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Week 4: Virtual and physical manipulatives

Part 1

Find and review a virtual manipulative for learning mathematics. How would you use it with your students? How would you improve it?

Part 2

What historical "tools of the trade" (such as abacus) are still relevant as learning manipulatives, circa 2013? Which tool(s) are you using, or would you use to help your students learn? Why?

Part 3

Take a physical manipulative you like, for example, Cuisenaire rods or graph paper. Redesign it to become a virtual manipulative. What features can you implement in computers or mobile devices that the physical manipulative does not have? If you are interested in blended learning or augmented reality, you can design a manipulative that goes into both computer-based and physical realities.

• Part 1

http://nlvm.usu.edu/en/nav/frames_asid_160_g_4_t_1.html

Here is the link for my manipulative. It deals with percentages; the part, the whole, and the percent. I would use it with my scholars to introduce the concept of percent and how it is related to fractions and decimals. I would have them enter random numbers into each category and record their results along with a visual drawing of what appears on the computer. I would then ask them what do they think percentages are and how they are related to fractions or decimals. I would then provide them with world problems where either the percent, whole, or part was missing and have them solve them by entering the information into the manipulative program. I would improve this virtual manipulative by allowing the scholars to type word problems where they would identify the whole, part, or percent and then the information would be transfered into the appropiate boxes so that they could study the visual aspect of it. I would add marking to the fraction tile or pie graph so they can see what 25%, 10%, and other percentages look in regards to 100% of an item. I believe they should be allowed to visualize what something great than 100% looks like as well because everything isn't always out of 100%.

Part 2

I believe that algebra tiles, fraction tiles, geometrical pattern blocks, money, and a host of other manipulatives are still relevant in present times. I believe that historical "tools of the trade" provide a strong foundation which scholars can build upon (especially in their developmental stages) and it also provides historical context and frame of reference. I still use pattern blocks, algebra tiles, and fraction tiles. I use these manipulatives because it assists my scholars who have a hard time visualizing concepts and need tangible items to make sense of things. Algebra tiles have always been a great tool for me to introduce the concept of using variables or integers, especially for my low level learners. Fraction tiles bridges the gap for scholars who quite don't get the concept of equivalent fractions or what the idea of fraction is in general. Pattern blocks are great for introducing geometry. I have used them to teach the differences and similarities of each polygon, games in which they would reconstruct a picture made up of various polygons, or to show which shapes are composed of other polygons. I believe that if we stop using these historical "tools of the trade", then we will lose an important aspect of teaching mathematics.

Part 3

I would redesign graph paper as a manipulative than is both physical and computer based. I guess it would be similar to the technology used for computer tablets that allow you to draw with a stylus. The scholars would be provided with graph paper designed tablets that would in turn record there drawings on virtual graph paper on a computer or technological resouce such as an iPad or Smart Board. I would also allow the teacher to be able to pull up a scholars work on the Smart Board, projector, or computer to point out any successes or challenges the scholar may have. This would also allow scholars to save paper because they would only print out their best work, once they have mastered the concept being covered. This could also be converted into an app for people that have smart phones or touch screen devices; with the incorporation of a stylus or their finger. Some options that you could incorporate into the technological side of the device is determining the best fit line one your plot points on your graph, identifying the minimum or maximum of a parabola, or determining the x- and y-intercept coordinates. There are a host of other options that you could implement by using this idea, but these are just a few I was able to think of.

• Very good point about going over 100%!

I like how you put money together with algebra tiles - they both can be used to learn about the positional systems and base ten. Of course, with tiles you can also explore other bases.

So the main idea for the design of your graph paper is the analysis of graphs, right? The coordinate system that also analyzes the data you put into it? Neat!

I am also thinking of some more mundane graph paper options, such as changing the scale quickly (which you can't do with paper), or switching to/from logarithmic scale.

• Part 1

This online manipulative has algebra tiles and a blank board so users can move the tiles around to model different equations and use inverse operations to cancel the terms and solve for the variable. The manipulative also has another option that generates equations for students to solve using the algebra tiles. It also checks their solution so students know if they are solving the equations correctly. http://my.hrw.com/math11/math06_07/nsmedia/tools/Algebra_Tiles/Algebra_Tiles.html

I would use this manipulative in order to model how to solve equations using inverse operations and also allow students to practice solving the equations using the algebra tiles. I think this would be a fun exercise to get student working with the terms of an equation and also deepen their understanding of inverse operations before they start solving equations on paper.

I really think that this site provides a lot of different options for teachers and students to use this software. I like that the equation is displayed and changes as the students put the different tiles on the board. The only thing I could think of that would improve this virtual manipulative is incorporating a tutorial when the program is opened to explain to students how to move the tiles, solve the equations, and check their solutions.

Part 2

A learning manipulative that I just used in class on Thursday was different three dimensional objects to explain surface area and volume. I think using these shapes that students can see, feel, and turn around in their hands is SO important in order to explain how we take the area of the base and then multiple it by how high the shape is stacked (V=lwh). I showed the students a Rubiks Cube so they could see the grid lines and then passed the object around the class for the students to see and turn the cube around to theorize how to calculate volume. I think online manipulatives do a great job of illustrating nets and showing how they fold up to be a 3-D object. However, it is still not REALLY 3-D since it is still constricted to a computer screen. Providing objects for the students to see and feel is an effective way for them to understand the different formulas and what they calculate.

Part 3

This is an online tool that I just assigned for my Pre-Algebra students to create a bar graph of their results in a probability experiment. They had to first calculate the theoretical probability of each possible sum when rolling two dice. Then the students actually rolled two dice 36 times and tallied their results. They were then to use the online bar graph tool linked below to graph the theoretical and experimental probabilities. http://www.mathsisfun.com/data/data-graph.php

What this online tool provides that the physical manipulative does not is efficiency and accuracy. I did not want my students, especially my perfectionists, to spend hours creating these bar graphs, especially when the objective of the assignment was not creating a bar graph, but just using it to visually analyze the data. It also gives accurate results, especially for those students who have trouble with following the straight lines on graph paper. The online tool provides students with a way to create an accurate and visually appealing graph in only a few minutes.

• Accuracy and precision - spot on for virtual manipulatives! You mention perfectionists, which goes with your territory of gifted ed, pretty much. There are several other populations who need help with accuracy. For example, some kids have physical issues that make their hands unsteady (or severely restrict movements). They can't draw or draft, but they can still use machines, with proper interfaces. A quadruplegic with some tiny motion in fingertips can still use a special stick with buttons to make great math art or diagrams.

Another target population is very young kids. Their fine motor skills can be way behind their intellectual prowess! With computer tools, toddlers and young kids can go deep and fly high in math, without crying over torn paper and wildly wrong lines.

• Part 3:

I was thinking about the rubber sheets that have different shapes on them that kids can pull out & then put back into the sheet. They are kind of like puzzles. So, if you are learning about shapes & you are literally holding shapes in your hands, you can go around a classroom & find things that are shaped like a circle & like a rectangle, etc. Then the students would write down what they found.

I would try to do a virtual maniplutive of this. So you've have a picture of say a neighborhood with houses, buildings, street signs, vehicles, people, playgrounds, a school building perhaps, maybe even a picnic area...basically the screen should be filled with REAL WORLD obejcts that have REAL shapes. Then the students would have the traditional shapes say on a left side bar. They'd have to drag the shapes onto the real world objects & find those shapes in their REAL WORLD virtual manipulative. ...For ex, houses would be pentagons, & stop signs would be octagons...maybe a picnic blanket would be a square...and so on & so on.

You could even put the kids in a car & they'd be driving through the neighborhood & have to stop every once in a while when they come upon a new shape. FUN!

• Lisa, I am thinking this would be an interesting app even for toddlers, if you have it on a touch screen device. But also for high school students and even adults - depending on how complex the shapes are! When I read about the car, I wasn't sure if you mean virtual car (with kids' avatars in it) or physical car. So I imagined kids looking at the world as they ride around in a real car, looking through their viewfinder, and taking a snapshot as soon as they see something resembling a shape! Then they can even share their found shapes by uploading it some common space. Sort of like a scavenger hunt.

All this from a simple shape sorter!

• PART 2:

Great question & got me thinking about how I've been approaching this class. Since I consider myself VERY old school, I have to agree that a pencil & paper are the MOST important thing to me in being able to work on math problems. It's funny because I've been taking this class & really trying to NOT use a pencil & paper while I navigate through this online technology class. I feel like I've been on a diet that I'm really fighting & seems unnatural.

Why shouldnt I also use a paper & pencil if I want to...along with all the new technology that I'm learning to also help me learn & teach? Its like I'm battling my natural tendency to WRITE...however...isnt this a good thign every now & again...get out of your comfort zone & try something different...expand my horizons?

So, thats what I'm doing for this class...I'm NOT using a pencil & paper. (oh how I miss it so) BUT- I do think its helping me learn in a different way!! As much as You don't want to hinder a students natural ways of learning, I do think its important to present NEW ways of learning & doing things. Helps us feel more comfortable navigating through life situations that arent comfortable & realizing, we can get through them & we do have ability to surge through.

For ex: I took an acting class in undergrad. In one class, we were NOT allowed to use our hands when we spoke. We literally had to SIT on our hands the entire class. This was SO difficult for me. I literally was speechless...mute...without my hands for a bit! However, it helped me learn to use my FACE to project what I was trying to get across. Its like a baseball coach havig the players throw with their less strong arm...I think these exercises keep a good balance for us...helps us to keep thinking OUTSIDE THE BOX & realize we CAN do things that we didn't think we could do!

• Interesting exercises, Lisa! Thank you for sharing your adventure in the no-paper land!

Maybe you can incorporate that design idea (what everyday thing NOT to use, so you can reflect on your processes) into your tech week? As I was reading your examples (paper, hands!) two other examples came to mind. First, touch screen technology - it does away with familiar peripherals, namely the mouse and the keyboard!

And second is an exercise I give teachers and parents. Prepare a mini-lesson on your favorite topic. But you are not to TELL students anything, and you are not to SHOW students anything. Can you teach if you don't tell, and you don't show?

• Virtual manipulatives:
PART 1:

Geoboard
Click on link above here & then when there, go to "Geoboard"

This website from Utah State has many examples of different math concepts & many virtual manipulatives to help students SEE what they are searching for. Its fairly simple compared to other virtual manipulatives. I'd change a few things.

-I'd like to be able to click on any 4 points on the graph & being able to see what the area & perimeter is of that NEW shape that you create.

-This is asking the question of area, but doesnt give the answer or explanation of how for ex: the AREA OF IRREGULAR SHAPE should be solved. I think this is important because if you have a child who really wants to know how to do something, but doesnt really have a lesson or instruction next to a manipulative where its EASILY found, the student may get discouraged & feel frustrated.

So, I'd add a link to see "LESSON" or "How do I figure this out?" kind of link available to see the proper steps of solving on each page where there is a questions initially.

• Part 1: I found this virtual manipulative which performs transformations such as rotations and reflections on geometric shapes in the xy-plane. What I like about this is that it easily performs transformations that can be difficult to sketch accurately and gives a great visual for what can be a tricky topic. I think this manipulative could be improved by offering more options/control with regards to the input shapes.

Part 2: I think one manipulative that will always be relevant is the original manipulative – your fingers. I read a book recently about the history of the number zero, and it talked about how early numeration systems were developed to count things – animals, tools, etc. It took time for the development of mathematics as something abstract rather than concrete. Five plus three equals eight whether or not you have eight cows to count, but before people understood that, there was no need for them to quantify the concept of nothing. I see this with my daughter who is four and in the very early stages of learning about math. She loves math puzzles and games designed for kids, which are all based on adding and subtracting by counting something – pictures, dots, fingers, and so on. She's learning important number skills, but the abstractness of mathematics is well beyond her maturity at this point. I think that the oldest manipulatives like fingers, beans, blocks, pictures of kittens, etc. will always be relevant because they bridge that gap from concrete to abstract.

Last year I observed in a special education geometry class. The teacher used manipulatives a lot. She had this whole collection of plastic 2d and 3d shapes, and she would hand them to students while she asked them questions. For example, she would hand a student a cube to hold and then ask about the number of faces, edges, and so on. I think that even though there are some very sophisticated graphics programs that can depict all kinds of complicated 3d figures, that is just not the same as holding a cube in your hands and turning it so you can touch each face and edge.

Part 3: I think that base ten blocks are an excellent manipulative because place value is such an important concept. I would design a base ten virtual manipulative that has a lot more functionality than physical blocks. For example, students could drag single blocks into groups of ten to automatically make a ten-block. Students could also easily work with larger numbers, such as numbers in the tens of thousands, which would be very cumbersome with physical blocks.

• Good point about accuracy. Precision is a math value, and speaking of anxiety, using virtual tools can help those who aren't so sure of their drawing and drafting skills.

Having worked with several designers of virtual manipulatives for place value... Showing large numbers is still a challenge, though easier than in physical space. How would you show 10,000 or 100,000 in your virtual manipulative?

At four, my kid sometimes could not talk about objects without physically touching them! But touching or holding the object brought out long and detailed stories.

The question of young children's abstract thinking is fascinating and complicated. Think about language development. All concepts - all WORDS - are somewhat abstract. "Dog" refers to a lot of different individual creatures, most of them not even looking like one another at all. To understand the word "dog" kids need to analyze similarities between those creatures, and to discard the differences. A couple of years ago, I interviewed a researcher who studied an Amazonian tribe. They had separate words for THREE FISH and THREE FRUIT, for example - and no word for just THREE. So, young kids are capable of some abstraction - but of course, in ways very, very different from what grown-ups do.

• My initial thoughts about showing larger numbers was that it is just a lot easier in a virtual environment where space is not an issue. With traditional base ten blocks, it’s just not practical to store and manipulate a lot of 1,000 cubes, but with virtual blocks, that’s not an issue. But your comments got me thinking, and I can see how it could still be tricky to show larger numbers.

I was thinking about how with traditional base ten blocks, a single block is treated like a point even though it’s actually a cube, so as you go from one to ten to a hundred to a thousand, you go from a point to a line to a square to a cube. Maybe larger numbers could follow the same pattern. Once you get to the 1,000 cube, build on that by treating it like it’s a point again. So 1,000 would be a single 10x10x10 cube, 10,000 would be 10 of those cubes lined up end to end, 100,000 would be a 10x10 square of 1,000 cubes, and 1,000,000 would be a 10x10x10 cube of 10x10x10 cubes. Each of these configurations would be a different color to make them distinctive. In addition to making it possible to manipulate larger numbers, I think it could reinforce some arithmetic concepts and relationships, like how 1,000 and 1,000,000 are both cubes because 1,000 is a thousand ones, and 1,000,000 is a thousand thousands, etc.

Thanks for the comments about language development. I remember when my daughter was learning to speak and she would point to every tall dark haired man with glasses and shout “Dada!” when we were out at a store or something, which made for some very interesting moments with strangers! It is really interesting how kids develop abstract thinking. My daughter loves doing simple addition and subtraction, but when I ask her two minus two, she laughs like it’s a silly question and says it’s nothing. She knows the number zero and she can identify it in print, but to her, it doesn’t have the same representation as a quantity that other numbers have. We’ve been talking about the value of different coins, and she thinks it’s outrageous that a dime is worth more than a nickel because a nickel is bigger. For her, bigger means more, and watching her wrestle with ideas that are counterintuitive (and constantly trying to answer her unending questions) is such a fascinating look at how kids make sense of the world.

• Part 1: The virtual manipulative I came across works with Venn diagrams. I like these are useful tools for showing the relationships between sets. They are simple to create by hand but it is nice to have them done virtually. I would use this to help the students explore on their own after I initially teach the topic. This would help students to better understand the difference between union and intersection. It also has worksheets to use in addition to the interactive Venn diagrams. http://www.shodor.org/interactivate/activities/ShapeSorter/

Part 2: I think that paper and pencil are still relevant. I am sure these seem obvious, but in our ever technological growing world these seem like things of the past in most areas. When was the last time you wrote a letter? (Not an email!) However, in math these are still necessary. Even though this is an online course, I have a notebook and pencil with me while I search. I work out examples like with Project Euler. Now a phone has an app for everything but I still think paper and pencil rule the day in the math world.

Part 3: I am not sure what these are called but they are different 3D shapes used for teaching students about area and volume. They can be taken alone, a single cube, or put together to make more complex shapes, a cube with a pyramid on top. If these could be made into a virtual manipulative (VA), I think it would be easier for the students to work with. Also it would be more cost effective for the teacher! With respect to talking about area, in the VA the unused sides could be faded out so it would be easy for the students to focus on the section they need. This is not something that could be done with the physical blocks. You could also make an object with many sides and very complex. You would not be limited to cubes and spheres. It would make a lesson more extensive to have them find the volume of a 3d hexagon! This would be hard to make be hand.

• Paperphiles unite! I also love 3d paper models, so I would add glue and scissors to the paper-related tool list. Origami uses neither, but kirigami is cool too.

The point you make about fading and highlighting is a great design principle for virtual manipulatives. Highlighting what is important is a simple, but very powerful way to scaffold mathematical learning.

And yes, you will never run out of parts, and you can make pretty complex objects. Even something 4-dimensional, maybe?

• Megan, I agree about a paper and pencil! I'm taking a geometry course this semester which is all based on Geometer's Sketchpad. I can construct the figures and answer simple questions right in the program, but I cannot work out a complex proof without a paper and pencil. It's like my brain just doesn't work the right way if I try to reason it out while typing instead of writing it by hand. It's funny because I can remember a time years ago when I had a hard time writing on a computer. Quick comments and emails were fine, but for something like a formal paper, I had to at least start out writing it by hand and then typing it in later. I graduated from high school in the mid 90s, so computers weren't as prevalent as they are today, so I had to sort of train myself to write on a computer. I can do it now, but I still find myself brainstorming my initial thoughts by hand.

• PART 1:

love the Fibonacci Sequence virtual manipulative at http://nlvm.usu.edu/en/nav/frames_asid_315_g_3_t_1.html?from=grade_g_3.html. This site provides a general explanation of the Fibonacci Sequence, as well as links for parents/teachers and applicable standards.

The table provided shows how the sequence eventually converges to the golden ratio.  However, it might be confusing for people new to the concept to understand the role of F1, F2, Fn, and so on.  Photographic depictions of the Fibonacci sequence as it occurs in nature (pinecones, sunflowers) would provide wonderful real-life examples.

PART 2:

Wow, what a thought-provoking question!  I think the most obvious (and enduring) example of a manipulative is people using their fingers (and toes) as counting aids. Personally, I find that non-tech manipulatives help me in stressful situations (such as when I'm hosting an extended family gathering and the number of guests changes by the minute - adding and removing plates/flatware/drinkware from the table allows me to know from a glance how much food will be eaten/drinks will be consumed, therefore how much to prepare).

I love technology - a calculator and the internet is an essential part of my daily life.  However, I become quite irritable when it's not available (frequently in our house for various reasons).  While I can certainly compute my bills and such 'by hand', I've always been slow with arithmetic (I have a compulsive need to triple-check everything).  I think that an abacus would diminish this a bit - I find it easier to see what I've done (as opposed to looking over handwritten problems and trying to find a mistake).  When I was in first grade, I was taught to be proficient in abacus use; I'm seriously considering using one again.  I would certainly teach my students how to use one - who knows what kinds of power outages we might face in the future.

I think that non-powered manipulatives still have a place in the world, despite society's technological advances.  Some societies do not have electrical power (or, if they do, it's not reliable or is temporarily unavailable (think Hurricane Sandy.  In these instances, non-powered manipulatives (such as the abacus)) can offer more computational efficiency than none at all.  This may translate into a significant business advantage.

PART 3:

The mention of graph paper caught my attention right away - it's something that I've seen all students use, though it often becomes quite a 'production' because students don't have graph paper or students don't have different-colored writing implements to graph separate functions,   I think that a phone or tablet app that's accessible to each and every student would make the learning process more efficient.  Physical graph paper often does not withstand erasing; subsequent graphs become smudgy.  Students often misunderstand scale when they are first learning to graph and wind up using a lot more paper than necessary (i.e. using 4 squares to represent 1 unit of measurement when a problem doesn't involve that).  Some students do not use pencil at all; mistakes made in pen may turn into an unintelligible mess from being scribbled out, or result in a waste of paper.  Computerized graph paper would eliminate all of these problems.

• Fingers 2.0 - use computer-based aliens with any number of fingers (and toes) to work in different bases!

The "undo" function is so easy on computers - and sometimes quite impossible in the physical world! Erasing graphs is a very good example of the "undo" function computers can provide. When it's easy to undo and to redo, students find it much easier to TRY many different things, just to see what happens. This is one example of computers LOWERING THE PRICE OF MISTAKES and thus encouraging experimentation.

• Thanks for summing everything up as "computers lowering the price of mistakes"!

Computers not only reduce material waste due to mistakes (paper and such), but also reduce the time spent correcting same.

I often feel that people who've grown up using computers don't appreciate the mangitude of the reduction in time and materials, and take computers (and, by extension, those who spent time and money developing the technology) for granted.  Maybe that's just my age showing (turned 44 last week and remember when calculators were only a concept item), but I do think this sense of technology entitlement can create challenges.

One challenge is that it might become increasingly hard to convince students the value of understanding the mathematics that create the foundations of the software that they use on a daily basis.  Some may feel that, unless they plan to become a programmer, why should they learn how to figure out how things work when they can just google a question, or use an app?

We can tell them that one might not always have access to such tools, but many people (young and old alike) do not accept this, as they feel that this technology has become a part of daily life (much like cars).  Yet, if one's car breaks down, people have the skills to find alternate transportation (whether it's asking a friend for a ride, hiring a cab, or using one's feet); these alternatives present themselves almost automatically without the need for higher-order thinking.  But, for whatever reasons, people seem to care less about having a backup plan in case their tech-based tools aren't available.  From what I've observed personally, this pertains more to younger folks than older.

I think that this is where math teachers might create a bit of motivation, as most people are frustrated by not being able to solve a problem, and problems may very well result if tech isn't available (i.e. having to do one's tax returns 'by hand', or trying to figure out how much mulch you'll need to cover your kidney-shaped flower bed when you're pitchforking the mulch at the township yard and your hands are too dirty to use the calculator on your phone).

Social problems can also result from taking math tech for granted - I was in the grocery store on Thursday and the older man who was in line ahead of me yelled at the cashier because the scanner was incorrectly programmed and he did not get the sale price (\$1 off) for his ground beef.  He obviously felt that technology was infallible and had no idea of the extent to which humans manipulate it.  It's good that he was able to calculate the mistake, but bad that he didn't understand its cause, blaming the mistake on the computers rather than the people who manipulate them and taking this out on the cashier.  Two keystrokes resolved the problem - in the past, it would have been a lot more complicated and time-consuming.  It's another example of how people mistakenly view computers as an entity unto themselves, rather than just the tools that they are, that lower the price of mistakes.

As teachers, we need to change that.  It might be easier with students, but we need to remember that our students live with older family members who might not always have the same perspective.

• I agree that "lowering the price of mistakes" encourages experimentation; well put and thanks for the concise thought!

I think that we can all relate to your statement.  We can use computers to experiment with various situations, whether it be monetary (household expenses) or aesthetic (superimposing a particular hair color onto one's face).   In each case, we can experiment at no cost aside from a few minutes of our time.  The risk is small when compared to the real-life alternatives (financial ruin or embarassment of self-created unflattering appearance).

Mathematical modeling lowers the price of mistakes on a larger, commercial scale as well; software such as AutoCAD (http://usa.autodesk.com/autocad/) gives users the ability to plan and visualize at a fraction of the cost that it would take to prepare this data without technology.   Such software allows the user to visualize many different outcomes before printing.  While printing might only be a few cents per print, this cost could certainly add up during the creative process, which could translate into production pressure for designers at financially-pressured firms.