers, can begin to build the skills necessary for children to understand algebra. As Seely stated, “algebraic thinking includes recognizing and analyzing patterns” and “studying and representing relationships” (as cited in Checkley, 2006, p. 17). According to the Alaska Common Core Standards, students in 3rd grade should be able to, "solve problems involving the four operations, and identify and explain patterns in arithmetic," (2012). As a second/third grade teacher I spend time working with my students on recognizing patterns and solving problems related to these patterns. I also devote time to understand different symbols; for instance, my students last year spent a great deal of time actually understanding what the equal sign means. Rather than always saying, “is equal to” our class began saying “is the same as”. Students began to understand that the equal sign meant that each side of the equal side had to balance.
Working on building skills as simple as understanding patterns and symbols like the equal sign can serve as building blocks for the study of algebra. As I stated in one of my Concept Description and Reflection papers,
“Furthermore, it is important for us to help children have a deeper understanding of mathematics by incorporating algebraic thinking early in children’s education. This will help them be successful in later algebra courses and in solving problems in the “real world” involving algebraic ideas.”
“Teachers need to thoughtfully consider how and when they incorporate algebraic thinking into their daily curriculum.” (Taken from EDMA 664 Concept Description and Reflection).
Geometries
Statement:
Reflects how to develop and assess student knowledge and skills of geometric modeling, structures and shapes and demonstrates some ability to foster exploration and analysis of geometry. Incorporates learning styles and abilities to foster spatial visualization and geometric relationships.
Prior to taking EDMA 655 I did not take a course focused solely on the content and pedagogy for geometry and measurement. During EDMA 655 I researched the van Hiele theory of geometry which is a “five-level hierarchy of ways of understanding spatial ideas,” (Van de Walle, 2007, p. 346). Before I had completed research on the van Hiele theory of geometry I often found it difficult to create community problems using geometry that were appropriate for my students. By studying the van Hiele theory of geometry I learned how to appropriately assess where my students are at and how to provide instruction based on my students’ level of understanding. Here is my teacher reflection section from my Concept Description and Reflection on the van Hiele theory of geometry.
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.
I currently teach in a third grade classroom; I have 23 students. 2 of my students are new within the last week, since I teach at a school on base this is a common occurrence. One point mentioned by Van de Walle, states that vocabulary instruction needs to be taught at a student’s current geometric level. When we introduce vocabulary that is too difficult for students they are not able to construct meaning for that term or idea. I have recently had difficulty with the copious amounts of vocabulary provided when one studies geometry. Even though we do complete math journals as a class, one of my new students is having difficulty because he is behind the other students. Today I paired him up with a peer, they worked cooperatively on their math journals and the boy did a better job. Sometimes students seem to do better when their peers present vocabulary or concepts previously introduced by the teacher.
Many of my third grade students are in level 1 in the van Hiele levels of geometric thought. This past week we worked on symmetry. The following is an example of providing students with an appropriate geometric experience for level 1: Analysis. The students watched a short clip from Illuminations describing symmetry. Then we wrote in our math journals; we included a definition and a few examples and non-examples of symmetry. Finally, each child was given a paper with different shapes on it. Then the students had to cut the shapes out and sort them based on their properties (symmetrical & non-symmetrical). The children folded each shape to see if it was or wasn’t symmetrical. The children did this, and they did a good job.
In addition to providing experience with the appropriate level of geometric thought, I also work hard to incorporate the Common Core Standards of Mathematics into my classroom. Alaska recently adopted the Common Core Math Standards in June of 2012, which require teachers to make certain that students experience geometry so they have a true “understanding” of the geometric object of thought being assessed. For instance, standard 3.G1 states the student will,
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. (Alaska Mathematics Standards, 2012)
By using van Hiele’s levels of geometric thought as a guide I will be able to create meaningful experiences for my students. My students will use the knowledge they have gained from the Visualization level, taught in previous grades, to help them notice and sort shapes based on properties and similarities between shapes, rather than just visual similarities.
Data Analysis, Statistics and Probability
Statement:
Incorporates learning styles and abilities to help students investigate and apply statistical methods.
Most of my teaching experience has been in 2nd and 3rd grade so prior to working on my master’s degree I had never really looked at the content strands in the Common Core for Statistics and Probability. As a result of EDMA 656 I have a deeper understanding of the standards taught in 6th-8th grade (and even into high school); additionally, I understand how I can foster statistical thinking in my elementary students. For instance, my students can participate in lessons that require them to collect and analyze data which is a precursor for statistics content my students may encounter later in their mathematics education. Here is an example of a community problem I designed: EDMA 656 Shared Problem.
Measurement:
Statement:
Incorporates learning styles and abilities, and helps students apply measurement concepts and tools to the local community and the real world.
Measurement experiences for students should include hands-on measurement activities. In other words, students should not be simply measuring a line on a piece of paper or estimating which weight is correct for an object on a multiple choice test. Learning measurement should be concrete especially for primary children. In my classroom I prefer to begin by using non-standard units of measure such as measuring the length of something using paperclips, graham crackers, or perhaps a student’s foot. Using a student’s foot is also a good lead into why standard units of measure are necessary if we are to have a consistent unit to compare measurements. My students always seem to have a better grasp on standard measurement when they understand why a standard unit is important.
Including measurement in the curriculum is fun for students and I can easily suggest ways for parents to incorporate measurement at home as well. For example, pbskids.org offers many activities for elementary age students to do with their parents.
Resources:
Alaska Mathematics Standards. (2012, June). Alaska Department of Education and Early Development.
Retrieved February 5, 2013, from http://www.eed.alaska.gov/tls/Assessment/standards/
AKStandards_ELAandMath_080812.pdf
Barton, Bill and Katz, J.Victor (2007). Educational Studies in Mathematics, Vol. 66, Issue 2, pp. 185-201. [On-line].
Retrieved March 14, 2013 from http://egandb .uas.alaska.edu:2048/login?
url=http://search.ebscohost.com/login.aspx?direct=true&db=aph&AN=26618755&site=ehost-live
Breyfogle M. & Lynch C. (2010) “van Hiele Revisited,” Mathematics Teaching in the Middle School. [On-line], Vol.
16, No. 4, pp. 232-238. Retrieved February 4, 2013, from http://lesage.blogs.uoit.ca/wp-
uploads/2010/09/van-Hiele-Levels- Assessment_2010-NCTM.pdf
Checkly, Kathy (2006). The Essential of Mathematics, K-6: Effective Curriculum, Instruction, and Assessment [On-
line]. Retrieved January 26, 2013, from http://egandb.uas.alaska.edu:2081/lib/uasoutheast/doc
Detail.action?docID=10120221
Clements, Douglas H. (1999). Subitizing: What is it? Why Teach it? Teaching Children Mathematics.
Cross, Christopher T. & Woods, Taniesha A. (2009). Mathematics Learning in Early Childhood: Paths Toward
Excellence and Equity. [On-line]. Retrieved June 21, 2014 from
http://egandb.uas.alaska.edu:2081/lib/uasoutheast/docDetail.actiondocID=10355555 &p00=cross%2C%20christopher%20t.%20woods%2C%20taniesha
Moyer, P. S. (2001). Are We Having Fun Yet? How Teachers Use Manipulatives to Teach Mathematics.
Educational Studies In Mathematics, 47(2), 175-197. [On-line]. Retrieved February 8, 2013 from
http://egandb.uas.alaska.edu:2303/ehost/pdfviewer/pdfviewer?sid=301653c5-3f17-4af1-9ea3
5ce25e8a0160%40sessionmgr104&vid=1&hid=104
NCTM Math Standards and Expectations. (2014). Retrieved July 25, 2014
from http://www.nctm.org/standards/content.aspx?id=4294967312
Parker, Thomas H. & Baldridge, Scott J. (2003). Elementary Mathematics for Teachers. Sefton-Ash Publishers.
Teaching Math Grades K-2. (2013). Annenberg Foundation, Inc. Retrieved online
from http://www.learner.org/courses/teachingmath/gradesk_2/session_05/index.html
Van de Walle, John A. (1998). Elementary and Middle School Mathematics Teaching Developmentally. (3rd ed., p.
11). New York: Addison Wesley Longman, Inc.