Show What You Know: Reviewing and Applying Knowledge with Debugging Activities

Toni and I led a number of 4-day Teaching Computer Science with TI-Nspire Technology workshops across the state of Texas this summer. Many of the sites offered days 1 and 2 consecutively, followed by days 3 and 4 at a later date. Sometimes day 3 of the PD happened months after day 2. In these instances, how might we structure an activity that helps participant review key computer science ideas from previous learning episodes, while also applying these ideas in new situations? One strategy that we found particularly helpful in this situation was a set of debugging activities.

Side note #1: The first two days of learning delved into most of the topics from the first four units of the 10 Minutes of Code activities, including user input (via parameters or Request statements), output (Disp, Text), variables, data types, conditionals (If, If…Else), and loops (For, While, and DoWhile). Additionally, we applied many of these same concepts to tasks with the Innovator through the first two units of the 10 Minutes of Code for the TI-Innovator Hub activities.

Side note #2: Our use of debugging activities was inspired by the work of Amin Lalani & Georgia Stuart at UT Dallas.

We started Day 3 with a Debugging Problems TNS file. The file was composed of 10 Problems. Each Problem contains a Program Editor page, like this one:

with a corresponding Calculator page, like this one, on which to run/execute the program:

Each program (named debug_1, debug_2, etc.) contained at least one error. Participants worked in pairs to debug/fix each program. In so doing, participants reviewed, applied, and discussed:

  • How to compile a program in the Program Editor (using Check Syntax & Store)
  • How to run/execute a program in a Calculator page (including the use of the var button to summon program names)
  • Syntax for the TI-Nspire programming language
  • Variable naming conventions
  • Data type considerations (including concatenation)
  • Control structures, including conditional statements and loops
  • Logic and conditional statements
  • User input mechanisms (including parameters and Request statements)
  • Error types (including compilation errors & execution errors) and error messages
  • …and more!

Some of the programs involved minor errors. For example, the debug_1 program shown above—participants determined that the string in the Disp statement should have been in double quotes (“) and not single quotes (‘). Here’s a quick GIF of troubleshooting this program:

Debug_1

Here are two other programs from the TNS file. What errors do you see? How might we debug these programs? What important CS topics connect to these errors and/or fixes?

It was incredibly valuable for participants to troubleshoot and debug these programs dynamically on the handheld calculators. They could fix errors as they found them and then compile and execute the edited program right away to test their “fixes”. On numerous occasions participants fixed one or more parts of the program, but then discovered another error when they ran the edited program.

Our brains often “fix the code for us,” helping us read the intention of the code, but perhaps missing some of the subtle errors specific to the coding language. This activity helped us review and apply programming skills specific to the TI-Nspire language, while also discussing larger computer science topics and content applicable across languages.

Additionally, we experienced the cognitive difference between writing a program from scratch and debugging an already written program. It feels different to debug–to find and correct errors in already written code–than to “code from scratch”; there was important learning to be had in this space.

Give it a try yourself! Here’s a copy of the TNS file with all 10 debugging programs in it. Each program does not necessarily have a unique solution—there may well be more than one way to “fix” the program so that it runs as intended, but with different code structures under the hood.

Also, Toni and I wonder…what would similar debugging programs look like that included TI-Innovator commands/components? What sort of errors would be helpful for learners to confront, discuss, and debug in regards to Innovator programs?

Here are some rough draft ideas for possible Innovator debug programs:

What other programs and/or lines of code might be valuable to include in a set of TI-Innovator debugging problems?

What other value do these debugging activities bring to the learning experience?

How To: Create Your Own (Digital) Foldables

Want to create digital versions of your own Foldables for use in your classroom or during professional development sessions? The process is so easy! Read on for an illustrated step-by-step tutorial. With a little bit of practice, you can whip up a professional, custom Foldable in less than 5 minutes!

If you want to learn about Foldables, definitely start with the master, Dinah Zike herself! Check out her website and her amazing resourcesguaranteed to spark your creative juices about how you can use 3D graphic organizers to help students organize their learning!

My professional journey with Foldables began several years ago when I had the incredible opportunity to attend a Dinah Zike Academy in Comfort, Texas. I cannot recommend this professional development experience enough. In just a few short days I went from knowing next-to-nothing about Foldables and journaling to feeling like an Interactive Notebook pro—fully prepared to take my Foldable learning and apply it in my high school math and science classes that fall. I owe so much of my Foldable work to Dinah and her wonderful team!

The first step to creating a digital Foldable is to sketch it out on paper. Might seem counterintuitive, but this sketch will give you a “feel” for how the Foldable will look, info on how many columns/rows you will need later, and will also help with correctly “double-siding” your Foldable if you choose to tackle that (see my upcoming post on creating double-sided Foldables). Your sketch doesn’t have to be fancy or to-scalejust include your tabs, your labels for those tabs, and at least one anchor tab (for gluing your Foldable down into its final homea journal, composition book, bound book, etc.).

Here’s a quick sketch of a Foldable we’ll create throughout this posta Solving Quadratic Equations Foldable for use in Algebra I:

This is a 4-tab Shutterfold Foldable that opens like a pair of shutters on a window.

Not sure where to start with your Foldable design? Check out the samples here to get started!

With your paper sketch complete, you’re ready to create your digital version. I create my Foldables in Microsoft Word. I like the robust suite of editing options provided in Word, as well as the precise design tools that aren’t yet available in GoogleDocs.

Here we go with some step-by-steps…

Open Microsoft Word and create a new, blank document. Click on the Insert ribbon, then select Table. Highlight the size of your desired table. Based on my sketch, I’m going to need a 2 row by 4 column table, so I’ll select that.

Here’s the inserted table, which will act as the “frame” for our Foldable.

This “frame” needs a bit of work to make it into our finished Foldable. We’ll start by selecting the whole table in order to make changes to it. Hover over the upper left corner of your table, making the four-way icon appear.

Left-click on this four-way icon to select the entire table.

Next, hover over and then right-click on the four-way icon to display a context menu. Select Table Properties from near the bottom of this menu.

This will open up the Table Properties menu. On the Table tab, click on Borders and Shading at the bottom.

From here, you can change the border width. A thicker border makes your Foldable much easier to cut out! Click the drop-down menu under Width to change the width from ½ pt to 6 pt. Now your Foldable will have thick borders for easier folding and cutting.

Next, you can resize the widths and heights of your rows and columns to match your needs. Hover over the column/row dividing line, then click and drag to resize the columns and/or rows. Here’s a short GIF overviewing this process:

Creating Foldable 1.gif

Worried that your rows aren’t exactly the same size? Highlight both rows, then right click to bring up the context menu. Use the Distribute Rows Evenly option to make your rows the exact same width.

Creating Foldable 2.gif

Confession here: I hardly ever resize my columns/rows “by hand” like this… Instead, I enter exact row and column values in the Row and Column tabs in the Table Properties menu. More on this in a follow-up post…!

To create cohesive anchor tabs, we need to merge the two cells in the first column and then merge the two cells in the last column. Start by highlighting the two cells you want to merge. Then, right click on one of the highlighted cells to display the context menu. Click on Merge Cells.

Now, our Foldable looks like thisit has the right shape, thick borders, and two anchor tabs for gluing!

Now we’re ready to enter our content.

The standard for solving quadratic equations in Algebra I (in Texas) states:

A.8A: The student is expected to solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula.

So, we’ll title the four tabs using the wording from this standard.

I think Foldables are much more powerful when they have a relevant image, graphic, or mathematical notation on the corresponding tabs. Here I’ll include a sample quadratic equation that we can use to illustrate each solution method. I used Word’s built-in Equation Editor to enter these equations. Here’s a quick GIF showing how to insert an equation:

Creating Foldable 3.gif

And the Foldable now has a relevant equation on each tab:

You could also include images, tables, graphs, geometric figures, etc. on your tabssomething that helps illustrate the big idea for that tab/topic.

One last finishing touch: I like to include the title of my Foldable written along the anchor tabs. Type the title into the anchor tab cell and then select the Text Direction options under the Layout tab in the Table ribbon.

Creating Foldable 4.gif

Repeat on the other anchor tab and voila…!

A customized, professional, digital Foldable for use with your learnersdesigned exactly the way you like it!

What kinds of Foldables will you create using these tips?
What other tips do you have for creating digital Foldables?

Foldables in Computer Science: Consolidating Our Learning

How might we help learners synthesize important computer science topics (including vocabulary, structures, and techniques) using Foldables? How might these three-dimensional graphic organizers help learners consolidate their learning, while also providing effective notes for later reference?

In designing learning experiences for new-to-CS educators, Toni and I considered how Foldables might dovetail nicely with the 10 Minutes of Code activities provided by Texas Instruments. The 10 Minutes of Code activities do a phenomenal job of crafting short learning episodes that provide students with bite-sized instructional episodes (see the Skill Builders) followed by opportunities to apply their new learning (see the Applications).

Toni and I wanted to build on these 10 Minutes of Code activities by layering additional content connections that deliberately drew out and discussed some of the computer science topics that are operating within the 10 Minutes of Code activities. Specifically, we were working to develop Foldables that formalized CS topics that were important for a beginning teacher of CS to know (as defined by the TExES Computer Science 8-12 certification requirements in Texas).

Our Foldable work is inspired by the top-notch work of Dinah Zike and her crew from Comfort, Texas. If you are new to the world of Foldables, be sure to check out Dinah’s site. Dinah has a ton of resources that support Foldable use and creation in all grade levels and subject areas. Here are some of my favourites: Notebook Foldables, Big Book of Math for MS and HS, and her Notebooking Central resources for Math.

Toni and I wondered: What would Foldables look like in computer science? This led us on a fun journey…! Here are some of the highlights.

Here is a Three-Tab Foldable that overviews introductory Program Protocols, including how to Create (using the Program Editor), Compile (using the Check Syntax & Store command), and Run (using a Calculator page) a program on the TI-Nspire handheld. These ideas connect to the work done in the Unit 1: Program Basics—Skill Builder 1: Introducing the Program Editor 10 Minutes of Code activity.

 

 

Here’s another example of a Foldable that uses a screenshot of a program and its output for a given set of actual parameters. This Shutterfold Foldable (with an inlaid screenshot) was used to annotate some details about Formal and Actual Parameters after we wrote our first program that used parameters (in the Unit 1: Program Basics—Skill Builder 2: Arguments & Expressions 10 Minutes of Code activity).

 

 

Here’s one of my favourites: a Matchbook Foldable that outlines how a For Loop operates. We used this Foldable to identify the four parts of a For loop, as well to practice tracing out a loop using a table. Notice the inclusion and annotation of a screenshot of the program. This Foldable formalized ideas from the Unit 4: Loops—Skill Builder 1: For Loops 10 Minutes of Code activity.

 

Here’s an awesome tweet from a learner in one of Toni’s workshops and his work with this exact Foldable! Love it!

We created similar Matchbook Foldables for While Loops and Do While Loops, but then also created this Three-Tab Foldable to compare and contrast the three types of loops (For, While, and Do While). This Foldable helped learners step back and consider the important similarities and differences across the three loop types that were studied throughout the Unit 4: Loops 10 Minutes of Code activities.

Here’s an example of using a Shutterfold Foldable to document and analyze two Computer Science techniques (specifically, using Counters and Accumulators). This builds on the ideas used in the Unit 4: Loops—Application: Bank Notices 10 Minutes of Code activity. Consider how a Foldable like this serves as a way to consolidate new information, but also as a helpful resource for later reference and use.

 

We have received really positive feedback about how these Foldables helped learners step back from the coding experiences and synthesize the ideas they were learning about and experimenting with in a meaningful way. The Foldables helped to organize and chunk the main ideas and were also really handy for revisiting ideas throughout the learning episodes.

The examples above are a sampling of the Foldables that Toni and I have created for formalizing CS topics. We love doing this work!

What other topics from computer science might be a good fit for Foldables? What other strategies might be effective for creating powerful Foldables for CS topics? How might Foldables dovetail into work with the TI-Innovator Hub and those 10 Minutes of Code activities?

Math Talk: ¼ of a Square (Part 2)

How might we use a Math Talk to help students connect fractions and a host of two-dimensional geometry concepts? How might this same Math Talk take learners into “a broad mathematical terrain of interconnected concepts, procedures, representations, and explanations”? (Kazemi & Hintz, 2014) How might a Why? Let’s Justify discussion routine overlaid on an extension Math Talk help deepen our understanding of why a particular geometric sketch is accurate? Let’s explore these ideas further…!

[Part 1 of this blog post discussed the original Math Talk and an Open Strategy Share discussion structure.]

Here is a “spin-off” Math Talk that emerged from the Open Strategy Share discussed in Part 1.

The group was presented with this sketch from one of their peers and asked to consider whether this image represented ¼ shaded. Note that the learner used a dot to mark the intended midpoint of the bottom of the square.

Triangle 1

Does the image above represent ¼ of the square shaded? How do you know?

We discussed this representation in pairs and small groups, then we shared some of our thinking whole group. For this particular Math Talk, we were using a Why? Let’s Justify discussion structure. Instead of trying to generate multiple solution pathways, we were considering one particular (potential) solution and working on justifying why this was or was not a solution.

Kazemi & Hintz (2014) summarize the goal of this structure as “to generate justifications for why a particular mathematical strategy works” (p. 3). Kazemi & Hintz continue: “During a Why? Let’s Justify discussion, the talk narrows upon a general claim in order to closely examine the mathematics and generate a discussion for it.” (p. 74).

The following justifications emerged from the group:

B suggested that the shaded triangle did represent ¼ of the square, since the base was half of the square’s side length and the height of the triangle was equal to the side of the square.

This verbal description resonated with some learners in the group, but others were unconvinced.

S suggested that so long as the bottom vertices of the triangle remain unchanged the top vertex of the triangle could actually lie anywhere along the top side of the square — and the triangle would always have ¼ of the area of the square. We sketched out some examples of S’s idea:

Triangle 2

S contended that all three of the triangles shown above represent ¼ of the square shaded in. The group agreed that the yellow triangle (a right triangle with its top vertex at the midpoint of the upper side of the square) definitely represented ¼ of the square, but some learners remained unconvinced about the other two examples.

A suggested that we draw in an altitude for each of the triangles we had sketched as a group. Our diagrams now looked like this:

Triangle 3

Adding in this additional structure (the altitudes) revealed something that wasn’t yet visible for all learners — namely, S’s contention that all three triangles had the same base and height, so must have the same area.

[Side note: This is a great example of what Jill Gough and Jennifer Wilson describe as “adding in an auxiliary line…to make what isn’t pictured visible” from their learning progression for SMP 7: I can look for and make use of structure. Check out their fabulous and thoughtful work on their blogs linked above.]

Consider how this Math Talk feels different than the Open Strategy Share version shared in Part 1 of this post. The Why? Let’s Justify discussion structure took us into another part of the mathematics — not merely considering a wealth of different approaches, but deeply considering one approach and focusing on creating convincing explanations to justify the mathematics used in that strategy.

In discussing the Why? Let’s Justify structure, Kazemi & Hintz (2014) contend: “Certain types of mathematical ideas lend themselves to a discussion that generates a justification for that idea” (p. 74). Not every problem or prompt is innately geared towards this particular discussion structure.

So what made this particular mathematical idea (specifically that the green triangle shown above does or does not represent ¼ of the square) a good fit for a Why? Let’s Justify discussion structure?

First, this particular representation did not look like ¼ of the square at first glance. A deeper analysis of the shape was required in order to determine whether or not it represented ¼ of the square–and deeper still to justify why.

Second, during the Open Strategy Share for the preceding Math Talk, most of the group’s sketches of ¼-shaded involved only congruent shapes. The approach used in this Why? Let’s Justify discussion did not subdivide the square into congruent shapes/pieces. Learners would have to rely on different reasoning (beyond subdividing a shape into congruent pieces and shading ¼ of the congruent pieces) to justify whether or not this specific triangle represented ¼ of the square.

Third, this particular triangle exposed that we (as learners) lose some of our intuition about triangles when they are obtuse triangles. We don’t often “see” the altitude of an obtuse triangle in the way that we see (and reason with) the heights of acute and right triangles. This drawing exposed some of that loss of intuition and made us dig deeper into why something was true.

For these three reasons (and perhaps others?), this prompt providing a meaningful task for learners to engage in a Why? Let’s Justify discussion routine.

Side Note: Kazemi & Hintz (2014) challenge that “During the discussion, we need to listen for procedural descriptions that students might give and be sure not to have the conversation stop there. When we press beyond procedural explanations into explanations that include reasoning, we are supporting students in justifying their ideas” (p. 56).

This paragraph still convicts me. Did I press far enough in the justification discussion outlined above? Why does any triangle with a base half the width of the square and a height of the square have an area of ¼ of the square? Maybe I should have pushed our thinking a little bit further here…

How does the Why? Let’s Justify discussion structure compare to the Open Strategy Share structure? In what ways does it open up a different part of the mathematics? What additional considerations might a teacher attend to when planning a Math Talk like this?

(See Elham Kazemi & Allison Hintz’s book Intentional Talk: How to Structure and Lead Productive Mathematical Discussions for more information on the Open Strategy Share and Why? Let’s Justify discussion structures.)

Math Talk | ¼ of a Square (Part 1)

How might we use a Math Talk to help students connect fractions and a host of two-dimensional geometry concepts? How might this same Math Talk take learners into “a broad mathematical terrain of interconnected concepts, procedures, representations, and explanations”? (Kazemi & Hintz, 2014) How might an Open Strategy Share discussion structure overlaid on this Math Talk help students see a range of possibilities within this interconnected landscape? Let’s explore these ideas further…!

Consider the following Math Talk:

Take a few minutes (say, time yourself for 3 minutes) and sketch squares where ¼ of each square is shaded in a different way. One ground rule first… Let’s say that the following two depictions represent the same “way” to shade ¼ of a square. Instead of sketching different iterations of the same “type” of ¼…can you shade ¼ in a different way? How many different ways? Set your timer and see where your brain takes you…!

Where did your brain take you? How many different ways of shading ¼ of a square did you generate?

After working on this task individually for 2-3 minutes, learners share their thinking with their tablemates. In so doing, they consider and discuss Did we all approach this task the same way? and What additional approaches did my peers use? and What additional ideas do we now have as a group because of discussing each other’s work?

We have led this Math Talk with multiple groups of adult learners. Here is a sample of the collective work that has come out of this individual think time followed by small group discussion time:

What would you do with this work? What is the benefit of orchestrating an Open Strategy Share Math Talk over some of these examples?

Kazemi & Hintz (2014) suggest:

“Open strategy sharing is typically the first way to get mathematical discussions going in classrooms. It’s like having a good, basic recipe for a soup from which you can make all kinds of variations. Open strategy sharing allows you to nurture the norms needed for a productive math-talk community. And you can use this discussion structure to model how students should talk with one another.” (p. 17)

I particularly like how the Open Strategy Share discussion structure provides an initial access point for teachers looking to introduce or enhance math talk in their classrooms. Also, as Kazemi & Hintz discuss, these structures provide wonderful opportunities for students to learn how to talk about mathematics.

But, if this is the “basic recipe for soup,” what would your variation be? What would you choose to highlight or spin off from for a follow-up Math Talk at some later time? Where else instructionally might this Math Talk go?

Check out Part 2 of this blog post to see a sample follow-up Math Talk that utilizes the Why? Let’s Justify discussion structure.

(See Elham Kazemi & Allison Hintz’s book Intentional Talk: How to Structure and Lead Productive Mathematical Discussions for more information on the Open Strategy Share discussion structure.)

“Cooking Show” CS Lessons / Presentations

What do you do if there isn’t quite enough time in a CS lesson or a CS conference session? What “cuts” do you make and why? What do you focus on and/or highlight? What do you downplay and/or skip? Would your decisions be different if the activity was for new-to-CS learners or experienced CS learners?

This is a dilemma Toni and I have faced on a number of occasions. One approach that we have found particularly helpful for these situations is what we call “Cooking Show CS”.

Consider a cooking show for a minute… Time is not spent equally on each part of the cooking experience. Rather, some parts—the really important parts—are slowed down and discussed and modeled in great detail. Other parts—the not-so-important parts—are sped up considerably or even skipped entirely. Like, seriously, when does a cooking show include “real-time footage” of a recipe baking second-by-second in the oven?? We don’t need to see that—we get it. We often do need to see the preparation and the active cooking and the tips and tricks and the cooking “theory” being applied to this particular recipe. So that’s where a cooking show spends its time.

What would the CS equivalent of this be? What would it look like for a CS “recipe”/program/project to slow down and focus deeply on the preparation and the active computational thinking and the tips and tricks and the computer science theory…while speeding up or skipping entirely the parts that we don’t need to see or do in their entirety?

Here are two vignettes of how Toni and I have tried to used “Cooking Show CS” in 1 hour conference sessions.

Side Note: We don’t think that this pedagogy is always effective or that it is necessarily the best approach in all instances. Rather, we contend that there are times (i.e. short conference sessions) where it might meet a need—allowing participants to “develop” (or co-develop) a program in one hour that otherwise might take them much longer.

Guess My Number

In this particular session, new-to-CS learners were learning about introductory CS control structures, including sequential, branching, and iterative pathways. Participant pairs worked together to make sense of block-based pseudocode for this particular program, then we collectively translated this pseudocode into the text-based language of the TI-84+CE (by hand — more on that decision later). (Toni or I will share more about the block-based pseudocode we used for this activity in a later post.)

We ended up with the following program coded by hand on paper:

:Disp “GUESS IS TOO HIGH”
:Wait 0.5
:Send(“SET SOUND 1046.5 TIME 0.5”)
:Wait 0.6
:
:Disp “GUESS IS TOO LOW”
:Wait 0.5
:Send(“SET SOUND 65.41 TIME 0.5”)
:Wait 0.6
:
:randInt(1,20)→N
:
:Input “GUESS? “,G
:
:While N≠G
:
:If G<N
:Then
:Send(“SET SOUND 65.41 TIME 0.5”)
:Wait 0.6
:
:Else
:Send(“SET SOUND 1046.5 TIME 0.5”)
:Wait 0.6
:End
:
:Input “GUESS? ”,G
:
:End
:
:Disp “YOU WIN!”

Getting to this point took time, as we carefully constructed, deconstructed, and discussed the various computer science principles at play in our program. How were we possibly going to have time to code this program into the calculators, as well?

Here’s where “Cooking Show CS” came into play.

Toni and I created a partially pre-made version of the program (called NUMGAME) and uploaded it onto all of the handhelds in the room. Instead of starting from scratch, participants opened up NUMGAME to find the first 9 lines of code already coded into the program—specifically, the lines highlighted below:

We helped participants navigate through entering the remaining lines of code into the “pre-prepped” program. Most participants finished coding these commands into their handhelds in time to test and run their programs, but a few did not.

We had considered this possibility, so we also preloaded a program called NUMGAME2 onto all of the handheld calculators. This program included all of the lines of code we (as a group) had collectively coded by hand.

This, to us, was akin to the moment in a cooking show where the chef puts a “raw” dish in the oven, but then a few minutes later pulls out a “finished” dish that had already been cooking for the full amount of time. Our choice here was purposeful: We wanted all participants to experience success with the program that they had helped write, even if they didn’t have time to “type in” each line of the code.

We included two additional “pre-cooked” programs for participants to consider, as well—NUMGAME3, which added in additional context for the user, visual (LED) outputs, and a celebratory tone when the user guessed the correct number; and, NUMGAME4, which abstracted the program so the user determined the upper bound on the range of possible numbers and also included a counter that tracked and displayed the number of guesses the user needed to guess the number. Participants were given a few minutes to run each of these programs and then predict what the changes in the code might look like in order to result in these “enhanced” versions of the program.

Respect the Beep

Toni has written here about the “Respect the Beep” session we have led where we use the TI-Innovator and the Ultrasonic Ranger to model a vehicle’s back-up sensor. I discussed this same program (but through a pedagogical lens) in this post about Introducing CS with Flowcharts.

When we led this activity in a conference session earlier this summer, we had a sequence of three programs we were planning to write as a collective group.

Here is the finished code that we wanted to arrive at for the first program:

Toni and I decided to create and provide participants with a pre-made file with the following parts of the program already “cooked”:

Including these commands allowed us to: save a few minutes of coding time in the conference session; and, focus primarily on the theory and syntax of sending and receiving commands from the Ranger, rather than just the digital connections.

Wrapping a While loop around the data collection part of this program doesn’t take a lot of extra time, so that’s something we had participants do themselves, as well, resulting in a program of:

P.S. Highlighting text using the Shift button and then inserting a While…EndWhile loop from the Menu will insert the While loop around the highlighted text. Here’s a little GIF showing this tip:

 

However, we did run out of time for participants to code in the final branching structures that were to be housed inside the While loop. Here was our end goal:

How might we have used another “Cooking Show CS” program here to help us meet our instructional goal in the short amount of time we had with these learners? Toni and I have considered using an “almost done” version of the file that looks like this:

where participants would fill in the output commands (represented above by the highlighted blank lines) within the various branches of the program.

What are your thoughts? How might using a “cooking show” approach for a CS lesson/activity/presentation allow us to focus more on our CS instructional goal and less on aspects that participants perhaps don’t need to see or do in their entirety?

Math Talk: Solving Equations Visually

How might we use a Math Talk to help students sense-make about solving equations? How might this same Math Talk help students work on showing what they know and making their thinking visible? How might a Compare & Connect discussion routine overlaid on this Math Talk help deepen our understanding of strategies for solving equations? Let’s explore these ideas further…!

Here is one of my favourite Math Talks:

(Note: The idea for this particular Math Talk came from a session I attended at NCSM 2016 led by Crystal Lancour & Jennifer Trievel of Delaware. Thanks, y’all!)

Take a minute and think about how you would respond to this prompt. Can you solve this problem without first “translating” it into algebraic notation and solving via algebraic manipulation? Rather, can you solve this problem visually?

Also, can you (as Jill Gough and Jennifer Wilson encourage us to do) “show your work so that a reader understands without having to ask [you] a question”? (See more on Jill & Jennifer’s #ShowYourWork progressions here.)

How might we make our thinking visible for how we solved this problem?

Here is how S, a recent conference session participant, made her thinking visible:

Elephants 1 Final

She explained her thinking aloud as:

I removed 4 horses from each side. Then I was left with the fact that 2 elephants were as strong as 8 horses. I divided these 8 horses into two equal groups showing me that each elephant is as strong as 4 horses.

When asked “Did anyone else solve the problem this way?” the majority of the participants indicated that they solved this problem the same way as S had.

L, however, had a different approach. Here is how she made her thinking visible:

Elephants 2 Final.png

Can you understand what L did without asking her any questions? How is her approach similar to and different from S’s approach?

L explained her thinking to the group in this way:

I started by dividing the problem in half. I knew that if 2 elephants and 4 horses were as strong as 12 horses, then 1 elephant and 2 horses must be as strong as 6 horses. Knowing that 1 elephant and 2 horses were as strong as 6 horses, I removed 2 horses from both sides of this situation, leaving me with the fact that 1 elephant is as strong as 4 horses.

How did L look for and make use of structure? And, does her method work all the time?

We quickly considered what the corresponding algebraic manipulation might have looked like for S and L’s approaches. We decided to represent the strength of an elephant with the variable e and the strength of a horse with the numerical value of 1. Here is what we came up with for the corresponding algebraic methods for these two visual solution pathways:

S’s Method

Elephants 1 with Algebra Final.png

L’s Method

Elephants 2 with Algebra Final.png

Do both of these methods work every time? When might S’s method be advantageous? When might L’s method be advantageous?

Why might it be important for students to Compare & Connect these two methods in a Math Talk? (See Elham Kazemi & Allison Hintz’s book Intentional Talk: How to Structure and Lead Productive Mathematical Discussions for more information on the Compare & Connect discussion structure.)

Also consider:

How does L’s method push back on well-intentioned but over-structured “procedures” we might teach our students for solving equations?

More on this idea in an upcoming blog post on Sense-Making for Solving Equations.