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# Week 2: Modeling software

Go on a scavenger hunt to find three pieces of software that uses modeling in (1) probability and statistics (2) calculus (3) number sense, number patterns, the number system. You can use modeling software or ready-made individual models other people created. Answer the following three questions.

Question 1

Do you think designing and creating models you found would be a good learning task for the students in a math class? Or would you rather students use ready-made models? Why?

Question 2

Multiple representations is a term that means different ways of describing the same object. For example, you can represent the same thing as the word "two" or the line with the slope of two or the digit 2 or an iconic image of two-ness, such as two eyes.

What representations (graphs, tables, equations, etc.) are used in models you found?

Question 3

Some objects model math ideas. For example, the abacus models the place value system. In other cases, math equations or graphs model life. For example, the equation x=4.9t2 models a free-falling object, in Earth gravity, with respect to time (in the metric system). What about examples you found? Are they about math models of some life phenomena, or about modeling math ideas?

•

Week 2 modeling:
for Probability & Statistics:

for calculus:

for numbers, etc:

I think kids will enjoy using websites from the UK or other countries and realizing math is very universal.

Question 1 Response:

I truly think you have to do both. The models/activities already out there are important for teaching & instilling those lessons. If it’s the first time students are learning a math concept or for review, its important to use ready made models. I think also, a combination of computer activities individually as well as group activities with & without technology.

I love when students work on teams to come up with their own activities or models/examples of any math concept. This is what really settles in the math from being numbers, equations, etc into real life. I think math really comes alive when the kids are thinking of their own scenarios (sometimes with a little nudging…but that’s ok !!)

Question 2 Response:

*So the prob & stat website link has many different math skills wrapped up in one models from rounding to probability & statistics & estimating. So , this model is using a word problem. I think any good word problem is really wonderful when you are helping students brush up on more than one skill. The model also talks about something everyone can relate to…pets… & how some people have cats and some don’t. I get so frustrated reading some of these test questions that an urban 7th grader is suppose to relate to (who’s writing those doarn benchmark tests anyway?) .. Any good lesson has to aim to have relatable models to the audience.

*The calculus model I’m not in love with, because its not very jazzy looking…but it does start small & build & gets to the eventual real life scenarios models of how some things are functions (or rely on) other things…like weather, to temperature & time of day, etc

*In the bbc. Link for number lines, etc, this shows the places tens, hundreds, thousands places of larger numbers which I think is a tough concept for kids that I’ve seen. I guess I’d call this a picturesque moving number line model. Seeing it as sort of moving & breathing in this way truly models WHAT those numbers are & WHY they are written the way they are written…which I LOVE!

Question 3:

I’d say that the probability example is definitely a life phenomenon with math model mixed in. The whole scenario is about the reality of how many people have cats & with the conversation comes math on many levels.

The calculus model begins as a math model, but the lesson quickly takes shape as a life phenomenon as the author speaks of functions in things we know to be true like weather.

The number line activity is a math model for the most part. I really hate to pigeon whole any math lessons/equations/activities, etc as “just math” though because on every level math is a life phenomenon as long as you take it there. (Feeling very philosophical tonight : )

ok-- & I have to ask. ..when doing a link...how do you get it to say " here's one" (in blue for someone to click on) instead of having the full website link there? Thanks much! - Lisa

• "Picturesque moving model" - this is a math term I can get behind! Do you write poetry?

This "Scale of the universe" comes to mind. Do show the kids :-) http://htwins.net/scale2/

Animated models do what static models can't. In particular, zoom in the example above shows powers like no other representation I know...

To turn some words into a link, have the web address (url) copied and ready. Then highlight the words and click the "world in chains" icon (a bit unfortunate - based on the literal interpretation of "links"), like so:

You can also paste pictures from the web into your comments, if you feel like it.

• Calculus
http://www.maplesoft.com/support/training/videos/maple12/Limits.aspx
http://www.maplesoft.com/solutions/education/solutions/college.aspx -> A straight forward approach to simulating math problems of the compuer. They also have a tutor assistance which will solve the problem step by step with the scholar. We used the program in our math lab at my university.

Number Sense, Number Patterns, The Number System
http://www.xpmath.com/forums/arcade.php?do=play&gameid=59 -> A game I have used to teach engaging lessons about the coordinate grid and points. This website has a lot of game adaptations that teach math.

Probability and Statistics
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5026&t=5078&id=17177 -> A calculator based simulation of rolling a dice, plotting the results (as sum of the dice), and compare calculations.

Question 1
I believe that creating and designing models would garner efficient results in a math class. Having the scholars creating their own models allows for them to establish their own ideas and theories concerning certain mathematical concepts. The world of mathematics was discovered by doing and modeling scenarios, objects, and etc... I would save the ready-made models for enrichment, re-teaching, or intervention purposes. I also believe that by having them create models gives the scholars a sense of ownership over their work and some input on the concept being taught.

Question 2
In the calculus link, the equations are entered using text or from the symbols pallette in the program. The equations are calculated once all the variables are entered. You even use a limit tutor in the program to assist you.

In the number sense link, in the "Simpson Homer's Donuts", points of the coordiante plane are represented by donuts.

In the prob. & stat. link, a dot plot is used to represent number of times a certain sum appeared from rolling two dice over a period of time.

Question 3
I found that many of examples found on the Texas Instruments website modeled life phenomena, while XP math had a mixture of life phenomena, games, and math ideas. Maple seemed to focus on math ideas but could easily be used to model life phenomena as well. I felt like the models that worked around life phenomena were more enjoyable becauase they dealth with tangible examples that scholars could either relate to or learn from. This also provided an opportunity for scholars to create models of life phenomena that they could relate to if resources didn't exist on other sites.

• Interesting you should mention the input on the concept being taught. How much input should students have over their education? There are extreme opinions. Unschoolers maintain it should be everything, 100%. Some stricter ed systems have no place for student input at all, believing in 100% guidance. Technology gives students some tools for input and freedom... but how much is too much?

Concrete objects or pictures (like donuts!) help students relate to abstract ideas (like points). In many cases, having students themselves represent something works great, too. For example, you can have a giant grid made in chalk, and students can hop around coordinates.

• Sorry I thought I had posted this when I posted my GeoGebra activity! I am so glad I checked!

The software I found for probability and statistics is at www.math.uah.edu/stat/ . This site has many interactive outlets for students to play with. I am a visual learner and really liked the Venn Diagram Applet. It allows the user to see the subsets of larger sets and the unions and intersections. These got me so confused when I first learned it and this would have been quite helpful.

For calculus I found, www.maplesoft.com/products/maple/students/ . This allows students to put in equations and easily differentiate and integrate. The can also find limits and graph equations. This software also has an app for ipads.

I found this http://www.funbrain.com/cashreg/index.html for number sense. This game has students working with money. They start with a dollar amount and then buy something. They then have to figure out what their change would be.

Question 1:

Being a new teach I feel like I would have to spend a lot of time planning an activity to allow students to create models like the ones I found. It might be interesting to have them design or think up a model without actually having to make it and make it functional. I think the creating of one would be more of a high school activity, for an advanced student that is interested in software and making it. I think it may be fun as a senior project that spanned the year. It would look good for college applications if they made one that could actually be used.

Question 2:

Results on the prob/stat software are displayed in actual numbers as well as graphs and charts. The calculus software shows equations and graphs. In the number sense game, the money is displayed on the screen and the students have to say how many of each coin they get. The money is not represented in any other way.

Question 3:

In the prob/stat software I found they used Venn diagrams, actual coin tosses and decks of cards. The students can see these real life situations. I also like the die rolling, because having several students actually rolling dice in the room can get load and the dice can end up everywhere! There is also a section on games of chance, where they show poker, and roulette, which at their age they may have seen. (At least know of poker).

Mainly math concepts are shown in the calculus software. On the maple soft site I did find other links to real world activities like dropping a ball on a net or throwing something with a catapult. With respect to the catapult, it shows trajectory and arc.www.maplesoft.com/products/maplesim/modelgallery It is worth checking out.

In the change making game, the students are working with a real life situation. They will be spending money for the rest of their lives and it is good to know how much change they will be getting back. This is a great real example in my opinion.

• It is wise to think of time tasks take, however interesting they are. Your idea of "just design" (without implementation) would work wonders. Sometimes, there are design competitions by companies such as Google and Mozilla. Students can submit their designs, and the company implements some of them. This can work well with your idea.

Another time-saver is to implement a scaled-down version of your model, or some aspect of it. With tools like GeoGebra or Scratch, very simple models can take hours rather than months to make.

I would add "decimal numbers for the money" to the list of representations, specifically - even though you already have "numbers" listed there. As you said, it's an important topic in life. And many kids learn money before they (formally) learn about the decimal points. For these kids, dime=.1 and penny=.01 makes a lot of sense - that's how they learn their decimals. So in this case, real objects (money) can model math ideas (decimals) - or vice versa!

• Scavenger Hunt

Probability and Statistics:

http://www.ds.unifi.it/VL/VL_EN/index.html - This website can be used as a web           quest since it includes three chapters of text and exercises that guide students through the basic understanding and underlying principles of probability. The website also uses applets to allow students to run experiments or to generate data quickly and easily in order to see how probability works with their own data.

http://www3.stats.govt.nz/games/Spinners-advanced-builder-2/index.html - This is a website where students can build their own spinners or build a spinner based on specific conditions under the “Challenge” tab. The site provides a tutorial, a user-friendly platform to build the spinner, and generates random results with the spinner the students built.

Calculus:

http://calculusapplets.com/limitsatinfinity.html - This calculus applet allows students to move the graph of a hyperbola to see what value it approaches as x approaches infinity. Students move the x slider to make x bigger and see what value f(x) approaches as x increases. There is a second example is set up so the students explore negative infinity as well. The third example is a linear function and the fourth example is a sine curve. The students will then see that the limits of the third and fourth graphs do not exist because as they slide the x value bigger and bigger, the value of f(x) continues to grow or oscillate up and down.

Number sense:

http://www.funbrain.com/linejump/ - My students LOVE FunBrain! This particular activity gives students practice with adding negative and positive numbers by “jumping” the number line. My students change all subtraction problems to addition by adding the opposite. Then we move up the number line if we are adding a positive and move down the number line if we are adding a negative. This is a great way to emphasize sign rules and remind students why the answer makes sense.

Question 1

I think that creating these models would be a very time-consuming task for my middle school students who are not very tech-savvy. It also may become a project where the student spends hours and hours just trying to make the technology work and not focusing on the goal of the project, which is enriching their math skills. However, if a student has an interest and talent with web design or programming, I would definitely give that student an opportunity to explore and create these type of online models. I have created technology options for chapter projects so students can complete the project within their own comfort and skill level. For example, I had a student in my Pre-Algebra course two years ago that created a computer game that incorporated fraction operations for my Fraction Game project. I had a student this year create a website for her advertising piece of the Create-a-Business project. I had two students this year use Sketch-Up to create scale models of the solar system using scientific notation. How cool is that?! I have learned so much from my students and they seem to really enjoy the projects because they are already comfortable with the technology they are using. So to answer the question, I think that these projects are not for everyone, but for those who have the interest and skill level, I think students can have a great experience creating these online models for certain math concepts.

Question 2

The sites that I linked above that include creating and running probability experiments show the data in multiple representations. It shows the results, displays it in a chart, tally results, shows percentages, and even shows a histogram of the results! The calculus site shows the hyperbola as a graph, equation, and a list of ordered pairs. This is very helpful to see the x values in the ordered pairs increase and watch the graph as a visual to see what value that f(x) is approaching. The FunBrain activity shows the addition or subtraction problem and then students represent that problem on the number line. The number line jumping is another way to represent adding a negative or positive number.

Question 3

The probability models that I found provide mathematical representations of real-life data. These models represent a math phenomenon since they are representing real-life data from student-generated experiments. The applet on infinite limits and the number line jumper are models that represent math ideas. The applet for infinite limits will allow students to play with the graph to SEE the limit. The number line jumper is a visual representation of addition and subtraction.

• "I had a student in my Pre-Algebra course two years ago that created a computer game that incorporated fraction operations for my Fraction Game project. I had a student this year create a website for her advertising piece of the Create-a-Business project. I had two students this year use Sketch-Up to create scale models of the solar system using scientific notation. How cool is that?!"

That is VERY cool! One way to have more students involved is to let them pair up, with more tech-savvy people having others as "apprentices" or "interns" who can help. This way, the love of tech may spread around some. As you said, none of that is for everyone, so it's nice to prepare some choices, as you do.

I like the number jumper as well. Sometimes we play with giant number lines - stairs or something drawn on the sidewalk or made with masking tape on the floor. This helps kids as young as three to five years old understand negative numbers and subtraction - both pretty abstract concepts.

• I LOVE the idea of pairing up the tech-savvy students with other students to help them learn! Thank you!

•

The modeling software that I found was:

Interactivate (http://www.shodor.org/interactivate/activities/Coin/), a model for probability and statistics (this activity used a coin toss, but activities that used spinners and dice are also available)

PhET at http://phet.colorado.edu/is a site that "provides fun, interactive, research-based simulations of physical phenomena for free" - lots of material available -LOVE IT!  For my calculus modeling software, I chose their http://phet.colorado.edu/en/simulation/calculus-grapher, as it starts with the basics (students draw a function and see a graph of its derivative and integral.

The MegaPenny project by Kokogiak Media (http://www.kokogiak.com/megapenny/) helps students visualize very large numbers by using pennies as models.

Question 1

I think that designing/creating models provides a great example of applied learning, as students need to know how the mathematics work before they program a computer to execute those steps.  However, since the task would blend programming and mathematics, I'm not sure if a math-only curriculum would have time available for it (in this age of teaching-to-the-standardized-test), or if all students in a given class would have the programming ability.  So, as a middle-school teacher, I would have to say that I'd prefer to have my students use existing software (that allows for user input), rather than create/design their own modeling software.

Question 2

The Coin Toss software allows students to display the results of their coin tosses using Lists, Tables, or Ratios.

PhET's Calculus Modeling software provides graphical representations of student manipulations, although the general formulas for the parent function, derivative function, and integral function are shown.

MegaPenny software provides results using graphing and tables.

Question 3

Coin toss models math ideas and real-life phenomena.  Math ideas involving probability of two possible outcomes are demonstrated.  People often use coin tosses to randomly choose one outcome out of two available (I used it as a teenager to see who would get first-ups in stickball; the NFL uses it to see who will kick or receive).

The Calculus Modeling software that I chose models mathematical ideas; it does not discuss the real-life applications of related rates or figuring out areas/volumes.

MegaPenny models both mathematical and real-life outcomes.  Everyone is familiar with the penny; its usefulness and perceived value varies with one's age and life experiences.  I've seen a lot of people at the bank hauling in heavy containers of pennies, only to be disappointed with the actual value.  Maybe if they could better estimate the volume of a particular amount of pennies (as the software shows), they might be a bit more realistic.

• Is programming math? Is it as necessary for math as, say, the ability to read and other literacies? These are big huge debates that have been raging from the seventies. Glad you are thinking about these issues.

I would add pictures and objects as representations of math ideas used in MegaPenny projects. The site, of course, only uses pictures - but they are of familiar objects (pennies, buildings) that kids can use in physical environments. Lovely project, isn't it? Large numbers are VERY hard to visualize, because humans are equipped with inner logarithmic scales. For us, a sound that is ten times louder sounds like "the next step" in loudness. Likewise, ten millions is just "the next number" after a million - that's how our brain hacks this idea, without additional training. This explains a thing or two about budget problems.

Good point about people at the banks! In fact, estimating the value of handfuls or jars of coins is a good classroom task, as well.

•

For probability and statistics, I found this model, which illustrates the basic concept of expressing probability as a fraction with the number of favorable outcomes over the total number of outcomes. I liked that it had several options; you can increase the number of colors to increase the difficulty, and you can also choose the option of creating a given probability.

For calculus, I found this model, which shows the graph of a function, the graph of its derivative, and the tangent line at a point on the original function. What I liked about this model is that you can drag the x coordinate and the tangent line moves, which lets you observe the relationship between the function behavior, the slope of the function, and the graph of the derivative as x changes, which is a very important concept in calculus.

For the number system, I found this model, which illustrates the idea of place value. There are several levels: whole numbers with 3 or 4 digits, or numbers with 2 or 3 decimal places. I like that it shows the running total to give instant feedback on the effect of adding the different values, and I also like that it gives the option of hiding the running total to increase the difficulty.

Question 1: I think there is value in both using pre-made models and designing new ones. Using ready made models does put limitations on the flexibility, which could be a drawback depending on what specific ideas I wanted to teach, but the models I found apply to broad concepts that would fit in well while studying a larger unit. Both the probability and number system models I found give instant feedback in a no-risk environment, which is something I think can be very important, especially when students have math anxiety or are struggling with a concept. A student could sit down at the computer and practice without the fear of giving the wrong answer in front of classmates or getting a poor score on an assignment. The models I found use the lower order thinking skills of Bloom’s Taxonomy, so they’d be a good choice for when students are seeing a new concept for the very first time, to gain familiarity and confident with the material before moving onto activities using higher order thinking skills.

As for designing models, that would fall into the higher order thinking skills of Bloom’s Taxonomy, so I think it would be a great task for students who have mastered the material and need something challenging to work on, or for after the whole class has had a chance to get comfortable with the material. Depending on the type of model and software, it could be an extremely challenging task, and I think that it would be a good addition to project based learning or interdisciplinary activities.

Question 2: The calculus model I found uses the equation of the derivative, the graph of the derivative, and the slope of the tangent line of the original function all to represent the same thing. The probability model uses a fraction such as ½ and a picture of 4 purple fireflies out of 8 total fireflies to show the same concept. The number system model shows the value of 425 both as a single number and as 4 hundreds, 2 tens, and 5 ones.

Question 3: The models I found all model math ideas. The calculus model relates a function to its derivative, and shows it both as a slope of the function and as a function itself. The probability model uses different colored fireflies to represent the concept of probability as parts of a whole. The number system model illustrates the concept of place value and shows what happens when you add hundreds, tens, ones, etc.

• "What I liked about this model is that you can drag the x coordinate and the tangent line moves" - this demonstrates something you can easily enough do with computer technology, but would have hard time doing without. Best computer models help students notice patterns and connections by making a lot of objects (in this case, a lot of tangent lines) available at once. You can observe the CHANGES in tangent lines as if they were living things.

Bloom's taxonomy is an interesting tool to use for looking at models. You are right, most models you find out there will be tied to lower levels. Some educators like to use high-order tasks to introduce topics, though - while others believe you should start with Understanding and Remembering and work your way up. This is a controversy among educators, with people arguing this and that way.

• That's an interesting point about what kind of tasks should be used to introduce new topics. I am a big believer in the idea that true mathematical fluency comes when students have the opportunity to make their own sense out of the material, and that is a journey that's going to be different for everyone depending on life experiences, background knowledge, learning styles, etc. I definitely think that the high-order tasks give students the tools and opportunities to get to that level of learning, however I think that the low-order tasks can be beneficial when introducing new material in certain circumstances. I do a lot of tutoring, and most of the students I work with have a history of struggling in math class, math anxiety, or both. Something I have found that often works is to have them practice a new skill on a task that is very straightforward or even a little below their level. Obviously this is not how they are going to achieve mathematical fluency, but I've had students who are convinced they are horrible at math become excited and motivated after getting a string of correct answers. I believe that all students are capable of high levels of learning, and if a low-order task can give anxious students a confidence boost that they can carry into high-order tasks, then I definitely think it's worth taking the time to do that.

Of course, students learn in such different ways. While some students might become anxious when they encounter a challenging task, other students might enjoy jumping right into that. I don't have the experience as an educator to give an informed opinion about which way is "better" but modeling software can be used across the spectrum of thinking skills. Using many pre-made models would be tied to low-order skills, writing step-by-step instructions for how to program a model for a given specific task would fall in the mid-range, and creating a model for a more open-ended problem would be at the high end. It seems like there would be a lot of ways to incorporate modeling software while differentiating instruction to meet the diverse needs of all students.