# Using Video Clips to Identify and Promote Children’s Rights as Mathematics Learners

Crystal Kalinec-Craig, University of Texas at San Antonio

When children have the chance to show their brilliance in mathematics, teachers can bear witness to sophisticated thinking that may otherwise go unnoticed or dismissed. But too often, children experience mathematics in a way that feels constrained, pre-determined, and dismissive. To push children’s ideas (whether finished or incomplete) to the forefront of instruction, teachers can commit to the Rights of the Learner (RotL).

Students can exercise at least four rights when learning mathematics: 1) to be confused; 2) to claim a mistake and revise their thinking; 3) to speak, listen, and be heard; and 4) to write, do, and represent what makes sense to them (Kalinec-Craig, 2017a, 2017b; Torres, personal communication, March 7, 2016). The RotL align with the notion of Rough Draft Thinking (Jansen, Cooper, Vascellaro, & Wandless, 2017), where students’ ideas are valued at whatever stage of completion. In the sections below, I briefly outline how I use a video from the IMAP program (San Diego State University Foundation, & Philipp, 2005) to help teachers learn how to teach mathematics with the RotL in mind.

## Gretchen Solving 70-23

The video begins with Gretchen, a 2nd grader, being asked by the interviewer to solve the problem 70-23, written in a vertical orientation. Gretchen says, “That’s easy,” and applies what appears to be a traditional, U.S. algorithm. Gretchen writes 53 as her answer (Figure 1).

Figure 1. Gretchen’s initial strategy and solution. Reprinted with permission.

Upon seeing Gretchen’s answer, the interviewer asks Gretchen, “Can you show me that problem, too, with these blocks?” and points to the base ten blocks on the desk. For her second method, Gretchen pulls out blocks and counts out 7 tens and then 2 tens and 3 ones to represent both quantities of the problem. She first separates 2 tens from the 7 tens (to show 70-20) and then takes 3 ones away from the 5 tens (to show 50-3). Gretchen then counts the remaining blocks (Figure 2) and states she has 47.

Figure 2. Gretchen using the base ten blocks.

Gretchen pauses, returns to her work, and reenacts her initial solution, but concludes by saying, “[sighs] Oh geez! I don’t get it.” When the interviewer asks her to consider her different answers, Gretchen realizes they do not match and says, “Ok, so 0 take away 3. Yeah, that’s 3. Ok. And then, 7 take away 2 equals 5. So, I put 3 there and 5 there.” After the interviewer asks, “but what did you get over there [with the blocks]?” Gretchen says “47, but I don’t get it.”

Sensing Gretchen’s frustration, the interviewer asks if there is another way she could solve the problem. Gretchen uses the transparent hundreds chart (Figure 3) and counts 23 spaces back from 70 and arrives at 47; again, confirming her second solution, but still not what she initially determined.

Figure 3. Gretchen using the hundreds chart.

Near the end of the video, Gretchen has not arrived at a final answer, but exclaims, “47 couldn’t be right because, like it has to be 53.” The video ends with Gretchen contemplating what the answer should be and the interviewer prompting her to follow up later.

## Gretchen Exercising her Rights as a Learner

I use this video nearly every semester, in a course for future teachers. First, it does not end with a tidy conclusion where children arrive at the correct answer. Instead, Gretchen claims that 53 as the correct answer and that she “doesn’t get it” even after arriving at 47 with two other methods. Many of my future teachers groan by the end because they want to know whether Gretchen finally learns that 47 is the correct answer.

Second, Gretchen exercises nearly all of her RotL and the interviewer supports Gretchen to exercise these rights. Gretchen says the phrase, “But I don’t get it” at multiple points to signal that she is exercising RotL #1 (the right to be confused). Instead of stepping in to clarify Gretchen’s thinking, the interviewer encourages Gretchen to use other methods to confirm or disprove her initial answer.

Gretchen also exercises the RotL #3 (the right to speak, listen, and be heard) and #4 (the right to write, do and represent what makes sense) when she says and records 53 despite finding 47 using two other methods. The interviewer inquires about Gretchen’s reasoning without quickly correcting her. Interestingly, Gretchen’s second and third strategies where she found 47 were conceptually different (e.g., using place value and a “counting back” strategy, respectively), but she was still convinced the answer was 53. When Gretchen returns to her algorithm, she says, “But three take away zero… that’s three.” Gretchen’s use of the traditional algorithm is a common approach as children develop their understanding of base ten, place value, and algorithms. Because Gretchen exercises her RotL, Gretchen’s teacher may know more about her thinking and help her reconcile the solutions.

## Conclusion

The case of Gretchen is not one that is unique to the IMAP video repository or of other similar video collections. I argue that our perception of children’s brilliance suggests a new approach—seeking opportunities for children to exercise their RotL rather than to passively replicate of efficient strategies.

Consider the students in your classroom. What if Gretchen were not a young, white child who felt comfortable exercising her voice in front of educational researchers, but a quiet Black child or a native Spanish-speaker who is learning mathematics in a new language? How can they exercise their RotL while showing their mathematical brilliance? How can we find opportunities to help each child exercise their RotL?

Because implicit bias and harmful stereotypes of Black and Indigenous children pose a real threat to their future success and advancement opportunities, the RotL might create a more equitable classroom for more students. I pose this goal: believe our students have rights as learners and create opportunities that highlight their brilliance.

Boaler, J., & Anderson, R. (2018). Considering the Rights of Learners in classrooms: The importance of mistakes and growth assessment practices. Democracy and Education, 26(2), Article 7. Retrieved from https://democracyeducationjournal.org/home/vol26/iss2/7

Hintz, A., Tyson, K., English, A. R. (2018). Actualizing the Rights of the Learner: The role of pedagogical listening. Democracy and Education, 26(2), Article 8. Retrieved from https://democracyeducationjournal.org/home/vol26/iss2/8

Kazemi, E. (2018). The demands of the Rights of the Learner. Democracy and Education, 26(2), Article 6. Retrieved from https://democracyeducationjournal.org/home/vol26/iss2/6

## References

Jansen, A., Cooper, B., Vascellaro, S., & Wandless, P. (2016). Rough draft talk in mathematics classrooms. Mathematics Teaching in the Middle School, 22(5), 304-307. Retrieved from https://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/2016/Vol22/Issue5/Rough-Draft-Talk-in-Mathematics-Classrooms/

Kalinec-Craig, C.A. (2017a). The Rights of the Learner: A framework for promoting equity through formative assessment in mathematics education. Democracy and Education, 25(2), Article 5. Retrieved from https://democracyeducationjournal.org/home/vol25/iss2/5

Kalinec-Craig, C. A. (2017b). Rights of the Learner blog posts. Retrieved from https://embracinglifewithmajorrevisions.wordpress.com/rights-of-the-learner-blogs

San Diego State University Foundation & Philipp, R. (2005). IMAP: Integrating Mathematics And Pedagogy To Illustrate Children's Reasoning (1st ed.) [CD-ROM]. New York, NY: Pearson Education.

## Credits

Author: Crystal Kalinec-Craig is an associate professor in the Interdisciplinary Learning and Teaching Department at the University of Texas at San Antonio. She examines issues of (in)equity in mathematics teacher education, primarily in K-8 settings. Kalinec-Craig is also interested in how pre-service teachers adopt democratic commitments to children’s rights as learners.

# A Productive Math Struggle: Questions and Assessment

Jenni Harding, University of Northern Colorado, Greeley, CO

Students need time to rationalize mathematics, play with it, and make it their own. Through a productive math struggle, teachers can create opportunities for students to conceptualize mathematics. Hiebert and Grows (2007) define a productive struggle this way: “students expend effort in order to make sense of mathematics, to figure something out that is not immediately apparent” (p. 387). Students can contemplate mathematical ideas, take risks, justify their own thinking, and evaluate other student ideas. Creating this environment in the K-12 classroom takes time and practice to establish. The purpose of this article is to focus on productive math struggle questions to enrich learning and guide assessment.

A productive math struggle can take place when students are answering problem solving questions within groups. In a productive math struggle, teachers focus students’ attention on ideas and sense-making and develop their math confidence. Also, teachers allow entry and exit points for a wide range of students, provide extensions and elaborations, and make connections between ideas, concepts, strategies, and procedures. Teaching responsibilities specific to a productive struggle include selecting high-quality tasks (some examples: Illuminations https://illuminations.nctm.org, YouCubed https://www.youcubed.org/tasks/, or Exemplars https://www.exemplars.com), asking high-level math questions, allowing students to verify and relate their strategies, listening to student responses, examining their work to extend and formalize their thinking, and providing targeted feedback. This structure of learning puts several of the Standards for Mathematical Practice into action, including: make sense of problems and persevere in solving them, construct viable arguments and critique the reasoning of others, use appropriate tools strategically, and attend to precision (Common Core State Standards Initiative, 2010).

I organized this article with questions and a rubric to support the structure of a productive math struggle. Teachers may use the first set of questions to support and challenge students’ understanding of mathematics content as they move between groups of students. The second set of questions guide student discussion, giving concrete stems to help students share their mathematics ideas and critique the ideas of others. The third set of questions model how to have students reflect upon their learning. Finally, the rubric demonstrates a specific way to evaluate students during a productive math struggle.

The teacher can extend learning by deciding what type of guidance is needed for students and then using their questions to scaffold mathematics thinking. Teachers can ask students questions to clarify students’ ideas, emphasize reasoning, and encourage student-to-student dialogue.

Teacher Questions: For a Productive Math Struggle

 Clarify Students' Ideas Did you use the red trapezoid as your whole? What parts of your drawing/diagram/web relate to the problem? Who could share what Julia just said, using your own words? Emphasize Reasoning Why does it make sense to start with these particular numbers? Can you give me an example? What connections do you see between Sara’s idea and Sam’s idea? Encourage Student-Student Dialogue Who has a question for Juan? Turn to your partner and explain why you agree or disagree with Shelly. Talk with Scott about how your strategy relates to his.

During a productive math struggle, teachers can give question stems to guide student discussion during math conversations. These question stems give structure to promote students’ participation. Because the stems help everyone to share their ideas, this can help more students to be heard, regardless of their background or status.

I suggest introducing one or two discussion stems each day that students participate in a productive struggle. Teachers can create a classroom anchor chart to hang in the room for student reference. Here are some sample math discussion stems:

• Explain why/how…

• What would happen if _____?

• How could _____ be used to _____?

• Why is _____ important?

• Did anyone think of this in a different way?

• Describe _____ in your own words.

• What are you thinking now?

• I agree/disagree with _____ because…

• That is good thinking because…

• I got different results because…

• My strategy is like yours because…

• My strategy is different than yours because…

• What I hear you say was…

By posing reflection questions for summative assessment, teachers can focus on the learning happening during the productive math struggle. This can exist in the form of math journals, exit tickets out the door, student self-evaluation, or group discussions to have students evaluate their own learning. Here are some sample math reflection questions:

• What were the main concepts or ideas you learned today?

• What questions do you have about ____?  If you don’t have a question, write a similar problem and answer it.

• Describe a mistake that you or a classmate had in class today. What did you learn from this mistake?

• How did your group approach today’s question? Was your approach successful?

What teachers assess and grade in their classroom demonstrates to students what is valued. For example, if teachers just grade homework, quizzes, and tests, it tells your students you only value formal assessments. I recommend grading students during a productive struggle session with a rubric to demonstrate the value of mathematics conversations happening in the classroom. This rubric gives students explicit guidance about what is expected of them during the productive struggle math group time. A group grade can demonstrate that conversation and understanding of mathematics is more important than the math answer. Teachers may use the rubric below by placing a tally mark each time they observe a group exhibiting one of the behaviors.

Productive Struggle Evaluation Rubric

 Group A Group B Group C Group D Leaning in and working in the middle of the table Equal air time (everyone takes a turn talking) Sticking together discussing each problem before going to the next one Explaining how they solved a task with justification and/or reasoning Listening to each other when someone is talking Asking each other questions to clarify and understand Providing solutions using multiple strategies Students persevere, persist, and don’t give up Following group roles or jobs Students encourage each other Overall Rating:____________

After teachers have all of the tallies recorded, they can give an overall group grade or they may use it as an informal assessment for that day's productive math struggle.  I use a 0-3 scale as an overall rating for all of the items on the rubric:

0 = No evidence during the observation
1 = A few isolated instances of evidence being observed (only a few items attempted; 5 out of 10 with at least one tally mark)
2 = Some evidence observed but does not seem frequent (many items attempted; 6 or more with multiple tally marks)
3 = Strong and frequent evidence observed; is regularly present (multiple tally marks in each)

Through questions, discussion stems, reflective questions, and assessment rubrics, teachers can create conditions for a productive math struggle. Through group conversations, students can delve deeper into mathematics. As a result, classroom math discussions can become more vibrant.

## References

Common Core State Standards Initiative. (2010). Standards for mathematical practice. Retrieved from http://www.corestandards.org/Math/Practice

Hiebert, J., & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F.K. Lester (Ed.), Second handbook of research on mathematics teaching and learning. (pp. 371-404). Charlotte, NC: Information Age.

## Credits

Author: Jenni Harding is a Professor of Education at the University of Northern Colorado. She teaches courses in education research, mathematics methods, and teaching to graduate and undergraduate students. Her students learn through productive learning struggles where they grow intellectually through multiple viewpoints of content, ideas, and concepts.

# Adapting an Instructional Routine: Stronger & Clearer Each Time

Dylan Kane, High Mountain Institute, Leadville, CO

Classroom routines can form a foundation for everyday teaching and learning (Kelemanik, Lucenta, & Creighton, 2016). Routines help to get a class started, to distribute materials, or to transition between activities. With routines, students can think less about what to do next or how to participate in an activity, which shifts the focus for students and teachers to mathematical ideas. Stronger & Clearer Each Time (S&C) is an instructional routine designed to support both mathematical understanding and language development (Zwiers, Dieckmann, Rutherford-Quach, Daro, Skarin, Weiss, & Malamut, 2017).

As a classroom teacher, I have adapted the S&C routine to fit my goals and better support my students’ learning. I adapt S&C by asking students to write notes summarizing the ideas of a discussion, often with the prompt to write as if the notes are for a future forgetful self. I use this framing to build on the idea of “meaningful notes” (Liljedahl, 2018), where I structure note-taking as an opportunity to occasion student thinking and create useful resources rather than copying information without thinking.

After giving students a few minutes to write, I use a spreadsheet randomizer (example) to put them into pairs. Then, I ask students to take turns sharing their notes with their partner, adding ideas to their own notes. I then randomize students a second time, and they share again with a new partner to allow students a second chance to share their ideas and hear a different perspective. An alternative is to ask students to talk to a shoulder partner, and then a partner in front of or behind them. Finally, students take a minute to summarize and clarify any additional ideas they want to have in their notes. I often finish by inviting students to share key ideas with the full class.

In a lesson on finding sums of arithmetic series, I asked students to take notes for their future forgetful self and gave them several minutes to write. When I assigned students partners to share and revise their notes, I walked around listening to their conversations. I noticed a few pairs talking about how when there are $$n$$ terms in a sequence, there are $$n-1$$ “jumps.” That is, if there are $$5$$ terms, the distance from the first to the last term is $$4$$ times the distance between consecutive terms. For instance, in the series $$11 + 13 + 15 + 17 + 19$$, there are five terms and four jumps. The total distance from the first to the last term is $$8$$, and the “jump” is $$2$$. This can help students to quickly calculate either the number of terms in a sequence or to calculate the final term given the number of terms.

The opportunity to listen in to students’ self-explanations allowed me to address a potential source of confusion in the moment. I asked one group to share their thinking with the class before putting students into new groups to summarize what they heard with a new partner. I encouraged students to add a similar example to their notes when they made their final revision to solidify their understanding of a challenging idea.

The adaptation of the S&C routine affords several opportunities. First, students have to self-generate explanations around key ideas and how they might use them to solve a problem. Self-explanations can help students integrate what they are learning with what they already know and support the transfer of their knowledge in the future (Lombrozo, 2006). By asking students to write notes to their future forgetful self, I hope to prompt them to go beyond summarizing what they learned, and consider how those principles might transfer to new problems. Giving students multiple opportunities to discuss with partners and revise their work increases the quality of their explanations, exposes students to more models of others’ high-quality explanations, and allows them to clarify misconceptions in the moment.

A second benefit of S&C is to teach students that revision is an essential part of learning and that students’ ideas are always valued, especially when ideas are partially formed. Jansen, Cooper, Vascellaro, & Wandless (2016) write:

If rough-draft talk is valued, brainstormed ideas are welcomed. More students are likely to take risks rather than freeze during challenging tasks. Valuing a wider range of contributions invites greater involvement, in contrast to the same students who participate frequently or not at all (p. 304).

S&C enacts rough draft thinking in practice. The teacher goes beyond telling students that rough drafts are valued; the S&C routine shows students the role rough drafts play in their learning by creating a resource through revision that is more likely to be useful to them.

Finally, S&C allows me to elicit evidence of student thinking. By listening in to conversations or looking at notes with revision, I can learn how students are thinking about a new idea and respond to that thinking. Transitions between partners can serve as an opportunity to address common issues, direct student attention to important features of a concept, or prompt students to add something to their notes and conversations. As I listen in to conversations and read student notes, I learn to understand content from a student perspective and see the many ways that students think about mathematical ideas differently than I expect. These opportunities help me to become a more effective teacher.

The S&C routine embodies the practice of mathematicians; rarely are initial ideas polished, and mathematicians revise their work and improve their explanations over time. I love this routine because note-taking can be a low-energy part of class, but with S&C students create useful resources while they rehearse and consolidate their thinking as they share and revise their explanations. My classroom becomes more equitable as I build in time and space for all students to reflect on their learning.

## References

Jansen, A., Cooper, B., Vascellaro, S., & Wandless, P. (2016). Rough-draft talk in mathematics classrooms. Mathematics Teaching in the Middle School, 22(5) 304-307.

Kelemanik, G., Lucenta, A., & Creighton, S. J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann.

Liljedahl, P. (2018, November). Building thinking classrooms. Presentation at the Regional Conference of the National Council of Teachers of Mathematics, Seattle, WA. Retrieved from:  http://www.peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-NCTM-Seattle-no-video.pptx

Lombrozo, T. (2006). The structure and function of explanations. TRENDS in Cognitive Science, 10(10), 464-470. https://doi.org/10.1016/j.tics.2006.08.004

Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the design of mathematics curricula: Promoting language and content development. Retrieved from Stanford University, UL/SCALE website: http://ell.stanford.edu/content/mathematics-resources-additional-resources

## Credits

Author: Dylan Kane is a National Board Certified high school math teacher in Leadville, Colorado. He blogs often about the intersections of research and practice in math education, and he is on the leadership team of the Teacher Leadership Program at the Park City Mathematics Institute.

# Investing Under Uncertainty: Mathematical Models and Computer Simulations

Alfinio Flores, University of Delaware

# Introduction

Mathematical modeling is stressed in the Common Core State Standards (Common Core State Standards Initiative, 2010) both as a content standard for high school and as a standard of mathematical practice for all ages. Modeling investment growth is one example mentioned for high school. Students need to experience mathematical models not only of deterministic situations but also of random phenomena. The following activities have students use computer simulations in the context of investing money with uncertainty. Students need to be able to interpret graphs of rational functions and have basic probability knowledge, such as if $$p$$ is the probability of winning and $$q$$ is the probability of losing, then $$p + q = 1$$. Students can download and run the simulations without needing any programming knowledge. By running many simulations, and comparing and contrasting the outcomes, students can learn that in probabilistic situations we can predict with some confidence the results for the long term, but that for small numbers of trials the results can vary widely.

## The Need for Bankroll Management

A particular investment may be a win or a loss for an investor. The uncertainty of investments can be better described and understood using probabilistic models rather than deterministic models. Investors, due to skill and experience, may have a favorable probability $$\left( p > .5 \right)$$, but even so, long losing streaks are not uncommon. Thus, it is important for investors to properly manage their bankroll, the total money that they can invest. For a long series of investments, is it better to invest each time a fixed amount, or a variable amount? Kelly developed a criterion that is used nowadays in investment theory and practice (“Kelly criterion,” n.d.). Students can use GeoGebra to develop a better understanding of the Kelly criterion, a real life long term optimal investment model, and Python programs to simulate and contrast two investment strategies.

## The Kelly Criterion

Kelly’s formula determines the optimal size of a series of investments to maximize the expected utility over long trials (“Kelly criterion,” n.d.). Considering simple investments with two possible outcomes, losing all the invested money, or winning with the corresponding payoff odds, the Kelly criterion indicates to invest the fraction $$f^* = \frac{bp-q}{b}$$ of the bankroll, where the quantities represent

$$f^*$$: fraction of the current bankroll to be invested.

$$b$$: net odds received ($$b$$ to 1).

$$p$$: probability of winning

$$q$$: probability of losing

Because $$p+q=1$$, students can rewrite the formula as $$f^* = \frac{p(b+1)-1}{b}$$.

## An Interactive Graph for the Kelly Criterion

Using GeoGebra, students can interact with the graph at https://beta.geogebra.org/m/CkxdBZ9n by seting the value of $$p$$ with a slider and graphing $$f^*$$ as a function of $$b$$. So, for $$p = 0.6$$ and $$b = 1$$, the Kelly criterion suggests investing 20% of the bankroll (Figure 1). For a fixed value of $$p$$, students can explore the relation between $$b$$ and $$f^*$$. For example, ask students to find a specific value that would suggest investing 35% of a bankroll. Let students discuss and describe how the fraction invested varies for different values of the odds. Students can identify the type of graph (a rational function, a hyperbola) and describe the two asymptotes. By dragging the slider, students can see how the graph changes for different values of $$p$$.

Figure 1. The Kelly criterion for b = 1 and p = 0.6

# Simulating Bankroll Management with Python

Python is a free programming language; different versions are available. We used Canopy, which has a friendly user interface (Enthought, 2014). Teachers or students can download Python. The file bankroll.py used in this article runs well with Python 3.x. After students download the file from http://udel.edu/~alfinio/bankroll.py, they need to right click and save it as a Python file (file extension .py). After opening the file with Python, in the “Run” menu select “Run File.” Students can then execute the different functions in the file by typing the name of the function with the desired value for the parameter, for example fixed(0.6), and pressing Enter or Return.

## Exploration of Investing a Fixed Amount

In this activity, students will learn that by investing a fixed amount, in the long term, investors can estimate their wins by using a linear function, but that for short sequences of investments the outcomes can vary widely.

Suppose a skilled investor can double the money invested 60% of the time ($$b = 1$$, $$p = 0.6$$). The other 40% of the time the investor loses the money. The starting bankroll is $1000 and$200 is invested each time. What is the expectation for this case? ($$0.6 \times 2 = 1.2$$) What is the expected net win for each investment? ($$\200 \times 0.2 = \40$$) What is the expected win after 200 investments? ($$200 \times \40 = \8,000$$)

We simulate the situation with the Python function fixed(p). The investor will make 200 investments or stop investing if the bankroll is depleted. Students can type fixed(0.6) to run the function for $$p = 0.6$$. Figure 2 shows two simulation examples for $$p = 0.6$$. Let students observe the graphs, discuss, and describe the results of different simulations, paying attention to the scale, identify interesting parts of the graphs, such as long losing streaks, and contrasting long-term predictions with short-term variability.

Figure 2a. Bankroll change with fixed investment amount, Example 1.

Figure 2b. Bankroll change with fixed investment amount, Example 2.

## Exploration of Using the Kelly Criterion

In these activities, students will learn that by using the Kelly criterion, in the long term, investors can make their bankroll grow much more than by investing a fixed amount, but that short sequences of investments can vary widely and include considerable losses.

We consider again two possible outcomes, one losing the entire investment and the other doubling the money ($$b = 1$$). The probability of winning is $$p$$ and $$q$$ of losing. For $$b = 1$$, Kelly’s formula is simply $$f^* = p - q$$. Thus, if an investor has a probability of 60% of winning, Kelly’s formula recommends investing 20% of the bankroll.

Students can run the function kelly(p) using $$p = 0.6$$. The initial bankroll is \$1000, and the investor will wager 200 times or stop investing if the bankroll is depleted. Figure 3 shows the results of two simulations for $$p = 0.6$$. When observing and discussing the graphs, students should pay special attention to the scale on the vertical axis.

Figure 3a. Bankroll growth with Kelly criterion, Example 1.

Figure 3b. Bankroll growth with Kelly criterion, Example 2.

Students can then run the program for $$p = 0.7$$ several times. What seems to be a modest increase in the favorable probability (from 0.6 to 0.7) can drastically change the outcomes.

# Final Remarks and Extensions

Bankroll management is also important in other contexts where people face uncertainty, such as professional gamblers. In this case too, bankroll management “should eventually reference the Kelly criterion.” (Luis Flores, personal communication, 2014). Other situations involving money and uncertainty can also be simulated using probabilistic models. Students can simulate the effect of different strategies for withdrawing money in retirement when the assets face unpredictable fluctuations, such as in the stock market.

# References

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

Kelly criterion (n.d.). In Wikipedia. Retrieved February 18, 2019, from https://en.wikipedia.org/wiki/Kelly_criterion

## Credits

Author: Alfinio Flores is the Hollowell Professor of Mathematics Education at the University of Delaware. He teaches mathematics and mathematics methods courses con ganas. His students use hands-on materials, visual representations, computers, and calculators to explore mathematical ideas and to develop a better understanding of concepts and relations.

# From soliciting answers to eliciting reasoning: Questioning our questions in digital math tasks

Heather Lynn Johnson, Gary Olson, Amber Gardner, Amy Smith

Students can interact with digital math tasks in different locations, on different devices, and for different purposes. What kinds of questions do students encounter when interacting with digital math tasks? And why might the kinds of questions matter?

We designed digital math tasks to provide opportunities for students to engage in math reasoning. Questions are a key component of the tasks. With our questions, our goal was to do more than solicit students’ answers. We intended to elicit students’ reasoning.

We share a digital math task and a question from the task. Then we provide three design principles guiding our questions.

We developed the The Toy Car task in collaboration with Dan Meyer and the Desmos team. The task begins with a video of a toy car moving along a curved path (Figure 1). Then students investigate and graph relationships between a toy car’s distance from a shrub, and its total distance traveled.

Figure 1. The toy car and the shrub

The Toy Car task is part of a group of tasks that we call Techtivities. The Techtivities include video animations and dynamically linked, interactive graphs. Students have opportunities to sketch different graphs to represent the same relationship between attributes. Then students reflect on what those graphs represent. To learn more about the Techtivities, see Johnson (2018).

In the Toy Car task, students sketch, then reflect on two different graphs, shown in Figure 2. Each graph represents the toy car’s total distance traveled as a function of the toy car’s distance from the shrub.

Figure 2. Two different graphs in the Toy Car task

Students might wonder how it is possible for two different looking graphs to represent the same function relationship. Furthermore, students might notice that the graph shown at right in Figure 2 does not pass the vertical line test, meaning that a vertical line would intersect the graph at more than one point.

Students can apply the vertical line test based solely on the shape of a graph, and they may miss how graphs can represent relationships between attributes in a situation (Moore, Silverman, Paoletti, & LaForest, 2014). In the Toy Car task, and across the Techtivities, our goal was for students to focus on relationships between different attributes in the situations. We worked to design questions that could help us to achieve our goals.

## A Question

We posed this question in the Toy Car task: Val says that both of these graphs represent the toy car’s total distance traveled as a function of the toy car’s distance from the shrub. Do you agree or disagree? Why or why not? (Graphs are shown in Figure 2.)

We purposefully posed this question as person’s (Val’s) claim, rather than as a claim devoid of human connection. Furthermore, we used precise language to clarify Val’s claim. In particular, we used the phrase as a function of, rather than the more general term, function. We did this so that Val’s claim focused on the function relationship that the graphs represented. Overall, we aimed to position Val as a capable doer of mathematics, who made a claim worthy of consideration.

## Three Design Principles

1. Provide opportunities for students to consider other students’ claims. Mathematics is a human endeavor (Freudenthal, 1973). In our questions, we decided to have students respond to another student’s claim. We could have asked students: Do both graphs represent the toy car’s total distance traveled as a function of the toy car’s distance from the shrub? By framing our questions as a response to another student, we aimed to humanize students’ interactions with the digital math tasks.

2. Allow for gender ambiguity when incorporating student names into task questions. Students can think that gender identity plays a role in mathematical ability (Boaler, 2002; Leyva, 2017; Rubel, 2016). In our questions, we aimed to use gender ambiguous names, and names we selected were often informal. We could have used a pronoun to assign a gender identity to Val, or selected a more gendered name. Instead, we intended to open possibilities for students to use a variety of pronouns, or no pronouns at all, when responding to the student claims given in the tasks.

3. Elicit sense making, rather than soliciting judgments of correct/incorrect. To promote students’ reasoning, we posed questions to elicit sense making rather than solicit judgments. We could have asked students if Val was right or wrong. Instead of asking students to judge another student’s claim as correct/incorrect, we chose to ask students to explain why they agreed or disagreed. We intended to offer students opportunities to consider possibilities, rather than rushing to judgments.

## Closing Remarks

Doing mathematics is so much more than finding answers. With our questions, we can work to create spaces for students to engage in reasoning and sense making. In designing questions for our digital math tasks, we are aiming to do just that.

Acknowledgments. This work was supported by a grant from the National Science Foundation (DUE-1709903). Opinions, findings and conclusions are those of the authors. We thank Dan Meyer and the team at Desmos for their work with us. We are grateful to our colleagues who provided feedback to help us to grow.

## References

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Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: D. Reidel Publishing.

Johnson, H. L. (2018). Helping students see how graphs work | Edutopia. Retrieved July 30, 2018, from https://www.edutopia.org/article/helping-students-see-how-graphs-work

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