21st century computer algebra literacies
By Murray Bourne, 06 Jan 2008
There was an interesting response to my Twenty Global Problems and Twenty Years to Solve Them post. So interesting, in fact, that I thought it deserved its own post so I moved it here.
MariaD, from NaturalMath, wrote:
Zac,
I come to you for help. I’ve been enjoying your blog, and it looks like you may know something that may help with answering a big question I have.
When industrialization was taking place, universal literacy became important for progress. What “literacy” meant for math, back then, was basic arithmetic. Now it looks like the civilization needs, among other things, mass algebraic literacy to progress and to solve its problems. Governments have been working toward that goal, by their usual means (influencing school curricula). Despite this mass algebra instruction, we don’t see mass algebraic literacy.
I think one of the reasons why we don’t see mass algebraic literacy is that algebra isn’t viewed as something you MESS with. Kids doodle or compose texts for their myspace, but they don’t MAKE (construct, create, build) any algebra entities, ever. Algebra is something professionals have made in the past. Look at the software you reviewed: MS Math 3.0 or Algebrator are SOLVERS. For all the web 2.0 hype, you still see blogs, wikis, podcasts with kids re-representing already created math that they look up somewhere, or at best solving problems professionals pose.
Now for the question. Have you seen good technologies supporting kids in creating their own math entities? Something that would help kids doodle in algebra and compose in calculus?
My Response
Hi, Maria
I come to you for help.
Sure, I’ll do my best. You’ve asked a great question.
I’ve been enjoying your blog...
Great! Thanks for the feedback.
Despite this mass algebra instruction, we don’t see mass algebraic literacy.
As you said, there has been a lot of effort put into improving test scores, but there is little real improvement in mathematical abilities.
I think one of the reasons why we don’t see mass algebraic literacy is that algebra isn’t viewed as something you MESS with.
I agree with you there. Math is seen as something that needs to be done for its own sake. It is something that is imposed from above. A lot of math educators also hold this view, so it is not surprising that many students and the public do also.
The ’normal’ situation is that students need to do the algebra (which usually means plugging a number into some formula), get the right answer, get a pat on the head, then move on in life. It is usually totally devoid of any application, any meaning or any connection to other things that they are learning. And it leaves most students wondering what any of it is about.
There are some wikis that I have come across where students are solving math problems collaboratively. There is a certain amount of "messing with" the algebra, but it is limited. These involve the use of a LaTEX-based image creator for mathematics. The limitation here is that students can’t do a lot with the algebra they have written. That is, the wiki doesn’t let them graph a result, or solve some system of equations, or whatever.
...you still see blogs, wikis, podcasts with kids re-representing already created math that they look up somewhere, or at best solving problems professionals pose.
Yes, this is another issue. There is always this feeling that mathematics has one neat answer that we can check in the back of the book. But ’real’ mathematical thinking and experience is often not like that.
Look at the software you reviewed: MS Math 3.0 or Algebrator are SOLVERS.
Yes, and they perpetuate the "math for math’s sake" mentality. There is so much more that can be done with computers to aid math learning than the ’automated tutor’ approach. Neither of them are Web 2.0, either.
Now for the question. Have you seen good technologies supporting kids in creating their own math entities? Something that would help kids doodle in algebra and compose in calculus?
My software of choice for such a task would have to be Scientific Notebook. It’s not Web 2.0, but it certainly allows the creation of ’user-generated content’.
The user interface for Scientific Notebook is similar to that of a word processor. You can easily enter math (one of the easiest input methods that I have come across) and manipulate it to your heart’s content. It looks like mathematics after you have entered it. It doesn’t look like computer code.
You can graph things (2D and 3D), solve equations, integrate and differentiate things, do matrix operations, etc, etc. You can also enter text, so adding explanations for the mathematics that has been "messed with" is possible.
You can output your work in useful ways, from TEX to image to HTML (which can easily be opened as a Word document).
A lot of the math software out there requires the students to manipulate something that some educator has designed before (including my own math manipulatives). But Scientific Notebook allows the students to start from scratch - and "mess with" the algebra as they go.
Of course, there are many other similar Computer Algebra Systems, but most of them have a steeper learning curve when it comes to math input. Cost is also an issue. The following 2 are free.
Maxima
I am currently playing with the open source computer algebra system Maxima. It appears to be quite feature-rich, but at the end of the day, it has several shortcomings if you want to use it with students (math entry looks more like computer code than math, for example). Output options are limited - you can save what you have done as text, but that is all.
Geogebra
I wrote about Geogebra in GeoGebra math software -a review. It is an interesting possibility for your application, but it is more geometry-based than algebra-based. Depending on how you design documents, it is possible for students to play around with ’what-if’ scenarios. A better possibility is to get the students to create their own documents.
The Activity
I think an important consideration here is the activity that students are being asked to do. Are we aiming to just cover an algebra chapter in some textbook?
Or if the aim is to excite students in mathematics, to help them see the usefulness, to let them develop the skills to apply it, and to help them to think mathematically, then what is essential is to give them open-ended and authentic learner-centred activities where they need to solve at least part of the problem from scratch.
Some ideas:
- Augmented reality - like the kind of thing done by the futurelab people and Harvard
- Authentic data gathering - and forming conclusions, speculating about future trends and the like
- Modeling - of a full range of variables
- Problem-based Learning - which is involved in each of the above and certainly in the Twenty Global Problems
- Building things - nothing like some construction to bring out the need for mathematics. The construction can be physical, or even the creation of computer games.
So Maria, I hope that helps.
Endpiece: I came across a discussion in the Physics Forums on computer algebra systems. One response went:
Learning how to use the computer for numerical calculations is all what the computer is about; I find the CAS [computer algebra system] to be pointless.
I don’t agree with him, but it is food for thought - does the importance of algebra shift a bit when we have powerful numerical tools available?
This goes to the heart of your question, Maria. What is the ’literacy’ that we are aiming for? I liked your question because you are looking for something that students can use to build mathematics, not just do a mathematics problem in a book.
And then collaboration is another aspect of 21st century IT math literacies that we need to think a lot more about...
Update (10 Jan 2007): Another possible solution is WebEQ, that claims that you can...
... create web-based learning environments that help educators engage students in math and science on the web.
I have played with this a bit in the past. Today when I looked at their site, the first WebEQ example I looked at completely froze up Firefox. While this is Web-based, it produces ’dead’ math at the end of the day.
See the 15 Comments below.
9 Jan 2008 at 10:07 am [Comment permalink]
Interesting response to Maria's question, Zac.
So are you saying that students don't need to learn a lot of algebra? Surely not - algebra is the building block of all mathematics!
9 Jan 2008 at 10:22 am [Comment permalink]
Hi Steven and thanks for your comment.
As I said, I don't agree with that forum writer that a computer algebra system is "pointless". And while I agree with you about algebra being the building block of math, I do feel that we get students to churn through a lot of useless algebra. Many times it is better to get the computer to do it for us and then we should spend the time seeing how to solve problems using that algebra.
Case in point: For most students, the topic of matrices involves hours of finding inverse matrices - and very little time is spent in setting up systems of equations from practical problems and gaining an understanding of why we are doing it in the first place. In my treatment of the topic (see Matrices and Determinants), I only deal with inverses of 2x2 matrices and leave anything higher to the computer.
But back to the topic of Maria's question. For students to be able to "mess with" algebra using some sort of computer tool, they will have to have a reasonable grounding in numerical and algebraic concepts.
You can't extract meaning from higher mathematics if you don't have some basics first...
9 Jan 2008 at 7:54 pm [Comment permalink]
I’m with you, Zac.
There is something very wrong with a curriculum that requires rote learning of formulas but does not include applications of those formulas.
But I also agree that "You can’t extract meaning from higher mathematics if you don’t have some basics first…"
So we have a dilemma. How much algebra should we teach, how much should we let the students figure out for themselves, and how much should we leave for the computer to do?
10 Jan 2008 at 8:01 am [Comment permalink]
All through school, I was good at maths but struggled with the fact that it all seemed so up in the air, with no application.
I once pressured my year 12 maths teacher: "Why are we learning calculus?". She told me that you need it to analyse rate of change systems. But then we did a question like this:
A water tank is filling at a rate of R=2t^2 + 4t -5 etc...
My issue was that you never buy a water tank with an equation written on the side! This problem was completely unrelated to any real world problem. They simply created a "just-so" scenario that required application of calculus to solve.
I always thought maths should be there to help you solve real world problems. But instead, my maths teacher took the real world, and adapted it to make it suitable for teaching the maths syllabus.
After getting the correct answer, I still had no idea how to estimate the amount of water in a real tank. I always suspected that with a REAL tank, you would need some degree of numerical-type approach.
My maths teacher responded by telling me to be quiet. No wonder so many people hate maths when they learn it, and never touch it again during the rest of their life.
-Michael
10 Jan 2008 at 9:06 am [Comment permalink]
Thanks for your input, Michael. I share your pain.
Mathematics teachers throughout the world have a problem - too much content to teach and not enough time to get through it. So they resort to "nice" problems that have "nice" solutions (like your unrealistic water tank) that can be solved in 4 or 5 lines on paper. And it is all so false and "up in the air" as you said...
Another issue is that a lot of teachers find themselves teaching mathematics even though they have little background in it. (Mathematics teachers are in short supply across most of the world). They stick to the "nice" problems because they are not confident in doing harder examples with the students. Even trained mathematics teachers often have not had experience doing any mathematical modeling or solving real problems from scratch, so they do not have that kind of thinking. You can't really blame them - there are a lot of wider issues involved. [You may be interested in the article on RAMP, where teachers go into workplaces to see what mathematics is really used. At least they are trying to do something in the right direction.]
So, as intimated above, 21st century computer algebra literacies should include solving real problems by modeling, analyzing, extrapolating and communicating. And this brings us back to Maria's quest for a good tool to achieve this end.
10 Jan 2008 at 12:09 pm [Comment permalink]
I don't know much about the tools that could aid this process. But I do believe they should be introduced to students MUCH earlier.
At primary school, they taught me the basics of arithmetic, then gave me a calculator. Perhaps we should follow the same method: Teach high school students the basics of algebra & calculus, then show them Mathematica or Maple in year 10.
For my Software Design class we used computers every lesson - and that was just for simple, linear algorithms. However all that complex math work was done on paper. In my opinion, math courses should spend some considerable time in the computer lab.
14 Jan 2008 at 8:38 pm [Comment permalink]
I know that it has been very exciting for me to "mess about" with Mathematics using technology. Some of my favorites so far are:
The Geometer's Sketchpad - the interplay between diagrams, measurements and graphs can be especially rich. For some examples see CLIPS (www.oame.on.ca/clips) which also includes Flash learning objects related to Fractions.
Fathom - another Key Curriculum Press product, and its younger cousin, Tinkerplots - that have made me think about effective data display much differently.
Computer Algebra Systems - like TI-nSpire CAS and Yacas (see my blog http://mathfest.blogspot.com for posts about Yacas and online calculators and CASs)
So, for me, a journey that began with Green Globs and Graphing Equations (Sunburst) and Mathematics Exploration Toolkit (MET from IBM) continues to enrich my conception of Mathematics and its connections within itself and to our world.
14 Jan 2008 at 9:00 pm [Comment permalink]
Hi Ross and thanks for your comment.
I am aware of how powerful Geometer's Sketchpad is, but I didn't include it in my recommendation to Maria because it is not an algebra scratchpad.
Now here's a question for you as a fellow "messer about" with computer algebra systems. Do you enjoy them because you are quite strong at algebra already and you know what it is doing for you and what the answer means?
I ask this because my observations of students who are new to CASs make me conclude that without a reasonably good grounding in algebra, the tool is not much good to you. (They tend to sit and look at the CAS without exploring much).
I also ask it because I suspect I like messing around with them myself for that reason, too.
14 Jan 2008 at 10:11 pm [Comment permalink]
One of the neat things about the PRISM-NEO Geometer's Sketchpad sketches [no longer available], especially the Grade 11 sketches, is how they do allow you to mess about with algebra and functions concepts. I know that Key Press has series of books with activities that that use GSP to explore algebra. There are sample sketches that ship with GSP, that have a virtual implementation of algebra tiles. At the PRISM-NEO site, there is a compilation of virtual manipulatives called the Ubersketch (which has its own entry on Wikipedia!) with a wide assortment of materials that you might associate with algebra and arithmetic not geometry. Interestingly, in my own learning, my interaction with Sketchpad has most impacted my notion of transformation.
I came to my interest in CAS after I had a classroom of students to try it out on. I think my interest in CAS relies on a wide range of previous experiences with algebra. Generally, getting students engaged with algebra is difficult, and CAS will be no different. I know that there are some interesting projects using CAS with early and struggling algebra students and that they show some promise.
One of the things that I have found interesting is how difficult teachers find learning a CAS. It has made me reflect on how difficult it must be for students to learn the classroom version of algebra. A reflection that teachers lose as they are so immersed and familiar with it.
I just finished reading a research paper in the JRME, that questioned to what extent students appreciate that when they are dragging a point in Sketchpad and watching its image under a transformation move, that they are sampling from the domain of the transformation (that is the set of all points in the plane) to deduce some sort of general properties. This is akin to the "pretty picture" objection to Dynamic Geometry Software. Some students didn't even think that dragging a point in the plane was effecting a change (the label was the same throughout).
So, as teachers, how do we really find out what our students are thinking and whether they are making any lasting, important conceptual links?
15 Jan 2008 at 9:53 pm [Comment permalink]
Zac,
What a treasure your response is! It really helps to move the thinking along. Huge thanks to everybody who participated. Let me try to develop some threads there. I will respond in several separate pieces to keep threads more visible.
Zac: Math is seen as something that needs to be done for its own sake. It is something that is imposed from above.
I used two dimensions to describe the paradigm shift I’d like to see. The first is job vs. hobby. I’d like to see people doing much more algebra CASUALLY, as a hobby or a game or an occasional pastime. Imagine if people mostly READ as a job, or for a class? When I pose this analogy, listeners usually respond, "But there are great read-for-pleasure books out there - what about math-for-pleasure?" Exactly! Gardner and Zoombinis notwithstanding ^_^ There are materials for more hardcore math hobbies, such as math Olympiads, but not as much for casual activities to do occasionally.
The second way to think about the same problem, or a need for a shift, is the social constructivist analysis of roles. WHO are people doing math? While doing math, what role does the person take upon himself or herself? Again, in the majority of instances we see people doing algebra, they will be in the role of students or relatively involved professionals. Avid hobbyists (math geeks), while I love them to death, are a small minority. Imagine if you had to be a student of literature, or a professional in a literature-related fields, to read! The shift "about roles" in creation of math materials for children that I’d like to see is from "math for students of such-n-such course" to "math for people."
Zac: The user interface for Scientific Notebook is similar to that of a word processor. You can easily enter math (one of the easiest input methods that I have come across) and manipulate it to your heart’s content. It looks like mathematics after you have entered it. It doesn’t look like computer code.
I think "what you see if what you get" and intuitive, drag-and-drop, visual interfaces have to be among principles of development of casual, everyday, mass math products. If the interface is too complicated for a six-year-old, it will scare away a sizable chunk of the population. I am now playing with the Scientific Notebook with an eye on implementing some of its capabilities online. As for Geogebra (thanks for the review!), or in fact the Geometer’s Sketchpad, or KaleidoMania from the same company, I think both can be used to create "visual algebra" applets and to explore algebraic ideas in the context of spatial reasoning.
The analysis of what it is users DO is crucial to figuring out a a way to help algebra enter the general culture more. Zac made a list of some activities that would be appropriate: Augmented reality, Authentic data gathering, Modeling, Problem-based Learning - which is involved in each of the above, Building things."
We have to be careful with the Augmented Reality bit, because it’s a medium more than an activity. Case in point: Timez Attack, a 3d virtual world for multiplication drill and nothing but the drill. To build on your list, I’d like to add a couple of my favorite group (adaptable to web 2.0) math activities:
* Making collections - say, of different representations of the same math object or concept; of function machines built by all participants; of examples from your personal life pertaining to a certain area of math. Having made a collection, you can engage in higher level, meta activities with it: make categories and create ways of sorting, evaluate, and so on.
* Playing "games by form" - creating objects using a particular common style or form for them. For a rowdy example, look at the "Mathematicians do it" joke collection. A wikipedia or a specialized dictionary is a result of a "game by form," namely, short explanations of terms given in a particular style (and a small personal gripe here - I can’t find a definition of "multiplication" I like!). There are many viral internet-games by form, such as lolcats. What about math forms?
* Humanistic mathematics - using math ideas as a basis for the arts. The Humanistic Mathematics Network isn’t very active anymore, but there are people here and there doing it in various disciplines, from dance to visual arts.
16 Jan 2008 at 12:50 am [Comment permalink]
Peter wrote: But I also agree that “You can’t extract meaning from higher mathematics if you don’t have some basics first…”
This is something of a keen interest to me, Peter. In 2005, I went to a CAME conference (Computer Algebra in Mathematics Education - their site is hiding from me atm, so not giving you a link) and started a big argument among researchers of computer algebra systems about this topic. The nature of the argument, and of my interest: if you explore algebraic ideas metaphorically and qualitatively, are you doing algebra? Here are a few particular topic examples of what I mean...
Functions. You can work with functions, inverses, composition of functions, iterations, domain and range, and other such ideas using the popular "function machine" metaphor. I’ve done it with very young kids (3-6 yo) using either very simple and visual numerical (how old will you be in a year, visual doubling and halving), or qualitative (turning baby animals into adult animals) functions that don’t require prerequisites ("basics") really.
Grids. I have done some research in the area called "grid reasoning." This is about using 2d grids to explore two-variable functions. Again, young children can develop pretty sophisticated reasoning about various aspects of grids, such as covariation, using qualitative grids such as facial feature combination (noses and smiles) grids. I have identified, or found in the works of others, about a dozen such particular concepts necessary for advancement of grid reasoning. The amazing thing is, you don’t need much in terms of "basics" to develop these concepts.
Equations. Some examples of big algebraic ideas related to equations are unknowns, equality, and logical equivalence or "if-then" structures (e.g. 2x=6 is equivalent to 10x=30; if z/(x+5)=0, then z=0 and x=/=5). You can explore a lot of these ideas playing hide-and-seek ("how many kids are now hiding?"), or manipulating objects - again, with either qualitative or concrete/visual basis of the activities.
Myself, and a few others, used this "early algebra" approach with other algebraic topics. What initially led me to think in that direction was work with struggling college students. From my experience, it looked like their problems originated, yes, from missing the basics - but not the basics of arithmetic or computational mastery as much. It was rather the basic, general, qualitative, metaphoric understanding of fundamental notions from more advanced math. When I helped the students, they often said, "But the way you explain it, even a little kid would understand!" - well, yes.
Proportional reasoning, to bring yet another example, is a major cornerstone of the algebraic thinking. Yet to require kids to master numerical proportions before moving on to algebraic ideas, in my mind, is quite often a mistake in planning. There are ways to develop proportional reasoning qualitatively (some studied by Piaget umpteen years ago, by the way), such as balances or image resizing, and also ways to develop the basics of algebra and beyond that don’t involve relatively advanced prerequisites and formal work with numerical proportions.
For a good collection of examples, there is the "Calculus for seven-year-olds" website http://www.mathman.biz/.
What I am trying to say, through the examples, is this. There is a need to analyze what we mean by "basics." There are basics in "higher up" math areas that are accessible without many, or any, prerequisites, even though math is not usually done this way. The lack of very general and qualitative understanding of these "advanced basics" is highly problematic, even for adult learners. So, it may be beneficial not to require too much advancement in one area before exploring basics from another.
When I am thinking about this group of topics, I am always reminded of a verse from Frank Herbert’s "Dune":
Here lies a toppled god,
His fall was not a small one.
We did but built his pedestal,
A narrow and tall one.
16 Jan 2008 at 10:56 am [Comment permalink]
Michael wrote, I once pressured my year 12 maths teacher: “Why are we learning calculus?”
I like to keep asking "Why?" questions again and again. There is a nice article about "The method of 5 whys" on Wikipedia: https://en.wikipedia.org/wiki/5_Whys However, it’s important to know that most people consider "Why" questions deeply intimate and/or a challenge, and tend to get emotional or defensive. There has to be a general atmosphere of trust and friendship before you can ask another person, "Why?"
Your question is, ultimately, about the meaning of life. You can also shorten it and ask, more generally, "Why are we learning?" Every person has their own answers, and it may take many whys to dig them up. Just a few days ago, I had the following conversation with a 7yo girl:
- So, why are you doing math?
- Because I want to do well in school.
- Why do you want to do well in school?
- To get good grades.
- Why do you want to get good grades?
- Because my friend P. gets good grades.
- Why do you want your grades to be as good as P’s grades?
- Because I love him!
While everybody in the room smiled at that, I do think personal love is an acceptable motivation for deciding to do something - to keep your loved one company in their endeavor. Zac calls us to be motivated by the greater good of the humanity - the big problems all people and our planet is facing. I know quite a few people who are motivated to learn by smaller and more local, but also noble, desires, such as the desire to create good theater performances, or the desire to be a good medical doctor. But at the end of the day, why you learn depends on why you live, in general.
I don’t know if schoolteachers can always assist students in their search of the meaning of life. It’s hard enough to help even your own children with something like that, let alone hundreds of strangers. It may be, in your case, the answer is that you have no personal reasons to know calculus beyond your government wanting to see more calculus literacy in the general population for the sake of the progress. I know what I would, personally, like to see people understand from calculus, for example: when a politician says, "Our pace of growth is slowing down" I’d like people to know what is meant.
16 Jan 2008 at 1:39 pm [Comment permalink]
Well said, MariaD.
You make a good point about the air of trust that must precede a Why question. When I was in high school I didn't realise how probing such a question can be. As you say, my maths teacher *was* becoming emotional (because I was being a pain).
In the years since I've left school, I've realised there are many good reasons to learn calculus, but I discovered them in my own time, for my own enjoyment. I just wish my formal education had helped me a little in this area, as I would have taken a personal interest many years ago. (And would have put some effor in at school as well)
I like why questions too. It just seems that the legislators in my country and others have lost the original purposes for teaching maths, and few people want to question this.
*Simpsons Quote*
"Mrs Krabapel, why do we have to learn Roman numerals?"
"Because otherwise you won't know the year that certain movies were copyrighted."
16 Jan 2008 at 1:53 pm [Comment permalink]
Hi Maria and thanks for your extensive responses.
Wow - you have moved the thinking on somewhat.
I’d like to respond to this:
"We have to be careful with the Augmented Reality bit, because it’s a medium more than an activity. Case in point: Timez Attack, a 3d virtual world for multiplication drill and nothing but the drill."
TimeZ Attack is a computer game environment that dresses up drill activities - "nothing but the drill", as you said. It has its place (I still believe knowing multiplication tables is important), but there is a lot more potential for learning in game environments than just posing multiplication questions before you can get through (virtual) doors.
The augmented reality that I was talking about (in the Harvard and Futurelab examples) is a different entity altogether.
Students use PDAs and physically walk through an environment that is enhanced with activities and/or problems that they must solve. It is generally GPS-driven, so that the PDA "knows" where the students are and the PDA will present more information (or clues, or questions) that are location-specific.
One example from FutureLab has students moving about the school oval (which is a "savanna" in the activity) and they take on roles (an important aspect) of predators or prey on the savanna. They learn about synergistic ecosystems by being immersed in roles within that augmented reality.
One of the Harvard examples involves students going out into their neighborhood and solving problems in math and science. As their blurb said (no longer available):
I would also like to see a shift to “math for people.” if more people were happy to mess around with math, there would be no sub-prime mortgage crisis...
Thanks for this thought: So, it may be beneficial not to require too much advancement in one area before exploring basics from another. And I like the "calculus for 7 year-olds" idea, because it follows the notion of exploring, which we don’t do enough.
Thanks also for the "5 Whys" reference. Learning is triggered by questions and I think it would be good to do a "5 whys" on why anything is included in a curriculum - and why other things are not.
I enjoyed my days doing curriculum review - we had to trim content but leave essentials there. It was interesting to get responses from my "why leave it in?" challenges to colleagues. A lot of the time, there was no good reason beyond that they liked it.
I would also like people to know what “Our pace of growth is slowing down” meant. But this is only likely if our emphasis is on application and meaning. I think Michael’s lament is not so much "why do we have to do this at all", but rather "give me some meaning for this learning". And that folks, is the crux of the issue.
And on a related note, financial mathematics is an area that I think we should emphasise more in schools. That’s why I wrote the Money Math chapter.
15 May 2008 at 10:09 pm [Comment permalink]
Wow... I just bookmarked my first blog.
Hope I have time to post some comments later.
stutt