If you only read the headlines, you might think that education is rapidly moving towards a monumental shift. Venture capitalists, tech start-ups, and Bill Gates himself hold the idea that with better data management, we can revolutionize education. Many of these ed tech companies employ some sort of “flipped classroom”, where students are autonomous beings who are given scaffolding to learn skills at their own pace. There are various reports to the efficacy of these practices, but they are drawn to the Mathematics world more than other domains.

## Pursuing the Dream

I want to discuss the methodologies, pros and cons of one such system: Dreambox learning. Dreambox promises to “meet students where they are” with respect to their mathematical abilities and then to “accelerate” their mathematical learning. Their methodology for accelerating students is a simple GUI where students are to complete various tasks, such as adding fractions or comparing numbers with decimals.

Let me say first that I do not wish to single out Dreambox or any other “flipped” classroom approach such as watching videos and taking subsequent quizzes at Khan Academy. However, the debates center around logical (and rarely empirical) arguments, and conclude that these approaches are good and that therefore we must spend money on them.

As I wrote last week, there seems to be a plethora of jargon on the systems used by many of these educational systems. Here are some jargon-y words from Dreambox’s FAQ: differentiated; adaptive; learning engine; gaming fundamentals; rewards; aligned (to Common Core standards); intervention; proven; and real-time reports.

Dreambox promises that for every *minute* that a student is engaged with their software, their engine analyzes 800 different pieces of information. Everything from the keys they press to the duration between those presses is analyzed and spit out to… somewhere. “The kids love it because Dreambox looks like a game”. I don’t really take issue with anything that they’re trying to do so far, with one large, glaring exception. The headline that Dreambox promises states explicitly that “Every child can think like a mathematician!”. Dreambox also markets itself like a teaching assistant with “unlimited patience and a perfect memory”.

The “game” I tried on Dreambox had students practicing identifying decimal values on a number line. Here’s a screenshot of the software in action.

And since I really am terrible at knowing my decimal places, I put the pin in the wrong place:

After I guessed incorrectly a couple of times, the narrator of the “game” said she would help me and gave me an entirely different problem. She led me through using the magnifier and basically solved a similar problem for me. I think as a teacher, this is exactly what I would want a teaching assistant to do: solve similar problems with the student giving scaffolding, assistance and problem solving. But how did this particular game meet me where I currently am? Did it analyze why I might have gotten a question wrong? As a human being that can ask questions that aren’t so binary, I *still* struggle with determining students misconceptions. Call it the experts’ (ha!) blindspot. But this system is no more adaptive than the “teaching machines” the behaviorists created.

(One more thought: Game like? I mean.. there’s definitely color? There’s a funny shaped character with what I believe to be a monocle. But I’m not sure where the playfulness of a game exists here.)

## Behaviorist Approaches

In the above video, we can see the emergent technology of the behaviorist: the teaching machine. Although it might be comical to watch the antiquated technology of the 1950s, they hold striking parallels to the emergent technologies of today.

Listen to Skinner’s opening line. “These young people are studying in a new way. The class is in spelling, but it might as well be in arithmetic, or in algebra or grammar, or anything involving the use of words or symbols.” This approach of content agnostic technology is what gets at me the most. The idea that mathematics should be taught in the same way as spelling or grammar is beyond fundamentally flawed – it loses the idea of divergence completely. Forget the connections of divergence to GDP, or economy or even being happy. If we’re dealing with a dichotomous world – wrong or right answers only – we lose the reason that we studied any of these topics at all. As Paul Lockhart said, we don’t study Mathematics for its usefulness; we study Mathematics because it’s so damned interesting.

Second, listen to the words that Skinner uses to describe these teaching machines: effective study; immediate knowledge (feedback); leads to *correct behavior*; motivating; free of uncertainty or anxiety about being wrong; scene of intense concentration; and quick report. I do not know the history of teaching machines or why they failed, but the connection between yesteryear’s and today’s “teaching machines” is fairly obvious.

So if this technology has been around for over half a century, why is it just now that their use is being so proliferated? Is it a political issue (grandstanding on the idea of improving education) or an efficacy issue (only now do we have the processing power to truly utilize these machines). I am unclear which it is. However, I do have one question: how would a teaching machine of the past or present handle a divergent question like this:

## Conclusions

Dreambox’s ultimate goal is to have every student think like a mathematician. Nowhere in their software could I find the use of thinking in extremes, solving a simpler problem or analyzing assumptions – all key components of how mathematicians think.

If there’s any message I could get across to these ed tech companies, it’s this: A skill-based approach will not a mathematician make. Mathematicians are interested in good questions, not how to follow instructions. Trying to predict what a student might say to a convergent question is easy – and you can get many data points. But is it useful? Will it help a student think creatively? These are the kinds of questions that will drive innovation in flipped classrooms and self-pacing. However, it doesn’t seem like we’ve made much progress there.

Thanks for giving DreamBox a critical analysis and sharing your thoughts. I’m the Curriculum Director at DreamBox, and before that I spent 10 years in public education as a high school math teacher and K-12 math director. I worked with our teachers, creative team, advisory board and programmers to develop the decimal number line you tried as well as many of the other lessons. This interactive number line is something we’re really proud of, and while it isn’t a “game” in some senses, it does engage students in a context with a clear purpose and a mathematical representation that develops conceptual understanding and prevents misconceptions from forming.

A while back I responded to another teacher who had tried one of our lessons and looked at it from the viewpoint of someone having difficulty with the concept (http://bit.ly/ZhvPDo). I’ll refer you to those thoughts in the comments sections on Dan Meyer’s blog because the hints, scaffolding and feedback our teachers put in our lessons are based on a knowledge of what the student has done previously in DreamBox. It’s like a principal observing a teacher for the first time on February 1st – a LOT has happened since school began that the principal isn’t aware of. The sample lessons we put on our website for review are not chosen because they’re the first experience a student has with the lesson or interactive tool. They’re ones we think will be most useful for teachers because they represent something much closer to the end learning goals we support.

We don’t consider DreamBox to be a part of a “flipped” classroom approach for a number of different reasons. Rather we are a supplement that develops fluency, number sense, and differentiates for students in partnership with classroom experiences. The flipped classroom approach generally uses a video explanation assigned at home as the first experience a student has with new content. DreamBox doesn’t even have video lectures, and instead always engages students in critical thinking first. That’s in line with the research in the book How People Learn. The lead author of that book – John Bransford – was one of the original advisors for DreamBox. Along with Cathy Fosnot, Skip Fennell, and others, they helped design our adaptive engine. Neither they, nor I, are behaviorists. What we do is much more constructivist, and in line with Freudenthal and Pappert than it is with Skinner. Fosnot is an expert in early numeracy, and as far as one can be from being a behaviorist. You can hear her describe DreamBox in these two videos: http://bit.ly/10Yyqm7 and http://bit.ly/YzEGx6.

I agree with you that the ed-tech hype is overwhelming. And there’s not nearly enough discussion of pedagogy. I wrote about that back in November (http://bit.ly/U0U42R). If I had been at SXSWedu, I would’ve joined you at a session on critical pedagogy if there was one. Maybe I’ll submit a proposal for a pedagogy session next year and hope it gets voted in. But I’m concerned many people – and many ed-tech people – don’t want to talk about pedagogy, curriculum, and learning pathways. That’s a large part of the reason why I came to DreamBox, where we have incredible advisors and classroom teachers on our team, and use technology that honors how children learn.

To the overall conclusions in your post, I think your definition of what a mathematician does is too narrow. As an example, the 1999 Berrick article on the AMS website notes induction, deduction, and abstraction as key tasks of mathematicians. DreamBox engages students in all of those critical thinking activities. We also help students look for and make use of structures, and reason quantitatively and abstractly (two of the CCSS Practices). Nearly all of our interactive tools have very open-ended lessons when they are first introduced to students. For more information and examples, you can read my blog posts and see some narrated lesson demos here http://bit.ly/15cvS4i.

We’re not about correct behavior. We develop skills, but not in a skill-based approach. We’re instructive, but we don’t give explicit instructions for how to solve problems. That wouldn’t work with kindergartners anyway. We’re about critical thinking. Thinking is how students make progress in classrooms, and that’s how they make progress on DreamBox.

Lastly, on an unrelated note, I appreciate your focus on divergent thinking. While the K-12 math director in my district, I also had the privilege of facilitating the development of our district’s Mission and 5-year Strategic Plan (http://bit.ly/140NdRw). We were committed to ensuring students were curious learners, and we decided divergent thinking was a key part of that. If you follow that link, check the bottom goal on page 13 about divergent thinking. I don’t know of any other K-12 districts with a specific goal about it. I hope you have similar goals in your district, or can work to get such important things on your school’s radar.

Thanks for giving DreamBox a critical analysis and sharing your thoughts. I’m the Curriculum Director at DreamBox, and I’ve tried a couple of times to post a comment, but it doesn’t seem to be working. So here’s a link to a Google doc with my thoughts: http://bit.ly/XU9FbH. My comments provide more context and perspective about how we engage students in critical mathematical thinking as opposed to behavioral conditioning. Our pedagogical approach is very different than what Skinner advocated in this video.

Hi Tim,

Thank you for your response. I love the fact that you worked with such a large swath of individuals for the software.

As far as the number line interaction: Can you clarify what the context is? From what I saw, I was dropped into an environment where I had to put a pin in a specific place. Maybe I missed the lesson that preceded the pin dropping, but from my perspective, it seemed fairly rote. I think the term context is used pretty loosely here, and would love to see how it could be strengthened in some way.

Apologies if I misconstrued Dreambox as a tool in a flipped classroom. I only was able to see a snippet of the interaction, and I can see how schools/administration would try to segue to using Dreambox as more than just a supplement – especially when the Dreambox navigation lists the program I played with under “Sample math lessons”. That word lesson is key. I understand that there is not a video “lecture” beforehand, so is the word “lesson” being used in the same way a teacher would use it?

I would love to be a part of a bigger discussion of pedagogy (and technology implementation). I was most drawn to testing out Dreambox because of the speaker who Bill Gates interviewed – it was a lot of hype that I want to analyze and critique.

As far as constructivism vs behaviorism: if we both share the basic premise that constructivism is defined so that: “The purpose in education is to become creative and innovative through analysis, conceptualizations, and synthesis of prior experience to create new knowledge.” (wiki), then it becomes very hard to see how this type of environment is based on constructivism. Again, feel free to say that I am uneducated on how data-mining or the adaptive system works, but I would love to know more.

(PS I tried clicking the link to your prior district’s page, but I do not have access).

Brandon,

I’m really sorry for the delayed reply. I thought I had checked the box to be notified of follow-up comments on this post and none never came through my email. So I just now checked back at your blog and saw your reply from a few weeks ago. I checked the box this time.

I really commend you for digging into DreamBox yourself based on the hype. And I’m happy to engage this dialogue. There’s not nearly enough critiquing going on in ed-tech. I’m an ed-tech skeptic myself – I know there are limits to what learning can be accomplished using technology. But I also know there are limits to what can be accomplished in a classroom with 25-40 students and 45 minutes of math each day. I won’t bore you with the story, but I’m actually at DreamBox because four years ago I put my 4-yr-old son on DreamBox precisely because I didn’t believe the hype (I was a life-long Missourian at the time, and we’re the “Show-Me” state). I watched my son for his entire first 90 hours on DreamBox, and if it were anything remotely like Skinner’s vision, I would have pulled him off immediately. But it wasn’t, and he now has far better number sense than I did at his age.

First, good point about the word ‘context.’ I’m using ‘context’ in a pretty straight-forward dictionary sense: “the set of circumstances that surround and give meaning to a particular situation.” I think often in math the terms “context” and “real world” get used synonymously, when they are distinct in some ways. Whether using technology or developing learning experiences for the classroom, what students need are situations that enable sense-making. Quite often, those situations do arise from the real world. But it’s not always a requirement.

The open number line in this DreamBox lesson is a model, but is also a context in itself by the definition above. Not a truly “real-world” context, but an accessible situation for young learners. Students using DreamBox work with the number line quite extensively beginning in 1st grade and all the way up to adding and subtracting integers and rational numbers. So students who are ready for this particular “zooming” lesson understand the number line. We built interactivity around this particular rational number line in order to help students understand it’s continuity and a sense of scale for different ranges on the number line. The magnifying glasses add context that helps students make sense of the real number line.

That said, locating a point on a number line is definitely a rote skill. And it’s one that’s being assessed by teachers in classrooms on paper and also by software. At DreamBox, our promise to teachers is that what students learn in DreamBox will transfer to student learning in the classroom. So some DreamBox lessons do involve rote skill execution, and demonstration of skill is required to be deemed proficient in DreamBox. But they key is how students come to be able to execute that skill. Simply being able to locate -8.24 on a number line doesn’t give much insight into what a student understands about rational numbers.

So in the DreamBox lesson progression, students must use the magnifying glasses and scrolling feature in order to demonstrate deeper understanding. In some lessons, students don’t see ten tick-marks, but only three tick marks. They may be asked to locate -8.24, but the only marked points are 300, 800, and 1300. By manipulating that number line display to locate -8.24, students show much deeper understanding than most number line assessments require. You can see this type of problem at 2:39 in a video I narrated here about this manipulative (http://bit.ly/10mVsSR).

What’s critical is that our teachers at DreamBox write lessons in ways that enable our intelligent adaptive engine to monitor how many times a student uses a particular magnifying glass and how often a student zooms out (along with many other things). So if a student zooms out too frequently, then they likely aren’t proficient at locating rational numbers on the number line. In DreamBox, you can’t pass lessons just by clicking ‘hint’ enough times or using our adaptive scaffolding. If a student is getting most of the problems wrong or requires a lot of scaffolding, he or she will be able to keep doing a few more problems (a good user experience), but in reality the assessment engine likely has already determined they aren’t proficient and it will respond in ways that our teachers have designed it to respond.

No need to apologize for the flipped classroom connection. It’s an important clarification, but not something that offended me. And related to that, the word ‘lesson’ is another good point you bring up. We use that term because it’s familiar for all educators and represents an increment of learning that is bigger than a problem and smaller than a unit. But it does bring along with it some preconceived notions – most notably that a lesson begins with students receiving some instruction and then practicing what they’ve learned. That’s probably influenced by Madeleine Hunter’s work. In reality, students in DreamBox are continually and seamlessly doing a combination of investigation, experimentation, fluency development and practice within the same unit or a mix of different units. In a 30-minute session, students usually complete somewhere between 3 and 9 lessons, depending on their age and the particular unit they’re working on.

Lastly, and perhaps the most important point, is the constructivism conversation. You and I are operating under the same premise of constructivism vs behaviorism. At DreamBox, we try to emulate what great teachers do to facilitate sense-making and transfer. The analytics empower it, but the student actions that the system is analyzing are critical. Someone not associated with DreamBox recently wrote this blog post about learning analytics (http://bit.ly/YBhHmr), and it’s a thoughtful description of many things our adaptive engine does.

In a nutshell, great teachers engage students with meaningful tasks, observe how they approach and solve the problems, facilitate inquiry and the testing of ideas, and give useful feedback – particularly not simply saying ‘you’re doing this wrong’ when a student shows evidence of misconception. As an example, in early grades, DreamBox engages students with the 10-frame and Math-rack, both which are useful manipulatives for students to understand the 5- and 10-structures of mathematics and to build on their innate abilities to subitize 2, 3 and sometimes 4. Great teachers who know how to use these tools strategically help their classrooms develop early number sense. But it’s difficult even for those teachers to know exactly what every single kindergartner is thinking about the tools and the quantities they represent.

Just having a math rack to explore with isn’t enough. Students need to have clear tasks and feedback as they work with them, and that’s what DreamBox provides in our lessons. That’s also where the constructivism wiki definition comes in. Students create, innovate, and conceptualize using these well-designed tools. For example, we ask them to build the number 6 using a Math-Rack with 20 beads. The question is not entirely open-ended because there’s valuable learning when we observe students as they complete a defined, accessible task. And the way a particular student composes the number 6 gives us insight into their thinking. It’s not about having a whole environment to empower constructivism, but rather presenting students with smaller, defined challenges they must analyze and reason through without first having someone show how it’s done. Perhaps you’re thinking of constructivism on a much larger educational scale, while we have constructivist pedagogy built into smaller learning situations.

I hope this helps describe what we do a bit better. I truly appreciate your thoughtful critique and analysis and am happy to discuss pedagogy and technology with you more. It’s a sorely missing part of most conversations. That’s why I gave a webinar about it last week in case you’re interested (http://bit.ly/16cscjo).

I’m really sorry the link to my former district’s goals didn’t work. Try this one: http://bit.ly/YpgnGV and click on “2011-16 Strategic Plan (pdf)” near the bottom.

Thanks,

Tim