Math on a Milk Carton?

Missing @ NECC -- MathI ran across an interesting posting on Hitchhikr yesterday attached to NECC 2007.  Lcrosswe, a Mt. Pleasant, South Carolina Math teacher, laments in NECC puts math on a milk carton

I have spent the last few days trying to refine my schedule for NECC and have run into a problem. I teach math at my school and while I am very interested in going to presentations on topics that are applicable in all disciplines … I would like to also attend some sessions that deal specifically with technology integration in mathematics. I am having a very hard time locating sessions that deal with math at this conference – they seem to be missing (thus the title of this entry.)

He (or she) has found only two sessions at this national/international technology conference that explicitly apply to math.

  • Technology and Mathematics: The Right Angle
  • The Art of Math: Cleveland Museum of Art Math Resources

This got me to thinking — because in the early days of personal computers in the classroom, these almost unapproachable machines were seen as the exclusive purview of Math teachers.

So what happened?  Why are we not spending as much time creating math applications for these almost exclusively mathematical devices?  This is the question that has been popping up in my thoughts ever since.   And it occurred to me this morning — because computers have changed.  They are no longer seen as computing machines, but instead, they have become media machines.  They are not about crunching numbers (though they’re stilled used for that).  Our cultural view of computers today is about machines that help us find, enjoy, mix, and make media — information-rich experiences.

But there are new possibilities for Math teachers — and once again,  I wish I’d known about this nine months ago when we were submitting our NECC proposals.  What’s got my head spinning (almost out of control right now) about Second Life is not so much the social and collaborative aspects as the technical workings that rest just beneath the surface, but well within our reach for study and play — scripting.

SL CodeHere is a piece of code that will move an object on any of three axis.  Apply any number of algorithms that change these numbers in desired ways, and the object can be programmed, using math, to train the object to behave like a car, or a hover craft, or an elevator, or a buzzing bee.

How much fun (not a bad word for learning) would it be to use algebra, trigonometry, calculus, whatever math you are teaching to make a virtual but working soccer-playing robot.  Fun?  Math? Could it happen?  Sorry about that.  I taught math for a year and I wasn’t very good at it.

But perhaps asking, “How might we use today’s computers to better teach math?” is actually the wrong question.  If I might be forgiven for once again stepping way off the edge here, what if the question we should be asking is,

How might we change what math teachers teach to reflect today’s computers and how they have changed the nature of information?

What do you think?


Image Citation:
Morgan Hsu, William. “Blur Milk Carton.” Williammmm’s Photostream. 18 Apr 2005. 18 May 2007 <http://flickr.com/photos/williamhsu/9839188/>.

26 thoughts on “Math on a Milk Carton?”

  1. I ran into this same problem this week as I was teaching a class on the use and integration of Inspiration in the classroom. I could find as many examples as I needed for using this tool (used for creating graphic organizers, concept maps, etc.) in a language arts or social studies class but math and science were very hard to find.

    In my two years as a Technology Coordinator the only real use for technology in a math classroom that I have heard discussed was to access review/remediation software. Drill and Kill I think they used to call it. Its a shame but in a State where testing is KING (is there any place where it isn’t anymore?) the teachers all say “We don’t have time, we need to get the kids ready for the test.” They don’t see the possibilities of technology as an assistant to the learning, they see it as an extra, a toy.

  2. I think the idea of using technology and math in a new way is really interesting. It seems like connecting math with something that students find interesting could be greatly beneficial and maybe even get them excited about learning math. I remember in my high school math classes that I always struggled to find the relevence. Yeah, AP Calc was teaching me to find the rate at which a pile of sand is growing, so what? Piles of sand and people finding the angle they need to place at ladder at (just two of the many recurring and seemingly ridiculous examples in all of my math classes) aren’t that interesting to students. I can see doing something like programming a robot as really engaging the students’ interest.

    However, I am currently in an educational technology class and one of the things we are learning is that the technology in a classroom should not define the curriculum but rather enhance it. The idea you present seems to be bordering on having the technology define the class. But maybe this is appropriate in some math classes, I’m not sure. I would say that such classes that are defined by technology should be electives and the core math classes (like Algebra, Geometry, and Calculus) should remain untouched by technology. These subjects are the basis that most students need for further math education and my fear is that by defining the curriculum around technology interests, some non-technology related topics could get missed or skimmed-over. But as electives I think this is a really great and interesting idea to use technology to liven the subject of math.

  3. I don’t think that any educational technologist I have met would ever ask that the technology define the class unless it is a pure technology course, such as a programming or information systems course. To think that technology should be shunned from a core class is teh absolutely wrong way to go in my opinion. Life doesn’t draw a distinction when it comes to the integration of technology. Our students will be faced with it, forced to use it. The more they have hands-on the more likely they are to be prepared for its effects on their lives.

    Technology has to be used as a supplement to the traditional curriculum. That is, until the powers that be realize that the way we teach today’s student is horribly outdated and overhaul our methodology to be 100% technology enhanced teaching. We can’t bury our heads in progressive era educational styles any longer or we are writing our own economic death certificate. Our future depends on our students being able to think and work with technology in a meaningful way and the classroom is where that will have to start.

  4. A very good point, Scott, about technology not driving curriculum. However, I would emphasize, that line, the changing nature of information, and wonder how that changes what it means to be a processor of data, whether it be numbers on a piece of paper, or the ones and zeros in a digital image.

  5. There is no greater area of failure than math instruction in our schools. We pride ourselves on using math as a way of sorting kids into winners and losers.

    None of this should come as no surprise since the national NCTM conference has ZERO tech-using sessions this year. Don’t even get me started on the recent back-to-basics capitulation by the National Council of Mathematics in Education.

    I’ve seen data from surveys that ask kids how often they use computers in each subject. The majority say “NEVER” in math class.

    There is no discipline where the gap between the actual discipline and its teaching is wider. There is “Math,” a bizarre school invention and mathematics that is based on beauty and making sense of the world.

    The graphing calculator’s popularity is based on the fact that it reinforces a time-honored tradition of ensuring that 5% of a school population learns what is taught.

    As my friend Seymour Papert says, “Instead of teaching a mathematics kids hate, we need to invent a mathematics they can love.” All the pedagogical tricks and technology in the world won’t make a damn bit of difference if we don’t change the content.

    You cannot have both rigor and engagement with the noxious math we currently teach. NCTM itself said in 1989 that “50% of mathematics has been invented since WWII,” as a result of computing and the social science’s demand for number. NONE of that mathematics is in most curricula. Cellular automata, number theory, fractals, chaos, topology, etc… are made possible by computing and allow kids to enter new sorts of more experimental, personal & concrete relationships with mathematics. It is such content that invites students to be mathematicians.

    David Thornburg shows a 16th century math text and the only difference from today is that it’s in Latin.

    When I was a 10th grader in 1978, I took a course called, “Algebra II with Computer Programming.” That course no longer exists. I still have no idea what Algebra II is, but I do know how to program. BTW: If Algebra II is so good, how come there is no Algebra III? I like to ask math teachers what Algebra II is because the question causes their head to explode.

    TECHNOLOGY DOES DICTATE EVERYTHING THAT HAPPENS IN MATH CLASS!!!!!! (Yes, I am screaming.)

    Everything we force down the throats of kids is based on what was possible with pencil, paper and the blue-armed teacher at the overhead projector.

    You might be able to TEACH arithmetic without computers, but learning mathematics is impossible. Loving it is even less likely.

    * The blue arm comes from writing and erasing over and over again five periods per day.

  6. In high schools, I’ve seen Maple used very well in a variety of math levels. I once asked a math teacher at the school I was at what benefit his students got from Maple and the response was interesting: He told me it helped students get past the arithmetic so that they could do mathematics. That sounded a little heretical to me at first, but he explained that his students acquired the concepts and developed an intuitioned understanding of the concepts earlier and more uniformly than they did without the software. Maple and tools like it are certainly visual and allow instant feedback to parameter adjustment (permitting development of intuitional understandings of how systems work), so it seems to me that there is validity in that point of view.

    We used Maple on notebook computers in math classes this way for a number of years, then found that TI calculators (they now use the TI-89) did all they needed from Maple in a more portable and less tech-needy form factor.

    On a lighter note, perhaps you saw today’s timely XKCD comic on math instruction: http://xkcd.com/c263.html

    Enjoy.

  7. If I had policy control, I’d consider de-listing math as a core subject after middle school. In high school, chemistry and physics are great math courses with far more interesting homework. As Policy Potentate, I’d dictate that every kid take a year of chem and another of physics. I’d insist, though, on a strong algebra foundation.

    That said, I have a hard time imagining such radical changes in math instruction. As an individual teacher working within these artificial boundaries (“Algebra II” and “Geometry”, for example), I favor projects that involve math, including robotics*, general computer programming, and carpentry.

    * My robotics students tied a marker to their robot this semester and set about programming art. They drew some amazing spiral designs using FOR loops and fiddling with the timing.

  8. I manage the NECC Web site. I just posted the following to the original blog that David cites:

    “If you use the Program Search at http://center.uoregon.edu/ISTE/NECC2007/program/search.php and search for “math” in the title field, you’ll get a found set of 28 events at NECC this year. Any of those strike your fancy?”

    Not sure if these 28 sessions address the issues you all have raised above, but just wanted folks to know that NECC _does_ have a number of math-related sessions this year.

    Barbara Hewick
    Web Marketing Manager
    ISTE/NECC

  9. Nods to Gary’s posting above where he also mentions arithmetic vs mathmatics. I’d intended to acknowledge it in my post, the the phone rang and disrupted the thought.

  10. I had the good fortune to participate as a Master Technology Teacher in Intel’s Teach to the Future program. Teachers were led through a 40-hour training program in which they designed a technology enriched unit plan for their use in their classroom. The Intel web site is loaded with sample unit plans submitted by other Master Teachers. I would recommend math teachers look through http://www97.intel.com/en/ProjectDesign/UnitPlanIndex/SubjectIndex/SubjectIndex.htm#1 to see if they might be able to derive something they can use in their own classes. These unit plans were made by individuals for their own use so it most likely wouldn’t be appropriate to use them verbatim but I tell teachers all the time that it isn’t exactly what they did, it is often how they did it that they should learn from.

    As a warning of sorts, these won’t be cutting edge types of technology uses. Most of it will be PowerPoint, Word and Publisher, but how many of our students are going to be motivated by cutting edge technology when they are struggling to read and write?

    I lead teachers in the integration of technology in a largly urban, under developed and under motivated district. I often advocate for simple, high probability for success activities. Students need to feel successful when using these technologies, not overwhelmed by flashing lights and pretty colors. For these students the cutting edge uses of technology can be found later, when they have mastered the basics and been given a firm foundation of success with technology on which to grow into those other uses.

  11. I just wanted to post what, in my experience, is a very good resource for using technology to teach math. It’s called Gizmos. It has math and science simulations. You do have to purchase a subscription, but it’s very reasonable and anyone can get into any gizmo for 5 minutes for free. One of the problems with how we teach math is that it’s so very abstract – kids (and adults) have a hard time relating it to the real world. These gizmos do a good job of making it visual and relate-able.

    The other one that I really don’t know anything about, but I saw it got a Codie Award this year is Tabula Digita’s Dimenxion game. I’ve downloaded a demo, but haven’t played it yet. Looks interesting.

  12. Goodness me. I almost didn’t bother writing this comment a second time. Silly me I typed into the comment box and when I submitted it I got that dreaded message telling me the security code was incorrect and to hit the back button and try again. Problem! I’d just written a huge post and it was all gone. Apart from doing what I am doing now and typing it in word so I can paste it when I’ve finished, is there a way to retrieve what I had lost?

    Anyway…. The tenor of my original post was that I agree with you David that the computer has changed significantly from a ‘computing’ machine to a communication machine. Although I do think this change, for most of us, was quite a long time ago. I’m not a specialist mathematics teacher but it seems to me that what is missing in maths classes often is relevancy. The pupils need to understand why a mathematical concept is useful to them in everyday life. Just because it is going to be on the test is not a good enough answer anymore.

    We have been playing with a couple of things in my maths lessons that I think are both fun and relevant. http://www.electrocity.co.nz/ is a game where the students have to build a city (a much simpler online version of sim city). Every decision they make in the game has an impact on population, environment, electricity supply, happiness and bank balance. My class have been playing the game for a couple of weeks now but I have just added the task of creating a spreadsheet to track the impact each decision has on the total score.

    Another group have been using google sketchup to create 3D models of the school buildings which we will post up to google earth when finished. Lots of mathematical discussion about how to create a scale model and how to measure the height of the hall without climbing on a ladder.

    I know many more teachers could add their ideas here about how they are using technology effectively in teaching mathematics.
    In New Zealand we have embarked on a numeracy development project across the whole country. See http://www.nzmaths.co.nz/numeracy/Intro.aspx for an introduction. As part of this project digital learning objects http://www.nzmaths.co.nz/LearningObjects/ have been created to illustrate mathematical concepts in an interesting way.

    The key thought I had in my head when I wrote my first attempt at this comment was that we need to be making maths relevant. If the children cannot grasp the purpose of learning a particular concept then we need to find other ways of presenting it until we do make it not only understandable but relevant. If I can’t answer the question “Why do I need to know this?” then I need to rethink my lesson.

    Maybe we need a movie/slideshow/website/whatever competition along the lines of… “You need to know this mathematical concept because…”

    That is my 10 cents worth anyway (inflation adjusted and taking into account the exchange rate and the fact that we no longer have 1, 2 or 5 cent coins in our currency I felt I couldn’t add 2cents to the debate) 

  13. I am constantly amazed at how teachers struggle to “find” technology to use in the math classroom. The technology is there, however, most teachers are never trained to use it and are very rarely presented with opportunities to learn it in a way that is meaningful to their professional development.

    The problem that I see with many math teachers is that they don’t know good application problems to relate to the math they are teaching. For me, finding good application problems often leads to good integration of technology.

  14. In my experience, no matter how well intentioned, “learning objects” are at best “teaching objects.” This is an important distinction.

    Demonstrating something for students online or in a software widget may or may not be better than a teacher teaching the same concepts.

    I don’t imagine that these “objects” do much to change the content of math education.

  15. The math subjects of high school were set in the 19th century and there’s been little discussion since. Colleges are complicit in this, since their requirements don’t change either. Yet in college and research institutions, the study of history is now dependent on statistics, game theory and computer simulations, all sciences are being transformed by recently invented math like chaos theory, and sociology, anthropology, economics are all being transformed by math and the ability to program computers. They are not simply using computers for communication and information transmission.

    So your question, “How might we change what math teachers teach to reflect today’s computers and how they have changed the nature of information?” confuses me. Information seems like a very tiny piece of the puzzle of how to make K-12 math more relevant in the 21st century. Or am I misunderstanding what you mean when you say “the nature of information”?

  16. Keith,

    I’m sorry, but I just cannot accept your hypothesis that “most teachers are never trained to use it and are very rarely presented with opportunities to learn it in a way that is meaningful to their professional development.”

    This is outrageous. Professional develop. If they do not develop, they are not professionals and should not be allowed near children. How did you learn to use a computer?

    Mathematics is dependent on computation. If a teacher does not appreciate this fact, then they are not mathematics educators.

    Let’s take technology out of the mix. A math teacher could go to any book store and find terrific mathematics books, written for a lay audience, and get all sorts of ideas for real mathematics activities in the classroom.

    The unpleasant truth is that not only is what is taught as “math” prehistoric and irrelevant, it is intended to cause most students to fail. The math education community has been delighting in the dazed and sullen looks on students’ faces since long before I was tortured by my secondary mathematics teachers.

    One high school math teacher was a notable exception. He told us that it was his personal obligation to reverse some of the confusion and battered self-esteem caused by his colleagues.

    The ed tech community has done little to improve this situation by ignoring Papert’s work and over-empasizing the computer as an information appliance.

  17. How might we change what math teachers teach to reflect today’s computers and how they have changed the nature of information?

    That’s exactly what we’re exploring at the Math Playground headquarters in Second Life. We’re very excited about the opportunities presented by the scripting capabilities of SL. We currently have a LOGO project in development as well as ideas for more student-driven projects. We’re meeting with teachers and holding brainstorming sessions. There have been many lively discussions about Papert, Mathland, and “hard fun”. Anyone interested should stop by and see what we’re doing. We are located on EduIsland II at 30/103/22

  18. Hey Colleen!
    I knew you would want to comment on this topic! I dropped back SL and showed a math graduate student from Korea your playground. He is going to come back around to talk to you sometime. He is a distant relative of mine:) Dong Kidd in SL. I hope people come and see what you are doing in SL!
    see ya,
    tansmom

  19. I was wondering if maybe the man/woman that posted “Math on a Milk Carton” simply did not recognize the possibilities when teaching math. A rose by any other name…and all. Just a different way of coming at a subject. Second Life and all.

  20. Interesting dicussion on this post … thought I’d add my two cents. 😉

    Gary Stager said:
    “Seymour Papert says, “Instead of teaching a mathematics kids hate, we need to invent a mathematics they can love.” All the pedagogical tricks and technology in the world won’t make a damn bit of difference if we don’t change the content.”

    Teachers don’t set curricula. Curriculum is mandated and we have to teach it. With that said there’s a good reason math curricula “still teach 19th century mathematics.” In order to understand fractals (fractional geometry, things that have not 1 or 2 dimensions but 1.5 or 1.6 dimensions) and chaos theory you have to understand 2 and 3 dimensional geometry and simple linear and non-linear equations. You don’t build a pyramid from the top down.

    As for how to use technology to teach math, well, I think that’s the wrong question. I teach math. My students use blogs, wikis, flickr, google docs, slideshare, podcasts, and more over the course of each semester. I don’t use technology to teach math. I use technology to teach and teach my students to use technology to learn.

    The content that we teach isn’t more or less amenable to being taught with technology. It’s our pedagogy that is more or less amenable to teaching with technology. It’s all about the teacher’s perspective, or attitude, towards teaching that results in students learning in a technologically infused classroom … or not. Two perspectives that have helped me focus on effectively teaching in a technologically infused environment that has lead to increased student engagement, retention and success are:

    (1) I say over and over again to my students that “Learning is a conversation. If you’re not talking to somebody about it, you’re not learning it.” and

    (2) I try to bring the med school paradigm of “Watch it. Do it. Teach it.” into my classes by having the students take on the role of teacher at every opportunity. No one knows better than teachers that we learn best that which we teach.

    Thinking about these two ideas as I plan how to deliver the curriculum I am mandated to teach has me constantly pushing the envelope of how I teach and how I expect students to demonstrate their mastery of what they have learned.

    When we talk about educational technology I think we have to move away from talking about what particular tools do (spreadsheets, graphing calculators, geometry software) and focus more on what pedagogies these new tools facilitate. How does having access to these tools change what we can do in the classroom and how we teach the things we are mandated to teach?

    In the words of someone who hangs around these parts often … that’s my 2 cents. For what it’s worth. 😉

  21. I’m a future educator of (hopefully) high school mathematics and am currently taking a course in using computers in education. Over the course of the semester we have been discussing how to use the technology not to teach the subject but to “enhance” the learning experience for the students. At the start of the semester I was a bit daunted to find applications for the computer that could be used in a math class. When I was in school, we did not use computers for much of anything except writing papers (and that was really only later in high school) The only technology I had used was the graphing calculator, which is now pretty much essential for any high school math class. However, with my course in using computers in education, we recently had to look at some e-portfolios to see what kinds of lessons can be done with the use of technology. I found teachers using applications from MS Excell and Inspiration to using Geometer’s Sketchpad in these lessons. It was very cool to see that more can be done in mathematics other than just using the calculator. So as Keith said, the technology is there, we just have to take some time (when we can actually find some), look for the applications, and then figure out how they will enhance our students learning experiences.

    I’d also like to comment on this love-hate relationship that we see between our students and mathematics. I don’t know why I love math, I just do. I also can’t say why students don’t like it. (Other than the “I can’t do it” claim) However, I have had 5 years of teaching experience where I taught regular, full-size elementary school classes topics from algebra over the course of a semester. You would think that it would drag and be misrable. If you thought this you would be wrong. I was teaching the same basic topics from algebra that all 9th graders need to know to 3rd and 4th graders (adding and multiplying integers–3rd, exponents–4th) and they were loving it. We didn’t apply it to where they would see it in the real world. But the students were engaged because of the way we taught it. Instead of lecturing to them and spoonfeeding the information to them (like most of us are used to) we taught them by using a more guided discovery approach. So, my point is that it’s not the content that makes students turn against math, I think it may come across in the way we present it. If we don’t show any excitement for any subject, will our students gain any excitement for it either?

  22. As a technology coordinator & mathematics teacher I find this an interesting discussion.

    First some information: Scott if you are looking to use Inspiration within the domains of math and science I might suggest “Learning How to Learn” by Novak and Gowin. Although they don’t speak of using the software, as the book was published before electronic graphic organizers, they do a lot of concept mapping throughout the book–they are science teachers and a good number of examples are in that realm.

    As far as NCTM and technology Gary was incorrect about this year’s conference containing zero technology sessions. I presented “Technology and Mathematics: The Right Angle” at both the NECC and NCTM national –both in Atlanta. I also attended several sessions on using Sketchpad, digital images, Tinkerplots, Fathom and math related websites at NCTM to teach math. So I can attest to the fact that technology is alive and well beyond the graphing calculator at NCTM.

    My dismay with the new NCTM Focal Points is its failure to mention technology at all. I also disagree with Gary’s contention that NCTM has moved “back to basics” but rather to a refining of the mile-wide curriculum toward a narrower curriculum, fewer topics at each grade level with more emphasis on each. This is long overdue.

    I have presented sessions on the uses of technology in the math classroom at the last three NECCs and 3 of the last 4 NCTM nationals. I am still amazed at the “wow” I get from teachers with some simple excel & sketchpad functions especially with iteration problems—fractals and chaos included. Or even the use of digital images within sketchpad to apply current curricular topics.

    This goes right to the argument that Keith presents, “most teachers are never trained to use it and are very rarely presented with opportunities to learn it in a way that is meaningful to their professional development.” This is true. Sad, but true.

    So Gary is both right and wrong in his assessment of these professionals. “If they do not develop, they are not professionals.” While I agree that professionals must continue to learn and develop, there is far too little professional development in the area of technology and mathematics for teachers to attend, beyond TI’s graphing calculators PD.

    So it is not totally the teachers’ fault. There just isn’t enough PD in math education and teacher are ignorant of what is possible, this in turns prevents them from demanding it.

    In a typical college program, few, very few, mathematics professors use technology—even calculators. Math ed students are lucky to get one course on the use of technology in math instruction.

    Where I do agree with Gary is that the math curriculum must change. I don’t, however, find it pre-historic, merely out of touch. What I find is that too many teachers simply do not understand how algebra and trig are used today. E.g., the rotation of weapon in a video game is a real application of (sin(A+B), cos (A+B)) as the coordinates of a point on a plane—the plane of the video screen.

    Teachers in NY for a long period in the 1980s and 1990s taught truth tables. I found very few who actually knew the application of truth tables to computer engineering. When I asked why they taught them, I always received the same answer, “It’s on the test.” 🙁

    We also teach math backwards, thank you E.L. Thorndike. Applications should precede the learning of the algorithms that solve them. Thorndike believed that all the parts should be taught in isolation, hoping that students could put them all together to solve problems later. Sadly we adopted this misguided notion and we learn and teach isolated facts/algorithms in the hope of applying them later.

    The right problem, i.e., one that is interesting to students, should light the fire for learning the math needed to solve it. There are curricular materials out there that support this approach that include technology, but they are slow to be adopted. By the way teachers who use this approach are never asked, “When are we going to use this?”

    I think Darren is right on with much of his comment on students and technology. We do need to bring learning to them with the tools that THEY have and use and as math teachers we need to learn to use their tools better. I think it is noteworthy that Darren also comes back to the conversation part of computers with his use of blogs, wikis, flickr, google docs, slideshare, podcasts, etc. in teaching math. However, I believe these may also fall short in terms of discovering the beauty of mathematics on one’s own not just discussing what was learned.

    To get back to David’s original question, “How might we change what math teachers teach to reflect today’s computers?” I think the question should be, “How might we change how math teachers teach to reflect today’s computers?”

    First and foremost, teachers need an awareness of what is possible. (Don’t however look at most of the math lessons that exist on teachertube.com—this is a very sad use of educational technology. Many of the lessons there are of very poor quality and lack the use of technology, except that of getting them there.)

    Once they see what is possible, then they will need to TIME to attend PD and we will need to provide good PD.

    I present a lot of ideas in my session about how technology can be used in the mathematics classroom and I believe I am providing the first part, i.e., awareness. How we move to the second part, I’m not sure. But that’s where the discussion needs to go.

    Frank

  23. To fsobiereajski:

    Regarding your comment about Thorndike. What is your source for your comments on teaching algorithms with the hope of later usage? I am working on a paper for class right now regarding Thorndike’s influence on the math curriculum, and I am having difficulty finding sources connecting the two. Is the comment your opinion or based on research or readings?
    With regard to technology in the math classroom, part of what’s sad is that, while there is a lot of very cool stuff out there, teachers don’t really know how to use it or have the time to figure it out. If anyone wants to see a good conference regarding math and technology, look for one of the TI conferences. While the purpose of these is mostly to sell TI products, they are still very valuable seminars.

  24. To STwanley:
    It’s been awhile since I’ve read it, but The Psychology of Mathematics for Instruction,by Resnick and Ford, I believe contains info about Thorndike.

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