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New Conference Paper: Time Tracking and Fact Fluency

I just returned from the 2019 AERA Annual Meeting in Toronto! It was a great meeting where I heard about lots of interesting research, met new colleagues, and best of all, got to present some of my own work.

My contribution to AERA this year was a paper I wrote with my friend and colleague, Dr. Meg Bates of UChicago STEM Education. A few years ago, we were given access to de-identified data from kids using an online fact practice game associated with the Everyday Mathematics curriculum. One of the most interesting features of the game is that students self-select one of three timing modes:

  • They can play in a mode with no time limit or tracking, where they take as much time as they need to answer each question and no information about their speed is reported to them.
  • They can play in a mode called Beat Your Time, where they still take as much time as they need to answer each question, but their total time is reported at the end of the round and compared to their best previous time. So, time is tracked but not limited.
  • Lastly, they can play in a mode with a 6 second time limit on each question.

When we noticed this feature of the game (and its user data), we starting digging into research on timed fact drills. It’s a highly discussed and controversial issue in elementary mathematics education. On one hand, several prominent researchers argue the potential connections to mathematics anxiety and inhibition of flexible thinking outweigh any benefits (e.g., Boaler, 2014; Kling & Bay-Williams, 2014). On the other hand, it’s well established that efficient production of basic facts is connected to later mathematics achievement (e.g., Baroody, Eiland, Purpura, & Reid, 2013; Geary, 2010). And arguably, even if it does not have to be discussed directly with kids, efficiency involves some amount of speed.

Overall, we were surprised at the inconclusive nature of the research when taken as a whole. There may be connections between timed fact drills and outcomes we don’t want (like math anxiety), but there has not been much unpacking of what features of timed testing are problematic. The game data — in particular, the contrast between the time limit mode and the Beat Your Time mode — gave us an opportunity to look at one particular issue: Does a focus on time always lead to detriments in fact performance, or is it specifically when time is limited?

Our analysis suggests that time limits may be the culprit. We compared students’ overall levels of accuracy and speed across modes, and found that students playing in the time limit mode had significantly (and practically) lower accuracy than when the same students played in the other two modes — but there was no practical difference in accuracy between the no time mode and Beat Your Time mode. So, in short, time limits were associated with lower accuracy, but time tracking was not.

When it came to speed, students were fastest in the time limit mode, but were still faster in the Beat Your Time mode than in the no time mode. So, Beat Your Time mode seemed to promote speed, without the detriment to accuracy associated with the time limit mode.

We were excited by this result. Although we can make no causal claims, the results do suggest that challenging kids to monitor their own speed when practicing facts could support the development of speed without promoting anxiety or other negative outcomes that can lead to lower accuracy (and general bad feelings about math). Although we did not see this result coming, it does make sense to us, upon reflection, that self monitoring could be helpful. In the future, we hope to do (or inspire others to do) more research on how metacognitive strategies could be applied to fact learning.

You can read the conference paper here and check out my slides here.

References

Baroody, A. J., Eiland, M. D., Purpura, D. J., & Reid, E. E. (2013). Can computer-assisted discovery learning foster first graders’ fluency with the most basic addition combinations? American Educational Research Journal, 50(3), 533-573.

Boaler, J. (2014). Research suggests that timed tests cause math anxiety. Teaching Children Mathematics, 20(8), 469-474.

Geary, D. C. (2010). Mathematical disabilities: Reflections on cognitive, neuropsychological, and genetic components. Learning and Individual Differences, 20, 130-133.

Kling, G., & Bay-Williams, J. M. (2014). Assessing basic fact fluency. Teaching Children Mathematics, 20(8), 488-497.

New Conference Paper: CT Implementation Profiles

Hi, all.

I just returned from the 2019 Annual Meeting of the Society for Information Technology and Teacher Education (SITE), where I made my first official presentation for the CT4EDU project. I wanted to share a bit about the paper and presentation for any interested folks who were not there.

We’re just finishing up our pilot year of the CT4EDU project. The project is an NSF funded research-practice partnership (RPP). Michigan State University (PI Aman Yadav and co-PIs Christina Schwarz, Niral Shah, and Emily Bouck) is working with the American Institutes for Research and the Oakland Intermediate School District to partner with elementary classroom teachers to integrate computational thinking into their math and science instruction. Over the spring and summer of 2018, we introduced our partner teachers to four computational thinking ideas: Abstraction, Decomposition, Patterns, and Debugging. Then we worked with our partner teachers to screen their existing mathematics and science lessons for opportunities to enhance or add opportunities for students to engage in these ideas. In the fall, the teachers implemented their planned lessons and we collected classroom video. (Note that in this first round of implementation, all of the lessons were in unplugged contexts.)

One of the first things we noticed was that there were some clear differences among teachers’ implementations of CT. In this work-in-progress paper, we share three patterns of implementation that we identified:

Pattern A: Using CT to Guide Teacher Planning
Some teachers were explicit within their plans about where they saw the CT ideas in their lessons, but did not make the CT ideas explicit to students during implementation.

Pattern B: Using CT to Structure Lessons
Among the teachers who did make CT explicit to students, some focused a lesson strongly on one particular CT idea. We described this pattern as structuring the lesson around a CT idea.

Pattern C: Using CT as Problem-Solving Strategies
Other teachers who made CT explicit in implementation seemed to reference the CT ideas more opportunistically. Rather than structuring opportunities to engage with one CT idea, they pointed out connections to one or more CT ideas as they worked through problems.

We’re looking forward to exploring how these different patterns of implementation relate to student thinking about CT as we go into our last year of the project — particularly as we begin considering ways to bridge students’ work in unplugged contexts to plugged activities.

You can find the conference paper here.

There is a version of the slides here.
(Sadly, the slides are missing the classroom video, which is clearly the best part of the presentation!)

Many thanks to everyone who came to my presentation.

New paper: Debugging LT

The LTEC project has a new paper out in the SIGCSE 2019 proceedings, authored by myself and my colleagues Carla Strickland, Andrew Binkowski, and Diana Franklin. It’s a new addition to our series of SIGCSE and ICER papers that detail learning trajectories that we developed through review of CS education literature. This time, the trajectory is about debugging (Rich, Strickland, Binkowski, & Franklin, 2019).

(If it’s helpful, you can read my description of what a learning trajectory is here.)

Although the overall approach we used to develop all of our trajectories was basically the same, we’ve tried to make a unique contribution in each publication by making particular parts of our process transparent through each paper. In our paper from SIGCSE 2017, we talked about the overall literature review and what we noticed as we examined the learning goals embedded in the pieces we read. In our paper from ICER 2017, we shared how we adapted our overall process from other work in mathematics education and focused on our synthesis of learning goals into consensus goals. In our paper from ICER 2018, we focused on one trajectory to give us room to discuss every decision we made in ordering the consensus goals.

This time, in addition to sharing a new trajectory, we also highlighted how we used the theoretical construct of dimensions of practice (Schwarz et al., 2009) to help us organize our consensus goals. We’re also really excited to be able to share more about the role that our learning trajectories played in the curriculum development we’ve been working on for two years now. We’re dedicating a significant piece of our presentation at SIGCSE to sharing an activity we are really proud of and how the trajectory shaped its development.

If you’ll be at SIGCSE, we hope you’ll come and check us out on Friday at 2:10 in Millennium: Grand North! (If you don’t come for me, come for Carla! She’s a great speaker whose PD facilitation is famous on Twitter.)

If not, please check out the paper if you are interested. Right now, the link above and the one on my CV page take you to the normal ACM digital library page. I’ll be switching the link in my CV to a paywall free version as soon as the Author-izer tool links this paper to my author page. At that time, we’ll also be sure to add a paywall-free link to the LTEC project page.

Although we have one more learning trajectory (on variables) that has been developed but not yet published, I suspect this might be the last conference paper from this work that I first author. The project is continuing to do wonderful work and you’ll be hearing more from us, but I’m into the thick of graduate school and not nearly as involved in the work any more. So, I just want to say that working with my colleagues at UChicago STEM Education on this line of work has been among my proudest and most gratifying professional experiences. I want to thank all of my collaborators, and also say a particular thank you to Andy Isaacs and George Reese, as without their graciousness I never would have had the opportunity to co-PI the project.

I’d also like to say thanks to all the folks in the CS education community who have been so receptive of our work and offered us such wonderful and helpful feedback. We’re particularly gratified for the shoutout that Mark Guzdial is giving us in his SIGCSE keynote this year.

From the bottom of my heart, thanks to all of you for making this longtime math educator who wandered into the CS education space feel welcome and like her contributions are worthwhile.

References

Rich, K. M., Strickland, C., Binkowski, T. A., & Franklin, D. (2019). A K – 8 debugging learning trajectory derived from research literature. In Proceedings of the 2019 ACM SIGCSE Technical Symposium on Computer Science Education (pp. 745–751). New York: ACM.

Schwarz, C. V., Reiser, B. J., Davis, E. A., Kenyon, L., Acher, A., Fortus, D., Shwartz, Y., Hug, B., and Krajcik, J. (2009). Developing a learning progression for scientific modeling: Making scientific modeling accessible and meaningful for learners. Journal of Research in Science Teaching, 46(6), 632–645.

 

 

Vocabulary, Part 3: What I Learned (3rd edition)

This is my last post for the semester! At the end of my last two semesters, I wrote posts listing five things I had learned. I figured since this post is also serving as Part 3 of my vocabulary series, I’d try highlighting five things I caught myself saying recently that illustrate some significant learning over the last year and a half.

 

  1. As our last research team meeting, in a discussion about a study my lab-mate is planning, I said: Oh, so if that’s when you’re doing the measure, that works. Then it’s a delayed treatment design. I entered my research design class rather skeptical at the beginning of this semester, but it’s clear I picked up some useful vocabulary for talking about research. Even if I understood the concept of delayed treatment before, I couldn’t articulate it.
  2. It has been similar with specific processes of data analysis. A few days ago, my office-mate asked me something to the effect of: You have three research questions but you’re using content analysis for all three, right? My response made very specific use of the terms content analysis, thematic analysis, linguistic analysis, and discourse analysis with a particular meaning behind each one. I can’t write perfect definitions, but I understand the difference. A year ago those would all have had the same connotation to me. (Essentially, they all meant to look at text and try to find patterns. Which is not entirely wrong, but overgeneralizes.)
  3. In a class recently, while talking to the professor about an assignment, he said, You always seem to anticipate me disagreeing more than I actually do. My response? Well, I mean, I’m just participating in discourse as I’m thinking. This was a joke — one you probably won’t get unless you have read some of Sfard’s work recently. I was joking that as I’m writing papers, I have a hypothetical discussion with my mentors in my head, trying to anticipate how they’d respond. This fits with Sfard’s (2008) notion of thinking as communicating with oneself, a theory we had discussed that day in class. The joke isn’t all that funny, even if you do know Sfard, but I did think it was interesting the way I was able to spontaneously use her definition of participation in context.
  4. This semester, I wrote and rewrote my practicum proposal justification at least three times from scratch. This was a rough experience in a lot of ways — wildly frustrating and anxiety-provoking. But I have to admit that when I finally landed on an approach that was working, I thought to myself, Oh my gosh. I think I know what a concept paper is! We talked about concept papers in one of my courses last year, and even after reading examples and attempting to write one, I really had no idea what it was. In the end, I’m reasonably sure the first half of my practicum proposal became a concept paper. I could use that term correctly now in conversation. That feels like a victory given how much I struggled with it last year.
  5. And now for a bit of sappiness (it’s the holidays, after all). When I started my master’s program, I can remember needed to learn the specific meaning of cohort used by academia. I knew of the word before that, but I knew it as an old-fashioned way to refer to a friend, collaborator, or partner-in-crime. I didn’t know the collective meaning of a group of students who enter a program together. I re-learned it again over the last year and a half, and it’s become even more meaningful as a PhD student. My cohort is my tribe, and I’m grateful for them.

Have a lovely break, everyone. Reflect on all you’ve learned. Try not to think too much about how much there is still to go.

Reference

Sfard, A. (2008). Thinking as communicating. Cambridge: Cambridge University Press.

 

Vocabulary, Part 2: Jingling Abstraction

In my last post, I talked about the value of using precise vocabulary in curricular resources. Today, I’m going to talk about (potentially) problematic vocabulary used in research and development.

There are twin problems of vocabulary in academia. Referring specifically to research on student engagement, Reschley and Christenson (2012) called these the “jingle, jangle” problems. The first problem is that we sometimes use the same word to refer to multiple ideas (jingle). The second problem is that we sometimes use different words to refer to the same thing (jangle).

I can think of plenty of examples of jangle — in particular, I think we use the terms real-world, relevant, contextualized, and authentic to refer to problem contexts when we really just interested in engaging problems. But I’ve been wrestling all year with an example of jingle. In short, I think that the word abstraction, as a learning goal, means rather different things to mathematicians versus computer scientists.

In general terms, abstraction can be used as a noun or a verb. As a noun, an abstraction is a representation that reduces complexity in order to focus attention on the core, essential elements of a situation or phenomenon. Usually an abstraction exists independently of any specific example or instance. The fraction ¾, for example, is an abstraction of three out of four equal-size pieces of pizza. The fraction can be used to represent a specific portion of any whole — it exists independently of the examples.

As a verb, abstraction is used to refer to the process of creating such representations.

For both the noun and verb meanings of abstraction, I do not think that mathematics and computer science differ too much. However, I do think there are subtle differences in the way the disciplines talk about abstraction as a learning goal. In short, I think mathematics is focused on the noun, and computer science is focused on the verb.

Admittedly, the Standards for Mathematical Practice in the Common Core State Standards for Mathematics (CCSS-M; Common Core State Standards Initiative [CCSSI], 2010) do highlight the process of abstraction. Standard for Mathematical Practice 2, Reason abstractly and quantitatively, says, “Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically …” (p. 6). (The other ability is contextualizing.)

However, when looking across grade levels within the content standards, I can’t help but notice a trend toward working with abstractions, rather than creating them.

Consider the following three standards about place value (CCSSI, 2010):

1.NBT.2     Understand that the two digits of a two-digit number represent amounts of tens and ones.

2.NBT.2    Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.

4.NBT.1    Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.

The first- and second-grade standards are focused on understanding numbers written in base-10 notation as abstractions of quantities. The fourth grade standard highlights an even higher level of abstraction. It generalizes the base-10 place value system, independent of the examples of two- and three-digit numbers that are the focus in first and second grade. So there are two themes here: Understanding abstractions, and moving from less abstract to more abstract understandings and representations of mathematics. The movement is in one direction, and because that movement is separated across grades, it’s not clear that students will even be aware of the process of abstraction happening.

Getting to the highest possible abstraction is an implicit goal in a lot of mathematics education. In addition to sequences of standards like the one above, some instructional frameworks highlight this specifically. Two examples are the concrete-representational-abstract framework (Agrawal & Morin, 2016) and concreteness fading (Fyfe, McNeil, Son, & Goldstone, 2014).

By contrast, my read of computer science education literature is that there is a greater emphasis on using multiple levels of abstraction, and that necessitates greater focus on the process of moving among those levels. For example, the Computer Science Principles (CSP) framework references the process of abstraction first (and the result last) in its description of Abstraction as a Big Idea: “In computer science, abstraction is a central problem-solving technique. It is a process, a strategy, and the result of reducing detail to focus on concepts relevant to understanding and solving problems” (College Board, 2017, p. 14). The process also is highlighted in the specific learning objectives (which I’d consider parallel in specificity to the mathematics content standards): “Develop an abstraction when writing a program or creating other computational artifacts” and “Use multiple levels of abstraction to write programs” (College Board, 2017, p. 15).

Specific instructional approaches for abstraction in computer science also emphasize moving between levels of abstraction. For example, Armoni (2013) outlined a framework for teaching abstraction to computer science novices. She emphasized being explicit about moving between levels of abstraction in order to help students learn to move freely and easily between levels as they problem solve.

Thus, mathematics focuses on abstractions. Computer science focuses on abstracting. Both refer to abstraction, but often mean different things in terms of the goals of learning.

I’m not claiming that one focus or the other is inherently better. But I do think the difference is important to keep in mind, especially when thinking about integrated instruction.

Reference

Agrawal, J., & Morin, L. L. (2016). Evidence-Based Practices: Applications of Concrete Representational Abstract Framework across Math Concepts for Students with Mathematics Disabilities. Learning Disabilities Research and Practice, 31(1), 34–44.

Armoni, M. (2013). On teaching abstraction in computer science to novices. Journal of Computers in Mathematics and Science Teaching, 32(3), 265–284.

College Board. (2017). AP Computer Science Principles Course and Exam Description. Retrieved from https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap-computer-science-principles-course-and-exam-description.pdf

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/Math/

Fyfe, E. R., McNeil, N. M., Son, Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25.

Reschly, A. L., & Christenson, S. L. (2012). Jingle, jangle, and conceptual haziness: Evolution and future directions of the engagement construct. In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of Research on Student Engagement (pp. 3–19). New York: Springer.

 

Vocabulary, Part 1: Why precision might matter in written resources

I’ve been thinking a lot lately about the role of vocabulary in learning, teaching, and research.

Until recently, I didn’t have a very well developed opinion on whether or not we need to worry about kids’ use of mathematical vocabulary, at least at the elementary grades. There were a few ill-articulated assumptions underlying my lesson writing style, though. Generally, I believed that:

  • Helping kids to learn definitions should never be the main point of a lesson. An idea is bigger than its definition.
  • Similarly, perfect use of the vocabulary shouldn’t be the main learning goal of a lesson. I don’t think imperfect expression should invalidate an idea, especially if it is coming from a young child.
  • On the other hand, if an idea is key to a lesson, there’s no reason not to introduce a term for it. I don’t believe in withholding words from kids because they are long or “difficult”. Words can be powerful tools.
  • Even though precise use of vocabulary isn’t an appropriate expectation for kids, I do think teachers should try to be precise in their use of the terms. And that means that curriculum materials should be precise in their mathematical language, too.

I was kind of a stickler about that last point in my curriculum writing days. I think it sometimes annoyed my coworkers, who believed that no teacher was going to notice or change her practice if we wrote The sides are equal in length instead of The sides are equal or said The angle measures 42° instead of The angle is 42°.

I had to admit at the time that they were probably right about that. Teachers have limited time to read curriculum materials and plan and I’m doubtful they spend that time paying close attention to subtle differences in language. Still, when I caught language that was imprecise, I was stubborn about changing it. I justified this mostly by arguing that if any mathematicians reviewed our materials, this would give them one less thing to pick at.

I read and article this semester, though, that made me wonder if there was a bigger reason for precise language than that. Gentner (2010) published a lengthy argument for the reciprocal relationship between language and analogical reasoning. The first half of the paper summarized research suggesting that making comparisons facilitates learning. The second half was a more specific argument about the role of language in learning and cognitive development. Gentner argued that:

  • Having common labels invites comparison of examples and abstraction of shared elements, and
  • Naming promotes relational encoding that helps us connect ideas that are learned in different contexts.

To illustrate her argument, Gentner cited research about learning to apply natural number labels to quantities. Studies of cultures whose languages do not contain specific number labels showed that people of those cultures were able to estimate magnitudes, but were not very accurate at assigning specific number names to quantities, especially as the quantities got larger (Gordon, 2004 and Frank et al., 2008, as cited in Gentner, 2010). Other studies showed that children who speak languages with number names (like English) first learn the count sequence by rote, but by comparing sets of objects (e.g. two trains and two dogs) that have a common number label attached, they gradually bind the number names to the quantities (Gentner, 2010).

This explanation makes perfect sense to me. This is why words are powerful — they are a means of connecting examples at their most fundamental, definitional level. They prompt looking for sameness in contexts where things feel very different.

This got me wondering whether my stubbornness was better justified than I originally thought. Abstract mathematical terms like equal (and its symbol) are known to be poorly understood (e.g., Knuth, Stephens, McNeil, & Alibali, 2006; Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012; McNeil et al., 2006). At least one study has concluded that the contexts in which the equal sign is used impact students’ understanding of its meaning (McNeil et al., 2006). I would not be surprised if a similar study examining how the word equal is used in sentences showed that these uses impact understanding of the word. According to Gentner (2010), we both consciously and unconsciously use words as labels, which invite comparisons, which invite conclusions about meanings. It seems reasonable to suggest that if we use the word equal with counts and measures, but use the term congruent with geometric figures, teachers could abstract more sophisticated and precise meanings of those terms than if we use the term equal sides.

I don’t know for sure, of course. But I don’t think precision in language, especially in resources that teachers can continually reference, can’t hurt the educative power of the resources.

Reference

Gentner, D. (2010). Bootstrapping the mind: Analogical processes and symbol systems. Cognitive Science, 34, 752–775.

Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 297-312.

Matthews, P., Rittle-Johnson, B., McEldoon, K., & Taylor, R. (2012). Measure for measure: What combining diverse measures reveals about children’s understanding of the equal sign as an indicator of mathematical equality. Journal for Research in Mathematics Education, 43(3), 316-350.

McNeil, N. M., Grandau, L., Knuth, E. J., Alibali, M. W., Stephens, A. C., Hattikudur, S., & Krill, D. E. (2006). Middle-school students’ understanding of the equal sign: The books they read can’t help. Cognition and instruction, 24(3), 367-385.

 

Integration, Part 3: Authenticity

The end of the semester is approaching, and I’m a bit crunched for time this week. But I do have one more brief thought about integration to share.

A few years before I left my full-time work as a curriculum developer, a freelance journalist interviewed me via email about mathematics word problems and my theories on why students often say they hate them. I told her that my years of curriculum development work really opened my eyes to just how inauthentic word problems can be. Is it possible to write word problems that target a particular mathematics concept and also are meaningful to children? Yes, definitely. Is it possible to write 50+ such problems targeting that same mathematics concept? I can tell you from experience that it gets really tough, really fast.

And that’s before applying the list of constraints that comes along with the task in large-scale curriculum development. During our latest round of development, we were not allowed to reference any junk food in our problems. We also had to stick to round objects when we talked about fractions, because our chosen manipulatives for fractions were circles. How many round items can you think of that aren’t considered junk food, but are round and make sense to divide into parts? It starts out easy: oranges, tortillas, cucumber slices. But then come the descriptors that help make semi-junky food sound ok: veggie pizzas, whole-wheat pita. By the tenth problem or so, I promise you’ll be grasping at straws. I’m pretty sure we wrote some fraction problems about cans of cat food.

My point is simply this: Starting with some form of disciplinary content and back-tracking to a reasonably authentic task is difficult after the first few times. And when tasks start to lose authenticity, kids notice. The activities they complete start to feel like busy-work (because they are).

The issue I’ve been thinking about this week is whether the task of contextualizing content becomes easier or harder when you’re thinking about two disciplines, as in integrated curricula. On one hand, it seems like finding a task authentic to both disciplines might be more difficult. But on the other hand, I think part of the difficulty of generating authentic tasks is that usually authentic tasks require multiple kinds of component skills. Finding one that gives kids exposure or practice to one particular thing, but does not require any other skills they don’t yet have, is a challenge. So I think it is possible that considering two disciplines might actually open up some space to move in task development.

Take the number grid activity I discussed last week. I’ve written other activities before asking kids to map out paths on a number grid. And I’ve asked them to limit their movements to moving in rows or columns — adding or subtracting 10s or 1s. But I never had a great reason for that restriction, other than a desire to focus on place value, so often the task felt inauthentic. But when I added the element of programming a robot, suddenly the restriction in movements had new meaning: Programming languages are made up of a limited set of commands. So a very similar activity became more authentic — along one dimension, at least — through integration.

I’m hoping to find more of these happy compatibilities as I continue to think about integrated curricula.

Integration, Part 2: Translation

Here we are at the end of the week, and that means that it’s time for Integration, Part 2!

Last week I wrote about shifting my views on the development of integrated curriculum. Rather than framing my efforts as trying to understand and consistently achieve a fully integrated curriculum, I started thinking about how a long-view of curriculum development might enable a different model: helping kids to walk across Kiray’s (2012) balance, rather than stay in the middle. This view doesn’t eliminate the need to find fully integrated activities, but it shifts their role. Rather than being the activity form that supports all kinds of conceptual development within an integrated curriculum, the fully integrated activities serve as a meeting point, or a gateway between the two disciplines. At other points in the curriculum, activities might make use of convenient connections between disciplines. In those key, fully integrated activities, though, I think kids probably need to look both disciplines straight in the face, note the similarities, and also wrestle with and make sense of some of the differences.

So. What might that look like? In my case specifically, the question is, what might that look like for elementary school kids working in mathematics and computer science?

I’ve spent a good bit of time thinking about this business of synergies and differences between mathematics and computer science and what they might mean for integrated curriculum. (I’m hopeful — please oh please oh please — that soon I’ll be able to point you to a published paper or two about this.) It’s a complex issue full of nuance and I always have a hard time coming up with a clear description or list of what’s the same and what’s different. For a while I thought I had to have a handle on that before I could think about writing an activity that asks kids to wrestle with the ideas.

But then I remembered a key idea that underlies my whole educational philosophy:

Don’t do the thinking for the kids. Let the kids do the thinking.

What does that mean for a fully integrated math + CS activity? First of all, it means I don’t have to have some definitive list of synergies and differences. Kids will notice things, and talk about them, and they may or may not map directly onto my ideas. And that’s ok, because there isn’t a perfect list.

It also means that I don’t have to generate a problem context that maximizes synergies and reduces differences as much as possible. I saw that as a goal, at first. Too many differences might either distract from the big ideas that lesson is meant to address or lead to confusion in one discipline or the other later on.

But I no longer think that’s true. The point isn’t to minimize differences, but rather to have kids think about them.

Based on this thinking, here’s my new idea for fully integrated activities: Instead of figuring out the best way to address both disciplines at once, we ask kids to translate the language of one discipline into another.

For example, take a common activity that happens throughout elementary mathematics: Counting, adding, and subtracting on a number grid. I’ve written plenty of activities in my career that have kids think about making “jumps” on a number grid. To add 23 to 37, for example, they might jump down two rows to add 2 tens, and jump to the right three spaces to add 3 ones. They land on 60, the sum.

NumberGrid1

They could think about this as a missing addend problem, too. How can I get from 37 to 60? Add 2 tens, then add 3 ones.

This activity, particularly the visuals associated with it, remind me a lot of the kinds of activities kids do in programming curricula aimed at elementary school. Kids direct on-screen characters and physical robots through pathways by programming them in terms of distance and direction of travel. When I first started thinking about this, it seemed like a superficial connection based on nothing but a common grid-like structure. But more recently I’ve been wondering if the similarities go deeper. In both cases, kids are giving directions to move from one point to another. The difference is in the way of communicating that information.

Is mapping the two kinds of language about directions onto each other something kids could do? I’m not sure, but I think at least upper elementary school students could. Not only that, but I think the kinds of thinking work it would take to translate directions like this:

Add 2 tens

Add 3 ones

… into directions like this:

Face toward increasing 10s

Move forward 2 units

Turn left

Move forward 3 units

… could be beneficial to kids. Assuming they start with a printed number grid and illustrate the mathematical directions with their fingers, to change that into spatial directions they’d need to engage in some perspective-taking. To orient themselves and “think like the robot,” much like Papert (1980) advocated using using the Logo turtle as an object to think with.

I think kids might come out of that translation experience with a different way of thinking about the structure of the number grid, and also a foundation for thinking about algorithms that would support other programming activities.

Maybe the “full integration” balance point isn’t about pointing out synergies and differences between disciplines to kids. Maybe it’s about allowing kids to translate their thinking from one to the other.

References

Kiray, S. A. (2012). A new model for the integration of science and mathematics: The balance model. Energy Education Science and Technology Part B: Social and Educational Studies, 4, 1181-1196.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc..

 

Integration, Part 1: Across the Balance

I’ve spent a significant amount of time over the last four or five years thinking about integration of mathematics and computer science. My efforts have been driven, in particular, by a general dissatisfaction with a lot of the available “integrated” materials for elementary school. Without calling out any programs by name (because criticism of any program is not at all my point), I’ll say that I see an awful lot of what I’d call “slapping a math standard on a CS activity.” The integration was superficial at best — in many cases, kids could (and probably would) complete the programming activities without really engaging with the mathematics. And that makes sense, because the math is not what the activities were claiming to support.

I just felt like there had to be a better way.

I looked as several models describing different ways to think about integrating mathematics and science to help push my thinking about what that better way might be.

While working on a paper with some colleagues, I came across something called the balance model for integration (Kiray, 2012). The article was about integration of math and science, so science was swapped for CS, but otherwise the diagram of the model looked something like this:

Kiray model

Kiray (2012) doesn’t use exactly those words (“Math with opportunistic CS” is “Math-centred science-assisted integration” and “Math with coherent CS” is “Math-intensive science-connected integration”), but I think my version captures the differences he is trying to get at. Integration is opportunistic when activities take advantage of any available connection, without worrying about whether the content that is connected is a part of the expected curriculum or related to anything else students have explored in the second subject. This is what I feel like a lot of the math connections in elementary CS materials look like. Yes, kids could count the number of blocks in a program that moves a robot around a maze, and technically that’s a math connection. But it’s opportunistic, and therefore likely not all that meaningful in terms of developing math learning.

And it works the other way, too. Full-time CS ed folks (I’m only a part-timer, really) feel the same way when I say kids could think about drawing a mathematical model in terms of  CS abstraction. It took me a while to understand the skepticism coming from some corners of the CS ed world when I would talk about such things, but now I think I get it. It’s as hard for them to see how abstracting numbers from a word problem is connected to meaningful CS learning as it is for me to see how counting blocks is connected to meaningful math learning.

The types of integration that are closer to the middle, math with coherent CS and CS with coherent math, still have one discipline or the other in the driver’s seat. However, the connections to the other discipline are made with careful attention to the development of both disciplines. The way each discipline builds on itself is considered, and how the two might be made to build on each other. Still, in all the cases when a choice has to be made for better development and opportunities for learning, the driving discipline wins. So, one discipline progresses at a typical, grade-level appropriate rate, and the other lags behind what might be possible otherwise.

Finding a way to plan for coherent progressions through CS/CT content as it gets integrated into mathematics was a big part of the goal of the LTEC project. By developing learning trajectories for CT concepts to use as a guide while working on a math+CT curriculum, we hoped to help make the learning in both disciplines meaningful.

It seems to be working (results coming soon!). Still, I think one of the biggest lessons I’ve learned as I’ve watched the work progress (I want to be clear that my colleagues are doing most of the work) is the delay in one discipline is unsatisfying, even for the people (like me) who claim to care more about one discipline over the other. My inner dialogues about these issues tend to go like this:

Well, we could introduce the idea of looping here. The connection seems really solid.

Yes, but the kids haven’t really been examining repetition anywhere else, so it’s going to take some build up to that.

But… the fit is so good! We can’t give up that opportunity! Maybe we do just a straight-up CT activity in here to get them ready?

We said we weren’t going to do that.

I know. Blargh.

We either choose to give up the good connection opportunity and let the CS lag behind, or put in a CS-only or CS-with-opportunistic-math activity. Then the curriculum feels like it’s got “extra” stuff in it, and it feels kind of jerky. So even though math with coherent CS is better than math with opportunistic CS, it still just feels like there’s got to be a better way.

That leaves us with the balance point: total integration. Here’s Kiray’s description of that:

In total integration, the target is to devote an equal share of the integrated curriculum to science and mathematics. Neither of the courses is regarded as the core of the curriculum. This course can be called either science-mathematics or mathematics-science. Separate science or mathematics outcomes do not exist. An independent person observing the course cannot recognise the course as a purely mathematical or a purely scientific course. One of the aims of the course is to help the students to acquire all of the outcomes of the science course as well as the mathematics course. (p. 1186)

Sounds great, right? I’ve conceptualized much of my work over the last five years to be a pursuit for what total integration looks like. Unsurprisingly, I have not figured it out. I can think of one or two example activities, maybe, but not a whole curriculum. Plus, no school is going to be happy with no recognizable math or CS outcomes. Total integration was supposed to be the solution I was searching for, yet sometimes it felt unachievable or inappropriate for real schools, teachers, and kids.

A week or two ago, though, I was doing revisions on another paper. We got some thoughtful comments that prompted me to step back and try to articulate some of the assumptions underlying the analysis in that paper. One of them that I included in my response to the editor was this:

We believe that curriculum development has to take a longer view than one activity at a time.

It’s true. This belief has underscored all of my mathematics curriculum development work for the last 10 years. (My colleague — shout out to Carla — left a comment on that sentence saying, “Actual snaps for this!”) Yet I had not really said it to myself for a while. And when I did, I realized I’d been thinking about this balance framework in the wrong way.

What if it isn’t about picking a kind of integration and staying there?

What if, instead, development of integrated curricula is about moving across the balance?

We start with math. (I mean, it’s already happening anyway!) We take some opportunities to dabble with CS in math (which mostly involve use of CT practices, I think). We find places to build reasonable, coherent activity sequences, still letting the math drive. We work on identifying those few, golden opportunities for total integration.

And here’s the part I thought I’d never say:

Then we keep developing the CS, and start being ok with letting the strong connections to math fade and maybe even disappear.

Integration is hard to achieve even one activity at a time, and it’s really, really, hard to maintain. Maybe a continual push for fuller integration isn’t really what we want. Maybe we just want to drive toward an example or two of really rich, total integration activities and make sure they’re functioning as good transitions into learning CS without the mathematics.

How do we find and develop those golden, fully integrated, strong transitionary activities?

I’ll talk about one idea I have — next week.

References

Kiray, S. A. (2012). A new model for the integration of science and mathematics: The balance model. Energy Education Science and Technology Part B: Social and Educational Studies, 4, 1181-1196.

Success and Failure, Part 3: How to Press the Submit Button

Welcome back for Part 3 of my series on success and failure.

To recap: I know from experience it how discouraging it can be to see rejections and failures pile up. But it’s important to keep in mind that just about everyone in academia has a Failure CV as long as yours. It’s also important to keep your CV+ in mind, and know you are more than a list of published articles and funded grants.

I do not claim to have figured out how to crack the publication code yet, so I can’t provide any advice on increasing your chances of getting articles accepted. Instead, in this last post, I’m going to share some thoughts about a very necessary (but not sufficient) step in getting there: getting manuscripts out the door and submitted.

I have come to the realization over the past six months ago that I submit a lot more manuscripts than your average graduate student, both to conferences and to journals. This is not a strategy I adopted to increase my chances. Rather, I think it’s a reflection of a particular set of opportunities I’ve had as well as some of my personality traits. I’ll talk about those a bit below. But first I want to make clear that I’m not claiming that acceptances are random or quality doesn’t matter. The value of a higher number of submissions, in my view, is twofold: (1) More submissions mean more writing, which means more practice; and (2) more submissions also means more feedback, both on your papers and on the way your work suits or doesn’t suit different venues. A lot of my work sits at the intersection of disciplines (math & CS; ed psych & learning sciences; student thinking & teacher thinking, etc.), and if nothing else, the past year of submissions and feedback has helped me learn about where different kinds of pieces might receive the best reception.

So, anyway… I think getting stuff out there is good for a lot of reasons as long as quantity is not pushed to the detriment of quality. Here are three things that I think contribute to my relatively high rate of submissions.

First, I’m involved in three different lines of work with three different sets of collaborators. There is my assistantship work, the work I’m continuing from my previous job, and a line of inquiry I’m pursuing with a favorite collaborator that is outside of either of these projects. This fact alone explains a lot of the reason why I’m able to contribute to so many papers: I have an abundance of opportunity. I know I am lucky this way, and I’m not suggesting everyone should seek to be involved in three projects as if three is some magic number. But I do thinking having at least a couple parallel lines of work can be helpful. More data and more collaborators always lead to more ideas, and it’s nice to have a writing project to work on when a project is at a stage when writing isn’t possible (e.g., during long periods of data collection).

Second, I write a lot and in a lot of different contexts. I write papers, yes. But I also write this blog every week. I write instructional activities for kids and educative materials for teachers as part of my assistantship work. I take a lot of notes as I read academic articles, trying to summarize important parts in my citation manager for better location and memory of the details later. All of this serves as practice that I think contributes to my ability to produce a workable manuscript in limited time when the need and opportunity arises. As I noted above, more manuscripts mean more practice — but my point here is that more writing of any kind is also more practice, and that matters.

Lastly, and probably the most importantly, I judge my own work in absolute, rather than relative, terms. I do not consider myself a competitive person. My upbringing almost certainly contributes to this — in my family, we played games, but we never kept score. My default mindset is therefore to think about whether something is good — not whether it’s good enough for x, better than y, or still not as good as z. I judge my own work by whether or not I feel like I’ve met my goal of writing a good paper worthy of someone else’s time to read. I don’t tend to think so much about whether I think it’s good enough for a particular journal or not as significant as something I read last week. Admittedly, I think this focus on my own standards for worth and clarity does me harm sometimes. More attention to cultural and methodological norms in my discipline, for example, is something I should work on. But my modes of judgement also help me let go of papers and move on to something else while I wait for feedback.

Different styles of research, collaboration, and writing will certainly call for different ways of operating, and so this list won’t work for everyone. But I hope it provides some food for thought.

As always, thanks for reading.