Careful Now: 21st Century Edition

Linda links over with her own “Careful Now” admonishment, probably best expressed by this poster, which she recommends her readership paste above its school copiers.If anyone needs me to explain the joke, let me know.

She describes, but doesn’t elaborate on, a set of decent handouts she made in her early career, which she recently discarded. However, it isn’t hard to infer from these bullets …

  • In years to come will you be stopped in the street by an ex-student and they will bow down before you and thank you for all the exciting worksheets they gave? I don’t think so!
  • Please challenge your students and teach them to think.
  • Please give your students a 21st Century Literacy skillset.
  • Please hang this poster next to your school’s photocopier.

… that she finds worksheets unchallenging and unrepresentative of the skills kids need in the 21st century.

Let me say, first, that I think there is a decent heart here, something that may rightly rattle those teachers who content themselves cranking out worksheet after worksheet, passing them out after a rote opener, and then receiving questions at their desks.

But I think her post also reflects:

  1. the 21st-century-learning crowd’s total misapprehension of how students learn mathematics, particularly of how students who don’t understand mathematics at all learn mathematics, and
  2. the 21st-century-learning crowd’s haste to throw out old mediums along with their bathwater.This blog hasn’t always been above confronting (c) the tendency of enlightened 21st-century-learning educators to alienate those they should support.

Unsurprisingly, Linda teaches (or at least taught) English, which lends itself so well to a substituted set of 21st-century activities (eg. instead of printing an essay out in hard copy, blog it and let your classmates comment; instead of taking hard copy notes on Chuacer’s Canterbury characters, set up wiki pages for each) that she’s developed a familiar myopia.

I mean, it’s gotta be that easy for other subjects, right?

But no. Set aside for a moment the hair-pulling difficulty of entering equations and math notation into a blog interface. Math is skill-based in a way that few subjects are. And skills demand practiceFeel free to notice, at this point, the disproportionate number of math teachers blogging. It isn’t (entirely) because words scare us..

Aside from that: worksheets are only a medium, empty pieces of paper, and anyone advocating that we chuck an entire medium in the name of progress would do well to justify it.

For example, what quality of paper prevents challenging exercises from adhering and allows only the lame, rote stuff to stick? What quality of paper insists on empty thought?

Once we exit that dim thought-corridor, the good times roll, and we can investigate the issue which deserves investigation: what we put on the worksheets.

Today’s worksheet is worth posting. We’re learning entirely new skills. By the end of today’s class, students will go from outright befuddlement at this …

… to a tentative ability to solve beasts like theseEquations which took five minutes to attach to this post. My thanks to anyone who can explain how the hell LaTeX works in here..

We did it with four carefully selected problems, problems which I delivered on a worksheet, each problem eliminating units of mental scaffolding so precisely that most of my desk-help topped out at the question, “How is this problem different from the last one?”

My ongoing question for the 21st-century crowd is:

  1. how do I perform that same feat (again from scratch) using blogs, wikis, podcasts, Skitch, VoiceThread, or whatever, or alternately
  2. should a student’s compulsory education even include that knowledgeFrom experience, I don’t anticipate much response to the first question. Ex-blogger Chris Lehmann recently put some screws to that second question, though..


I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I’m coming at this from an elementary school point of view, which may make all of my comments completely absurd to high school teachers – or completely irrelevant.

    I laughed at the poster because it immediately brought to mind those teachers who copy many page packets of work for students every week. I make an effort to use different web 2.0 tools, but I use lots of paper as well.

    As a teacher of all subjects, I think you do English a disservice by ignoring all of the skills involved in it. English is not so obviously skill driven as math, but there are plenty of skills in mechanics, composition, etc. (I’m sure you’ll call me on this and it will push my thinking even more.) The blogging of their writing and getting comments and working together on a wiki allows for collaboration in ways that are more challenging using just paper. It doesn’t change the skills or goals, it gets students thinking at a higher level when looking at each other’s work.

    This can’t always be done in math (at least not until the technology can include the things you want, like the equations). But, students could use wikis to problem solve in depth problems or brainstorm different strategies and ways to work problems. My students keep math journals reflecting on their thinking and problem solving. This year they’re doing it on blogs (along with writing about reading, science, etc). In this way their classmates and parents can read it as well. They are learning from each other in ways that I can’t facilitate in the short amount of time I have them. It’s been surprisingly powerful.

    None of this is to say that what’s happening in your classroom isn’t phenomenal. It is. You know that because tons of people comment in that vein on your blog daily. You are not the audience for that poster above the copier.

    (Do you use Math Type for equations?)

  2. I’m intrigued by your worksheet from today — moving from two-step to five-step solutions in one class period sounds like great fun! But somehow I’m missing a link to your worksheet; that is, if you really did mean that “Today’s worksheet is worth posting.

  3. Whoa, there. I just found your blog recently, and I’ve appreciated thinking about your ideas through a lens that is slightly different than my own.

    As an English teacher, I can’t help but take exception: “Unsurprisingly, Linda teaches (or at least taught) English, which lends itself so well to a substituted set of 21st-century activities…that she’s developed a familiar myopia.”

    It seems a little like you are missing the skills that go into English, which are not as concrete as skill in math but are still important, and also that you are missing some of the greatest things about Web 2.0, that is, the collaborative and multimedia possibilities. Blogging or wiki-ing and editing do lend themselves to writing, but do you ever use collaborative teaching live in your classroom? Web 2.0 broadens the scope of that collaboration. I love watching kids help each other to answers, and they could do it on the web, too.

  4. It was funny to see your post about 21 Century Classrooms and the particularly making the transitions in Math classrooms. My own blogging for the past two days has traveled similar paths.
    I’d like to first comment about Dan’s statement that the post reflects “the 21st-century-learning crowd’s total misapprehension of how students learn mathematics, particularly of how students who don’t understand mathematics at all learn mathematics.”
    I think the 21st-century-learning crowd sometimes comes across this way because they’ve become accustomed to defending their positions against those who simply resist change. I’m sure there are some tech-nazis out there who insist that everything done without technology is a disservice to our students. But, most are level headed folks that encourage us to explore ways to enhance what we do using technology.
    Judging from the staff development I’ve attended over the past few years, Math teachers must be the bain of the guest speaker’s existence. 95% of the sessions I’ve attended finds at least one Math teacher stating, “That’s great! But, how do I use that in Math?” There really isn’t a viable substitute for paper in Math. Not only is it difficult to write equations with fractions and exponents, there are the process steps of carrying the one and simplifying the fraction that require you to move around the problem to annotate steps you’ve taken. Interactive whiteboards, school pads, and notepads permit you to write with a “pen” just like on paper. But if its just like on paper, how is it “better”? At first, students might be more engaged because they’re playing with the new toy. When the new toy loses its new, we’re left where we started. All these items are wonderful tools and contribute to classrooms in amazing ways. As the “21st-century-learning crowd” we should never imply that teachers shouldn’t use handouts any more. And, as teachers we can’t assume that integrating technology into our classrooms means we have to use it for everything.
    Jenny made an excellent point that there are aspects of other subjects, including English, that don’t lend themselves well to the use of technology. Math generally gets the most attention because it has fewer areas that can be completely taught using technology alone.
    The similar Blog postings mentioned in the first paragraph are:
    Does the School of the Future Buy Textbooks?
    Interesting Discussion Regarding “21st Century Classrooms”.

  5. In between comment drafts, Curtis swooped in and snaked basically everything I was gonna say.

    A few resonant echoes, then:

    I don’t want to imply that English & Social Studies & the other liberal arts have it perfectly easy integrating technology into their classrooms, just that it’s much easier.

    Beyond the mechanical difficulties of integrating math notation into the web, the time lost trying to accomplish the same objectives with tech is a barrier to my entry. It’s my Important Ratio #1: if there are multiple ways to learn the same skill, I choose the one that absorbs the least time (assuming fun & engagement are kept constant).

    e.g., I won’t take my students to the computer lab for a period to set ’em up with Wikispaces accounts just so I can have them brainstorm problem solutions. We can do that in class, face to face. Inter-continental collaboration intrigues me a bit so long as I’m not spending twice the time to accomplish the same learning objective. Does that make sense?

    Anyway, since the inception of this blog, I have felt a distinct lack of empathy for how far away math teachers are from this particular fire, which wouldn’t be a big deal if the 21st-century crowd weren’t also so judgmental about our continued use of stuff like worksheets. And pencils.

    But on behalf of all math teachers, let me apologize for our intractability. We resist technology like we resist literacy. May your zealotry and our intractability meet through empathetic dialogue.

  6. We’ve all seen that the majority of worksheets are far less elegant and contain far more busywork than yours. A goofy poster reminding teachers to think before they print hardly seems outrageous.

    However, I do share your doubts about the heavy promotion of Web 2.0 tools for “education” when they really mean 90% language arts. I know people will say that it’s useful to record instruction, or encourage student collaboration, but that seems to just barely qualify as pushing the envelope for math.

    I do appreciate your concern that the pro-2.0 crowd doesn’t seem to understand math. In my mind, the answer is to do real math using the computer, like programming and working with real data. I’m at a loss to understand how this is not considered an essential skill for kids to learn. It’s perfect for a collaborative, scafolded experience for kids that helps build all kinds of math and logic skills.

    I think the answer to your first question is that these tools won’t help, so skip it. It seems like you are doing just fine using other tools. (as you described in your “Hot or Not” lesson linked your #5 footnote).

  7. It is basically a defining characteristic of the “new literacies”/”information whatever”/”21st Century blah blah” genre that the disciplines are simply ignored. In particular, the hard-won body of professional knowledge in each discipline is disregarded, in English as much as Math.

  8. Dan, who are these 2.0 people so vociferously judgmental about your use of handouts (ugh!)?

    And what the hell is a ‘pencil’?

    You’ve got this free-spirited, wiki all the way English teacher totally befuddled.

    Thank goodness we english teachers don’t be teachin anything ‘dat requires praktice like grammer and speeling.

  9. at my previous school, there were a *lot* of p.d.s on differentiation, and the techniques were sold as being content neutral.

    not one single one of them was useful or even applicable to math.

  10. Reap the whirlwind, Dan.

    You can’t belittle the level of skill-based instruction in the ELA world and expect to get away with it.

    Reap the whirlwind. Reap it.

  11. > the 21st-century-learning crowd’s total misapprehension of how students learn mathematics, particularly of how students who don’t understand mathematics at all learn mathematics.

    i’d be interested in hearing more of about this. i’ve got a vague idea of what you’re trying to say (or at least i have some variation that runs through my own head).

    how *do* we explain how (or why) students have difficulty with math to people who proudly proclaim “i never got math”?

  12. Greetings, Dan,

    Glad to see you haven’t lost sight of the use of rhetorical hyperbole in order to dramatize your point :)

    A few comments that stand out as worthy of highlighting:

    “if there are multiple ways to learn the same skill, I choose the one that absorbs the least time” — absolutely, with one addition: assuming that it also is the most accessible method for a meaningful number of students in the course.

    “In particular, the hard-won body of professional knowledge in each discipline is disregarded, in English as much as Math.” — Amen!

    “I do share your doubts about the heavy promotion of Web 2.0 tools for “education” when they really mean 90% language arts.” — this is another example of bad practice. The people delivering this type of message deserve the criticism, not the medium. Lest we throw the baby out with the bathwater, it needs to be recognized that tech advocacy that masquerades as professional development that ignores the actual skills necessary to excel across disciplines is simply bad professional development — and, by extension, bad technology advocacy.

    Dan — I’m also surprised that no one has addressed some of the more difficult skills required in teaching language — things like recognizing sentence structure, and knowing how to vary sentence type (ie, when will a simple sentence make your point more effectively than a complex, compound, or compound-complex sentence). This gets us to verb choice within the sentence, which quickly leads us to word choice throughout the sentence, and balance between sentences. Once we had engaged in that discussion (and there would be no better medium than conversation/discussion for communicating those specific ideas) we might actually be ready for a bloggable exercise: how to analyze sentences for effect, and examine the structure for a greater sense of meaning.

    So, for example, we could look at a short paragraph like:

    “But on behalf of all math teachers, let me apologize for our intractability. We resist technology like we resist literacy. May your zealotry and our intractability meet through empathetic dialogue.”

    and break down how verb choice, loaded language, irony, and balanced phrasing helped summarize and clarify the preceding arguments. This analysis would be bloggable, but IMO it would have more use for more students if preceded by the scaffolding to introduce the grammatical and rhetorical concepts.

    And this isn’t a technology issue. These ideas could be conveyed effectively on a worksheet (a list of terms with definitions and examples) or a wiki (a list of terms, with links to examples) or a blog (like a wiki, but less editable over time). Personally, I’d choose the wiki, because that would have the greater potential for reuse by more teachers and students.



  13. Since I’m a computer lab teacher in elementary, I’m teaching both language arts, math, and many other subjects in the lab. I think the problem I run across with math on the web/computers is that the idiom/symbols for math have less manipulability than words and letters.

    When I was teaching in a regular classroom last year, the one subject I still used an overhead projector for was mathematics. I just couldn’t get the visuals correct on slides, and didn’t want to take the time to learn Latex. I hate sending kids to some math programs because there is nothing more ridiculous than have a child at a computer station trying to add/subtract/multiply numbers greater than 100 and having to get a sheet of paper and a pencil to do it.

    Some of this will likely be addressed by having Smart Boards, where the teacher has a “note book” to write on. Also, some have class-sets so that students can also write.

    OTOH, I’m starting to use spreadsheets to explain input/output tables and simple equations with two variables. This is EXTREMELY difficult developmentally to explain to students in paper form. I’m hoping the spreadsheets will help with this. I’m using this game: to work with students on coordinate graphing since many have a very hard time remembering what to start with the horizontal or vertical axis.

    I’ve also used blogs for discussion of math problems to have students explain their reasoning behind a given answer.

    I think worksheets are a problem if they are poorly designed (many at the elementary level are banal and pointless), and poorly used (sit silently in your seats, do your own work). Unfortunately, this is often how they are used.

  14. I teach high school math and have been recently exposed to web 2.0 tools. I so want to integrate technology into my classes, but it is so hard. I have sought out blogs written by math teachers to learn from their example. When you factor limited lab space and low socio-economic backgrounds into the equation you exponentially increase the difficulty level. Perhaps, the best use of web 2.0 tools is by teachers for teachers. I know that these digital conversations have been beneficial to my own skill set.

  15. In response to A. Mercer:

    “When I was teaching in a regular classroom last year, the one subject I still used an overhead projector for was mathematics. I just couldn’t get the visuals correct on slides, and didn’t want to take the time to learn Latex. I hate sending kids to some math programs because there is nothing more ridiculous than have a child at a computer station trying to add/subtract/multiply numbers greater than 100 and having to get a sheet of paper and a pencil to do it.”

    My school purchased Interwrite Schoolpads for Math teachers this year. It took a little getting used to, but I haven’t turned on my overhead since September. It requires a data projector connected to a computer, but costs a lot less than a notebook. I love it because I can give examples to the class from any location in the room. I’m no longer chained to the front of the class. I think there are plenty of similar products on the market. Thought you might want to investigate this option for your school. (Students can also present their solutions without leaving their desks)

    For Erin Remple:
    I’m in the same camp regarding how to use web 2.0 tools in Math. My first attempt is a blog that I’ve started (only recently) offering extra credit to students who post homework questions as a comment for the given week. I also offer extra credit to the student that correctly answers someone else’s question. I’m spending a lot of time promoting it in my classes, but it’s not taking off like I had hoped. I’m hoping that more students will use it as they become more accustomed to posting comments. I may need to schedule lab time to teach the students how to post comments and what the process involves. You are welcome to visit my classroom blogs to see how I’ve set it up.

    Executives Math Web Page click on Algebra Blog, Pre-Algebra Blog, or Integrated Math Blog to visit the various Blogs. (No students have posted to the Pre-Algebra Blog yet)

  16. Dan, been wanting to ask this of you for awhile. This conversation thread seems as good as any other for pulling the trigger.

    Question 1:

    What would it take to have your current students craft acceptable (merit based on intention and preparation at first…) set of math-oriented, video-based “Graphing Stories” (which, of your work, still has me returning over and over to think through the possibilities: as a small-toe-in-the-water 2.0 move?

    Question 2:

    Could a series of student-developed “Graphing Stories” videos be placed within the context of a collaborative class blog where students defend the thinking-process of both mathematical concept and audience-driven execution, all tied together via a student developed (and teacher-mentored) wiki, for instance?

    Would there be value?

    I am learning to appreciate the challenge (or at least value-question) of using blogs (et al) in the day-to-day instruction of math (at all levels). I am beginning to appreciate how direct instruction, overheads, etc are vital elements in the teaching arsenal in a profoundly skill-driven pedagogy. And I am definitely at peace with the idea that Language Arts (which pays my teaching salary) is more aligned with the default assumptions/benefits of 2.0 tools at this point in time?

    But I am really curious what happens when we do not limit — on either side — the conversation to ‘teachers teaching’ vs. ‘students studenting’ (forgive that) to define or defend the practical uses of any tool (including emerging 2.0 elements).

    At the end of the day, every subject needs both practice and performance to justify that a student did indeed ‘learn’. And inevitably the highest form of teaching is based on expertise and a rigorous commitment to “by any means necessary” to engage/launch our students to be both competent (skills/knowledge) and further interested (vision/application).

    If such a premise offers enough leeway to connect Language Arts (et al) and Math (et al) at the purpose level, what happens when we concentrate our shared efforts on merging tools of all shapes and possibilities across a spectrum of experience with our kids complete success in mind? (Ours, too)

    What would a Math classroom ‘look’ like that employed handouts (well-designed), overheads, problem sets, continual practice, and skill-leads-to-skill expectations WITH the opportunity for students to be brought into the PRESENTATION game as well, using a myriad of classic and 2.0 tools to help them a) defend their thinking, b) interact with a larger audience, and c) engage their long-term interests?

    And what would it look like if — and here’s where I diverge into a crazy little wonderland of probably-ain’t-possible since guys like this don’t exist, right? — a guy like DAN MEYER led a classroom with the absolute commitment to the best of classic instruction and assessment merged with an vision for helping his own students (along with educators of all stripes paying attention to his blog) demonstrate their emerging knowledge to a wide array of audiences with a wide array of tools and a designer’s instinct for presentation?

    Instead of an either/or, is there an and/and proposition that sits somewhere ahead of the classic curve (or parabola)?

    ’cause if there were — and I’m only daydreaming here — a teacher like a Dan Meyer who actually ‘got it’ on both ends of the classic and emerging line, a guy who could demonstrate what it looked like in a real-live Math classroom with real limits of time/space/resources, a guy who had an audience and a vision, it might be liable to spark a profound shift in both the larger edu-conversation and get-r’done execution case study set as the rest of us try to figure out how to live up to such a model in whatever classroom or discipline we call home.

    I know I don’t come here for math-based instruction, per se. But I also know that I’m stealing ideas like mad from you for my decidedly English/Language Arts curriculum at every possible turn.

    What is missing for me — and maybe for others — is the ability to learn from a guy who gets both the rigor of instruction/knowledge and design vision while ALSO demonstrating how to bring the kids into the possibility fold as far as presentation, demonstration, and audience engagement.

    Perhaps its a foolish, quirky wish, but what I wouldn’t give for a guy like you to be a pied piper in terms of bringing it all together — with our kids working in concert with us — so that ALL classrooms, subjects, and teachers were employing the full range of tools to both a) instruct based on our expertise and b) challenge their kids to engage a larger audience when defending ‘how they got there’ themselves.

    For what it’s worth, I’m curious what such a math classroom would look like. And also what it might do in terms of inspiring the rest of us to stop arguing the either/or coin flip until time stands still.

    Following your lead here —

    BTW, have you considered sending Stanford a letter at some point in the next few years?

    If I were you, and explained to the good folks at Wallenberg Hall (the education school gem) and the (oh, you know what I’m talking about here, my friend!) explaining what a dual masters degree would look like if given the freedom to combine programs, I can’t imagine that the likes of Tom Kelly (IDEO) and Diego Rodriguez don’t hand-walk your application to the “Accept!” line.

    Just a thought.

  17. The conversation’s been plenty interesting here without my continued involvement, which is for the best since I can’t address much right now.

    It’s obvious to me that the capital-C conversation could only benefit from more empathy from all over the content areas.

    For my part, I’m glad to know that the ELA content standards (wait, you said ELA has standards, right?) extend past finger painting and color-by-numbers. (Thanks, everyone, for correcting my pedagogical chauvinism.)

    But I’m still convinced that, at least as the tools exist now, language & tech courses enjoy easier access, more one-to-one 21st-century substitutions, a fact which has gotta receive more than lip service on a blogosphere dominated by language & tech teachers.

    To anyone suggesting these tools can revolutionize the lives of math teachers, if not their students: absolutely, yes, I agree. Emphatically. I’d like to believe, given the time I invest to make my practice public, that my commitment to that ideal is beyond question.

    Cherrypicking a few of Christian’s comments:

    There was a moment, back when I was investigating Moodle for my school, when I thought I’d be the salesman you describe, the skeptic entering the fold.

    Excepting material constraints (the fact that we don’t have a class set of camcorders, much less a class camcorder in the singular) inasmuch as I’d like to have my kids make their own Graphing Story videos, produce a DVD compliation, organize a mailing, invite another school to an online screening with regional specific events, etc etc etc etc, I have to contend with time. Constantly.

    The sheer breadth of material the state asks me to cover (and I don’t begrudge ’em that goal) prevents a lot of 21st century fun.

    The class you describe, where students share in the teaching as much as you’d like, would have to integrate tech in more than just microscopic instances. Two hours in the lab every other week kills too much time (in a state of perpetual introduction) for us math teachers.

    You’re talking, instead, about a complete reinvention of the curriculum, particularly at the lower levels.

    I saw potential in information design earlier this semester. I accomplished a lot of the instructional goals I had last year only with more student engagement and without letting an hour slide off my bottom line.

    I’m uncertain where to take it next but time will always be my first reality check.

  18. Hello, Dan,

    It seems that last night, as you were composing your response, I was reading (and re-reading) Christian’s text.

    You say that

    time will always be my first reality check and that what Christian describes would require

    a complete reinvention of the curriculum, particularly at the lower levels.

    To which I ask: would this help students learn more effectively?

    If time is an obstacle, then what are the factors placing demands on your time? If these factors are things that impede learning (and I’m talking the life-long variety, not the pick-the-right-bubble variety) how can they be overcome?

    And yes, if that requires a complete re-write of the curriculum, so be it. If that requires policy advocacy to effect change within state standards, so be it. From what I hear you saying, you are caught between the

    sheer breadth of material the state asks me to cover


    a lot of 21st century fun.

    While I’m not going to address the inherent misnomer of “21st century fun” there does seem to be a tension here. Blogs can be used as a reflective tool, a presentation tool, a way of tracking time spent working on a problem, a way of asking questions, etc, etc, etc — all skills that are useful in math, lit, and, oh, yeah — life — the thing they are going to school for in the first place.

    If the time required to cover a curriculum put in place by people who have likely never set foot in your class, your school, or even your ESD impairs your ability to deliver lessons that would be of educational value, then something is wrong.

    Break some rules, Dan. Have some fun.



  19. Tempting. The same thought occurred to me composing that reply last night. Stories have crossed my path lately (one here and one here) of public service workers at the end of their careers deciding to kick dirt at the system and act wholly on principle. I don’t want to wait that long.

    I can claim these principles:

    I don’t want to bore my kids. I ask them questions that I want them to eventually ask themselves without my help. I want them engaged. I want them working in a rigorous place for as long as I’ve got ’em.

    But I can’t stand on principles I don’t have:

    eg. I know there are skills I’m obligated to teach but I don’t know how much I want to poke at that. (I’m not so selfish I can’t see that choices I make about curriculum affect my colleagues.) I know that tech skills and media literacy are crucial but I don’t know how much business math has leading that charge.

    At this point I’m trying to find the 21st-century fun that won’t interfere with the breadth, a feat which requires a lot of one-to-one translations, a one-hour 20th-century activity swapped for a one-hour 21th-century activity.

    It’d be easy to blow off the second half of Algebra 1 and build math games online or reflect on interesting problems with kids in Bangalore. I find it very difficult to walk the beat I’m walking now. But, to your last directive, I am having fun.

  20. How about a plausible middle fulcrum that supports both ends of the spectrum:

    Could your kids craft “Graphing Stories” (for example) problems and story board concepts based on the math being studied? Could they vote for which one that they would like you to take a swing at behind the camera away from school? Could they then see the visual/video fruits of their inspiration/labor and your vision/solution as one of the opening class moments soon after? Could their passion for the subject matter be positively influenced without time lost in developing the tangible skills needed at the end of the day?

    If so, it would seem that they would be doing the mental & practice work inside the classroom — honoring both the systemic standards and legit time constraints — and still have a direct influence on what you create on new playing field. At the end of the process, both teacher and students would have tested whether their ideas/solutions have actual merit (beyond the answer key). No need for a trip to the computer lab, nor shelling the 2nd half of Algebra I, nor putting future teachers at risk by sending incomplete students.

    Again, the “Graphing Stories” model is being used as an example, not as the literal must-do suggestion. Given what you come up with on a daily business for your kids (and this blog), you are certainly not going to be drinking from a dry well anytime soon.

    Undoubtedly, Dan, you are due NO reprimand for pushing hard on the 24-hour clock. You are relentless with your timepiece and clearly in love with both discovery and the teaching alike. And your relentless desire to bring a modest sampling of that big-bad-emerging world of tools/media into the classroom is to be commended. Heck, followed. Regardless of subject or experience.

    And to that, keep on keepin’ on…

  21. Teaching math, I usually assign some of those problems in the textbook that call for a written exposition or explanation: Describe the steps for graphing the solution of an inequality. Explain why you can add the two equations. Discuss the advantages and disadvantages of presenting data in a table versus in a graph. The few students that do a serious job of these problems do seem to develop clearer ideas of the math involved. However, a large number of my students either skip these problems or produce incomprehensible, incoherent sentences – so vague and diffuse that it’s hard to determine whether they are anywhere near accurate. I do not know what to do about that, how to break down the skill of precise mathematical writing into teachable components. I’ve wondered whether asking the students to send these responses electronically would be useful – then I could at least project sample responses up on the screen and we could discuss what makes the better ones better. But for a combination of technical reasons (when I gave students the option of e-mailing these responses instead of writing them on their paper together with the computational problems only two students opted to do it this way) and for sheer lack of knowledge of how to talk about what makes some sentences precise and clear and others just confused and confusing – I’m rather at a loss as to how to teach decent mathematical writing. And if they don’t learn that in math class they won’t learn it in high school, because I’m pretty sure they won’t be learning it in ELA either!

  22. So now and then I think dy = 2.0dAn. And then you come up with something like this.

    Sketchpad and websites and purplemath are all fine, but at the end of the day, you need to come to class with a pencil.

    We shouldn’t exclude the “stuff” on principle, nor should we use it just to use it.

    Which brings me back to pencil, chalk, blackboard, paper, almost every day.

    I just avoid worksheets, mostly, since they have such a bad rap. (but when I write my own, they ‘develop’ a topic in a really engaging sort of way. Figure it’s the same idea as your equations work sheet.)