Hope I have time to post some comments later.

stutt

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 says:

As the students move around a physical location, such as their school playground or sports fields, a map on their handheld displays digital objects and virtual people who exist in an augmented reality world superimposed on real space. This capability parallels the new means of information gathering, communication, and expression made possible by emerging interactive media (such as Web-enabled, GPS equipped cell phones with text messaging, video, and camera features).

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.

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.”

I like to keep asking “Why?” questions again and again. There is a nice article about “The method of 5 whys” on Wikipedia: http://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.

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.

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.

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?

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.

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.

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.