Finding Math

Planning Your Lesson

Great afterschool lessons start with having a clear intention about who your students are, what they are learning or need to work on, and crafting activities that engage students while supporting their academic growth. Great afterschool lessons also require planning and preparation, as there is a lot of work involved in successfully managing kids, materials, and time.

Below are suggested questions to consider while preparing your afterschool lessons. The questions are grouped into topics that correspond to the Lesson Planning Template. You can print out the template and use it as a worksheet to plan and refine your afterschool lessons, to share lesson ideas with colleagues, or to help in professional development sessions with staff.

• Lesson Planning Template (PDF)
• Lesson Planning Template (Word document)

Lesson Planning Template Questions

Grade Level

What grade level(s) is this lesson geared to?

Duration

How long will it take to complete the lesson? One hour? One and a half hours? Will it be divided into two or more parts, over a week, or over several weeks?

Learning Goals

What do you want students to learn or be able to do after completing this activity? What skills do you want students to develop or hone? What tasks do they need to accomplish?

Materials Needed

List all of the materials needed that will be needed to complete the activity. Include materials that each student will need, as well as materials that students may need to share (such as books or a computer). Also include any materials that students or instructors will need for record keeping or evaluation. Will you need to store materials for future sessions? If so, how will you do this?

Preparation

What do you need to do to prepare for this activity? Will you need to gather materials? Will the materials need to be sorted for students or will you assign students to be "materials managers"? Are there any books or instructions that you need to read in order to prepare? Do you need a refresher in a content area? Are there questions you need to develop to help students explore or discuss the activity? Are there props that you need to have assembled in advance of the activity? Do you need to enlist another adult to help run the activity?

Think about how you might divide up groups?who works well together? Which students could assist other peers? What roles will you assign to different members of the group so that each student participates?

Now, think about the Practice that you are basing your lesson on. Reread the Practice. Are there ways in which you need to amend your lesson plan to better address the key goal(s) of the Practice? If this is your first time doing the activity, consider doing a "run through" with friends or colleagues to see what works and what you may need to change. Alternatively, you could ask a colleague to read over your lesson plan and give you feedback and suggestions for revisions.

What to Do

Think about the progression of the activity from start to finish. One model that might be useful—and which was originally developed for science education—is the 5E's instructional model. Each phrase of the learning sequence can be described using five words that begin with "E": engage, explore, explain, extend, and evaluate. For more information, see the 5E's Instructional Model.

Outcomes to Look For

How will you know that students learned what you intended them to learn through this activity? What will be your signs or benchmarks of learning? What questions might you ask to assess their understanding? What, if any, product will they produce?

Self-Evaluation

After you conduct the activity, take a few minutes to reflect on what took place. How do you think the lesson went? Are there things that you wish you had done differently? What will you change next time? Would you do this activity again?

Sample Lessons

Scavenger Hunt (1-2)

Students hunt for hidden geometric shapes and work together to recognize the properties of shapes.

Duration: 30 minutes

Learning Goals

• Understand the basic properties of shapes (number of sides, corners, square corners)
• Recognize regularities, similarities, and differences among shapes

Materials Needed

• Different colored cut-outs of squares, rectangles, circles, ovals, and triangles in different colors
• Chairs

Preparation

• Cut out squares, rectangles, circles, and ovals from construction paper (in various colors). You should have enough for each student to find a shape. Cut out sets of three shapes that have something in common (straight sides, four corners, rounded, etc.).
• Save one example of each shape and color to review, then hide the shapes around the room for students to find.

Teaching Tips

• For very young students, review the shapes and colors, then give each student a shape and ask him or her to find a similar shape hidden in the room.
• Use guiding questions to help students use what they know about the properties of shapes (straight sides, number of corners, etc.) and colors to figure out what makes another shape similar or different. For example, you may ask them what makes their shape like another. Encourage them to justify their thinking.

What to Do

• Review each shape and color with students. Ask them to describe each shape, its color, the number of sides and corners, or state that it doesn't have any corners. Which shapes are similar? Which are different? Why
• Explain to students that there are shapes hidden throughout the room for each student to find. As they find the shapes, they need to find two other students to sit with (groups of three) whose shapes have something in common with their own.
• Give students time to find the shapes and two students whose shapes have something in common with their own.
• Give groups time to discuss the properties of their shapes.
• Ask each group to report in, describing where they found their shapes, what shapes they found, and how they are both similar and different.

Outcomes to Look For

• Student participation and engagement
• An understanding of the basic properties of shapes
• Answers that reflect an ability to describe shapes
• The ability to compare and contrast specific properties of shapes, identify similarities, and highlight differences
• The ability to work together in small groups

ELL Enhancements

Teaching Tips for ELLs

• Post a graphic representation of each shape and color along with the written word that students can refer to when discussing their shapes and colors. A text-rich environment can help ELLs develop both oral and reading fluency. Terms used to describe the various shapes such as sides, corners, and rounded should also be highlighted and illustrated on the word wall.
• Many English words for geometric shapes have cognates, or similar translations, in other European languages. For example, "triangle" is triángulo and "oval" is óvalo in Spanish. Pointing out cognates allows students to activate background knowledge and to take more risks when speaking English. However, the possibility that ELLs may recognize the word "triangle" as triángulo does not necessarily mean that they can distinguish it from rectángulo, or rectangle.
• Clarify the difference between the words "same" and "similar." Using shapes in different sizes and color to highlight that distinction will promote critical thinking and create discussion using some of the terms and concepts in this activity.

Language Goals

• ELLs often have distinct areas of difficulty when learning English depending upon the linguistic features of their native tongues. Try to listen for areas of difficulty with each student and language group as they describe their shapes. For example, Spanish speakers of English may need to practice the correct word order of adjectives and the nouns they modify (e.g., "red triangle" is triángulo rojo,in Spanish). Speakers of some Asian languages, on the other hand, may have difficulty including a final "s" to indicate a plural. Provide feedback to individual students or groups of students rather than calling attention to these areas in front of the large group.

What Is the "Best" Snack? (2-5)

Students use nutritional information to analyze data about different snacks and survey their peers to determine the "best" snack.

Duration: 1 hour

Learning Goals

• Predict the "best" snack, meaning one that is healthy, inexpensive, and tasty. Then test the prediction by collecting data
• Organize data using a bar graph and a table
• Find the median
• Understand that data represent specific pieces of information

Materials Needed

• Three different snack foods (raisins, pretzels, and peanut-butter crackers), enough for each group of students
• Nutritional information for each snack
• Ziploc bags
• Copies of the Data Analysis and Probability Handout (PDF)

Preparation

• Divide snack food into plastic bags for each group of students.
• Make copies of the nutritional information for each snack, including the price of each.

What to Do

• Ask students to pick a partner to work with. You may want to assign partners.
• Provide space and give out materials (snacks and handouts) to each pair.
• Ask students to think about the snacks they have and to make a prediction about which snack is best.
• Working with students, develop a list of criteria for evaluating the snacks. For example, the best snack might be one that is low in fat, inexpensive, and tasty. Ask about factors they might consider in determining quality. If the two students working together don't agree, they can find the average between the two ratings.
• Take time to talk about nutrition and healthy amounts of calories, fat, and sodium.
• Using the nutritional information provided, ask each pair of students to complete the data collection table and rate each snack.
• Post all the data on one large table, using poster paper, an overhead transparency, or a chalk board. Refer to the sample table in the handout.
• Ask each pair to describe how they determined the rating for each snack. For example, if one student rated a snack a 3 and the other student rated it a 4, the median would be 3.5.
• Next, conduct a survey to find out what the most popular best snack was among all students. Work with students to create a chart. Refer to the sample survey chart in the handout.
• Using the data from the survey chart students create, ask each pair to make a graph of the results. Refer to the sample graph in the handout.
• Based on the data, decide which snack is the "best." Give students a chance to talk with their partner, and bring the class back together to share ideas. Each pair explains, based on the data, their decision about the best snack. Encourage students to use both sets of data when making a decision. Students can revisit the predictions they made before the data were collected.
• Ask if any of their ideas about what "best" means have changed. Ask, What is the relationship between the quality rating and your favorite snack?
• Finally, discuss why the quality rating either matches or doesn't match their favorite snacks.

Outcomes to Look For:

• Answers that reflect an understanding of information as data
• An ability to collect and organize data in a sensible way (for example, a bar graph or table)
• Answers that reflect an ability to explain reasoning based on data
• An ability to find a measure of center or median
• Students work together to solve problems and discuss strategies and solutions

ELL Enhancements

Teaching Tips for ELLs

• Pair beginning ELLs with strong English-speaking students. Students often perform better when they have strong models to follow.
• Write short and specific directions for each part of the activity on the board or on chart paper. Read the instructions with students before beginning the activity.
• Make use of cognates (i.e., words that are similar in other languages). For example, "calories," "protein," "sodium," "price[[comma]]" and "quality" all have similar equivalents in Spanish and other European languages. Even though the words may be similar, this does not mean that students understand the meaning of the words.
• When introducing nutritional information, serving sizes, and other information for this activity, take time to explain and demonstrate relevant criteria and abbreviations to ELLs. For example, ELLs may not be familiar with ounces or the abbreviation "oz." ELLs are often more familiar with metric measurements. Have ELLs count out or weigh snacks in ounces and grams to demonstrate characteristics of a typical serving.

Language Goals

Provide ELLs with the sentence structures or frames that they can use to explain their choices and reasoning. If ELLs can then listen to classmates, use vocabulary from their activity sheets, and practice a familiar form to express their opinions, they will be more likely to participate in discussions. For example,

I like peanut butter crackers because they have more protein.
I like raisins because they're not salty.

Students may struggle with the use of "fewer" and "less" to compare nouns that are either countable or uncountable. Nouns that form a plural with "s" are countable and use "fewer," as in "fewer calories." Nouns that do not have a plural form ending with "s" are uncountable and use "less," as in "less salt." "More" and "a lot of" can be used to describe both countable and uncountable nouns (e.g., "more salt," "more calories"). These distinctions are more appropriate for advanced ELLs.

Largest Number Race (3-5)

Students design a relay race to compare whole numbers and compete to create the largest 10-digit number.

Duration: 30 to 45 minutes

Learning Goals

• Compare whole numbers
• Understand the value of single numbers in a larger number
• Use specific skills in the context of a variety of physical activities
• Work cooperatively to solve problems

Materials Needed

• Two 10-sided number cubes
• Masking tape
• Paper
• Items for relay race (balls, jump ropes, etc.)

Preparation

• For each team, place 10 strips of masking tape on the floor with each strip representing a placeholder for a 10-digit number, and small pieces for the commas. For example:
  _____, _____ _____ _____, _____ _____ _____, _____ _____ _____

Teaching Tips

• Begin by reviewing the objective of the game and making sure that the steps are clear to students. The goal of the game is to come up with the largest number. See if students can figure out that larger numbers should go at the beginning of the number (9,000,000,000) and smaller numbers at the end. However, this is a game of chance and strategy. Students can't move the numbers once they have placed them.
• Encourage students to come up with fun obstacle courses that include activities that are entertaining, challenging, and accessible to all students. If students are physically challenged and can't participate in the obstacle course, they can roll the dice.
• Use guiding questions to encourage students to justify and discuss how each team came up with its number. Use these sample questions or develop your own: What strategy did you use to make the largest number? Was it a good strategy? Why or why not? What strategy would you like to try in the next round?

What to Do

• As a class, design an obstacle course relay that incorporates at least two physical activities—jumping rope and throwing a basketball through a hoop, for example—while moving toward a finish line. The final activity in the relay involves receiving a randomly generated number between 0-9, and working together with your team to determine where to stand in a string of 10 digits to make the largest number.
• Divide students into two teams.
• Choose a person to be the number roller (a good choice is someone who can't participate in the physical activity or who would prefer not to). This person will roll the number cube, write the number on a blank piece of paper, and hand it to each team member after he or she has completed the other relay tasks. Both teams will receive the same number during each turn. Encourage students to write big.
• Tell students that the first team to complete all tasks and make a 10-digit number gets one point. The team who makes the largest number gets one point. Play can continue as long as time allows. The team with the most points wins.
• Have each team read its number to determine who has won. In order to read the numbers, have students put their numbered pieces of paper on the floor, stand back, and look at the number. You might ask, How do you know which number is the largest?
• Once teams have completed their first round, discuss how each team came up with their number, the strategies students used, and what they learned.

Outcomes to Look For

• Student participation and engagement
• Students play fairly together
• Students work together to problem solve
• An understanding of the value of whole numbers
• Answers that reflect an understanding of how base-ten (0-9) numerals can be arranged to make the largest possible number

ELL Enhancements

Teaching Tips and Language Goals for ELLs

• To give ELLs verbal and visual representations of the procedures, have a student from each team demonstrate how to proceed through the obstacle course and place the numbers
• To incorporate oral language into this role, have the number roller shout out the number he or she has rolled at each turn.

Olympic Races (6-8)

Students examine Olympic athlete profiles and use algebraic skills to calculate average speed and calories burned per minute in different events.

Duration: 45 minutes (can be ongoing)

Learning Goals

• Understand a calorie as a unit of energy
• Use algebra skills to figure out average speed and calories burned per minute
• Understand the basic concept of a function (describe how changes in one quantity result in changes in another)
• Problem solve using a variety of strategies, approaches, and math skills

Materials Needed

• Algrebra: Olympic Races (PDF)
• Notebooks or journals
• Calculators

Preparation

• Print out copies of the Algebra: Olympic Races PDF

Teaching Tips

This lesson asks students to take data about an athlete and use algebra skills to figure out more information (calories burned per minute and speed). Students should be able to work together to come up with the formulas that will lead to the answers. Rather than giving them an answer, use guiding questions and the Teaching Tips below to help students use what they know to find an effective formula. Approaches and formulas will vary from group to group. Allow students enough freedom to discover their own methods for approaching the problem, and use each group's discussion to evaluate students' understanding of the algebra skills involved. Remember that the approach students take and how they use variables are more important than the answers they come up with.

To find the calories burned per minute:

• One way to approach the problem is to present a simpler problem. If someone burns 60 calories in 60 minutes, how many calories are burned per minute? What about 120 calories in 60 minutes? What does this tell you about the formula needed to find the calories burned per minute?
• Now, let's look at the male cross-country skier who burns 2,050 calories in a 90-minute event. To find the calories burned per minute, divide 2,050 by 90.
• You may want to use a calculator to find the answer (2,050 calories/90minutes = 21.6 calories per minute).

To find the speed:

• Speed equals distance divided by time. However, rather than giving students this formula, consider the tips below to help them figure it out.
• One way to approach the problem is to ask students to estimate if the skier's speed is going to be faster or slower than 30 kilometers per hour. (We know that the skier went 30 kilometers in 90 minutes, so he must be traveling slower than 30km per hour.)
• Some students may see a way to break down the problem into parts. For example, students may figure out that 90 minutes represents three half-hours, and that the skier went a total of 30 kilometers. Then you can ask them how many kilometers the skier went per half hour (10km) and per hour (20km).
• Another way that students can find the speed is to divide the distance (30km) by the time (90 minutes), for a total of .333 kilometers per minute. They can then multiply the speed per minute (.333) by 60 to find the speed per hour (20km).
• Students may want to use a calculator to solve this problem (30/90 = .333 or one-third kilometer per minute).They then multiply the speed per minute (.333) by 60. Note that if students do use a calculator, they will come up with an answer of 19.98, and should round up.
• To get a number that isn't rounded off, students can convert time in minutes (90) to time in hours (1.5) before doing this last calculation (30/1.5 = 20).

What to Do

• Assign students to small groups of four or five, based on skill level and which students work well together.
• Introduce the activity and distribute the athlete profiles. Explain that students will use the information given about the athletes' performance to calculate average speed and calories burned per minute.
• Review a calorie as a unit of energy.
• Ask students to read the profiles and then choose one athlete per group. As a group, students should note the data or information given and come up with a strategy for calculating two averages: calories burned per minute and speed per minute. Use the guiding questions on the handout to help students figure out the formula for each answer. See the Teaching Tips for this lesson for more information.
• Move among the small groups, asking guiding questions or giving simple examples to help students come up with the formula. Avoid giving them a formula outright. Instead, see if a series of guiding questions can help students use what they know to figure out the formula as a group. Be aware that there are a number of ways they may set up formulas.
• When students have finished, ask them as a group to discuss how they found their answers, what strategies they used, and how they decided whether their answers made sense. You may also wish to ask them which events appeared to burn more calories or result in faster overall speeds, and why. Are there other differences that matter? Do you think the athletes' speeds and calories burned would vary from day to day or race to race? Why? How might students account for that in the average?

Outcomes to Look For

• Student participation and engagement
• Students work together to understand, define, and solve problems
• Answers that reflect an understanding of a calorie, data, and the information provided in the athlete profiles
• Answers that reflect an understanding of a function (how quantities affect other quantities)
• Answers that reflect an understanding of how to find an average of calories burned per minute and speed per minute
• The effective use of variables

ELL Enhancements

Teaching Tips for ELLs

• In order to set the context, show students a picture of cross-country skiers or racers, as many students may be unfamiliar with the sport.
• Point out the cognates, or similar words, that may exist in both English and ELLs' native languages. For example, "minutes" is minutos and "calories" is calorias in Spanish. Even though the words are cognates, that does not mean that students understand their meaning.
• Spanish-speaking students may also appreciate knowing that "per" (as in "calories per minute") is the same as por in their native language.
• American-born students may have difficulty visualizing 30 or 50 kilometers. Since the metric system is used in most other countries, many ELL students may be able to visualize this better than their American-born peers. In order to encourage ELLs' participation, ask them to share their expertise.
• Encourage all students to use the board or overhead projector to explain their problem-solving approach. Using a visual aid will enhance comprehension for ELLs, and perhaps encourage them to share their own solutions and reasoning.

Language Goals

• Encourage all students to read and write numbers followed by their respective measuring units, such as 43 minutes or 15 kilometers.

All About Money—Does it Pay? (9-12)

Students use computational and problem solving skills to analyze data, and algebraic skills to represent the information they generate.

Duration: 1 hour

Learning Goals

• Communicate about mathematics (e.g., use mathematical language, compare their own thinking with other students' thinking)
• Use strategies to understand new math content
• Develop efficient solution methods
• Extend solution methods to other problems
• Explain relationships among different representations of problems
• Use a variety of computation procedures to find answers
• Use formulas to model and solve real-world problems
• Solve problems involving rate
• Select and use the best method of representing and describing a set of data
• Use a variety of models to represent patterns, and relationships

Materials Needed

• Teacher's Guide (PDF)
• Student Worksheet (PDF)
• Pencils
• Paper (graphing and other as needed)
• Calculators

Preparation

• Print out the accompanying PDFs and familiarize yourself with the scenarios students will be involved with. If necessary, share the lesson with a day-time mathematics instructor to talk about the standards and mathematics involved.
• Organize students in small groups (to allow for discussion partners, if necessary).
• Make sure all materials are available for all students.
• Prepare a brief introduction of the scenarios and the problems the students will be involved with. Explain how they are directly related to life outside of school. The goals of this discussion are to clarify the task and pique students' interest in the problem.

What to Do

• Introduce the scenarios and problems. You may begin the discussion by asking, How many of you plan to use credit cards when you are 18 and how many of you have a plan for staying debt free?
• Ask students to begin reading through the scenario of Mr. Opportunist. Allow students to ask clarifying questions before they begin. Provide an example of how students might move forward if they get stuck. You may decide that students can refer to each other for clarification.
• Circulate and stay involved while students work. Ask questions that generate student thinking, but try not to give away answers or lead students down a particular line of thinking. For example, you can ask, How did you arrive at that answer? or What is one way you can get this solution from this number? Other guiding questions include:
• What patterns do you see?
• How might you generalize this pattern?
• In what ways might you convey this information in a graph?
• I see that you are stuck. Can you explain how you got here? What might be a next step?
• Be prepared for some students to move forward to the extensions (featured in the PDFs for this activity) and/or save the extensions for another day. You may decide that some students can take the extensions home to work on if they wish.
• Save this part of the activity for another day if there is not enough time for students to prepare reports and discuss what they have learned with the class. Do not skip it, as it is ultimately the most important aspect of the lesson. Students need to affirm what they have learned.
• If time allows, lead a class discussion on credit myths and misconceptions and/or the best way to handle a credit card.

Outcomes to Look For

• All students are engaged and actively seeking answers
• Students communicate effectively about mathematics (e.g., use mathematical language, compare their own thinking with other student thinking, gain clarification from each other)
• Students use strategies effectively
• Students explain relationships among different representations of problems
• Students use a variety of computation procedures to find answers
• Students use formulas to represent, generalize, and solve real-world problems (during the extensions)
• Students effectively represent and describe data
• Students use a variety of models to represent patterns and relationships

Cooking (K-2)

Students use measuring cups to compare and estimate volume as they make a healthy snack.

Duration: 1 hour

Learning Goals

• Compare and order measuring cups according to volume
• Use tools to measure
• Make comparisons and estimates among different measurements

Materials Needed

• Measuring cups (1 set per small group)
• 50 small marshmallows (per small group)
• Cup of orange juice (1 per student)
• Frozen yogurt (enough for each student to have a scoop)
• Spoons
• Plastic cups
• Paper and pens/pencils
• Bags for supplies

Preparation

• Assemble the materials into the supply bags (1 bag per small group).
• Be prepared to review the meaning of the word "prediction," which means to make a reasonable guess about something that hasn't happened yet. You may want to model strategies for forming predictions with marshmallows for younger students.

Teaching Tips

• If students haven't used measuring cups or learned about volume and measurement, you may want to begin by reviewing each measuring cup, how many halves, thirds, and quarters make a whole, and answer any questions that students have.
• If time allows, ask students to describe their process. What did they do first? What did they do second? You might number the process on the board, and ask students to write a "how-to" piece as an extension activity. Younger students can draw pictures.

What to Do

• Divide students into groups of four or five, based on who will work well together. Give each student a bag of supplies, and each small group a set of measuring cups.
• Ask children to line up the measuring cups from smallest to largest.
• Ask students to predict which cup will hold the most. Try to get students to articulate that the big cup will hold more than the smaller cups, and that the amount each cup holds is called "volume."
• Ask students how they found their answers, and how they know which cup holds more.
• Next, pour orange juice into the one-cup measuring cups, and have students pour the orange juice from the measuring cup into their plastic cups.
• Ask students to observe any similarities or differences in the juice in each container. What did they notice as they poured the juice? Younger students may think that there are different amounts, based on how the juice fills each container. Ask them how they know. It may help them to clarify the volume by pouring the juice back into the measuring cup to test their predictions.
• Add a scoop of frozen yogurt into each of the plastic cups with juice.
• Next, ask students to predict how many marshmallows will fit into half-cup measuring cups. Have them write down their predictions and then describe how they made their predictions.
• Have students test their predictions using the marshmallows and then discuss their answers. Answers will vary depending on how the cup was packed. Finally, students can add marshmallows to their juice and yogurt and eat it!

Outcomes to Look For

Much of the assessment for this lesson comes from listening to and watching the children. Listen for use of vocabulary, facility with ordering measuring cups, understanding of conservation of matter, and estimation skills and strategies.

• Student participation and engagement
• The productive use of measuring tools
• An understanding of different measurements and how to measure
• Answers that reflect an understanding of size and volume
• Answers that reflect an understanding of comparison and prediction, as well as strategies to make and test predictions

ELL Enhancements

Teaching Tips for ELL

• It is unlikely that ELLs are familiar with the U.S. measurement system since most countries use the decimal-based metric system. However, this is a great hands-on exercise for ELLs to begin to develop some familiarity with measurement, volume, and fractions in the English system.
• Point out the fraction names on the sides of the cups and write the symbols and words on the board or chart paper.

Language Goals

• ELLs may be unfamiliar with fraction names and may have particular difficulty pronouncing the final "-th" sound. The afterschool instructor should model and have students repeat the fraction names for each volume: one eighth, one fourth, etc. Write the fraction words on the board and show the related cup to which each fraction word refers as they repeat the words.
• Write, model, and practice sentences which use comparative words to describe relative volumes. Have ELLs select two different cups and create their own sentences using this model:

  1/3 cup is greater than 1/4 cup (bigger)
  1/4 cup is less than 1/3 cup (smaller)

Resources

Technology Tip

Using data from the real world helps bring math to life. The use of graphing calculators to collect and analyze such data allows students the opportunity to reinforce mathematical skills and concepts through application. The Texas Instruments Calculator-based Ranger and other data collection devices exist that can interface with standard graphing calculators. The Texas Instruments Activities Exchange provides a collection of K-12 mathematics activities that allow students to use real world data to solve mathematical problems.

Web Resources

• National Council of Teachers of Mathematics Illuminations 
• National Council of Teachers of Mathematics 
• AIMS Educational Foundation 

Text Resources

• Housman, L. B. (2000). The Use of Situations Outside the Math Hour. In Young Children Reinvent Arithmetic: Implications of Piaget's Theory (Second ed.). New York, NY: Teachers College Press, Columbia University.
• Lauer, P. A., Akiba, M., Wilkerson, S. B., Apthorp, H. S., Snow, D., & Martin-Glenn, M. (2004). The effectiveness of out-of-school-time strategies in assisting low-achieving students in reading and mathematics: A research synthesis. Aurora, CO: Mid-continent Research for Education and Learning.
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, Virginia.
• Policy Studies Associates for the US DOE. (1995) Extending the learning time for disadvantaged students: An idea book. Volume 2, Profiles of Promising Practices. Washington, DC.
• Policy Studies Associates for the US DOE. (1995). Extending the learning time for disadvantaged students: An idea book. Volume 1, Summary of Promising Practices. Washington, DC.
• Sutton, J., & Krueger, A. (Eds.). (2002). ED Thoughts: What we know about mathematics teaching and learning. Aurora, CO: Mid-continent Research for Education and Learning (McREL).