**Math Topics**- Common Core
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- Manipulatives
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Math Manipulatives contains three pages of resources:

**About Virtual Manipulatives (Page 1 of
3)**:

- The definition of a virtual manipulative
- The role of virtual manipulatives for learning mathematics
- Cautionary statements about using and overusing manipulatives and calculators

Virtual Manipulatives on the Web (Page 2 of 3): a list of resources

Math Manipulatives (Page 3 of 3): Calculators and PDA resources, including calculator tutorials, activities, software enhancements, and calculator apps for mobile devices

In *What are Virtual Manipulatives?*, Patricia Moyer, Johnna Bolyard, and Mark Spikell
(2002) defined a virtual manipulative as "an interactive, Web-based visual
representation of a dynamic object that presents opportunities for
constructing mathematical knowledge" (p. 373). Static and dynamic
virtual models can be found on the Web, but static models are not true
virtual manipulatives. Static models look like physical concrete
manipulatives that have traditionally been used in classrooms, but they are
essentially pictures and learners cannot actually manipulate them.
"...[U]ser engagement distinguishes virtual manipulative sites from those
sites where the act of pointing and clicking results in the computer's
providing an answer in visual or symbolic form" (p. 373). The key is
for students to be able to construct meaning on their own by using the mouse
to control physical actions of objects by sliding, flipping, turning, and
rotating them.

Virtual manipulatives have a range of characteristics, such as pictorial images only, combined pictorial and numeric images, simulations, and concept tutorials, which include pictorial and numeric images with directions and feedback (Moyer-Packenham, Salkind, & Bolyard, 2008). Virtual manipulatives have been modeled after concrete manipulatives such as base ten blocks, coins, pattern blocks, tangrams, spinners, rulers, fraction bars, algebra tiles, geoboards, and geometric plane and solid figures, and have been in the form of Java or Flash applets. Patricia Deubel of CT4ME developed the figure above to illustrate virtual manipulatives found on the Web, which are useful for mastery of basic skills and conceptual understanding of K-12 mathematics and calculus.

In 2016, Moyer-Packenham and Bolyard revised the definition of virtual manipulative owing to the rise of technology tools containing virtual manipulatives. For example, sometimes you can also find virtual manipulatives embedded in gaming environments. They are no longer only web-based and manipulated by a computer mouse. "Today, virtual manipulatives are presented on computer screens, on touch screens of all sizes (e.g., tablets, phones, white boards), as holographs, and via a variety of different viewing and manipulation devices." Manipulation can occur via a "mouse, stylus, fingers, lasers," and other modalities in years to come (Abstract section). Hence, the updated definition of a virtual manipulative is "an interactive technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge." This revision implies that "a virtual manipulative may: (a) appear in many different technology-enabled environments; (b) be created in any programming language; and (c) be delivered by any technology-enabled device" (Moyer-Packenham and Bolyard, 2016, section 1.8).

Virtual manipulatives can be used to address standards, such as those in
*Principles and Standards for School Mathematics* (NCTM, 2000) and the
Common Core Standards (2010) for mathematics, which call
for study of both traditional basics, such as procedural skills, and new
basics, such as reasoning and problem solving and an emphasis on understanding. Using manipulatives in
the classroom assists with those goals and is in keeping with the
progressive movement of discovery and inquiry-based learning. For example,
in their investigation of 113 K-8 teachers' use of virtual manipulatives in
the classroom, Moyer-Packenham, Salkind, and Bolyard (2008) found that
content in a majority of the 95 lessons examined focused on two NCTM
standards: Number & Operations and Geometry. "Virtual geoboards,
pattern blocks, base-10 blocks, and tangrams were the applets used most
often by teachers. The ways teachers used the virtual manipulatives most
frequently focused on investigation and skill solidification. It was common
for teachers to use the virtual manipulatives alone or to use physical
manipulatives first, followed by virtual manipulatives" (p. 202).

Virtual manipulatives provide that additional tool for helping students at all levels of ability "to develop their relational thinking and to generalize mathematical ideas" (Moyer-Packenham, Salkind, & Bolyard, 2008, p. 204). All students learn in different ways. For some, mathematics is just too abstract. Most learn best when teachers use multiple instructional strategies that combine "see-hear-do" activities. Most benefit from a combination of visual (i.e., pictures and 2D/3D moveable objects) and verbal representations (i.e., numbers, letters, words) of concepts, which is possible with virtual manipulatives and is in keeping with Paivio and Clark's Dual Coding Theory. The ability to combine multiple representations in a virtual environment allows students to manipulate and change the representations, thus increasing exploration possibilities to develop concepts and test hypotheses. Using tools, such as calculators, allows students to focus on strategies for problem solving, rather than the calculation itself.

According to Douglas H. Clements (1999) in
*"Concrete" Manipulatives, Concrete Ideas* there is pedagogical value of
using computer manipulatives. He said, "Good manipulatives are those that are
meaningful to the learner, provide control and flexibility to the learner, have
characteristics that mirror, or are consistent with, cognitive and mathematics
structures, and assist the learner in making connections between various pieces
and types of knowledge—in a word, serving as a catalyst for the growth of
integrated-concrete knowledge. Computer manipulatives can serve that function"
(The Nature of "Concrete" Manipulatives and the Issue of Computer
Manipulatives section, para. 2).

Christopher Matawa (1998, p. 1) suggested many
*Uses of Java Applets in Mathematics Education*:

Applets to generate examples. Instead of a single image with a picture that gives an example of the concept being taught an applet allows us to have very many examples without the need for a lot of space.

Applets that give students simple exercises to make sure that they have understood a definition or concept.

Applets that generate data. The students can then analyze the data and try to make reasonable conjectures based on the data.

Applets that guide a student through a sequence of steps that the student performs while the applet is running.

Applets that present ''picture proofs''. With animation it is possible to present picture proofs that one could not do without a computer.

An applet can also be in the form of a mathematical puzzle. Students are then challenged to explain how the applet works and extract the mathematics from the puzzle. This also helps with developing problem solving skills.

An applet can set a theme for a whole course. Different versions of an applet can appear at different stages of a course to illustrate aspects of the problem being studied.

While the research is scarce on mathematics achievement resulting from using
virtual manipulatives, Moyer-Packenham, Salkind, and Bolyard (2008) found,
overall, results from classroom studies and dissertations "have indicated that
students using virtual manipulatives, either alone or in combination with
physical manipulatives, demonstrate gains in mathematics achievement and
understanding" (p. 205). Generalizability might be a concern, however, as
found in Kelly Reimer's and Patricia Moyer's action research study (2005),
*Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom
Study*. The study provides a look into the potential benefits of using
these tools for learning. Interviews with learners revealed that virtual
manipulatives were helping them to learn about fractions, students liked the
immediate feedback they received from the applets, the virtual manipulatives
were easier and faster to use than paper-and-pencil, and they provided enjoyment
for learning mathematics. Their use enabled all students, from those with
lesser ability to those of greatest ability, to remain engaged with the content,
thus providing for differentiated instruction. But did the manipulatives
lead to achievement gains? The authors do admit to a problem with generalizability of results because the study was conducted with only one
classroom, took place only during a two-week unit, and there was bias going into
the study. However, results from their pretest/posttest design indicated a
statistically significant improvement in students' posttest scores on a test of
conceptual knowledge, and a significant relationship between students' scores on
the posttests of conceptual knowledge and procedural knowledge. Applets
were selected from the National Library of Virtual
Manipulatives.

**Resources**:

Visit the Math Forum for more on the role of manipulatives.

MathBits.com developed an online-PowerPoint, Working with Algebra Tiles. Algebra tiles can be used to factor numbers; add, subtract, multiply, divide signed numbers; make simple substitutions; solve equations; illustrate the distributive property; represent polynomials; add, subtract, multiply, divide, factor polynomials; investigate polynomials; and complete the square. Slides also show how to make your own tiles.

In order to effectively use virtual manipulatives in the classroom, "teachers must have an understanding of how to use representations for mathematics instruction as well as an understanding of how to structure a mathematics lesson where students use technology...Teachers must also be comfortable with technology and be prepared for situations where computers may not be available or Internet connections are not working properly" (Reimer & Moyer, 2005, p. 7). My own experience (P. Deubel) confirms that virtual manipulatives may take a while to download, and in some cases, the wait time might be frustrating. Imagine the frustrations for a learner anxious to begin. Plus, even when successfully downloaded, they might not work fast enough for learners who are accustomed to playing high speed, interactive video games. In some cases, the footprint on the screen might be too small for learners with poor mousing skills or for those with limited dexterity to click on relevant icons or to perform the spins, rotations, flips and turns required.

Teachers should be aware of problems that might arise from overusing both
concrete and virtual manipulatives. In *The State of State Math Standards 2005*, David Klein (2005) discussed nine
problem areas in which state standards come up short. Among those was
concern for an overuse of calculators and manipulatives in that students might
come to depend on them and focus on the manipulatives more than on the math.
"[M]any state standards recommend and even require the use of a dizzying array
of manipulatives in counterproductive ways" (p. 11). Again, in my
view (P. Deubel)
such a reliance might have its roots in the quality of instruction, in part, and
failure of the math educator to explicitly state and reinforce the link between
the use of the manipulative, and development of concepts for understanding and
properties of mathematics to be learned. Such might be the case, for
example when using algebra tiles for multiplying and factoring polynomials, if
the educator failed to explicitly link the knowledge of the distributive
property to that action.

In more recent research to identify evidence regarding the effectiveness of different strategies for teaching mathematics to children aged 9-14, Hodgen, Foster, Marks, and Brown (2018) found the strength of evidence was high regarding calculators and concrete manipulatives. Their synthesis revealed:

- "Calculator use does not in general hinder students’ skills in arithmetic. When calculators are used as an integral part of testing and teaching, their use appears to have a positive effect on students’ calculation skills. Calculator use has a small positive impact on problem solving. The evidence suggests that primary students should not use calculators every day, but secondary students should have more frequent unrestricted access to calculators. As with any strategy, it matters how teachers and students use calculators. When integrated into the teaching of mental and other calculation approaches, calculators can be very effective for developing non-calculator computation skills; students become better at arithmetic in general and are likely to self-regulate their use of calculators, consequently making less (but better) use of them." (p. 10)
- "Concrete manipulatives can be a powerful way of enabling learners to engage with mathematical ideas, provided that teachers ensure that learners understand the links between the manipulatives and the mathematical ideas they represent. Whilst learners need extended periods of time to develop their understanding by using manipulatives, using manipulatives for too long can hinder learners’ mathematical development" (p. 11).

I have an interesting personal story to relate on the use of calculators. One day our newspaper person, who was a middle school student at the time, knocked on our door to collect our monthly payment for the newspapers. He took out his calculator to multiply the weekly payment by four, which he should have been able to do mentally. I asked him what he would do to figure out my bill, if his calculator no longer worked. He said, "I'd go buy new batteries!" Klein (2005) stated that manipulatives are useful for introducing new concepts to elementary students, but, "In the higher grades, manipulatives can undermine important educational goals" (p. 11). Among those are for students to develop skill fluency, conceptual understanding, and mathematical reasoning. Many states' standards documents overemphasize calculator use, for example.

I agree with Klein (2005) that educators should not overly rely on calculator use at the expense of having students master basic skills and memorize basic facts, which are essential for higher order learning in mathematics. In this sense drill and practice still have a role in teaching and learning mathematics. According to E. D. Hirsch (1999), drill and practice may have a disparaging connotation as a pedagogical tool to teach skills and runs contrary to the progressive movement, but the method should not be slighted as low level. It is just as essential to complex intellectual performance as drill and practice are to the virtuoso violinist or the athlete on the playing field.

**Bottom line**: According to the National Mathematics Advisory Panel
(2008) in its *Foundations for Success*:

Despite the widespread use of mathematical manipulatives such as geoboards and dynamic software, evidence regarding their usefulness in helping children learn geometry is tenuous at best. Students must eventually transition from concrete (hands-on) or visual representations to internalized abstract representations. The crucial steps in making such transitions are not clearly understood at present and need to be a focus of learning and curriculum research. (p. 29)

With this being said, CT4ME has a number of virtual manipulatives that can serve you well in the classroom. As one educator recently told me at one of my own conference presentations on this topic, "I don't have to worry about students flicking rubber bands at each other any more!" She was using virtual geoboards.

Clements, D. H. (1999). Concrete' manipulatives, concrete ideas. *Contemporary
Issues in Early Childhood, 1*(1), 45-60. [Update online]. Retrieved from
http://www.gse.buffalo.edu/org/buildingblocks/Newsletters/Concrete_Yelland.htm

Common Core State Standards. (2010).* Standards for Mathematics*. Retrieved from
http://www.corestandards.org/Math

Hirsch, E. D., Jr. (1999). *The schools we need and why we don't have them*. New
York, NY: Doubleday. ISBN: 0-385-49524-2.

Hodgen, J., Foster, C., Marks, R., & Brown, M. (2018). *Evidence for
review of mathematics teaching: Improving mathematics in key stages two and
three: Evidence review.* London: Education Endowment Foundation. Retrieved from
https://educationendowmentfoundation.org.uk/evidence-summaries/evidence-reviews/improving-mathematics-in-key-stages-two-and-three/

Klein, D. (2005, January). *The state of state math standards 2005*.
Washington, DC: Thomas B. Fordham Foundation. Retrieved from
http://www.edexcellence.net/publications/sosmath05.html

Matawa, C. (1998, August). *Uses of Java applets in mathematics education*.
Paper presented at Asian Technology Conference in Mathematics, Tsukuba, Japan.
Retrieved from
https://web.archive.org/web/20120912105550/http://www.atcminc.com/mPublications/EP/EPATCM98/ATCMP016/paper.pdf

Moyer, P., Bolyard, J., & Spikell, M. (2002). What are virtual
manipulatives? [Online]. *Teaching Children Mathematics, 8*(6), 372-377.
Retrieved from
http://mathed.byu.edu/kleatham/Classes/Winter2009/MthEd308/MoyerBolyardSpikell2002WhatAreVirtualManipulatives.pdf

Moyer-Packenham, P., & Bolyard, J. (2016). Revisiting the definition of a
virtual manipulative. In P. Moyer-Packenham (Ed.), *International
perspectives on teaching and learning with virtual manipulatives* (pp.
3-25). Springer International Publishing Switzerland.

Moyer-Packenham, P., Salkind, G., & Bolyard, J. (2008). Virtual
manipulatives used by K-8 teachers for mathematics instruction: Considering mathematical, cognitive, and pedagogical fidelity. *Contemporary Issues in
Technology and Teacher Education, 8*(3), 202-218. Association for the
Advancement of Computing in Education (AACE). Retrieved from
https://www.learntechlib.org/index.cfm?fuseaction=Reader.ViewFullText&paper_id=26057

National Council of Teachers of Mathematics. (2000). *Principles and standards
for school mathematics*. Reston, VA: Author. Retrieved from
https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/

National Mathematics Advisory Panel. (2008). *Foundations for success: The final
report of the National Mathematics Advisory Panel*. Washington, DC:
U.S. Department of Education. Retrieved from
http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

Reimer, K., & Moyer, P. S. (2005). Third graders learn about fractions using
virtual manipulatives: A classroom study. *Journal of Computers in Mathematics
and Science Teaching, 24*(1), 5-25.

Durmus, S., & Karakirik, E. (2006, January). Virtual manipulatives in mathematics
education: A theoretical framework. *The Turkish Online Journal of Educational
Technology, 5*(1), article 12. Retrieved from
http://www.tojet.net/articles/v5i1/5112.pdf [Note: CT4ME is cited in this article.]

Young, D. (2006, April). *Virtual manipulatives in mathematics education*. Retrieved from
http://plaza.ufl.edu/youngdj/talks/vms_paper.doc [David Young presents a
review of the literature.]

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