Thanks for your question, Jacqueline. There are really two different answers, based on the focus and purpose of the professional development. In terms of using the software itself, we have provided teachers with a brief introduction prior to their using CLP, then been present to provide technical support during the class. We’ve put a lot of effort into making the system interface intuitive and easy-to-use, and we’ve found that most teachers get comfortable with the software itself within a few days.

However, choosing and using student work for productive class discussion is a different skill, one that relies on a teacher’s deep understanding of the math and “pedagogical content knowledge.” We have not undertaken professional development around this topic as part of the INK-12 project. Some teachers we have worked with have had training in the use of student work for class discussion and/or are using a curriculum that includes explicit support for this practice (e.g. Investigations in Number, Data and Space, which was developed here at TERC). Others, however, have little experience with this approach. In cases where teachers do not have good strategies for building class discussion around student work, we have taken an active co-teaching role, looking over student work with them after class, helping them design class discussions, and occasionally leading class discussions ourselves.

Successful implementation of any innovation in math instruction that requires teachers to analyze and respond to evidence of students’ mathematical reasoning requires considerable professional development. I have no doubt such PD could be developed and offered – but school districts also need to provide the time and financial support for teachers to engage in such PD.

Thanks, Sean; good question, and one on which we’re focusing at the moment. Our findings so far are in the form of case studies. We’ve seen students make conceptual progress after using the tools. A student, for example, who was quite fluent in using a traditional division algorithm, didn’t understand place value in division: After using the division tool, he had an aha moment when he said that he hadn’t realized that a 1 in the 10s place was actually a 10 rather than a 1. For other students, we’ve seen increased independence and confidence. In one case, a student who was not able to work without a teacher’s help, and who was routinely given smaller numbers than the rest of the class, was able to use the tablet tools to independently work the same problems as her classmates. This student also participated in class discussion, which the teacher said was a rare occurrence. We’ve also seen when working with students who struggled with organizational skills that the tools provided enough structure for the students to be able to start a problem and see a path to a solution, something many of them weren’t able to do on paper. We’ve seen students in a class using the tablet software create more representations on an assessment than students in classes working with paper and pencil, but we’re still analyzing that data—-clearly classroom culture and different teaching approaches make a big difference. One of our summer projects is to more definitively figure out what we’ve learned. So stay tuned!

Thank you, Kimberle. I definitely will stay tuned. The case study evidence you cite above is compelling. I’m guessing that the technology is more than just another mode of engagement, that it has affordances that go beyond traditional modes of engagement. It sounds like the embedded scaffolding of organization might be one of those features.

Good observation. I’d say that the labeling of the jump size on the number line is another form of scaffolding. There’s a similar feature, not shown in the video, that enables students to turn on a label that shows the size of the group represented by a stamped image they’ve created. We debated about whether to display the group size below a stamped image, thinking that it might be best in some cases for kids to write the repeated addition or skip counting themselves, but opted to let the kids decide when/if they wanted to be reminded of the group size. It certainly makes designing the UI easier if you can let the users make some of their own decisions! But it’s also interesting from a research point of view to look at when kids decide to use labels and what sorts of tools they choose for particular types of problems. We’re looking at these sorts of issues now as we analyze our data.

Good question, Gilly. We’ve been pleased to find that it takes about 5 minutes for students to learn to use the tools. Kids are very facile with technology these days, and we’ve also worked to make sure that the tools are easy to use, match the task at hand, and use familiar analogies. So the stamp tool resembles a physical stamp; the way the array tool is used resembles the TERC Investigations array cards and associated games; the division tool had its beginnings in the Investigations factor cards. We’ve worked with another curriculum as well—-one that didn’t rely quite as much on students creating their own representations. The students were able, though, to use the tools quite effectively. (Adapting a curriculum for use on the tablets, especially when there is a mismatch between pedagogical approaches, is a whole different conversation, as you can imagine!)

Thank you, Arthur, and good questions. The software can be used with a mouse (or finger) instead of a pen, though drawing and writing with a pen is much easier and more intuitive than with a mouse. We’re working to get the software running on inexpensive tablets, so that more kids can have access to the use of a pen. Some of the tools can be used fairly easily with a mouse though—-the array and division template, for example. It’s just harder for students to annotate their work. We don’t have a web-based version mainly because browsers currently don’t support pen interaction. We do have some anecdotal evidence that the tools help ELL students in learning math: We worked with a class of seven 4th and 5th grade students who had been identified as having learning difficulties, and several of them were ELL students. We saw the ELL students, especially one boy who had very little knowledge of English, being able to articulate their thinking using the tools, and we saw increased independence and confidence as a result. We have some ideas about a proposal to fund work with that population, but nothing specific yet. We’d be happy to get back in touch when we think more about it, though.

A p.s.: The software currently is written in C# and runs on Windows machines. The students and teacher have machines, and a machine is connected to a projector. The machines all communicate via a wireless network; we typically use a local peer-to-peer, but any network will work. We’ll be making the software available at some point this next year. We’re also considering implementing a version that would run on Android machines.

Thanks for your questions, Davida. I hope some of the other replies have given you some ideas of how students and teachers use the software. One of things we’ve learned is that individual students tend to gravitate toward particular tools – and that particular types of problems tend to evoke the use of different tools. One fairly obvious example is that students are more likely to use the stamp with a word problem that has objects that are easy and useful to draw – like the problem in the video about the number of legs on spiders and butterflies. Some students, though, will use a stamp even for problems that aren’t as obviously visual – e.g. a problem about bags of rice that sell for $7 each. Several students (all of them girls, it turns out) created a stamp with a brown bag-like lump with the number 7 on it. These students were more likely to use stamps on other problems, as well. In contrast, there were several students who often chose to use number lines – and other who preferred not to use any representational tools. For us, these findings argue for providing students with choice in the kinds of representations they can use, knowing that they will have individual preferences.

I also wanted to address in particular your question about models of multiplication that are different from repeated addition. Our tools generally support thinking of a multiplicative structure as a number of equal-sized groups. We don’t have tools, for example, that would lead students to the idea of multiplication as a scaling operation. However, our array tool, in particular, helps students think about how numbers can be decomposed and recomposed in multiplicative structures – i.e. how partial products can be combined. This allows them to visualize both the associative and distributive properties. For example, because students can cut and recombine arrays, they can see how 25 X 12 can be thought of as (25 X 4) X 3 = 100 X 3 = 300. I’d be happy to talk more about how other tools could extend the ways students think about multiplication, if you have some particular ideas to share!

A good point, Gerald. We have definitely seen students use representations to solve problems – but not always! We are currently coding the student work we collected in one third grade class to get a sense of the ratio of times students used the representation to solve the problem vs. to “show their thinking” – although sometimes it’s hard to tell the difference. On the one hand, we’ve seen students write an answer, create a representation, discover that their first answer was incorrect based on the representation and correct their answer. On the other hand, we’ve seen students who created a representation they could only have specified if they already knew the answer – e.g. a student who created a 4 X 5 array in response to the problem 20 / 4. One question we’re pondering: Is it useful for students to create a representation, even if they don’t use it to find the answer? What do you think?

Of course we ask students to show representations often to assess their understanding of a concept or procedure, so it is important, beyond using it as a tool to solve the problem. Being able to flexibly move between representations, symbolic, graphic, verbal is a clear indication of understanding. On a slightly different note, we try to provide representations for students in an attempt to support and develop understanding. It seems we know less about the types of mental representations students use in their thinking. Is a number line or array, etc the picture that is in the kid’s head? Probably not. But these representations are important to learn as part of their mathematical “vocabulary.”

Thanks very much, Kathy. Both power and flexibility have been high priorities for us. We did interview students, both informally during class while they used the software, and also more formally before, during, and after using the software. The classroom observations also gave us a sense for how the students used the tools. In addition. we’re able to look at each student’s process of using the tools because we can play back the history of interactions (as shown in the video). It’s been fascinating to see the underlying differences in students’ creation of what look to be very similar final artifacts. We’re analyzing that work now, but what we’ve learned so far in terms of how students use the tools is that, not too surprisingly, some students create a representation then fill in an answer; others fill in an answer first, then create a representation. In some cases, we think that creating the representation did help a student solve a problem; these situations are most obvious when a student writes in an answer that happens to be wrong, creates a representation, then corrects the incorrect answer. In quite a few cases students try out multiple representations, but leave only one as their final product. In some cases it looks as if students are verifying their answers using the different representations. In other cases, it looks as if students started out with one type of representation, e.g., a number line, then before finishing the representation switched to another. So your question is a good one, and we’re working on answering it “as we speak”.

As for availability, we’ll be making a version of the software available sometime this next year; it’ll be downloadable from our project website.

Fascinating to hear about the differences in interaction. It sounds like having the back-end data / playback ability is really providing a nice insight into their learning pathways, as well as evidence of the role of the multiple representations. Thanks for the detail!

And thanks for your insightful questions!

Hi Teon – thanks for your question. Whether students have to show a representation even if they haven’t used one to find their answer is more a matter of classroom culture than technology. We tend to encourage students to do so for several reasons: sometimes they discover that their answer is incorrect when they create a representation; even if their initial answer was correct, we may find out more about their mathematical thinking if they’ve created a representation; modeling a mathematical situation with a representation may push a student’s thinking to deeper levels. Also, see the response to Gerald Kulm’s question above, in which I give a few examples of the ways in which students use representations.

Thanks very much for your comments, Kelly. As I mentioned in an earlier reply, we have worked with students identified as having learning difficulties, and we were pleased to see that the tools did seem to help them. The stamps in particular seemed to help students who had fine motor issues: They could draw just one image and easily make multiple copies of it. They also were able to use the array tool quite effectively. We didn’t have the number line tool implemented at the time, so they didn’t try that one out. We have seen some students with fine motor issues have trouble drawing the number line arcs, though, e.g., missing the end point of an arc and having to erase and start again multiple times. We’re implementing a new version that we’ll try out with the same students in a few weeks. The idea with the new design is to let them just make a single tick mark on a number, and the software will draw an arc for them. As you can imagine, we’ve had lots of conversations about the tradeoff between what students do themselves and what software does for them. We hope that for students with fine motor issues, the new version of the number line tool will strike a good balance between student agency and software help.

Thanks for your comment, Janet. We’ve worked hard to create a tool that leaves as much agency in the hands of the students as possible – yet provides some analyzable data for teacher and researcher. This is an interesting and challenging line to walk. One thing we’ve learned – which is alluded to in several other responses – is that it’s useful to provide a variety of tools and representations for students. We’ve found that students often have particular representations that they find easier to think with – or representations that they feel don’t fit certain kinds of problems. Having multiple tools also creates the possibility of asking students to compare across representations, which can generate new insights for students who have become comfortable with one particular approach.
Our design has been informed by research on mathematical reasoning and by the excellent Investigations in Number, Data and Space curriculum materials, which are themselves based on years of research. We’ve always started with the grounded knowledge of the ways in which students think – and then asked, “What can technology offer that goes beyond the manipulatives and techniques students already use?”

Thanks for making these very good points, Laurie. We didn’t want the “virtual” manipulatives to completely replace physical ones either. When the curriculum employed physical manipulatives, the students used those, then segued to the software tools. Also as I mentioned in an earlier reply, we used physical analogies to design the software tools. For the tag work, we’re currently analyzing student work—-both final product and process of creating it—-to figure out what particular characteristics and patterns of use can tell us about student understanding. We’re also using data from interviews and classroom observations. In the case of the “trouble with remainders” example, it was important to use that extra data to determine if the student was actually having trouble with the concept of remainders or just with the software tool. The analysis work is being done by hand at the moment, but we’re also working on machine analysis of the student interaction histories, with the goal of getting the software to add the tags that we identify as important.

Agreed, Jeremy! Digital tools are too often used to restrict students’ expressivity, rather than expanding it. And computer-based assessments tend to focus on questions that can be quickly and automatically scored, which usually means multiple-choice answers. We’ve made some progress in automatically understanding students’ work – and inferring something about their underlying reasoning – but it’s a research challenge to combine free expression with automatic understanding.

Thank you, Wendy! We’ve been committed to letting kids create their own representations, and you’re right, most really enjoy using the tools. (“Who knew that math could be so fun?” commented one boy.) We’ve had a few students who just wanted to write out answers, but once they see that they can identify mistakes by creating representations to check their work, they generally do use the tools. And most kids love the easy access to colors. We’ve seen some creative use of color for mathematical meaning, e.g., showing which partial products map to particular areas of an array, as seen in the video; or using a number line for two-part problems and using different colored arcs to distinguish the different parts. We should confess, though, that with some kids who like to draw we’ve had to encourage them to not obsess, e.g., over creating the perfect image for a stamp. The teachers assure us, though, that the same students would be obsessing on paper and in some cases would get upset over the state of the paper after many erasures. At least the tablet pen lets them erase easily and quickly!

Thanks, Jessica! We’ve seen all of the tools we’ve developed help students learn at times – what’s been interesting is how different tools seem to provide support for different students at various points in their developing a sense of multiplicative structure. Each tool highlights certain potential issues in students’ reasoning, so they have turned out to quite valuable as research resources as well. For example, watching students use the number line for division gave us new insight into the ways that they distinguish between two kinds of division: partitive (number of groups known, group size unknown) and quotative (group size known, number of groups unknown). Because a jump on the number line is more easily interpreted as a “group,” some students would only use the number line with quotative problems, where the group size was known. The representation didn’t map as easily to their model of partitive division.