Math teacher looking for modeling tool

[Quantumduck]

Hello,
I am a High school algebra teacher looking for a modeling tool for algebra tiles. I have four sources of online "virtual" manipulatives, but none of them do what a set of algebra tiles on an overhead can do.

First there is the National Library of Virtual Manipulatives. http://nlvm.usu.edu/en/nav/category_g_4_t_2.html

The problem with that model is that it has no negative 1. I can use the X and Y to do positive and negative variables, but there is no way to model (x-1)(x+1). That is essential in using Algebra tiles.

Next there is http://www.shodor.org/interactivate/activities/ or Interactivate which has no algebra tile modeling, but huge amounts of other math models.

There is also http://www.x-power.com/Flash/Tools/AlgebraTiles.html or Xpower. They are a pay website, but the color background is black. Ever tried to keep High School students awake in a darkened room so we can see red on black?

Finally there is Dr. Strader's website: http://strader.cehd.tamu.edu/Mathematics/ . He has the Java source code available to manipulate the what he has, but I don't know java to reprogram it to just allow me to choose the equations.

So, I am asking for help. Does anyone know of any modeling tool that would allow a teacher to stand at the smartpanel and actually model algebra tiles and have the kids do the same equation on their desks with the tiles?

Help! And thank you for any suggestions! I am trying to instill a whole new generation of kids loving math!

 

[ice109]

i don't understand the point of this?

 

[D H]

Algebra tiles appear to be a mechanism for making simple algebra more understandable. Out of curiosity Quantumduck, are these used for teaching basic or gifted and talented algebra? (I went to school with the dinosaurs; I think my high school had two overhead projectors total.)

Qduck, you might be addressing the wrong crowd in this forum. We do have a few K-12 teachers here, but most of us are college students, professors, or non-teaching professionals. Have you tried your question on a forum aimed at teachers?

 

[Quantumduck]

Algebra tiles appear to be a mechanism for making simple algebra more understandable. Out of curiosity Quantumduck, are these used for teaching basic or gifted and talented algebra? (I went to school with the dinosaurs; I think my high school had two overhead projectors total.)

Qduck, you might be addressing the wrong crowd in this forum. We do have a few K-12 teachers here, but most of us are college students, professors, or non-teaching professionals. Have you tried your question on a forum aimed at teachers?

Yes I have. Unfortunately, most high schools don't have the technology to display the modeling. They use the overhead projector and colored plastic tiles to do the modeling.

As to the question of who the students are, they are normal high school students. Algebra tiles are a way to differentiate instruction in the teaching of math concepts, specifically "like terms", binomial multiplication, trinomial multiplication and factoring trinomials into binomial terms.

They are highly effective at all levels of algebra education, but the shift from "old school" to technology has not caught up.

If someone has a straightforward resource on learning Java programing, I would be up for that. I have the source code from one person's site, and I could modify it to suit my needs (If I know Java Programming! Big IF).

Thanks for all your suggestions and questions. I appreciate them all.

 

[ice109]

no offense but this seems soooooo dumb to me. are you wanting to do this because your students don't understand what you're trying to teach them or because this is how you prefer to teach them? i fail to see why this would at all be necessary for teaching this stuff.

 

[symbolipoint]

no offense but this seems soooooo dumb to me. are you wanting to do this because your students don't understand what you're trying to teach them or because this is how you prefer to teach them? i fail to see why this would at all be necessary for teaching this stuff.

The use of actual physical items can help certain students understand the symbols and expressions in a realistic practical way. The actual tiles can be manipulated, which is more difficult if done in software form; but if a software program is available, then this brings the potential to show several students at once while they sit at their seats. The actual physical tiles should be easier to manipulate than tiles only in software.

Hey, QuantumDuck, did you yet find any program that allows you to actually rotate the tile pieces?

My own sensibilities are that a good drawing can or should work well, and no computer technology be needed.

 

[D H]

no offense but this seems soooooo dumb to me.

Don't knock it. I can see how techniques like this could be quite beneficial to average students. Symbolic techniques lead to deeper understanding and work fine with gfted and talented students. Basic students (the ones who graduate high school without even knowing how to count money) need help. Boys, in particular, need help. We have an impending crisis because boys are going on to college in ever smaller numbers and dropping out of high school in ever larger numbers. The current school structure does not suit the natural psychology of boys. We tag them far too many boys ADDH, but many of these supposed ADDH boys are just acting naturally. Boys tend to be visually oriented. View these visual techniques as a good thing.

 

[ice109]

The use of actual physical items can help certain students understand the symbols and expressions in a realistic practical way. The actual tiles can be manipulated, which is more difficult if done in software form; but if a software program is available, then this brings the potential to show several students at once while they sit at their seats. The actual physical tiles should be easier to manipulate than tiles only in software.

Hey, QuantumDuck, did you yet find any program that allows you to actually rotate the tile pieces?

My own sensibilities are that a good drawing can or should work well, and no computer technology be needed.

i still don't understand. i looked at the algebra tiles webapp on the first page he linked to and from it i get the impression that whole point is to make the xs and ys appear as objects, which they are though. a monomial is an object, i don't understand why that needs to be belabored with a tile.

 

[Quantumduck]

Hey, QuantumDuck, did you yet find any program that allows you to actually rotate the tile pieces?

My own sensibilities are that a good drawing can or should work well, and no computer technology be needed.

Thanks for the support to you and to DH. The use of modeling in bridging the gap between the concrete thinking of younger ages and the more abstract thinking required to do algebra is essential. Especially when we are dealing with 13 and 14 year old kids (ie. Freshman Algebra in high school.)

I have not found anything yet.

The goal of this is for me to be able to model to a class, while they do the same thing at their desks with actual pieces. Then I give them some problems and they model the problems with their pieces. Then I give them so problems that can not be modeled (numbers are too large, they don't have enough tiles) and they then have to bridge the gap from concrete to abstract.

The bridging actually very difficult for some learners to achieve. Asking them to jump from adding, subtracting, multiplying and dividing numbers to then dealing with doing functions with X and X^2 is not easy. They will add the X and X^2 because they both have X's. Then they will get X^3, because they know something has to happen with the X's.

I have thought of just going with a drawing program straight up. The hard thing about that is then you have to 'draw' 6 different types of shapes / color combinations and do multiples of them all very quickly. Difficult to do in the classroom.

Thank you for all the suggestions (even the challenges!)

 

[Quantumduck]

i still don't understand. i looked at the algebra tiles webapp on the first page he linked to and from it i get the impression that whole point is to make the xs and ys appear as objects, which they are though. a monomial is an object, i don't understand why that needs to be belabored with a tile.

Because you are an adult with a firm grasp of the ability to do abstract thinking. If you go talk to a 13 year old kid (typical age of an 8th grader or freshman in algebra I) you will find they don't have that ability to think abstractly. They are very concrete in their thinking.

Numbers relate to things. Negatives are weird, because they don't. X's are placeholders for any number. But we can still do things with them even though we don't know what it is. How does that happen?

These are abstract thinking patterns that you and I take for granted. A 13 - 14 year old learner needs something to help bridge that concrete / abstract divide. We only need to use them for a few classes at the beginning of adding and subtracting expressions, multiplying binomials, and factoring trinomials. Those few days are the difference between a whole classroom of lost learners, and three lost learners.

 

[ice109]

i'm sorry this boggles my mind. i refuse to believe that the students need this. i really think you're underestimating them.

adolescents graduate to abstraction ~11-12. pre algebra is taught to many students in middle school.

edit

i went to a crappy elementary school and i remember solving a linear system in 3rd grade. and I'm dumb so i solved it guess and check.

 

[D H]

Do a google search on algebra tiles. That's how I found out what these things are.

 

[Quantumduck]

i'm sorry this boggles my mind. i refuse to believe that the students need this. i really think you're underestimating them.

adolescents graduate to abstraction ~11-12. pre algebra is taught to many students in middle school.

I beg to differ, but obviously I should defer to your significant grasp of educational theory. I teach 180 kids (six classes) of algebra, so could you come to my school and give inservices on how to educate the learners better? I obviously could use your significant knowledge of high school educational practices.

 

[D H]

i'm sorry this boggles my mind. i refuse to believe that the students need this. i really think you're underestimating them

Really. Have you noticed how math scores are plummeting in the US compared to other countries? Have you noticed all the nasty statistics on a growing divide between educated and less-educated? Between boys and girls? Have you noticed the growing tendency toward anti-intellectualism and pseudoscience in this country?

Qduck, go for it.

Does anyone have suggestions on how to learn Java? An alternative is Ruby, which is very easy to learn. I don't know if Ruby has any graphical capabilities.

 

[Quantumduck]

Really. Have you noticed how math scores are plummeting in the US compared to other countries? Have you noticed all the nasty statistics on a growing divide between educated and less-educated? Between boys and girls? Have you noticed the growing tendency toward anti-intellectualism and pseudoscience in this country?

Qduck, go for it.

Does anyone have suggestions on how to learn Java? An alternative is Ruby, which is very easy to learn. I don't know if Ruby has any graphical capabilities.

Thanks DH. I will go to the Barnes and Nobles this weekend and get a Java book. It seems that if I need a tool, I am going to have to build it myself. I have no problem doing that!

 

[ice109]

Really. Have you noticed how math scores are plummeting in the US compared to other countries? Have you noticed all the nasty statistics on a growing divide between educated and less-educated? Between boys and girls? Have you noticed the growing tendency toward anti-intellectualism and pseudoscience in this country?

Qduck, go for it.

Does anyone have suggestions on how to learn Java? An alternative is Ruby, which is very easy to learn. I don't know if Ruby has any graphical capabilities.

let's take a look at how they teach algebra over seas then. i don't think that people overseas are all have average iqs higher than here either. i think these crutches are exactly the reason scores here are plummeting. I'm not an elitest either, I'm all about educating everyone but

if they're adding monomial and binomials it's because they've been poorly defined for them.

and considering negative numbers just what will the tile $-1$ look like? you'll bring it close to $x$ and not only will it disappear but $x$ will shrink as well? how concrete is that?

and what about complex roots to the quadratic formula?

 

[D H]

ice109, you are not contributing to this thread. The polite thing to do would be to bow out.

Qduck, if you need to learn a new language I recommend Ruby over Java. Ruby is a very friendly and easy to grasp language. Did I mention that its free?

A fifteen minute starter on Ruby: http://tryruby.hobix.com/

I don't see any algebra tiles in Ruby, but this http://puzzlemaps.rubyforge.org/ is educational software in Ruby that incorporates graphics.

 

[Quantumduck]

let's take a look at how they teach algebra over seas then. i don't think that people overseas are all have average iqs higher than here either. i think these crutches are exactly the reason scores here are plummeting. I'm not an elitest either, I'm all about educating everyone but

if they're adding monomial and binomials it's because they've been poorly defined for them.

and considering negative numbers just what will the tile $-1$ look like? you'll bring it close to $x$ and not only will it disappear but $x$ will shrink as well? how concrete is that?

and what about complex roots to the quadratic formula?

Thank you very much for your opinions on teaching in the US.

You make some blanket assumptions, such as "because they've been poorly defined for them." You certainly must have heard the one about "when you assume things". Take out the You when you say that one.

In other countries, they use things like Algebra tiles to help learners understand the concepts. Gee, I guess that is why we started using them too. IT WORKS.

As to how the negative and positives interact.

A positive X is green. A negative X is the flip side, and it is red. Anytime you have two opposite colors (units are generally tan, red, X^2 are blue, red, and X's are green red) you get a zero pair. Why? Well -X +X = 0. They don't shrink, they add to zero because that is they are additive inverses of each other, and when you add the inverses you get the additive identity.

See, we use those big words in class, and now they are not just concepts to grab out of thin air and hope it sticks in their brain, I can have something solid and tangible for such definitions as identity, inverse, commutative, associative and distributive properties.

Yes, they didn't have this 'new fangled stuff' when we were in math class. But it works. It works in a matter of hours instead of days and days of memorizing facts and boring drill and kill. It works because it helps them understand the concepts on tests. It works because they are able to understand the words with something in addition to me telling them.

I am sorry you can't understand that. I offered you the chance to come to my district and share your expertise on secondary teaching methods with me and my department. Will you accept?

 

[Quantumduck]

ice109, you are not contributing to this thread. The polite thing to do would be to bow out.

Qduck, if you need to learn a new language I recommend Ruby over Java. Ruby is a very friendly and easy to grasp language. Did I mention that its free?

A fifteen minute starter on Ruby: http://tryruby.hobix.com/

I don't see any algebra tiles in Ruby, but this http://puzzlemaps.rubyforge.org/ is educational software in Ruby that incorporates graphics.

Thanks DH, I will check it out tomorrow. I appreciate your help.

 

[ice109]

I am sorry you can't understand that. I offered you the chance to come to my district and share your expertise on secondary teaching methods with me and my department. Will you accept?

of course i accept but it's not really feasible is it.

 

[symbolipoint]

ice109 serves like the challenger to the use of algebra tiles; if ice109 took his algebra instruction without any significant use of visual or graphical methods to help explain, and if he understood without those methods, then the use of mostly symbolism worked well and ice109 would not understand the advantage of graphical representations to explain. At least a few students in a class will have some difficulty learning to deal with the symbolism, and for these students, a graphical method could help. Who of us remembers using algebra tiles? Anyone? Sotware for this? More likely just pictures, if anything.

One day several months ago, there was surprisingly in a Algebra 2 textbook, a picture showing the Completion of the Square for quadratic expressions. This made the idea very very clear. I had the process well understood from studying, but I never had seen a picture to show a visual-graphical interpretation. Everything made sense now more easily. Still, this was a picture; not a piece of software.

 

[Chris Hillman]

Algebra tiles: looks like a good idea to me!

Hi, Quantumduck,

Welcome to PF! And don't pay ice109 any mind. If you go to the top of this page and click on "user CP" and then "Buddy/Ignore lists" you can add ice109 to your "Ignore List", which functions somewhat like a "kill file" in UseNet newsgroups. Bad attitude is fairly uncommon at PF, BTW.

I couldn't surf the first website you cited without changing my browser settings, which I am generally loathe to do, so I still don't understand what algebra tiles are. (EDIT: I got it now; see below.)

(My first guess was pretty funny; I thought it might have something to do with certain root lattices--- which are associated with certain Lie algebras--- which arise in the theory of almost periodic tilings, e.g. Penrose tilings; for example http://www.arxiv.org/abs/0709.1341)

So, just be sure: did you try the mailing lists mentioned at mathforums? mathforum.org/teachers/high/ E.g. math.teaching.technology? http://mathforum.org/kb/forum.jspa?forumID=183 Oops, you did, never mind!

University math faculty have to deal with the product of math high school classes (and know how difficult teaching math to a large and diverse group is), and some of them have figured out that's worth their while to liase with high school teachers to try to find some cure for the problems we are all familar with, rather than simply pointing fingers. I would imagine that mathforum.org might be helpful if you are not already a member of one of these informal networks. When I was a graduate student at UW, we used to have Brown Bag lunches to discuss teaching issues, including at high school and even elementary level. One faculty member, Neil Koblitz, who happens to be a very well known researcher on elliptic curves, has "guest-lectured" to kindergartners all over the world!

Anyway, keep up the good fight, I know very well that this kind of algebraic manipulation is a serious stumbling block for many students, and if they can't overcome it, all the doors which college calc would open for them are more or less welded shut, which is a great pity.

EDIT: OK, I found this http://regentsprep.org/REgents/math/teachres/ttiles.htm which suggests making algebra tiles from cardstock; I vaguely understand that you want something which works with something else you do, though, so that you need a computer simulation (?) of manipulating these tiles. Hmm... for example http://regentsprep.org/REgents/math/polymult/Smul_bin.htm contrasts four methods of multiplying two polynomials each having two terms:

1. FOIL (yuk)

2. Aligned columns method (evidently I've been using this without knowing it had a name!)

3. algebra tiles (heh! ingenious! I like it!)

4. grid (yuk)

Yeah, they're all in some sense mnemonics, but the goal is to get the students to internalize this, and I can see that (3) would be most popular. Hmm... and it can be generalized to more elaborate products--- that's another advantage.

In fact this is very much like the very first proof I did (a long long time ago): to compute the sum $1 + 2 + \dots n$, arrange n unit squares in a "right triangular" array: a column with n squares, next to a column with n-1, and so on. Then draw the obvious diagonal. The total area is the area of a big isoceles right triangle, $\frac{n^2} {2}$, plus the area of n small ones, $n \cdot \frac{1}{2}$, which (ahem!) adds to $\frac{n^2+n}{2} = \frac{n \, (n+1)}{2}$. Which is easily generalized to the problem of computing $1^m + 2^m + \dots n^m$, by using (m+1)-dimensional cubes.

Gauss's proof (age 10) is cleverer:

$$\begin{array}{ccccc} 1 & 2 & \dots & n-1 & n \\ n & n-1 & \dots & 2 & 1 \\ \hline n+1 & n+1 & \dots & n+1 & n+1 \end{array}$$

which gives $n \, (n+1)$ divided by two (since we double counted). But his proof doesn't generalize to higher powers. I'm just saying

So now I'm curious--- do you know who invented these algebra tiles? Can you name some countries where they are in common use in math classrooms?

Sweet! http://regentsprep.org/Regents/math/factor/facttiles.htm

Hmm... I think I prefer vertical alignment for addition and subtraction, though http://regentsprep.org/REgents/math/polyadd/Tadd_til.htm

I guess the idea is to try to get the students to get some intuitive grasp of what they are doing using the tiles, before trying to wean them into a more "formal" or "algebraic" symbolic method? I.e. lining up powers of x or whatever? (This feels like trying to figure out exactly how one rides a bicycle, while one is riding a bicycle, long after one has in fact learned to ride a bicycle!)

 

[robphy]

Just to add a programming suggestion...
try http://vpython.org which uses Python, which (in my experience) has a very gentle learning curve... and you'll be quickly be writing programs with real-time 3D graphics.

Here's an entry I wrote for the PF blog:
https://www.physicsforums.com/blog/2005/10/19/vpython-3d-programming-for-ordinary-mortals/

 

[D H]

Python is another good suggestion.

robphy, you didn't mention that python is free. Given the huge amounts of financial support (typed with tongue firmly planted in cheek) teachers receive from their administrators, free is a big deal.

 

[D H]

Quantumduck:

This site wasn't on the list in your original post: http://seeingmath.concord.org/Interactive_docs/FA_Warmup.htm

The more I think about it, the more I like the idea of visual techniques for teaching math. We are a very visually oriented species. It is our primary sense, and massive portions of our brains are dedicated to interpreting imagery.

 

[Chris Hillman]

Python is popular in the linux world. By the way, there are some eduware live CD demos out there somewhere, some of which might be of interest to high school students.

Speaking of highly visual subjects which are (if I am not mistaken) taught at high school level in France and some Asian countries, I'd really really like to see linear algebra and solid/spherical/hyperbolig trig done in American high schools.

 

[robphy]

The more I think about it, the more I like the idea of visual techniques for teaching math. We are a very visually oriented species. It is our primary sense, and massive portions of our brains are dedicated to interpreting imagery.

I like to use visual techniques to teach physics, as well.
In many examples, I find that a geometrical argument is a little more tangible, compared to a purely algebraic one... for example, teaching relativity.

 

[Chris Hillman]

The more I think about it, the more I like the idea of visual techniques for teaching math. We are a very visually oriented species. It is our primary sense, and massive portions of our brains are dedicated to interpreting imagery.

Sometimes a table of mathematicians* discuss styles of mathematics, and someone often recalls Lagrange's boast that he had no need of figures in writing his Mecanique Celeste. A poll then reveals that almost everyone at the table admits to thinking visually I know I do.

Rob, have you seen Ludvigsen, General Relativity? Features some nice stuff on connecting vectors and Lie dragging. Short and sweet, good choice of material given the length of the book, e.g. features Raychaudhuri equation (both for timelike and null congruences).

*I think I just discovered the collective noun for "mathematicans"

 

[Quantumduck]

I want to thank all of you for your great suggestions.

My original degree is in Physics, so the idea of using physical - hands on learning is very intuitive for me. It is kind of like doing a lab in math class.

The geometry of what we are doing is also interesting. The fact that every time you square something (x+4)^2 is actually a square, starts to mean something to the kids.

"Oh, that is why you call it 2 squared! That makes sense now!" is a statement I hear when working with these tiles.

And we only work with them for a couple of days in each chapter. It builds the physical, visual knowledge so that when we have problems that can not be done, they have the idea and steps and skills to do all problems.

Again, thank you all! I will be checking out Python and Java and Ruby today. (although it seems like Ruby and Python are at the top and I will do those 2 first.)