- Deborah Spencer
- http://www.linkedin.com/in/deborahbspencer
- Managing Project Director
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Education Development Center (EDC)

- Katie Chiappinelli
- Senior Curriculum Associate
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Education Development Center (EDC)

- Kate Coleman
- https://www.edc.org/staff/kate-coleman
- Senior Curriculum Associate
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Education Development Center (EDC)

- Jim Galdos
- https://www.linkedin.com/in/jim-galdos-a1556442/
- Videographer and Editor
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Galdos Pictures

- E Paul Goldenberg
- https://www.edc.org/staff/e-paul-goldenberg
- Distinguished Scholar, Mathematics and Computer Science
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Education Development Center (EDC)

- Zak Kolar
- Digital Learning Teacher
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Waltham Public Schools, Education Development Center (EDC)

- June Mark
- https://www.edc.org/staff/june-mark
- Managing Project Director
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Education Development Center (EDC)

- Kristen Reed
- https://www.linkedin.com/in/kristenreededc/
- Managing Project Director
- Mathematics through Programming in the Elementary Grades
- https://elementarymath.edc.org/programming/
- Education Development Center (EDC)

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Public Discussion

Continue the discussion of this presentation on the Multiplex. Go to Multiplex

## Kristen Reed

Co-PresenterManaging Project Director

Welcome to our Math+C page and thank you for viewing our video!

In collaboration with elementary school teachers, technology specialists and mathematics coaches, this early-stage design and development project explores the promise of integrating programming into the elementary mathematics curriculum in a way that is mutually supportive of both subjects. Within the block-based programming language, Snap

!, we are developing mathematical microworlds. Each microworld contains a customized mathematical stage (e.g. a number line, a coordinate grid, an “array building” grid), at least three sets of mathematical puzzles to solve, and a clean user-interface design that presents only the features and programming blocks that children need in order to focus on and explore the mathematics and solve the puzzles.Our team is interested in your feedback and having a discussion related to these topics:

Check out our website https://elementarymath.edc.org/programming/ to access our microworlds and learn more!

Kristen Reed

## Emmanuel Nti-Asante

I thank you for this wonderful research. I do not know how you might capitalize on the mathematical language over that of the programming.

E Paul Goldenberg

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Hi, Emmanuel.

I'd been so busy writing my comment (next in this list) that I didn't see yours until just now.

I'm assuming that you are asking how we connect what children do on the computer with the kind of notation they use in math classes. (I hope that assumption is right. Otherwise, please correct me.)

Conventional mathematical notation—the normal written forms we use—is a brilliant (and hard won) invention, a language that is precise and concise and has very few ambiguities. It's

essentialfordoingmathematics, and so it's essential for kids to acquire. But that concision also makes it is a hard place tostart. Especially for little ones, reading and writing is still a developing skill and learning mathematicsthroughthat written link is hard. And mathematical notation isverydifferent reading and writing from the text reading-writing that kids are learning: every character matters, it's not strictly left to right, and even vertical position can matter. We recognize prose writing as a way torecordideas, and teachers of young children increasingly realize that the expression of those ideas is primary and getting the expression into proper form takes time. By contrast, mathematics is often treated as if itwerethe marks on paper, not a collection of ideas and a manner of thinking about and manipulating those ideasfollowed bya way to record that thinking precisely. Even when everything has been figured out, the math isn't "done" until there's something on paper.Our project hasn't been studying the transition from doing the mathematics in the programming context to recording that mathematics. (We've done a few experiments with that, but this grant cannot support serious research in that area.) But our tentative hypothesis would be that if people can get the ideas and strategies and processes straight in their heads, the notation becomes a way to summarize and record, and is easier to acquire in that order. And, of course, every other bit of mathematics instruction is

focusingon that notation, so....Kristen Reed

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Hi. I’m part of the team that has been developing the ideas you see in this project. Of the many things that have made the work on this project particularly intriguing to me, here are two that almost always come first to my mind.

Kids explore mathematics and solve problems using a language that is clear, informative, and forgiving. Beginning writers can misspell an entire sentence and we still know their intent and can decide where to focus our attention. But conventional mathematical notation is extremely terse with no redundancy to help clarify intent. If a child writes 7 + 5 = 2 we can’t tell whether the child

thoughtthe wrong thing or just slipped in the writing, omitting or changing just a single character. And looking back later, even the child may not know. Coding gives instant feedback, letting children learn the mathematical language.They also learn to evaluate their ideas and work for themselves, giving them autonomy and independence in problem-solving. We regularly hear the little yip of “I

didit!” Weneversee kids looking to the teacher to ask “is that right?” They do sometimes ask small questions about how to use the tool, but mostly figure that out by experimenting and are just as inclined to look over at another child as to ask an adult. Children gain control and confidence, visibly increasing the number of children actively engaged in the work.I’ve also found it fascinating—and fun!—to see the enormous contribution that both teachers

and childrenhave made to the evolution of the puzzles and tools. These contributions are, to us, clear evidence that the environment is “clicking” with teachers and kids, both interesting and useful. Here are just two examples of contributions from the kids: A 2^{nd}grade girl near the beginning of the year spawned several puzzles when she figured out how to reach all the numbers on a number line, explaining “I just go plus 3, plus 3, minus 5. Then do that [a lot].” Mathematical induction from a 7-year-old. Several second graders asked for abstractions we hadn’t yet provided so that when they build scripts like the one that little girl explained, they can get the effect ofcombiningthose bits of arithmetic to make itlooklike just a single jump, not several. This spawned two new tools and several puzzles.We'd love to answer any questions you have about the microworlds or the ideas behind them. And we should also note that the elementarymath website linked above is evolving, with new microworlds being added as soon as they are ready, and with other mathematical resources for teachers.

Kristen Reed

## Myriam Steinback

FacilitatorConsultant

This project is fascinating - it's a great example of children having mathematical ideas! The students' words describe their learning best, and ending with "I found that coding helps me learn" was perfect. The example you cited of the 2nd grader was just great.

I watched some of your videos about the microworlds and found them very interesting. I'm curious to hear about the support for teachers - in addition to the materials that are specific for them, do you offer them PD? I understand that it is intended to be used as a module - how are you finding teachers' use of it - I realize they already have a full plate with what they need to do; is it doable?

E Paul Goldenberg

Kristen Reed

## Katie Chiappinelli

Co-PresenterSenior Curriculum Associate

Hi Myriam! Thanks so much for your interest in our project. I'm so glad you like the student's description about how coding helps them learn–it really is such a powerful and exciting point. I'm also glad you had a chance to check out some of the microworlds posted on our website. We plan to upload some additional microworlds this week on arrays, fraction number line and decimals as well, so please stay tuned.

Our microworlds are designed to be easily accessible for teachers, as well as children. Teachers need not be experts in programming, or inintegrated with the core content being explored in the mathematics curriculum.

Snap!to use these resources. Because the microworlds limit the palette of programming blocks that children have access to, the interface is easy to learn and use. Meaning that teachers do not need to explain many technical details to children, or know how to deal with the complexities of the programming language itself. Use in classrooms is intended to be a multi-day sequence of instruction–like a mini-module–that can be fullyMost of our microworlds have an accompanying teacher guide that offers guidance on mathematics and programming content, suggestions for implementation, and lesson plans. However, several teacher guides are still in development and we can certainly keep in touch when they are fully available. Many teachers have shared that playing in the microworld, and trying each of the puzzles themselves, provides the best preparation for teachers who want to introduce the tools in their classrooms. A digital learning teacher can also be a supportive partner in the classroom for implementing these activities, as can a math coach, if either are available as a resource – and may be especially helpful when first getting started.

Of course, we're always happy to connect directly if you have additional questions and our contact information is posted on our website as well. Thanks again for your interest and we always love to hear from teachers, math coaches and facilitators about how the microworlds work for them and their students.

Best,

Katie Chiappinelli

kchiappinelli@edc.org

E Paul Goldenberg

Kristen Reed

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Addendum: We are gradually and informally learning what a TG really needs. As Katie pointed out, the startup load for a teacher is small—we've designed for easy accessibility and will have super-short (60-second) intro videos for each microworld to help—but want to be more helpful to teachers about

howthe MWs might best help their classroom. Teachers are already enormously over-busy, their teaching time is already obligated and crowded, and their planning time is also full, so any planning or classroom time they invest in having kids use the MW had better pay itself backbothin advantages to the kidsandtime repaid-saved for the teacher. The TG needs to be helpful but very, very short and easy to use for exactly the same reason; teachers don't have time for much more.Myriam Steinback

## David Kung

FacilitatorDirector of Policy

Love hearing about these students' positive experiences! As with any intervention, I'm sure this one works better for some students than others. What is the experience like for kids who aren't as engaged by the coding?

E Paul Goldenberg

## Katie Chiappinelli

Co-PresenterSenior Curriculum Associate

Hi David. One of the great things we’re hearing from teachers is that student engagement is high with these microworlds. Students often ask if they can do it again tomorrow, or whether they can work on the microworlds at home. However other students may take longer to engage with the microworlds and benefit from spending time experimenting on their own before digging into the puzzles. This level of experimentation means students are free to test out ideas as they gain familiarity with the how the microworld works. Often times, a student may then move directly into working on puzzles or even opt to create challenges of their own. Programming becomes a logical way for students to explore and share their mathematical thinking whether they jump right into the puzzles or take some extra to time experiment.

I hope this answers your question but please know that we're always happy to continue the conversation. Thanks again for your interest in our work.

Best,

Katie Chiappinelli

kchiappinelli@edc.org

## Zak Kolar

Co-PresenterDigital Learning Teacher

Hi David!

Speaking from the teacher end of this experience, I completely agree with Katie. We sometimes see a handful of students who spend more time experimenting with the blocks and other microworld features prior to engaging with the actual puzzles or between puzzles. For the most part, students who explore and experiment organically make their way (back) into the puzzles and the few who don't can be redirected with a gentle nudge. One way to do this is to ask them about the discoveries they've made and encourage them to use these discoveries to create (and solve) their own puzzles. Some of these student-created puzzles have included trying to create (or avoid) certain patterns, trying to

efficientlyproduce very high or very low numbers, trying to create their own blocks (using existing blocks) to help solve future puzzles, and more. In the 4 grades of students we've worked with over the past 3 years, I have yet to see anyone who doesn't engage in some way!## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Thanks, David. One more thought. We know that some of the (enormous!) engagement we see is just because this is "different." Kids like new experiences and if

everythingwere taught this way some of the excitement and novelty would be lost. We also assume that some kids are captured by the idea of coding but wedon'tassume that's a primary source of their initial attraction. (We hope, of course, that positive experiences with coding "invites" more children to see themselves as able and interested. The fact that we put this inregularclass time, not pullout time, is strategic and deliberate in order to make that invitation to a broad population of kids.)But it also seems that a big part of what is attractive to the children is that they can experiment in a way that typical mathematics experiences (even with manipulatives) does not provide. They can be

activewith manipulatives, but the objects, just like notations on paper, then just sit where they were placed. Kids may or may not find themselves engaged bycodingbut pretty much all kids likeinteraction. And the feedback from the interaction gives them information. Near as we can tell—and there's lots, of course, that we can't easily tell yet—their engagement is because they are genuinely puzzling through the tasks we suggest in much the way they puzzle through so many new things and situations they encounter out of school.The rationale behind this project is not that coding would attract kids but that the ability to

interactwith mathematics—not just repeat or practice it—will support their learning and increase their engagement. At least so far, that seems to be the case.## Cynthia Stenger

Thank you for this explanation of your program.

One statement succinctly describes what we strive for and have seen in our program as well (CPR2): programming gives students the “ability to

interactwith mathematics”.E Paul Goldenberg

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Yes, interaction is crucial. For people who are already "in" the game—enamored of mathematics, with good mathematical habits of mind and skilled in the manipulations—true

with mathematics may require nothing but a sheet of paper, sometimes not even that. But tointeractionenterthe game—tolearnhow to think mathematically and love it and gain the skills to interact with it—one needs more lively tools. Dynamic geometry tools like Sketchpad, Cabri, Geogebra let one create interactive geometric experiments. CAS sort of does that with algebra, but only sort of because the user still has to be able to compose and read fairly complicated structures. Programming has the potential to be an easier entryway.Kristen Reed

## Noelani Ogasawara Morris

FacilitatorDemonstration Teacher

I'm curious to know how teachers are using this in their classroom as well! Is this a supplemental technological piece like ST Math, Dreambox, or iReady where students can log on at home and work at their own pace? Or do teachers choose from a collection of math units and students are assigned tasks?

## Kate Coleman

Co-PresenterSenior Curriculum Associate

Hi Noelani,

Great question! Thanks so much for your interest. Our microworlds are designed to be embedded into the core mathematics instruction (for grades 2–5) in the classroom. They are not designed as a supplemental add on. This is so that all children gain experience with programming, in developmentally appropriate ways– and it increases access to mathematical ideas by providing an engaging approach to learning critical mathematics content that offers different affordances than pencil-and-paper activities.

Use in classrooms is intended to be a multi-day sequence of instruction – like a mini-module – that can be fully integrated with the core content being explored in the mathematics curriculum. Because our microworlds are focused on core mathematics content, teachers align their placement into the unit of study on that given topic. We have seen teachers use our microworlds as a way to introduce a new unit of study, for use during the unit, or even towards the end of a unit. There is a lot of flexibility given to teachers as to where within a unit of study it make sense to place these microworlds—and this is one piece feedback that has been very positive. Teachers have shared that because the microworld content is so closely aligned to the common core state standards, it is a very natural fit to include our microworlds as a two or three day experience within their unit of study.

During the microworld, students often work at their own pace through the sets of puzzles. While puzzles are offered in a highly intentional sequence, the microworlds do not enforce that sequence, allowing children to customize their path through each puzzle set. Each microworld is designed to tap children’s curiosity and spark their engagement in problem solving. We build and sustain that engagement through simple but elegant design, compelling puzzles, and children’s delight with their own discoveries and growing competency in the programming environment. There's very little preparation on part of the teacher—because they microworlds are really guided by each child's unique experience in working through the various puzzles.

I hope that this addresses your question—and we are always happy to connect directly if you have additional questions too. Our contact information is posted on our website as well.

Thank you again for your interest. We always love to hear from teachers, math coaches, and facilitators about the microworlds and their use in classrooms!

Best,

Kate Coleman

kcoleman@edc.org

Kristen Reed

## Noelani Ogasawara Morris

FacilitatorDemonstration Teacher

Yes, this definitely answers my question is peeks my interest! Thank you for expanding and explaining further how some teachers have utilized this program in their classrooms.

Kate Coleman

## Jessica Stovall

I enjoyed watching the video for your project and reading your comments. I am part of a team that has also been researching the impacts of using programming to teach mathematical reasoning. Our current project has been focused on 7

^{th}and 8^{th}grade classrooms, though past projects have had us in elementary classrooms as well. In our projects, we have also seen that students “learn to evaluate their ideas and work for themselves” as Paul stated. The nature of programming makes it appear more game-like to many and the students seem to enjoy working through the processes themselves. They don’t mind when a computer tells them they are wrong in the same way that they do when a teacher may tell them the same thing.In another post, Paul also mentioned conventional mathematical notation. I agree that computational thinking helps lead to an easier transition into formal mathematical writing. We use Python in our CPR2 Instructional Model, and the general expressions are in the code directly. Once the students have successfully generated a program that does what they are wanting, the mathematical notation is right there. One cool thing we see, as you mentioned in your video as well, is the computer allows multiple ways to arrive at the same result. This leads to great discussion about mathematical concepts.

I think introducing students to computational thinking at an early age is very important. I believe that having students explore mathematics in this way would lead to a very smooth transition into what we do using Python. Thanks for sharing your work!

Kristen Reed

## Kristen Reed

Co-PresenterManaging Project Director

Jessica, I'm so glad you commented here. I just had a chance to watch your video and read a little more about the CPR2 project. Our work seems very well aligned and it would be great to connect to learn more.

## Jessica Stovall

I agree and I look forward to continued discussion about our projects with each other!

## Kristina Anna Kramarczuk

Thank you for this project! The relationship between math and computing, especially how math tends to create barriers for students to pursue computing later on in high school and college, intrigues me. I was wondering if your team has thought of ways to present this research in a way that promotes district leaders and policy makers to reconsider math as a gatekeeper to computer science and instead see blended environments like those you are creating (or computing classes in general) as alternative pathways for students to pursue computing later on? Your findings are powerful and I am curious how they may all fit into the broadening participation in computing movement, especially for altering a system that historically marginalizes many students in both math and computing education.

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Hi, Kristina. Thanks for your comment and question. Just a moment ago, I commented briefly (and parenthetically) on this issue of broader participation in a response to David (above).

Absolutely right. Mathematics is a gatekeeper, and not just to CS but to higher academic or vocational education broadly. That's partly because so many disciplines and professions rely on clear mathematical thinking (whether or not they rely on any of the myriad facts and content details taught in K-12 math classes). Computing is quickly becoming another gateway to good careers. Providing access more broadly and eliminating conscious or unconscious filtering—the common (wrong) perception that people "have" or "don't have" mathematical minds and neither a teacher nor the individual can do much to change that—is a serious

socialissue, one part of the historical marginalization you mentioned, and an issue that broadly robs our entire economy by wasting talent and differentially decides which talent to waste.Teaching "regular" mathematics via programming is, we believe, good for the mathematics; it lets kids

expressthe mathematical ideas in ways that allow them to experiment with the ideas and hone them; it lets themdothe mathematics visibly, rather than just writing down what their heads have already finished doing and told their fingers to write. Testing that belief was a principal aim of our project. And our deliberate design to test it in the context of regular mathematics lessons, not special pullout classes, was to ensure thatifwe are right about the value of this approach for mathematical learning, it happens in a context that can potentially reach any student. And another very important "deliverable" is a model that other people can use to invent and build more of this kind of educational opportunity.We have already seen that happening a bit, as a group at CitiLab in Barcelona created two microworlds built on one of our early prototypes. And our own microworlds are already in English and Spanish, with some in Portuguese and German as well.

We hope that the models we can provide and test inspire other projects to go further not just in inventing more education of this kind but doing the vastly important, but vastly bigger, job that you point to.

## Paul Adams

Great video, and the voices of the students provides great insight on the impact of your project. My question - what do you see as the next steps for these students as they move forward to different grades? It appears that the work shows the power of using coding as a manipulative to learn mathematics. What would you recommend to help students continue the journey of learning mathematics in this manner?

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

The longer journey is a great question, one we hope someday to be able to pursue. Even as we put "coding" at the service of mathematics—mathematics the driver and coding a tool—we do need to pay attention to the growth of students' programming knowledge. As the students' mathematical ideas grow, so must the tools for exploring it.

With the youngest children, the programming is sequences of commands to cause visible effects—a move on a number line, the creation of an array, the location on a grid, some manipulation of shape. But even early on we give experiences with functional programming (in our multiplication microworld)—creating

expressionsthat calculate and return a result that can then be used in more complex expressions. That naturally arises early and advances quickly long before algebra.We hope that the models we provide by the end of this project will inspire inventions by others that we don't have time on the project to invent ourselves, and we have deliberately put effort into making the building of these microworlds (and documentation for building them and providing them in multiple languages) accessible to others.

## Paul Adams

Thanks! Your project is sparking new ideas for others to follow.

## Victor Minces

That is such a wonderful idea. We are developing a flow-based programming language connecting CS with music. I hope someday we can also connect with math.

E Paul Goldenberg

## June Mark

Co-PresenterManaging Project Director

Hi Victor, Thank you for watching our video and sharing about your work. I believe that our projects have the shared goal of providing children experiences engaging in mathematics, science, etc. themselves and seeing themselves as capable of doing mathematics and science. And that can lead to children considering STEM pathways in the future. Look forward to learning more about your work.

## Kelly Powers

This is pretty awesome. Have you or any teachers, cross-walked when they should perhaps use these modules, to a curriculum for pacing purposes? Our teachers often desire a crosswalk or when best to introduce the "microworld" in this case. A module alignment to a curriculum the school is using could be useful.

Can the microworlds be used in Scratch ? I notice the choice was Snap ?

Also wondering how we can build this experience into an applied project in the future :)

## June Mark

Co-PresenterManaging Project Director

Hi Kelly, Thanks for your interest in out work. We have been working closely with the elementary mathematics coach, digital learning teacher, and the elementary teachers in our development and testing of these microworlds. The discussions include considering the mathematics standards and computational thinking standards addressed in the microworlds, and we work closely to think about how the mathematical focus of the microworld aligns with the elementary mathematics curriculum that is being used.

For example, in our recent testing of a decimal microworld, conversations about when best to use the microworld with students led to providing students with the opportunity to explore the decimal number line microworld before the class started their decimal unit. It led to teachers learning how much students already knew about decimals. The knowledge gained about pacing and alignment will inform the continued development of our teaching guides.

The microworlds are developed in Snap

!which is very similar to Scratch. We used Snap!because of some of the mathematical capabilities it affords and because the Snap!developers were interested in the idea of creating microworlds and have worked with us to facilitate our development of them in Snap!.For students who have used Scratch, working with the blocks will be similar.As to your question of applied projects, the digital learning teacher in a school worked with teachers to build on students' experiences in our microworlds to develop other projects that integrate programming into other subjects, including some lessons where students created programs that shared their learning about turtles.

Hope this answers some questions, and we are happy to continue the conversation.

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Hi, Kelly! So nice to hear from you!!

I had two small addenda to June's quite thorough response. In our proposal to the NSF, we did say we'd use Scratch. After all, Scratch would already be more likely to be familiar to elementary school teachers. The specific issue that changed our mind after we started was some of the ways the two languages (more like two dialects) are optimized. Scratch is optimized for stage plays (whence the name "stage")—protecting kids from losing their sprites off the edge of the world, and giving them lots of control over drawing and a suitable amount of mathematics to make proper measurements for their drawing. It is wildly successful. And the "limitations" it imposes are deliberate to make it inviting and accessible and successful for introducing coding, and offering tons of room for growth. That is the deep rationale for a microworld—everything you need and no more, in order to limit distraction.

But, for mathematics, a different optimization is needed. I haven't checked recently to see how Scratch may have evolved, but at the time we started, Scratch's finite stage protected us from losing the sprite, but also meant that the simple

repeat (move turn)algorithm for drawing a circle or square worked only if executed far enough from the stage wall. And Scratch's decision to avoid the complexity involved in letting users define new functions (reporters) would keep not only our team from building the behind-the-scenes tools we needed to provide for the children, but would also limit what mathematical objectstheycould create.The pacing question fascinates me because of the surprise this has been for

us. The first microworld we built was the second-grade number line. It matches standards, was tested with kids, and the second graders find it easy enough to be accessible and hard enough to be puzzling and fun. If it weretooeasy, it would bore them. So perfectly tuned to second grade! Right?BUTwe discovered quite early that first gradersalsofound it both accessible and fun. And much later, we learned that third graders did, too. That then turned out true for our 2nd grade map microworld. I don'tknowwhy this is the case, but my conjecture is that the learning is justdifferentin a programming environment than it is in pencil-paper learning. The fact that kids at three different ages and levels of mathematical competence find these equally accessible and challenging must (I'm guessing, so take that "must" with a grain of salt) mean that they are gettingdifferentchallenges, seeing different things, mentally asking themselves different things or interpreting the problems in different ways at the three ages.In any event, what we're seeing is that pacing and crosswalking isn't quite as straighforward as we, ourselves, would have guessed. Most recently, the decimal number line was, in one fourth grade, the very first contact they had with decimals—

noprior instruction or (known) experience with decimals. And they came out of it having learned a lot. So, introductionbeforeinstruction?Withthe unit? As a review or applicationafterthe unit. All three, with different puzzles just to avoid repetition? We're not able to say which is the optimal use. That'll take some feedback from the field.Myriam Steinback

## Janet Stramel

I love hearing the elementary students saying, "I love doing math with coding!" I have the same question as Paul... what are the next steps with these students? And do you plan to track their progress to the next grade level?

## June Mark

Co-PresenterManaging Project Director

Hi Janet, My colleague Paul Goldenberg responded to Paul's question above about next steps with these students, and shared a bit about ideas and microworlds that we are developing for upper elementary and beyond. I can add a little more here. With some of our microworlds, there are natural extensions of the mathematics such as extending students' experiences working with small integers focusing on addition and subtraction to explorations with fractions and decimals on the number line. Other microworlds focus on mathematics content that includes place value, area and multiplication, angles, the coordinate plane, and factors and multiples.

We have been fortunate to work in a school for multiple years where the children who tested early microworlds with us as second graders are now fourth graders and soon to be in fifth grade. It has been fascinating to see the growth in their comfort with the programming,

andto see their thinking about the mathematics develop. Hopefully in future work, we'll have opportunities to more systematically track students' progress across grades. Thanks for watching our video and for your interest in the work.## Cynthia Stenger

What an amazing opportunity to follow these students. I am going to do more digging into your microworlds. It reminds me of mapping from one set to another, like working in the integers then expanding to rationals.

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

Extension is one of the motivators of our puzzles. In the simplest 2nd grade integer microworld are seeds of linear combinations (a later version of the integer world for kids who are studying common factors gives kids ±9 and ±15) and negative numbers (we don't

posepuzzles that involve the use of them, but 0 is not at the far left of the line and we've learned thatnochildren, even in first grade, are puzzled when they accidentally or deliberately land on those numbers). And we zoom in on that same structure for fractions and decimals.It's been fun for us. I wish we

couldfollow these kids and keep building for them (and, of course, everyone else—these are all freely accessible) as they move into middle school and algebra!Thanks for your interest. It'd be fun to think together more about these ideas!

## Cynthia Stenger

I am excited about by quote from your abstract, “Programming gives students a language with which to express their thinking precisely and have that thinking enacted by the computer, making it easier for students to assess, refine, and extend their thinking.” … how elegantly put! This is what I have observed over 11 years working with students in our project CPR2. Humans need some sort of language to communicate (even with their own minds and imaginations) . Programming language and even formal mathematical language such as “ for any”, and “for all” give students a vehicle to access, and also express, general and abstract math and science concepts.

## Cynthia Stenger

The part of your comment to Paul about “creating

expressionsthat calculate and return a result that can then be used in more complex expressions” has been productive for us. (CPR2) When possible we try to find the general expression in the code, but also to give students the mathematical language and symbology to accompany the expressions they are using in their code.E Paul Goldenberg

## Kate Coleman

Co-PresenterSenior Curriculum Associate

Hi Cynthia!

Thank you for sharing your thoughts. I just had the opportunity to watch your video and learn more about CPR2 and it certainly seems like there is a lot of alignment between our two projects. I agree with you that programming gives students a language with which to express either thinking and one unique aspect to it is that it is an active language. The approach takes advantage of students' native way of learning: doing, getting feedback, building experience, then naming and systematizing. I think that it is so powerful that students can debug their work as they go.

In your video, one of your partner teachers makes a comment that students "didn't want the lessons to end." That was our experience as well! The level of engagement with our microworlds was incredibly high in the classroom. Teachers also reported that students asked to do more programming outside of the mathematics block.

We'd love to connect more on this important work. Thanks again for sharing!

## Cynthia Stenger

Yes, we have also had students asking their teachers, “when do we get to program again”. Since they rarely ask, “when do we get to work with general expressions and learn how to use them to write proofs?” This is a win!

E Paul Goldenberg

## E Paul Goldenberg

Co-PresenterDistinguished Scholar, Mathematics and Computer Science

!! and a broad smile!

## Joyce Malyn-Smith

Excellent video and explanation of what is important about this project. Congratulations All!!!

## June Mark

Co-PresenterManaging Project Director

Thank you Joyce, I also appreciated the opportunity to learn more about your work through your projects' videos. Such exciting projects!

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