Mathematics
Representations Model adapted from Lesh,
Post, and Behr.
The delivery of most school subjects has traditionally been geared toward the left brain, even though not all students learn in this way. When in charge of a classroom of 30 individuals it is difficult to target all the individual needs that exist. For children to really understand a concept, it is important that they make as many connections as possible to that concept, and to allow enough practice for those connections to be strengthened.
When teachers present mathematical problems in a variety of ways, we are not only increasing our chances that we are reaching each student with at least one way, but we are providing opportunities for multiple connections to be made. Symbols and spoken language, for example, target the left hemisphere whereas pictures and manipulatives target the right hemisphere. When we allow children to make connections between the left and right hemisphere, we are strengthening the connections in the corpus callosum, which in turn improves the communication between the two sides. Allowing children to discuss their strategies with each other help them to make these connections. Children who need to first make sense of the mathematics in a visual way (right brain processing) before attaching that meaning to a standard algorithm, struggle when it is only presented in a symbolic way (left brain processing). Allowing children to discuss their strategies with each other help them to make these connections. Click on image to download article which describes each of the five representations shown in the diagram.
When teachers present mathematical problems in a variety of ways, we are not only increasing our chances that we are reaching each student with at least one way, but we are providing opportunities for multiple connections to be made. Symbols and spoken language, for example, target the left hemisphere whereas pictures and manipulatives target the right hemisphere. When we allow children to make connections between the left and right hemisphere, we are strengthening the connections in the corpus callosum, which in turn improves the communication between the two sides. Allowing children to discuss their strategies with each other help them to make these connections. Children who need to first make sense of the mathematics in a visual way (right brain processing) before attaching that meaning to a standard algorithm, struggle when it is only presented in a symbolic way (left brain processing). Allowing children to discuss their strategies with each other help them to make these connections. Click on image to download article which describes each of the five representations shown in the diagram.
Elementary Mathematics is Anything but Elementary
My colleague Dr. Damon Bahr and I wrote this text to prepare pre-service teachers for today''s classroom, ELEMENTARY MATHEMATICS IS ANYTHING BUT ELEMENTARY: CONTENT AND METHODS FROM A DEVELOPMENTAL PERSPECTIVE is a comprehensive program that delivers both a content and a methods text. Serving as a professional development guide for both pre-service and in-service teachers, this text''s integrated coverage helps dissolve the line between content and methods--and consequently bolsters teachers'' confidence in their delivery of math instruction. A strong emphasis on the National Council of Teachers of Mathematics five core standards provides key information common to most state curricula relative to NCTM standards for pre-K through sixth grade. In addition, text content is based on thorough elementary mathematical scope and sequences that have been shown to be an effective means for guiding the delivery of curriculum and instruction.
The text also has a companion website which houses over 150 videos of classroom lessons, student math interviews, and teacher interviews. They are separated by chapter and highlight the pedagogical practices and mathematical examples from the text. To purchase a premium website instant access code or a digital version of the book, click here.
My colleague Dr. Damon Bahr and I wrote this text to prepare pre-service teachers for today''s classroom, ELEMENTARY MATHEMATICS IS ANYTHING BUT ELEMENTARY: CONTENT AND METHODS FROM A DEVELOPMENTAL PERSPECTIVE is a comprehensive program that delivers both a content and a methods text. Serving as a professional development guide for both pre-service and in-service teachers, this text''s integrated coverage helps dissolve the line between content and methods--and consequently bolsters teachers'' confidence in their delivery of math instruction. A strong emphasis on the National Council of Teachers of Mathematics five core standards provides key information common to most state curricula relative to NCTM standards for pre-K through sixth grade. In addition, text content is based on thorough elementary mathematical scope and sequences that have been shown to be an effective means for guiding the delivery of curriculum and instruction.
The text also has a companion website which houses over 150 videos of classroom lessons, student math interviews, and teacher interviews. They are separated by chapter and highlight the pedagogical practices and mathematical examples from the text. To purchase a premium website instant access code or a digital version of the book, click here.
How to get Students Talking: Generating Math Talk that Supports Math Learning
This is an article that I wrote, which was published by Math Solutions, that describes steps a teacher needs to take in order to set up a strong culture of discourse. I followed two primary (kinder and second grade) teachers as well as reflected on my own practice in the upper grades. I outline specific teacher moves throughout the year to help facilitate academic talk, something that does not necessarily come naturally to children or teachers.
This is an article that I wrote, which was published by Math Solutions, that describes steps a teacher needs to take in order to set up a strong culture of discourse. I followed two primary (kinder and second grade) teachers as well as reflected on my own practice in the upper grades. I outline specific teacher moves throughout the year to help facilitate academic talk, something that does not necessarily come naturally to children or teachers.
Math Teaching Strategies is a website that my pre-service teachers put together. There are descriptions and resource links connected to a variety of content and pedagogy strategies for teaching math.
Joshua's Math Journey
A few years ago I did an action research project for myself on my son, Joshua. I was working with him faithfully in mathematics and for a period of a few months I recorded what I taught him, and many sessions I videotaped. In this study, I focused on teaching him division from the beginning. Since Joshua has very little expressive language, it was my challenge to figure out how to teach him a concept as complicated as division. I started with basic division understanding and worked to division with remainders. This website has the teaching notes, videos, as well as an unpublished paper that I wrote, entitled "Refining the Concrete-Representational-Abstract (CRA) Instructional Approach for Teaching Division to a Language-Impaired Child with Autism: A Case Study."
A few years ago I did an action research project for myself on my son, Joshua. I was working with him faithfully in mathematics and for a period of a few months I recorded what I taught him, and many sessions I videotaped. In this study, I focused on teaching him division from the beginning. Since Joshua has very little expressive language, it was my challenge to figure out how to teach him a concept as complicated as division. I started with basic division understanding and worked to division with remainders. This website has the teaching notes, videos, as well as an unpublished paper that I wrote, entitled "Refining the Concrete-Representational-Abstract (CRA) Instructional Approach for Teaching Division to a Language-Impaired Child with Autism: A Case Study."
XtraMath.org is a wonderful program that will assess your child's fact skills and will follow his progress as he practices. You can change the settings to different operations, but the default setting is to start with addition.
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Moby Max is another adaptive program which works with all subjects, not just mathematics. Again, the child is assessed and given a grade level and from there is assigned lessons. Teacher/parent can control lessons that are provided. As a parent, you may need to sign up under homeschool if having difficulty registering.
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Electronic Books
The following are electronic books that were made by some of my pre-service teachers. They range from kinder through 6th grade. Many of the books can be used in multiple grades. Select on the picture of the book to download. Use with your students through projector or interactive white board.
The following are electronic books that were made by some of my pre-service teachers. They range from kinder through 6th grade. Many of the books can be used in multiple grades. Select on the picture of the book to download. Use with your students through projector or interactive white board.
Math Games
Playing games is the best way to learn, as the children do not even realize that they are learning. The internet is full of ideas for games for each of the math standards and most curricula have a game component to them that often get ignored. The following are games that were created by my pre-service teachers. I welcome any additions of math games created by teachers.
Mental Calculator
Each student has 10 tiles with 0-9 written on them. Have them put them in front of him/her in order in a row. The teacher, or designated student, will provide math problems that students have to mentally compute and then put their fingers on the digits of the correct answer. For example, if the problem is 6 + 8 and the answer is 14, the student puts the left finger on the 1 (tens) and the right finger on the 4 (ones). String several problems together before checking for the final answer to see how many students are on the correct numbers. Alternative: assign half the class to do the mental calculations and half the class to compute on the calculators to see who is faster (and to check for the final answer so the teacher doesn't have to necessarily keep track).
Least Common Multiple Game (click for pdf)
Greatest Common Factor Game (click for pdf)
Playing games is the best way to learn, as the children do not even realize that they are learning. The internet is full of ideas for games for each of the math standards and most curricula have a game component to them that often get ignored. The following are games that were created by my pre-service teachers. I welcome any additions of math games created by teachers.
Mental Calculator
Each student has 10 tiles with 0-9 written on them. Have them put them in front of him/her in order in a row. The teacher, or designated student, will provide math problems that students have to mentally compute and then put their fingers on the digits of the correct answer. For example, if the problem is 6 + 8 and the answer is 14, the student puts the left finger on the 1 (tens) and the right finger on the 4 (ones). String several problems together before checking for the final answer to see how many students are on the correct numbers. Alternative: assign half the class to do the mental calculations and half the class to compute on the calculators to see who is faster (and to check for the final answer so the teacher doesn't have to necessarily keep track).
Least Common Multiple Game (click for pdf)
Greatest Common Factor Game (click for pdf)