Whole Number Place Value

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Title (Whole Number Place-Value Pedagogical Content Knowledge Project) Elementary K-6

  • By [Nicole Armendariz, Amy Whiteside] (UCIrvine, August 2009)
  • Revised By: Brittany Maynard, Kara Gregory, Katie Lund, and Zareen Charna (2010)


Contents

Mathematical Background

Place Value is the value given to the place a digit has in a number. Numbers are made up of digits, and each digit has a special value depending on its place in the number. Place value concepts may be applied for both whole numbers and decimals. In an elementary school classroom, students might use the following chart to break multidigit numbers down into their component place value parts.

Placevalue.gif

Students in California explore conceptual understanding of whole number place value beginning in first grade. Even in kindergarten, students practice counting sets of objects using concrete manipulatives. By the end of fourth grade, they should understand place value with both whole numbers and decimals to two decimal places.

First Grade Number Sense:

1.1 Students understand and use the concept of ones and tens in the place value number system.

Second Grade Number Sense: 1.0 Students understand the relationship between numbers, quantities, and place value in whole numbers up to 1,000:

1.1 Count, read, and write whole numbers to 1,000 and identify the place value for each digit.

Third Grade Number Sense:

1.0 Students understand the place value of whole numbers:

1.3 Identify the place value for each digit in numbers to 10,000.

Fourth Grade Number Sense: Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions.

1.1 Read and write whole numbers in the millions.

1.2 Order and compare whole numbers and decimals to two decimal places.

1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand.

Cognition and Learning

Although students may be able to identify the place value for whole numbers using rote memorization, that does not mean have conceptual understanding. Later in their mathematics careers, they may struggle with higher order place value concepts. Teachers often assume that students understand place value concepts because they can place numbers correctly in a place value chart. This actually reflects more procedural knowledge than conceptual knowledge.

Five “Big Ideas” regarding whole number place value development (Van de Walle, 2007):

1. Base ten numerations: sets of ten objects can be counted and used to describe the quantities. E.g. 55 is five sets of ten and five single objects

2. Place value numeration: the positions of digits in numbers determine what they represent

3. There are patterns to the way that numbers are formed.

4. Ones, tens and hundreds groupings may be taken apart and regrouped in different ways; they are flexible. E.g. 50 can be fifty single ones, five tens, a combination of ones and tens, etc.

5. “Really big” numbers (1000 or more) are best understood in terms of familiar real-world referents. E.g. the number of people who can fill the Staples Center

Five Components of Place Value which are necessary in order to teach for understanding (Gorlikov, 2000):

1. Learning number names (and their serial order) and using numbers to count quantities: Students need to be able to say, read and write single and multidigit numbers. Multiple practice opportunities must be given in the classroom with these skills. Many students can write multidigit numbers, but struggle with correctly reading the numbers aloud. Young students need practice not only in counting objects or sets of objects, but also with knowing “how many” are in the set.

Asian countries use a more transparent number naming system than we do in the U.S., and this may account for why their students do better on standardized tests which assess place value concepts. Those countries use a base ten number system with names that clearly communicate the base ten relationships: e.g. thirteen is ten-three (Fuson, 1998).

2. "Simple" addition and subtraction: Students need practice in adding and subtracting pairs of numbers which don’t require regrouping until they become automatic. After that, they must practice regrouping with addition and subtraction until they are comfortable with the process. Students who are not comfortable with the regrouping algorithm cannot recognize when they have made a mistake, because they are unable to recognize that their answer is not reasonable.

3. Practice with counting by groups and multiples of groups, e.g. counting things by fives. This is much faster than counting each individual item. Students should also have practice with counting by multiples of ten to practice with arithmetic that is based on tens.

4. Representation (of groupings): Students should be able to represent single and multidigit numbers using concrete manipulatives, e.g. counters, chips, etc. Once students are able to show a quantity using single counters, they may exchange individual counters for counters representing larger quantities. E.g. ten single counters may be exchanged for a red chip to represent ten. In a classroom, students may practice with money, poker chips, base ten blocks, etc.

5. Specifics about representations in terms of columns: This is a way to designate groups, or represent multidigit numbers using columns. When we express numbers orally, we skip over place values which are not represented. E.g. 5004 is read aloud as “five thousand four” and not “five thousand no hundreds no tens four.” With place value, we use columns to represent a new group of ten. For example, after we have used 0-9, we need to add a new column to start the next group of ten. Students sometimes use “place value mats” to practice representing these columns.

According to Gorlikov, skills 1, 2 and 3 require teacher demonstration and repetitive practice for students. Skills 4 and 5 require more conceptual understanding.

Cognitive Obstacles and Common Misconceptions

Teachers assume that students who show the procedural ability to be able to identify the place value for numbers in a multidigit string understand the concept of place value.

Students may show mastery for basic fact addition and subtraction for the single digit numbers 0-9. However, they become confused when asked to perform addition and subtraction with regrouping. They do not understand that they need to begin adding with the ones column, and then regroup if the answer is bigger than nine. Instead of “giving the ten” to the next column, they believe they have made an arithmetic error. They may try to “fix” it by transposing the numbers, e.g. subtracting the smaller number from the larger number.

Students do not have conceptual understanding of the regrouping process, and so they cannot recognize their errors with multicolumn addition and subtraction.

When given a sample number such as 16, 50% of fourth grade students were unable to correctly speak to its face value. That is, they could not explain that the 1 in 16 stands for 10 and the 6 in 16 standards for 6 ones. This leads to more confusion regarding regrouping. Students do not realize that moving the digit one place to the left makes the number ten times bigger. For example, adding a “0” to 10 makes a new number 100. This new number is ten times bigger than the old number.

English counting words above ten hide the place value meaning of the numeral. For example, eleven means 10 + 1. In order to connect the values of numbers to their place value meaning, base-ten oral and written names can be used. 32 can be said as “three tens and two” or “thirty-two” (Van de Walle, 2010).

Pedagogical Tools and Strategies

Due to misconceptions between face value (the numerical symbol, ie. 123) and the complete value (10, 100, 300) students need a scaffold between the concrete and abstract concepts of place value. Teachers need to give students concrete representation of place value through the use of tangible manipulatives. Once students have a complete understanding of both the face value and complete values of numbers they can work solely with standard numerical values. (Baroody, 1990) The following is a list of tools and strategies teachers can use to create a conceptual understanding of place value, which will aid students in their progression to addition and substractin of whole numbers.

Learning/Teaching Approach: A physical embodiment is associated with place value. The use of base-ten manipulatives, word form and standard form are combined to give students multiple ways of thinking about given numbers. A student would be given a number in standard form and would have to build the number using base ten blocks. In this approach, students are able to make a connection between the face value of numbers and the complete value represented with the base ten blocks. Additionally, the student has to identify the word form that is associated with the given number. Usually, word form and standard form are taught seperately without a connection to concrete values. (Fuson, 1990)
Face Value and Complete Value

An intermediate system between the standard written place-value system and the base-ten manipulative system of teaching place-value. In the system there are two-sided pieces and a board. The upper-sides of the pieces are all one color and the under-sides of the pieces are another color. The upper-side of the pieces represents the "face value" and the under-side stands for the "complete value" Students build numbers, in which the face value and the complete value are correct. For example, students would be asked to build 125. Students would have to find a 1, 2 and 5, but the back of the pieces would have to represent the complete value. The "1" would have a 100 on the back; the "2" would have a 20 on the back; and the "5" would have a 5 on the back. (Varelas, 1997)

Academic Language

  • standard form: a way to write numbers by using the digits 0-9, with each digit having a place value
  • expanded form: a way to write numbers by showing the value of each digit. Ex: 832 = 800+300+20
  • word form: a way to write numbers in standard english
  • place value: the value of a place, such as ones or tens, in a number (Houghton Mifflin, 2002)


Using Cognitively Guided Instruction (CGI) to teach Place Value

Cognitively guided instruction, or CGI, is a classroom method that is guided by student thinking. CGI centers on students “discovering” new mathematical concepts versus being told what they are. In a CGI classroom, teachers should not immediately demonstrate solution methods, or suggest that any mathematical strategy is preferred over another (Hiebert et al., 1997). Teachers should expect their students to become thoroughly involved with the posed math problems and to develop varying methods to reach solutions to posed problems. Teachers using this method work with the prior knowledge of students and build upon it to solidify mathematical concepts (Carpenter et al., 1999).

Questioning strategies are a primary method used in CGI classrooms. These strategies should be used to illicit students’ independent thinking and encourage them to talk about the math concept at hand. By using questioning strategies to initiate student verbal responses, teachers can also better assess student understanding and misconceptions.

CGI classroom methods for mathematics are based on the best current research-based ideas on how to design classrooms that help students learn mathematics with understanding. Students need flexible approaches for defining and solving problems. In CGI or “problem-centered learning classrooms”, students should be presented with problems that are meaningful and interesting to them, but which they cannot solve with ease using routinized procedures or drilled responses (Hiebert et al., 1997).

In order to manage a successful CGI classroom, expectations should be set. Students should be able to take over and determine correctness by questioning peers’ solutions, use reasonableness as a guide and become comfortable with the idea of living with temporary uncertainty (Hiebert, et al., 1997). CGI can provide a strong base in mathematical language use for students related to place value.

Teachers should focus on their language use when describing whole number place value, so that students may better understand the written number’s complete value as it pertains to the face value name we give it. For example, “In the number 435 there are 4 hundreds, 3 tens and 5 ones. Who can tell me another way that we might be able to represent the same amount.”

Guiding Math Conversations

Teachers can guide mathematical conversations by developing a classroom culture that supports the opportunity for students to use math-talk. Students should be encouraged to talk about their place value ideas. Mathematical conversations can strengthen student conceptual understanding. Teachers should allow time for students to share their place value ideas and strategies. Students may benefit from using their own language to describe place value and other students might benefit from hearing their classmates’ ideas.

Teachers can engage student in productive discourse in three areas (Chapin et al., 2003)

1.Whole Class Discussion

2. Small-group Discussion

3. Partner Talk

Some teachers prefer starting their CGI work with a small group of students. In other classes, an entire class might work on the same problem or idea. Children who work more quickly than others are often asked to find two and three ways to solve a problem or define an idea (Carpenter et al., 1999).

Examples of Possible Questioning Strategies and Concept Devleopment for Place Value

Have students offer their “definitions” of the word value. Write student ideas on the board. Keep student ideas and strategies posted in the room so that they can reference the concepts they built on. See image below.

Show students the number 39. Point to the two different number in 39, the 3 and the 9. Have students explain “how they know” that this number is 39.

PlaceValue.png

CGI classrooms also encourage students to write about math by using a personal math journal. Offer a number example to students. “We know our numbers 0 through 9. Show me what nine looks like by drawing a picture in your math journal. How did you represent the number nine? Now, what about the number 74? How do we know the value of 74?” CGI should primarily be used to guide student-centered math discovery within whole number place value, as well as encourage students to use mathematical language when describing their thinking.


Manipulatives

Manipulatives are visual representations of numbers. When used in the introduction of a topic they help create a conceptual understanding versus a procedural/algorithmic understanding of numbers. Place value materials can be teacher created or purchased, but should be focused on creating a conceptual understanding of the base-ten system. (Fuson, 1990). Exchanging 10 ones for 1 ten and 10 tens for 1 hundred establishes the place value system for thousands, millions, and billions (Kennedy, Tipps, & Johnson, 2008). It is important to remember that models for place value do not “show” the concept and that students must mentally construct the concept to impose on the model (Van de Walle, 2010).
Baseten.jpg
  • Base-Ten Blocks
    • Wooden or plastic units, longs, flats, and blocks that are durable and easily handled. While expensive, base-ten blocks are the only groupable model with 1,000.
    • For students in the early stages of developing whole number place-value concepts, base-ten blocks allow them to easily and efficiently model large numbers. However, students may potentially use these blocks without reflecting on the ten-to-one relationships. It is important that students understand that 10 ones is the same as a ten (Van de Walle, 2010).
  • Place Value Mat
    • Simple mats that are divided into two or three sections to hold ones and tens or ones, tens, and hundreds pieces.
    • Used to link base-ten models and the written form of numbers. Students see that the left-to-right order of the pieces on the mat is also the way that numbers are written (Van de Walle, 2010).
    • It is recommended that two ten-frames be drawn in the ones place so that the amount of ones is always clearly evident, eliminating the need for frequent counting (Van de Walle, 2010).
  • Hundreds Chart
    • Can be used to help young children count, find patterns, and recognize two-digit numbers before they develop a base-ten understanding of these numbers (Van de Walle, 2010).
  • Bundles of sticks (wooden craft sticks, coffee stirrers)
    • Students make a set of base-ten materials by bundling sticks to show groups of 10, then bundling groups of 10 to show 100 (Kennedy et al., 2008).
    • If bundles are kept intact, they can be used as a pregrouped model.

Virtual Manipulatives

Virtual manipulatives for place value are free, are easily grouped and ungrouped, can be shown in a whole-class setting using a projector, and have no supply limit. Virtual models can be printed, which allows students to keep a hard-copy record of what they have done. (Van de Walle, 2010). Research suggests that students may also develop more complex understandings of concepts when using virtual manipulatives (Moyer, Niezgoda, & Stanley, 2005). However, it is important to note that a virtual model is similar to a physical model in that they are only used as a representation for students who understand the relationships involved (Van de Walle, 2010).

Assessment (Kennedy, Tipps, & Johnson, 2008)

Most students construct place value meaning by the third grade by working at the concrete level. To assess place value understanding of ones and tens in individual students, ask students to complete the following tasks in an interview.
1. Give a student 12 to 19 cubes and ask them to count out 16 cubes and draw a picture of the cubes on a piece of paper.
2. Ask the student to write the numeral for the number of cubes.
3. Ask the student to circle the number of drawn cubes shown by the numeral 16.
4. Point to the 6 in the numeral 16 and ask the student to circle the number of cubes that goes with that numeral.
5. Point to the 1 in the numeral 16 and ask the student to circle the number of cubes that goes with that numeral.
Evaluate students’ understanding of the number. For question 5, many students only circle one block instead of ten. Kammi (as cited in Kennedy et al., 2008) found in her interviews that misconceptions about place value persist into third or fourth grade. Concrete models for trading are critical to illustrate the nature of place value.

Response to Intervention (RTI)

Students who struggle with learning and meeting grade-level expectations may benefit from an intervention program tailored to their specific learning needs (Scanlon, 2010). Students who demonstrate limited proficiency in understanding whole number place value can benefit from Response to Intervention (RTI) programs. RTI is a method of academic intervention designed to provide early effective assistance to students who are having difficulty learning. There are 3 tiers in this model, the Universal Level (Tier 1), the Targeted Level (Tier 2), and the Intensive Level (Tier 3). Tier 1 students, or those at the Universal Level, represent 80-90% of students. These students are taught in a regular classroom setting, without academic intervention. Tier 2 students, or those at the Targeted Level, represent 5-15% or students and receive small group instruction. Tier 3 students, those at the Intensive Level, require intensive one-one-one instruction due to intensive academic needs.
RTI2.png

Tier 2 students should receive supplemental small group mathematics instruction aimed at building targeted mathematics proficiencies, in this case, whole number place value. Student progress should be monitored by assessment throughout the intervention on a weekly basis. Interventions are provided to these students based on them being identified at being at-risk and who require specific supports to make adequate progress in their general education.

Tier 3 students requiring a more intensive intervention should be tutored one-on-one. On-going analysis and assessment of student progress is critical for the success of these students. At this level, instruction on whole number place value should be designed with the student’s specific needs and misconceptions in mind (Rimbey & Anderson, 2010). Examples of some common misconceptions with whole number place value are that students do not understand that the 1 in the number 14 represents 10 ones. This is problematic as students begin to learn addition and subtraction with regrouping, as they only see the 1 as “1”, instead of 10. The understanding of place value is foundational as students progress in mathematics.

Intervention for both Tier 2 and Tier 3 students should provide models of verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review. The most obvious difference is that Tier 2 students are taught in small groups, and Tier 3 students receive one-on-one instruction. (Henry, 2010)

Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas. Physical models for base-10 concepts can play a key role in helping students develop the idea of a “ten” as both a single entity and as a set of ten units. However, children must mentally construct the concept and impose it on the model (Van de Walle, 2010, p. 191). Please refer to the “Manipulatives” section for specific instructional ideas in creating learning opportunities utilizing conceptual representations specific to the needs of your students.

Concrete-Representational- Abstract (CRA) Instructional Strategies

Theoretical Framework
The sequence of instructional steps in Concrete-Representational-Abstract Models differs from other pedagogical approaches as it relates to teacher activities and instructional methods. CRA instruction is primarily focused on developing students’ conceptual understanding and demonstration of mastery. It has been shown to be very useful in the instruction of struggling students (Flores, 2009; Butler, et al., 2003). CRA is differentiated from other methodologies because it uses a combination of teacher demonstration, guidance, and student demonstration of mastery over three separate lessons (Flores, 2009).

Concrete-Representational-Abstract models can be used in order to successfully teach whole number place value concepts to students. CRA provides students the opportunity to learn mathematics concepts in a graduated, conceptually supported framework. The goal of CRA is to create meaningful connections among concrete, representational, and abstract levels of understanding (The Access Center: American Institutes for Research, 2004). The CRA approach provides opportunities for students to expand their understanding through pictorial representations before moving to abstract concepts. Students can come to a greater understanding of place value through visual, tactile, and kinesthetic experiences.

There are 3 Key Steps in the CRA process: Concrete is the “doing” stage, Representational is the “seeing” stage, and Abstract is the “symbolic” stage. The following description is provided by The Access Center: American Institutes for Research in Washington, D.C.

Stage 1:Concrete. In the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base- ten blocks, pattern blocks, fraction bars, and geometric figures).
Stage 2: Representational. In this stage, the teacher transforms the concrete model into a representational (semi-concrete) level, which may involve drawing pictures; using circles, dots, and tallies; or using stamps to imprint pictures for counting.
Stage 3: Abstract. At this stage, the teacher models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the skill of focus. The teacher uses operation symbols to indicate addition, multiplication, or division. (The Access Center: American Institutes for Research, 2004).

Example of using CRA to teach Whole Number Place Value
Note: The following example comes from an excellent online educational resource: “Math VIDS: Video Instructional Development Source”, additional ideas can be accessed at: http://fcit.usf.edu/mathvids/strategies/cra.html#
STEP 1: Concrete
Materials
• Base 10 cubes/blocks
• Beans and bean sticks
• Popsicle sticks & rubber bands for bundling
• Unifix cubes (individual cubes can be combined to represent "tens")
• Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Instruction: Students can be shown to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side.) Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds" (Math VIDS, 2010).
STEP 2: Representational

At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. Have students draw models of the combinations of ten blocks and other manipulatives they have been working with (Math VIDS, 2010).

STEP 3: Abstract
Students who are ready to problem-solve at the abstract level, do so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with only numbers and math symbols. In the case of whole number place value, that means students need to be identify place value of digits by simply looking at the number (Math VIDS, 2010).

Curricula and Technological Resources

Curricula Resources

  • Houghton Mifflin: Math Expressions

A website associated with the curriculum company Houghton Mifflin. The website provides helpful resources to parents, students and teachers. The resources are grouped by grade level and include: challenge masters, accessible alogorithms, visual support, problems sets and an e-glossary.

  • Base Ten Manipulatives

A website that sells base ten manipulatives, individually and in concept kits. http://www.basetenblocks.com/

  • Printable Base Ten Manipulatives

http://www.eduplace.com/math/mthexp/

  • Online example lesson that emphasizes the complete value of numbers.

http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/3_Place_Value/index.html

Technological Resources

  • Learning Box Base-Ten Activity

Activity.png

http://www.learningbox.com/Base10/BaseTen.html

  • Place Value to 100 Thousands Game

Game.png

http://www.toonuniversity.com/flash.asp?err=503&engine=15

  • eManipulatives Hundreds Chart

Hundreds.png

http://www.eduplace.com/kids/mw/manip/mn_k.html

Lifeguard

SF67~Lifeguard-On-Duty-Posters.jpg

The lifeboat will only move in steps of 10 or 1. Click the boat's pedals until you have reached the drowning swimmer, then click the lifebelt to rescue them. Covers place value for 2-digit numbers.

http://www.ictgames.com/LIFEGUARDS.html


Dinosaur Place Value

DinoPlaceValue.jpg

Players are given a number and must correctly identify the correcty tens and ones unit, which are the complete value for that number. For example, players must identify that 10 and 5 are components of 15. This game allows players to think about place value in terms of groups of numbers, instead of the traditional face value.

Place Value Pirates

Pirate.gif


In this highly interactive game, players must identify correct locations of numbers in designated place value positions. For example, the player will be asked to find "the 3 in the tens place" and the player must correctly identify the pirate who is on the platform with "the 3 in the tens place". This game gives players an opportunity to practice standard and word form of place value.

http://www.mrnussbaum.com/placevaluepirates.htm


Place The Penguin

Penguin.jpg

This game provides players with both the face value and complete value of numbers. Players must recognize the place values that make up numbers in order to solve problems. For example, 80 and 2 equals the number 82. This game is an excellent transition into early addition teaching.

http://www.bbc.co.uk/schools/starship/maths/games/place_the_penguin/small_sound/standard.shtml

Annotated References

  • Baroody,Arthur J. How and When Should Place-Value Concepts and Skills Be Taught? Journal for Research in Mathematics Education, Vol. 21, No. 4 (Jul., 1990), pp. 281-28
  • Butler, F.M., Millar, S.P., Crehan, K., Babbitt, B., & Pierce, T. (2003) Fraction Instruction for Students with Mathematics Disabilities: Comparing Two Teaching Sequences. Learning Disabilities Research & Practice. (18) 2 99-111
  • Carpenter, T.P., Fennema, E., Franke, M., Levi, L. & Empson, S.B. (1999) Children’s Mathematics: Cognitively guided instruction. Portsmouth, New Hampshire: Heinemann.
  • Chapin, S., O’Connor, C., Andersen, N. (2003) Classroom Discussions: Using math talk to help students learn. Sausalito, California: Math Solutions Publications.
  • Concrete-Representational-Abstract (CRA) Instructional Approach Summary Report. (2004) The Access Center: American Institutes for Research (AIR). Retrieved from http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.as.
  • Flores, M. (2008). Using the Concrete-Representational-Abstract Sequence to Teach Subtraction With Regrouping to Students at Risk for Failure. Remedial and Special Education. (31) 3, 195-207.
  • Fuson, K. C. (1988). Teaching adapted to thinking. Journal for Research in Mathematics Education, 19. 263-267.
  • Fuson, Karen. (1990). Using a Base-Ten Blocks Learning/Teaching Approach for First-and Second-grade Place-Value and Multi-Digit Addition and Subtraction. Journal for Research in Mathematics Education, Vol. 21, No. 3 (May, 1990), pp. 180-206
  • Garlikov, Richard. (2000) The Concept of Teaching Place Value in Math. www.garlikov.com/PlaceValue.html
  • Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier, A. & Human, P. (1997) Making Sense: Teaching and learning mathematics with understanding. Portsmouth, New Hampshire: Heineman.
  • Kennedy, L.M., Tipps, S., & Johnson, A. (2008). Guiding Children's Learning of Mathematics, 11th Edition. Belmont, CA: Wadsworth Publishing.
  • Math VIDS: Video Instructional Development Source. (2010). Retrieved from http://fcit.usf.edu/mathvids/strategies/cra.html#.
  • Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. J. Masalski & P.C. Elliott (Eds.), Technology-supported mathematics learning environments: Sixty-seventh yearbook (pp. 17-34). Reston, VA: National Council of Teachers of Mathematics.
  • Thompson, R Bramald - Newcastle upon Tyne: University of Newcastle upon Tyne, 2002 - atm.org.uk
  • Van de Walle, J. (2010) Elementary and Middle School Mathematics: Teaching Developmentally, Fifth Edition. Boston: Arlington
  • Varelas, Maria and Becker, Joe. (1997). Children's Developing Understanding of Place Value: Semiotic Aspects. Cognition and Instruction, Vol. 15, No. 2 (1997), pp. 265-286