Advantages and Disadvantages Of Virtual Manipulatives in Middle and High School Math

 

Virtual manipulatives are digital tools that simulate the use of physical objects in mathematics. They offer a dynamic and interactive approach to learning while engaging students in a more hands-on way. However, like any educational tool, they have their own set of advantages and disadvantages.

We'll begin with the advantages of using virtual manipulative. They increase engagement since the interactive elements, animations, and the ability to manipulate objects on screen make learning more engaging and exciting for students. This can be particularly beneficial for kinesthetic learners who learn best through movement and interaction.  

Virtual manipulatives can be accessed from anywhere with an internet connection, making them ideal for remote learning, independent study, and differentiated instruction. They can also be easily adjusted to different learning paces and individual needs.  Many virtual manipulatives offer dynamic visualizations, allowing students to explore mathematical concepts from multiple perspectives. For instance, they can rotate 3D shapes, zoom in on details, and observe patterns that might be difficult to see with physical manipulatives.  

Some virtual manipulatives can collect data on student interactions, providing valuable insights into their understanding and identifying areas where they might be struggling.  In addition,  virtual manipulatives eliminate the need for purchasing and storing expensive physical manipulatives, making them a cost-effective solution for schools and individual learners.  

Virtual manipulative have some definite limitations, especially if you live in an area with unreliable internet.  Access to technology and reliable internet connectivity can be a barrier for some students. Technical issues, such as slow loading times or software glitches, can also disrupt the learning process. While some virtual manipulatives strive for realism, they may not always replicate the tactile experience of handling physical objects. This can be a disadvantage for students who learn best through touch and manipulation.

 The interactive nature of virtual manipulatives can sometimes be distracting for students. The potential for off-task behavior, such as browsing the internet or playing games, needs to be carefully managed. Unfortunately, over-reliance  on virtual manipulatives may hinder the development of essential spatial reasoning and problem-solving skills that can be gained through the use of physical manipulatives.

What virtual manipulative are available for middle school and high school students?  Let's look at the following:

• Geometry:
  • GeoGebra: A powerful and versatile platform for exploring geometry, algebra, and calculus. It allows students to construct geometric shapes, perform transformations, and explore dynamic relationships.   
  • SketchUp: A 3D modeling software that can be used to create and manipulate 3D objects, providing a valuable tool for visualizing and understanding geometric concepts.   
• Algebra:
  • Desmos: An online graphing calculator that can be used to explore functions, solve equations, and visualize data. It also offers interactive activities and lessons for students.   
  • Algebra Tiles: Virtual versions of algebra tiles can be used to model algebraic expressions and equations, helping students understand concepts like factoring and solving systems of equations. 
• Statistics and Probability:
  • StatCrunch: A web-based statistical software package that allows students to perform statistical analyses, create graphs, and explore data sets.  
  • Random Number Generators: Online tools that can be used to simulate random events and conduct probability experiments.  

Virtual manipulatives offer a valuable tool for enhancing mathematics education in middle and high school. By carefully considering the advantages and disadvantages, educators can effectively integrate these tools into their teaching practices, creating engaging and effective learning experiences for all students.

Physical versus Digital Manipulatives.

 I've been wondering which type of manipulatives are better to use in the classroom.  I see advantages for both and usually end up using the digital ones because it's impossible to lose pieces, break them, stab them to death, or throw them at others.  I do know that the feel is not the same so today, I'm looking into both to see if one is better than the other. 

For definitions, physical manipulatives are the ones that students can hold in their hands, move around, and feel.  Digital or virtual manipulatives are web-based or app based representations of the physical ones. The virtual ones allow students to interact with to a higher degree than the physical ones.

As you know, math manipulatives offer students the opportunity to interact with concepts and can be any thing from base 10 to fractions to clocks. Research as far as using physical manipulatives helps students learn better than those who do not use them but note that it is not the manipulatives themselves but the context in which they are used.

The use of digital manipulatives has not been explored as much as the physical ones but early results indicate one should use both forms for the best results.  There are advantages to either one.  Digital manipulatives offer flexibility in that the virtual ones often can be broken into smaller parts or put together in ways physical ones cannot.  Digital ones also offer the opportunity to create more if there are not enough since physical manipulatives come in a predetermined number so if the student needs more, they have to find a second set.

In addition, the digital version is accessible from home or school and only require a tablet or phone to get to. It is easier to keep sets whole because one doesn't have pieces to lose, account for, and the teacher does not have to make sure sets are complete. The classroom always has the right number of sets.

Physical math manipulatives work well for students who need to "play" with something.  It allows them to have a kinesthetic experience and they do not do well with the digital version.  In addition, many students find the use of physical manipulatives more motivating than using digital ones because the physical ones provide immediate feedback while improving their use of time. Consequently, physical manipulatives also engage students more, especial boys.  When boys use the digital ones, they are less confident and less involved. 

Although physical manipulatives come in specific sets, if one used homemade manipulatives, one can create as many as needed.  The pieces can be placed in sandwich bags and more can be made if students lose pieces. The bottom line is that both types of manipulatives can be used effectively to improve conceptual understanding.  Let me know what you think, I'd love to hear.  Have a great day.

Does Using Manipulatives Really Help Students Transfer Knowledge?

  When I first read about concrete examples, my first thought was manipulatives.  It has always been recommended students use manipulatives to help them learn mathematical concepts but does it really do this?  Do manipulatives actually help students transfer knowledge? 

Well, lets see!  It was discovered that students may not picture the concept the same way the teachers do who already know the concept.  Students may not interpret the action in the way the teacher hopes and they may not carry the meaning of the mathematical idea.  Unfortunately, students are often taught to use manipulatives in a rote manner.

Manipulatives work best if reflect on their actions with manipulatives.  This helps them build meaning.  This is integrated concrete knowledge which combines separate ideas into an interconnected web of knowledge.  In other words, mathematical concepts become integrated concrete by how well they are connected to other ideas and situations.

It does not matter whether the manipulatives are virtual or real, the students who use them need to build the ideas from the manipulation of the representations and thinking about their actions. 

The one idea that comes up again and again in literature is that giving the students the manipulatives does not guarantee they will understand the concept.  It requires careful planning and good transitions.  The teacher needs to remember the purpose of the lesson when using manipulatives and must create a carefully thought out sequence of steps.

It was found that the higher the level of guidance the teacher provides, the better the learning outcomes in retention and in problem solving.  When they reach a point where they require a lower level of guidance, their are better able to transfer their knowledge.

So all in all, if we use manipulatives in math, we need to make sure that we carefully plan their use so the students gain the knowledge we want.  If we do not, then we may just be using them in a rote manner so students may not learn.  I'd love to hear about your experiences.  

Virtual vs Concrete Manipulatives

  Recently, I've been wondering how effective virtual manipulatives are because I know that typing uses different muscles and does not provide the same kinesthetic experience that handwriting does.

After quite a bit of reading it appears that both have merits but it appears the bottom line in any manipulator use is that the teacher must be comfortable with the use of them.

First of all, the problem with concrete or physical manipulative is the individual pieces that can get lost.  On the other hand some students need to feel the physical piece in their hands.  Its like some of us love the feel and smell of paper in our hands.  It helps our memories.  On the other hand, virtual manipulatives can take a while to download. If the internet is messed up, it can make use of the virtual manipulatives difficult to use and they become frustrating.  I know from personal experience, students often see either one as a toy to play with.

One issue that come up is simply making sure you don't over use manipulatives including calculators to the point that students loose track of the math.  It was noted that sometimes the lessons focus more on the manipulatives than the math.  I've had my students in Algebra I using an app to help them learn to factor basic trinomials.  I realized my students have not figured out that the c and b go in specific places in the diamond.  They had the connection on paper but not on the app.  This gives me something to work on with them on Monday.

The interesting thing about virtual manipulatives is that most papers were not looking only at virtual representations of the physical manipulative such as the base 10 boards etc but as including visualizations of the concept which may be why it is often suggested both types be used in the classroom.  

A commonality among everything I read comes down to three or four main points.
1. Manipulatives either virtual or concrete should be used in all grades from K to 12 to help students develop a greater understanding of the mathematical concepts.
2.  We need to monitor the use manipulatives so students do not focus only on manipulatives but on the concept.
3. In general, manipulatives are not used enough in the upper grades/middle school/high school.
4.  Many teachers do not have proper training in using manipulatives effectively in the classroom.

I fit the last one.  When I went through teacher training, there was nothing on using manipulatives in high school.  Only elementary teachers receive that training.  I've had to teach myself about them.  I suspect there are others out there in the same boat.  Have a nice day.

Barron's Algebra App

I had a shock yesterday when I had my Algebra I class do the first quiz in Barron's Algebra App.  In order to unlock the game, the individual student had to score an 80 or better answer 20 problems that only used signed numbers and the four basic operations.  An example of a problem was -2 + 5 -(-3) =
The highest score was 65%.  These are 9th graders who took pre-algebra last year and took a basic college math class last year.  This app gave me a way of obtaining student data in a quick and easy manner so that I know what I need to scaffold.  I think the next time I have them do a quiz such as this on an app, I will record student scores.
I tried using nearpod in my last class yesterday and we had issues with it.  I found out later that there are at least two other classes using the internet at the same time because only 5 students were able to log on out of a class of 20.  Now I have to figure out a way of being able to use it or give it up.
As far as virtual manipulative websites, I am looking at the individual websites to see if they work on the iPad.  It appears that due to several using Java applets, there may not be any apps out there that would allow me to access any of the sites.

Virtual manipulaties vs real manipulatives.

I have been wondering about if there is a difference in student learning when they use virtual manipulatives rather than real manipulatives.  This question came about from watching a student play solitaire on his iphone and when I asked him if he could play the same game using real cards, he gave me a funny look and said no as if he didn't need to know how to actually deal the game.  One of the apps I've had them use requires them to use a virtual balance for solving multi-step equations and some of them had a fair bit of trouble.  I think the trouble was not moving pieces around but transferring the information from the tablet to the problems on the paper. 
The nice thing about virtual manipulatives is that students do not loose the pieces and can easily redo the activity again and again.  
I think on Thursday, I am going to have them use real manipulatives (the squares for one and the long ones for the variables) and see if the transference is any better.  This is a topic I need to research to find information on.  The other thing about virtual manipulatives is that most websites I would use are not usable on the iPad due to the site needing flash. 
Furthermore, there are few apps that allow the iPad to use flash and I have to finish checking that out. Usually these apps run the flash on a remote server and stream it to the iPad.  This can create a bandwidth issue.
So over the next few days, I am going to see which virtual manipulative websites currently work with the iPad and I will post a list as I find them

Infographics

I've been reading about infographics and I think it is something worth trying.  Since my college prep is in the middle of a project, I think on Monday I will give them a choice of actually writing the short report or preparing an infographic on logs and natural logs and their use.  I think they might have more fun doing the infographic rather than a report since it is a bit more creative.  From what I've read about infographics, these would be great for the probability unit as students can see the real life applications of probability.  Bingo is a huge business out here and parents will often go play to 3 or 4 in the morning.  Maybe an infographic on the probability of Bingo would be a good place to start the unit. 
I am hoping to discuss a couple math apps I discovered on iTunes that will help students with the distributive property and with the special binomials such as (x^2 - 4).  I even have ideas of how to do a graphical representation on Geoboard but I want to do more research on other apps that might help me with the graphical representations.
If anyone knows of one, let me know.

Three apps

I downloaded three apps I found on a list of apps you should have if you are teaching math.  One seems to have disappeared and I've been unable to find it.  The three I found are Ooops, Virtual Manipulatives, and iSpy-x.
The basic premise of Ooops is that you get several numbers and operations that are equal to something.  You insert parenthesis to make the problem come out right.  It has 5 levels and uses all four operations.  Sometimes it uses one set of parenthesis, sometimes it uses two.  When you think you have the answer, you hit the check button and it shows the work with the answer your problem has and tells you if it is right or wrong.  I like this app in that it gives the students a better conceptual grasp of PEMDAS and how it can effect the results of the problem.
Virtual Manipulatives has actual manipulatives for fractions, decimals and percents.  It uses the bars that start at 1 and then are subdivided for 1/2, 1/3, 1/4, etc.  The 1/3 for fractions, decimals and percents has all the same color and you can move pieces so you can have students explore 1/2 = 1/4 + 1/4 or 50% = 10% + 10% + 10% + 10% + 10%.  Virtual Manipulatives is great for the upper elementary or middle school grades.  I could see using in high school with students who struggle with basic math or in a pre-algebra class.
Finally iSpy-x which is supposed to help students learn to solve simple problems such as 2 + x = 7. You can choose the one of three levels and on each level you can select the operation or used mixed operations.  I downloaded the program 3 different times and each time it had a different bug.  I could never get it to work on more than the easy level for addition.
I know that I am going to load Ooops on my iPads next year to use in the classroom.  I like it and it can provide great differentiated instruction due to the 5 levels it offers.

Virtual manipulatives in the classroom.

Currently I use two different ones in my math classes.  One is Geoboard for the iPad.  It is just like the real life geoboards we have in our classrooms except this app comes with two different boards and lots of colored rubber bands but no chance of loosing either the board or the rubber bands.  I've used it for students to create various types of triangles such as those classified by sides, or angles, finding the number of triangles in a polynomial as a starting point to find the equation telling the interior measurements of the angles, tessellations, transformations, etc.  If you use a geoboard in real life, you can use this app to do the same thing.
The second app I've used in geometry is Geometry pad.  It allows students to use a coordinate plane for geometry.  It allows students to make all sorts of quadrilaterals and polygons, triangles, circles with radius, arcs, chords, etc.  There is an option for points, angles, lines, rays, segments, perpendicular bisectors, medians, altitudes and angle bisectors.  you can draw, type, use a ruler, and do certain functions.  I've used this one when the book has a coordinate plane problem, transformations, midpoints, diagonals, etc.  If I have to resort to worksheets for certain activities, I will have my students use this app to help them complete the worksheet. 
Due to iPads not having java, I am unable to use many of the virtual manipulative websites on the internet.  I found another app that has great ratings and I hope to down load it tonight to play with.  Its virtual manipulatives and focuses on decimals, percents and fractions.  Although it is rated for middle school, many of my lower performing high school students are weak in that area.  I think this could be used in that class to help them learn how decimals, percents and fractions relate to each other.  I hope to give more information on this app on Monday or so.