Numeracy
Objectives
I use these goals to self-assess my own learning. I use a *** for excellent understanding, ** for satisfactory learning and * for needs further exploration. I hope to come back to those areas throughout the course.
- investigate strategies for understanding the nature of a student’s difficulty with mathematics **
- develop a repertoire of strategies to address math deficits experienced by students **
- design adaptations to address the mismatch between the student’s strengths and needs, and the demands of the curriculum **
checklist for inclusive mathematics instruciton
This checklist was a real eye opener to the lack of cultural inclusion within my classroom. In general, I need more training in teaching math. My first step in changing this is to collect resources and ideas of math teaching strategies. I also plan to dedicate a lot of my planning time to self-learning math strategies. My other step is that I am hoping to take an AQ next summer in math inquiry based learning.
Memory,language, attention, and production
http://www.pbs.org/wgbh/misunderstoodminds/mathstrats.html#memory
Memory
Provide the technology tools needed for problem solving. Encourage children to think mathematically, even if they have not mastered basic skills. For example, let them use computer spreadsheet programs and calculators when the goal of the math activity is to develop problem-solving skills as opposed to calculation skills.
Teach basic math facts. Use explicit instruction to promote student mastery. Put a few selected unknown facts on index cards. Put strategies for remembering on the back of the cards. Cards can be put on notebook rings. Add new facts as previous ones are learned. Build practice into lessons. Also, routinely conduct cumulative reviews of skills and knowledge to help children develop automaticity with math facts.
Use rule books. Ask children to keep a notebook in which they write math rules in their own words. Encourage children to use rule books with classroom or home assignments by looking up the rule in the book and talking about it. Rule books could have a math vocabulary section and a strategy section for recording "tricks" that help with the operations.
Teach subvocalization as a strategy. Show children how to quietly repeat sequences (such as numbers and procedures) under their breath while working. Practice the strategy by giving them a sequence of numbers or directions and having them quietly repeat them back to you.
Practice subskills. Help children recall math subskills (like multiplication) more automatically with the use of flashcards and drills. Play a game in which you quiz a child about math facts and record how many he answers correctly. To build motivation, have the child record her own progress each day. Together, review progress periodically.
Teach math in more than one mode. Children respond well when math is taught in a variety of ways -- visually (such as demonstration), verbally (such as using oral explanations), and experientially (such as setting up a mock store) -- so that children have an opportunity to process and use math information in multiple ways.
Use games. To enhance active working memory, play mental math games. For example, "What two numbers can be multiplied to get 24? How many different combinations can you find?" Gradually build up a child's ability to hold a long problem (How much is 4 + 2 - 1 x 3?) in memory. Make sure the child understands the reason for playing the game.
Review patterns. Use flash cards to review patterns, such as key words that provide clues to the operation of a word problem, or geometric patterns or shapes within complex visual designs.
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Language
Focus on the information provided in word problems. Have children separate the necessary information for solving the problem from unnecessary details.
Teach mnemonic strategies for solving word problems. Choose strategies that suit the child's learning style. One strategy is TIPS: Think (read and paraphrase), Information (what numbers and information do you need in order to solve the problem), Problem (write equation), Solve.
Encourage children to put problems into their own words. Teach children to read for meaning when trying to identify the operation to use for solving a math problem. Have them verbalize the problem before trying to solve it.
Teach math vocabulary. Review the meaning of key words and phrases commonly used in mathematics problems, such as "all" or "total" in addition problems ("How much money did they spend in all?" "What was the total amount of the grocery bill?"). To help children identify key terms in problems, ask them whether a problem requires a particular procedure, and have them underline the word or term that gave the answer away. Include new vocabulary in their rule books (see above).
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Attention
Teach children how to preview an assignment. Help them to see the importance of thinking ahead before beginning the task. For example, cue them to ask, "Which math operations will I need next?"
Teach children how to self-monitor. During a task, show children how to stop and assess how well they are progressing. For example, tell them, "Every 10 minutes you will need to stop and check your answers." Teach children to ask themselves questions such as "How is it going?" and, "Do I need to make changes?" "Does my answer make sense?" and "Does my answer match my estimate?"
Help children maintain mental energy. Allow them to take frequent breaks while completing math assignments. Suggest that they get up and walk around during these breaks.
Teach self-checking strategies. Have students change to a different color pen when they have finished their work, becoming a "test checker" instead of a "test taker." This will help them notice their errors. For students who continue to make attentional errors in calculation, despite instruction and practice with self-checking, permit the use of a calculator for checking.
Help children stay focused. Let them choose the best place to do assignments, or allow them to listen to music if that helps their concentration.
Provide a model. Work through the mathematical problem with the child, verbalizing or demonstrating each step. Especially with homework, assist the child by doing the first problem together.
Identify topics of interest to children. Explore mathematical concepts in relation to motivating topics, such as building a skateboard ramp, tracking a satellite's orbit around the earth, discovering how the pyramids were built, or saving money in an interest-bearing account. Ask children to help you identify topics for mathematical problems.
Build a foundation for multi-step problems. Be sure the child understands basic one-step problems (problems requiring only one math operation) before advancing to those that require multiple operations.
Isolate steps. Have children focus on one step at a time. For example, provide mathematical activities in which children identify only (1) what the question is asking them to find, (2) which information is necessary to answer the question, and (3) which operations should be used in solving the problem.
Complete each step. Explain to children that even good problem solvers rarely skip steps when solving problems, though they may appear to.
Reduce the amount of data on a page. Children with spatial problems often become overwhelmed by large amounts of visual data on a page. Reduce the number of math problems or the number of diagrams to interpret per page. Remove unessential visual features.
Have children draw pictures to represent what is going on in a math problem. Suggest they draw representations of objects from the problem (for example, three shirts, a 6-by-12 foot garden plot).
Make auxiliary tools available. Provide calculators, graph paper for aligning numbers, or templates for tracing geometric shapes.
Production
Because math difficulties can affect a child's performance and ability to get work done, the following strategies are designed to help children improve their organization skills, work habits, and overall production.
Use assignment books. Teach children to use assignment books and "To Do" lists to keep track of their short- and long-term assignments, tests, and quizzes. Use peers to help monitor other children's assignment books. Most schools have a "homework hotline" on voicemail or homework posted on the school Web site. These resources provided by the school can help you support a student who does not yet record assignments consistently without reminders.
Provide models of assignments and criteria for success. Give children a clear sense of how a final product might look by showing examples and sharing exemplary products (such as providing a workbook of sample problems completed correctly). You might make work from last year available and draw the children's attention to specific qualities of the work (for example, "Notice how lining up the columns makes the problem easier to understand."). Do not, however, compare children's work with that of peers or siblings.
Build in planning time. Give children five minutes of planning time before beginning an assignment. Provide guidance in effective planning when necessary.
Use stepwise approaches. Require children to break down tasks into parts and write down the steps or stages. Compile steps of frequent tasks into a notebook for easy reference during work assignments. For long-term assignments, provide a due date for each step of the assignment.
Teach proven strategies. Provide children with specific age-appropriate strategies to use in checking work. For example, use TIPS: Think (read and paraphrase),Information (what numbers and information do you need in order to solve the problem?), Problem (write equation), Solve. Children can create a reminder card to keep on their desk or in their assignment book for quick reference to the strategy.
Stress the importance of organization. Have children preview an assignment and collect the materials they will need before starting it. Guide children in keeping their materials and notebooks organized and easily accessible. In middle and high school, conduct intermittent "notebook checks" and grade organization and completion. At the beginning of the school year and a week before each check, give a list of requirements. Emphasize the positive impact that organization and preplanning will have on the completed project or assignment. By grading organization, you will emphasize its value in the learning process.
Let children wait to turn in work. The day before an assignment is due, have children review their work and check it with a parent. This will give the children enough perspective to catch errors or add more details and produce better results in the end.
Encourage self-evaluation. Set a standard of work quality or criteria for success for children to follow, and allow them to self-assess the quality of their work before turning it in. If the grade matches the child's appraisal, give extra points for good self-assessment. Rubrics are one way for students to assess their own work.
Set goals and record progress. Have children set a short-term goal, such as completing all homework for the week. Record their daily progress toward the goal for children to observe. Graphic recording, such as plotting their own line graphs, may be particularly reinforcing for some children. Reward improvement at home.
Practice estimating. Children may benefit from estimating answers to math problems and science experiments. Stress the real-life benefits of estimating and understanding what the correct answer might look like.
Eliminate incentives for frenetic pacing. Remove any positive reinforcement for finishing first. State the amount of time a task should take. This will slow down children who work too quickly and will speed up children who work too slowly.
Provide consistent feedback. Create a feedback system so children understand which behaviors, actions, or work products are acceptable and which are not. Use specifics to praise good work and recognize when children use strategies effectively. Say, for example, "I like the way you drew a table to help explain the problem," or "Asking to take a break really seemed to help you come back and focus."
Try a mentor. Some children may benefit from a mentor who will work with them to analyze their academic progress, brainstorm alternative strategies, and provide recognition of progress. The mentor must be seen as credible, and may be an individual from either inside or outside the school.
Memory
Provide the technology tools needed for problem solving. Encourage children to think mathematically, even if they have not mastered basic skills. For example, let them use computer spreadsheet programs and calculators when the goal of the math activity is to develop problem-solving skills as opposed to calculation skills.
Teach basic math facts. Use explicit instruction to promote student mastery. Put a few selected unknown facts on index cards. Put strategies for remembering on the back of the cards. Cards can be put on notebook rings. Add new facts as previous ones are learned. Build practice into lessons. Also, routinely conduct cumulative reviews of skills and knowledge to help children develop automaticity with math facts.
Use rule books. Ask children to keep a notebook in which they write math rules in their own words. Encourage children to use rule books with classroom or home assignments by looking up the rule in the book and talking about it. Rule books could have a math vocabulary section and a strategy section for recording "tricks" that help with the operations.
Teach subvocalization as a strategy. Show children how to quietly repeat sequences (such as numbers and procedures) under their breath while working. Practice the strategy by giving them a sequence of numbers or directions and having them quietly repeat them back to you.
Practice subskills. Help children recall math subskills (like multiplication) more automatically with the use of flashcards and drills. Play a game in which you quiz a child about math facts and record how many he answers correctly. To build motivation, have the child record her own progress each day. Together, review progress periodically.
Teach math in more than one mode. Children respond well when math is taught in a variety of ways -- visually (such as demonstration), verbally (such as using oral explanations), and experientially (such as setting up a mock store) -- so that children have an opportunity to process and use math information in multiple ways.
Use games. To enhance active working memory, play mental math games. For example, "What two numbers can be multiplied to get 24? How many different combinations can you find?" Gradually build up a child's ability to hold a long problem (How much is 4 + 2 - 1 x 3?) in memory. Make sure the child understands the reason for playing the game.
Review patterns. Use flash cards to review patterns, such as key words that provide clues to the operation of a word problem, or geometric patterns or shapes within complex visual designs.
Back to Top
Language
Focus on the information provided in word problems. Have children separate the necessary information for solving the problem from unnecessary details.
Teach mnemonic strategies for solving word problems. Choose strategies that suit the child's learning style. One strategy is TIPS: Think (read and paraphrase), Information (what numbers and information do you need in order to solve the problem), Problem (write equation), Solve.
Encourage children to put problems into their own words. Teach children to read for meaning when trying to identify the operation to use for solving a math problem. Have them verbalize the problem before trying to solve it.
Teach math vocabulary. Review the meaning of key words and phrases commonly used in mathematics problems, such as "all" or "total" in addition problems ("How much money did they spend in all?" "What was the total amount of the grocery bill?"). To help children identify key terms in problems, ask them whether a problem requires a particular procedure, and have them underline the word or term that gave the answer away. Include new vocabulary in their rule books (see above).
Back to Top
Attention
Teach children how to preview an assignment. Help them to see the importance of thinking ahead before beginning the task. For example, cue them to ask, "Which math operations will I need next?"
Teach children how to self-monitor. During a task, show children how to stop and assess how well they are progressing. For example, tell them, "Every 10 minutes you will need to stop and check your answers." Teach children to ask themselves questions such as "How is it going?" and, "Do I need to make changes?" "Does my answer make sense?" and "Does my answer match my estimate?"
Help children maintain mental energy. Allow them to take frequent breaks while completing math assignments. Suggest that they get up and walk around during these breaks.
Teach self-checking strategies. Have students change to a different color pen when they have finished their work, becoming a "test checker" instead of a "test taker." This will help them notice their errors. For students who continue to make attentional errors in calculation, despite instruction and practice with self-checking, permit the use of a calculator for checking.
Help children stay focused. Let them choose the best place to do assignments, or allow them to listen to music if that helps their concentration.
Provide a model. Work through the mathematical problem with the child, verbalizing or demonstrating each step. Especially with homework, assist the child by doing the first problem together.
Identify topics of interest to children. Explore mathematical concepts in relation to motivating topics, such as building a skateboard ramp, tracking a satellite's orbit around the earth, discovering how the pyramids were built, or saving money in an interest-bearing account. Ask children to help you identify topics for mathematical problems.
Build a foundation for multi-step problems. Be sure the child understands basic one-step problems (problems requiring only one math operation) before advancing to those that require multiple operations.
Isolate steps. Have children focus on one step at a time. For example, provide mathematical activities in which children identify only (1) what the question is asking them to find, (2) which information is necessary to answer the question, and (3) which operations should be used in solving the problem.
Complete each step. Explain to children that even good problem solvers rarely skip steps when solving problems, though they may appear to.
Reduce the amount of data on a page. Children with spatial problems often become overwhelmed by large amounts of visual data on a page. Reduce the number of math problems or the number of diagrams to interpret per page. Remove unessential visual features.
Have children draw pictures to represent what is going on in a math problem. Suggest they draw representations of objects from the problem (for example, three shirts, a 6-by-12 foot garden plot).
Make auxiliary tools available. Provide calculators, graph paper for aligning numbers, or templates for tracing geometric shapes.
Production
Because math difficulties can affect a child's performance and ability to get work done, the following strategies are designed to help children improve their organization skills, work habits, and overall production.
Use assignment books. Teach children to use assignment books and "To Do" lists to keep track of their short- and long-term assignments, tests, and quizzes. Use peers to help monitor other children's assignment books. Most schools have a "homework hotline" on voicemail or homework posted on the school Web site. These resources provided by the school can help you support a student who does not yet record assignments consistently without reminders.
Provide models of assignments and criteria for success. Give children a clear sense of how a final product might look by showing examples and sharing exemplary products (such as providing a workbook of sample problems completed correctly). You might make work from last year available and draw the children's attention to specific qualities of the work (for example, "Notice how lining up the columns makes the problem easier to understand."). Do not, however, compare children's work with that of peers or siblings.
Build in planning time. Give children five minutes of planning time before beginning an assignment. Provide guidance in effective planning when necessary.
Use stepwise approaches. Require children to break down tasks into parts and write down the steps or stages. Compile steps of frequent tasks into a notebook for easy reference during work assignments. For long-term assignments, provide a due date for each step of the assignment.
Teach proven strategies. Provide children with specific age-appropriate strategies to use in checking work. For example, use TIPS: Think (read and paraphrase),Information (what numbers and information do you need in order to solve the problem?), Problem (write equation), Solve. Children can create a reminder card to keep on their desk or in their assignment book for quick reference to the strategy.
Stress the importance of organization. Have children preview an assignment and collect the materials they will need before starting it. Guide children in keeping their materials and notebooks organized and easily accessible. In middle and high school, conduct intermittent "notebook checks" and grade organization and completion. At the beginning of the school year and a week before each check, give a list of requirements. Emphasize the positive impact that organization and preplanning will have on the completed project or assignment. By grading organization, you will emphasize its value in the learning process.
Let children wait to turn in work. The day before an assignment is due, have children review their work and check it with a parent. This will give the children enough perspective to catch errors or add more details and produce better results in the end.
Encourage self-evaluation. Set a standard of work quality or criteria for success for children to follow, and allow them to self-assess the quality of their work before turning it in. If the grade matches the child's appraisal, give extra points for good self-assessment. Rubrics are one way for students to assess their own work.
Set goals and record progress. Have children set a short-term goal, such as completing all homework for the week. Record their daily progress toward the goal for children to observe. Graphic recording, such as plotting their own line graphs, may be particularly reinforcing for some children. Reward improvement at home.
Practice estimating. Children may benefit from estimating answers to math problems and science experiments. Stress the real-life benefits of estimating and understanding what the correct answer might look like.
Eliminate incentives for frenetic pacing. Remove any positive reinforcement for finishing first. State the amount of time a task should take. This will slow down children who work too quickly and will speed up children who work too slowly.
Provide consistent feedback. Create a feedback system so children understand which behaviors, actions, or work products are acceptable and which are not. Use specifics to praise good work and recognize when children use strategies effectively. Say, for example, "I like the way you drew a table to help explain the problem," or "Asking to take a break really seemed to help you come back and focus."
Try a mentor. Some children may benefit from a mentor who will work with them to analyze their academic progress, brainstorm alternative strategies, and provide recognition of progress. The mentor must be seen as credible, and may be an individual from either inside or outside the school.
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