RYSK: Dewey's The Child and the Curriculum (1902)

This is the 20th in a series describing "Research You Should Know" (RYSK).

In my last RYSK post, I joined some other math teachers in discussing Richard Skemp's Relational Understanding and Instrumental Understanding (1976). Skemp's is a classic article that wrestles with a duality; in Skemp's case, the distinction between math for procedural skill versus a deeper mathematical understanding. For this meeting we turned the clock back further to Dewey's The Child and the Curriculum (1902), another classic article struggling with a duality in learning.

As D. C. Phillips (1998) noted in his review of The Child and the Curriculum, Dewey had a particular passion for dualisms, addressing more than three dozen of them in Democracy and Education (1916) alone. As Skemp and many others have shown, dualisms can be a starting point towards building a more nuanced understanding, as "neither is the world divided into a series of polar opposites, nor is it one" (Phillips, 1998, p. 404). Somewhere in between the opposites and the same lies the understanding many of us seek.

The Child and the Curriculum presents a particular dualism that very much persists to this day: should education be rooted in content, or in the needs and wants of the child? In Dewey's time, the push for a content focus was seen in the report by The Committee of Ten, not totally unlike how we currently push for content with documents like the Common Core State Standards. Dewey, like Skemp, also uses the metaphor of the map, using it to describe the logical versus psychological ordering of subject matter. Again, we still struggle with this duality today; last November Jere Confrey remarked at a conference, "There are some parts of the common core standards that I would express as mathematicians’ thought experiments," meaning we often guess how mathematical understanding is developed based on the structure of the mathematics instead of research on how children actually learn. These, of course, are not opposites, but they aren't the same, either.

Most of our discussion used Dewey as a prompt for thinking how Dewey's words more than a century ago frame modern challenges in education. (Reading Dewey seems particularly good for this kind of activity.) I was joined by +Nik Doran+Bryan Meyer+Nat Banting, and +Chris Robinson was feeding us ideas in the chat as we went along.



I plan to have more of these discussions, and hope we can get into some literature that really addresses research in math education versus some of these more theoretical or philosophical pieces. If you have suggestions for articles to read, please add them and vote them up or down in Google Moderator!

References

Phillips, D. C. (1998). John Dewey’s The Child and the Curriculum: A century later. The Elementary School Journal, 98(5), 403–414.

NCTM Denver 2013: Final Thoughts

Despite my flurry of blog posts during and immediately following this year's NCTM Annual Meeting, I fell short of recapping the last of the sessions I attended or summarizing some of my thoughts about the week. After my last post two months ago I needed to refocus on finishing a very busy semester — in less than two weeks I managed to give a presentation, write six papers, and tackle a digital pile of backlogged grading. Following that I just wanted to enjoy some calm and quiet, and I've done just that. Now on to the recapping:

Meyer's Tools and Technology for Modern Math Teaching


Annual Meeting - Saturday, April 20, 11:00 am

Dan Meyer - Stanford University and mrmeyer.com

I was one of many who turned out for Dan Meyer's session, which focused on why teachers should be using technology to capture, share, and resolve perplexity. What's perplexity? Dan described it as "not confusion" but a "wanting to know, thinking you're able to know." Dan is careful to differentiate perplexity from engagement, as we've all been engaged in something that was simply tedious or boring.

"I'm about THIS big."
This kind of perplexity describes a particular state of mind, one with more promise than the traditional definitions that describe perplexity as full of uncertainty and difficulty. However, when Dan speaks of "capturing" and "sharing" complexity, he's not so much describing a state of mind as he's describing the kinds of phenomena that provoke the asking of mathematical questions accompanied by an eagerness to mathematize. I'm hoping as Dan and others go forward we develop some sort of theoretical basis for these phenomena, or at very least, a useful classification system that can aid in task design. For example, I see the phenomena of scale frequently in Dan's work, provoking questions like "How much might the big blue bear weigh if it were a real, live bear?"

For capturing perplexity, Dan showed various tools for finding and saving things from the web and the world. These tools included an RSS reader for following blogs and news sites, a tool for downloading and saving YouTube videos, a note-taking application, a tool for capturing audio memos, and the camera on your phone. The particular tools here don't matter as much as knowing why to use them — they key is finding the tools that work well for you.

For sharing perplexity, Dan included technology like a computer with speakers and a projector, a document camera for showing student work, slideshow software, editors for photos and video, and a personal blog to "share the best stuff you do publicly."

For resolving perplexity, Dan made some connections to the Common Core State Standards. Standards aren't technology like computers and smartphones, but the CCSSM — particularly the Standards of Mathematical Practice — can be seen as tools for mathematical task design. There's a lot in the world that could be mathematized, but having a set of standards can help make sure it's done with the right content and practices in mind.

You can access Dan's shared resources for the session at nctm13.mrmeyer.com.

Hart and Hart's Viral Math Videos: A Hart-to-Hart Conversation


Annual Meeting - Saturday, April 20, 12:30 pm

Vi Hart - Khan Academy
George Hart - georgehart.com

I think Christopher Danielson said most of what I was thinking during this somewhat odd father-daughter session. It's difficult to describe the vibe that was in the room, with the presenters casually and sometimes clumsily taking turns describing then showing their videos. Near the end Vi grabbed a guitar for a rather brave musical performance that filled me with some kind of vicarious embarrassment, as if Fiona Apple had gone on stage thinking she was singing for lovelorn teens when in fact it was just those teens' math teachers. Then again, I feel embarrassed for others quite easily.

George Hart and Vi Hart

Perhaps I shouldn't be too critical. Some of the videos were pretty cool and who among us hasn't had at least one "Hey guys, check out this thing on YouTube" kind of moment?

Reflection


Having a big conference in your backyard is very nice, although I spent far more time on the bus or waiting at bus stations than I would have ever imagined. I take conferences seriously and believe in attending as many sessions as possible. Just as I never skipped a class in college -- and felt guilty about missing anything even when it was absolutely necessary — I'm not one to turn a conference visit into my personal vacation. So my NCTM experience turned into an 80+ hour grind investment not only in my own education, but as a proxy for the many who couldn't attend.

I tried to seek out a balance of sessions that were personally beneficial, high quality, and of wide interest. In a conference of this size there is plenty to choose from, but the downside of that is that session proposals are almost comically short and descriptions in the conference program don't provide much detail. The sessions are also of varying length and they overlap, which I think adds to the variety of sessions a person can attend. This year a new 30-minute session called a "burst" was introduced into the schedule, but I didn't attend one. My colleague +Ryan Grover attended a burst session, but was disappointed that many bursts happened at the same time. That made it difficult to schedule several in a row, and attending a 30-minute burst meant not attending a longer session offered at the same time unless you didn't mind sneaking in halfway through. I don't know what kind of feedback the program committee has gotten, but I hope they find a better approach for the bursts, perhaps something modeled like the paper sessions at the research conference. There, three authors briefly introduce their papers, and then session attendees have the option of sitting at a roundtable for 20 minutes with their choice of two of the authors. Perhaps in the future the bursts could be grouped in a similar way.

It's tricky to consistently find good sessions, and session titles like iPad Games for the Flipped Classroom seem far more likely to attract a standing-room only crowd than something based solidly in both research and practice. Then again, I attended some sessions because the presenter was well-established in the field of mathematics education, and frankly, that didn't always translate into a session that was engaging or helpful to me.



Some people used Twitter to try to improve their chances of finding the best the conference had to offer:







Shaunda McQueeney addressed a particular pet peeve of mine: Those who would rather spend time in the exhibit hall instead of attending sessions:



Over the past year or so I've become more and more aware and annoyed by how Twitter's limitations constrains our ability to communicate complex ideas or have fluent conversations. Unfortunately, the Twitter use at this conference didn't do much to change my mind. As I've written about elsewhere, Twitter is very good at covering events with short summaries of something that is happening or just happened, such as:



But when we try to use Twitter to share and engage in ideas, it's harder to scratch the surface. For example, this Tweet was far and above the most retweeted of any of my Tweets during the conference:



To me, the above statement doesn't really mean much of anything. By itself, I can't imagine it having an impact on a teacher's practice at all, and even if it did have an impact, there's nothing in the Tweet that describes how this is done. This is a platitude and little else, and I knew it when I tweeted it. I have a theory that these things are retweet bait, and I was testing out the theory. Many of the most retweeted Tweets appear as platitudes to me. That doesn't make them all bad, and it's a phenomena certainly not limited to Twitter, but there's a tempting superficiality there that I'd like to think people are aware of. That's why I'm thankful for efforts like MathRecap that at least offer an opportunity to sink some conceptual roots into solid ground.





I certainly could have tweeted Leinwand's talk — he may have been the most tweetable speaker at the conference — but I thought that recording his talk and taking notes would be the better approach in the long run. Conferences reach a limited audience for a limited amount of time. Twitter widens the audience, but is minimally helpful to the person wanting to revisit the event a day, week, or month later. Blogging helps remove limitations of both geographic and temporal limitations, and including audio and video is even better.

I've submitted a proposal to present next year, but I'm concerned about conference and travel expenses. As much as I enjoy conferences, it's difficult on a grad student salary to justify spending hundreds of dollars to present to 25 people, when I could reach many more staying home and assembling the presentation for the web for nothing. I'll wait to see if my proposal is accepted first and then decide from there.

NCTM Denver 2013: Lester's Whatever Happened to Problem Solving in the Math Curriculum?

Annual Meeting - Saturday, April 20, 9:30 am

Frank K. Lester - Indiana University

Frank Lester was the editor of the 1234-page Second Handbook of Research on Mathematics Teaching and Learning. This was an interesting talk, not so much for any specific content but for how it was put together. Lester began the talk by demoing Algebra Touch, an iOS app that promotes fluency with symbol manipulations in solving equations. He asked, "What will the math classroom of the future be like?"

Frank K. Lester

Then Lester went into problem solving, something he feels has slowly slipped out of most mathematics curricula. Problem solving, says Lester, is "What you do when you don't know (or aren't sure) what to do." That leads to the teacher's role in the classroom:


From here, Lester showed some of his favorite problem solving tasks. The first is probably familiar to most of you:

A snail is at the bottom of a well that is 10 meters deep and it wants to get out. Every day it climbs up 4 meters. It then slides back 2 meters when it rests at night. If it does this day after day, how many days will it take the snail to reach the top of the well?

As part of the discussion, Lester related his problem solving heuristics, harkening back to Polya's How to Solve It. Suddenly a talk that began with a discussion of technology and the future was using a (good?) problem that felt like it was from the 1980s and (good) strategies that were from the 1940s. Lester's choice of heuristics to apply here were "Draw a picture/diagram" and "Be skeptical of your solutions," since many initially reason that the snail reaches the top of the well in 5 days.

Lester then looked at finding the square root of 12,345,678,987,654,321. His heuristic -- one he called a "super heuristic" -- was to look for a pattern. I couldn't help but feel like this was a trivial problem with a trivial answer.

Things got better with the next problem: "On a European river cruise, 2/3rds of men are married to 3/5ths of the women. How many men and how many women are on the cruise?"

Lester joked that this problem predated talk of same-sex marriage, and I found it to be a bit out of touch. Lester said the problem could be adapted to involve pairings of animals or objects. Another heuristic here was "make reasonable guesses, not as final answers, but to get you started." After discussion of this problem, we moved on to one more: "A club has 500 members. At the Spring dance, tickets for new members were $14 but $20 for longtime members. All of the new members attended but only 70% of the longtime members attended. How much ticket revenue was collected?" It seems like there isn't enough information, but solving this plays off the fact that $14 is 70% of $20. That might elicit some interesting reasoning, but again I think this trivializes the problem.

Lester returned to technology at the end, mentioning strategy games like Math Dice and Rush Hour. He advised that teachers have an important role to play when students play games. Prior to the play, teachers need to help students be clear about the rules for playing, model how to play, and discuss special situations. I think that depending on the game and the goals, this could turn into too much guidance. During student game play, teachers need to watch students play, attend to their thinking, help and point out misunderstandings. It's important, says Lester, to not suggest strategies for playing. Kids should be left to figure those out for themselves. After gameplay, reflection is important, just as with other classroom activities.

Lester's slides can be found at the conference planner.

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NCTM Denver 2013: Danielson's They'll Need it for Calculus

Annual Meeting - Saturday, April 20, 8:00 am

Christopher Danielson - Normandale Community College, Bloomington, Minnesota

As Steve Leinwand noted in his Thursday talk, math teachers are a relatively conservative, risk-averse bunch. Perhaps our conservatism comes from the perceived slow but steady progress of math over millennia where it's easy to take comfort in the old because the new can seem so difficult to obtain. Some of this rubs off in the way we teach, the activities we choose for students, and our judgement about what's important for students to know.

Chris Danielson's session kicked off by calling out some mathematics that gets taught in the name of "needing it for calculus," despite no widespread need for it anymore. Simplifying radicals. Rationalizing the denominator. Simplifying rational expressions. Factoring quadratics. Composition of functions. The binomial theorem. It's not that someone, somewhere doesn't have a use for these things, but what is increasingly becoming the exception should not prove the curriculum rule. Mediocre proficiency with these topics is not what leads students to be successful in calculus. What students really need for calculus is a deep understanding of slope as a rate of change and accumulation.

Christopher Danielson

This is a familiar story for some of us. We cringe when we ask students "What's slope?" and they parrot back, "rise over run" without knowing much beyond that. Yes, that might be one way to describe slope, but there are other, and arguably more important ways to describe slope. Danielson's focus on slope as a rate of change not only is most fundamental for calculus, but it is in alignment with the research on teaching slope (Lobato & Thanheiser, 2002; Peck & Matassa, 2012; Stump, 1999, 2001).

Danielson led the well-attended workshop through a number of middle-school appropriate tasks involving rates of change. Because the tasks were set in informal contexts, students would be most likely to work in terms of "dollars per bicycle rental" or "enjoyment per piece of candy," depending on the context of the problem. Time was spent not just looking at rates, but doing simple calculations to compare changes in rates over time, a fundamental conception needed for calculus. The problems in the workshop were adapted from tasks found in Connected Mathematics, a popular NSF-funded curriculum for the middle grades.

For Christopher's presentation, related tweets, and participant notes, see his post at http://christopherdanielson.wordpress.com/2013/04/21/the-goods-nctmdenver/.

References

Lobato, J., & Thanheiser, E. (2002). Developing understanding of ratio-as-measure as a foundation for slope. In B. H. Litwiller (Ed.), Making sense of fractions, ratios, and proportions (pp. 162–175). Reston, VA: NCTM.

Peck, F., & Matassa, M. (2012). Beyond “rise over run”. RME in the classroom. Workshop at ICME-12, Seoul, South Korea. Retrieved from http://rmeintheclassroom.blogspot.com/2012/07/icme-12-workshop-and-sharing-group-on.html

Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

Stump, S. L. (2001). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2), 81–89. doi:10.1111/j.1949-8594.2001.tb18009.x

NCTM Denver 2013: Fennell and Wray's Math Specialists Get Ready Now: Common Core Assessments Are Coming

Annual Meeting - Friday, April 19, 3:30 pm

Francis (Skip) Fennell - NCTM Past President; McDaniel College, Westminster, Maryland
Jon Wray - NCTM Board of Directors; Howard County Public Schools, Ellicott City, Maryland

Skip Fennell, Jon Wray, and Beth Kobett (who was absent for this presentation) are the leads on ems&tl, the Elementary Mathematics Specialists & Teacher Leaders Project. As the name implies, the focus here is on supporting math specialists, such as district-level curriculum directors, instructional coaches, and anyone who is in a position to support mathematics teachers.

For this presentation, Fennell and Wray looked at the upcoming Common Core assessments, PARCC and Smarter Balanced (SB), and suggested ways math specialists can help teachers prepare for the tests.

Francis (Skip) Fennell

The challenge Fennell and Wray presented was essentially to focus on the upcoming assessments and respect the influence they will have on curriculum and instruction, without focusing too narrowly on the assessments and cause instruction and learning to suffer. This means, for example, not turning classroom practice into test prep, and using sample items from both PARCC and SB wisely.

Fennell and Wray used the concept of assessment literacy to describe the ability for teachers and specialists to understand a testing program. Many teachers have no formal training in assessment, so math specialists must be able to help them build their assessment literacy. Part of this is simply becoming more familiar with the schedules and formats of the upcoming PARCC and SB assessments. Both consortia offer more than just an end-of-year test, and teachers are going to need to help students interpret new kinds of technology-enabled assessment tasks.

Jon Wray

Fennell sees great potential in the CCSSM, but said, "If the Common Core becomes political, it's dead." Teachers and specialists need to work with the standards in ways that doesn't reduce them to a checklist of vaguely connected ideas. Using a number of items and task prototypes, Fennell and Wray showed examples of sample items from PARCC and SB and showed the many ways these could be used richely in lessons if the teacher provides the right support and instruction. "There are a lot of ways sample items can be used as instructional gems, " said Fennell. A list of potential questions and strategies for various tasks can be found in their slides.

The presentation wrapped up with an urging to better understand the role of formative assessment around these sample tasks. Also, encouragement was made to use materials from both PARCC and SB, regardless of the test your state has adopted. More task resources were linked to, including Illustrative Mathematics, the Institute for Mathematics and Education (especially the progressions documents), The Mathematics Common Core Toolbox, the PARCC Educator Leader Cadre Portal, and the Smarter Balanced Scientific Sample Pilot Test Portal.

The slides for this presentation are available here.