# 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

Boaler, J. (2002). Paying the price for “sugar and spice”: Shifting the analytical lens in equity research. Mathematical Thinking and Learning, 4(2-3), 127–144.

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

Leyva, L. A. (2017). Unpacking the male superiority myth and masculinization of mathematics at the intersections: A review of research on gender in mathematics education. Journal for Research in Mathematics Education, 48(4), 397–452.

Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157.

Rubel, L. H. (2016). Speaking up and speaking out about gender in mathematics. The Mathematics Teacher, 109(6), 434–439.

Heather Lynn Johnson, CMT Editor

I ask questions about writing. What are you writing today? Have you written that down? What are your writing goals? How are you nurturing your writing process?

With my questions, I intend to encourage others to grow their writing practice. Yet, writing can be difficult. Even intimidating.

I find getting started to be one of the most challenging aspects of writing. Have you thought about sharing your ideas in writing? Do you have research findings that you want to share with a broader audience? Have you given (or attended) a presentation that sparked conversation? Do you have fresh insights into mathematics teaching and learning? Then the time for getting started is now.

As the new editor for a new CMT journal, my goal is to cultivate a space for community, connection, and conversation. A space with synergy between research and practice. A space where people invested in mathematics education in Colorado (and even beyond) can share their expertise to learn and grow from each other.

Interested in writing for the CMT? Here’s how to get started.

# What Kinds of Articles are Suitable for the CMT?

CMT articles should address relevant issues in mathematics education. Relevant issues can span research and practice. Share your stories, your insights, your struggles, your innovations, or your new findings. The CMT editorial team is particularly interested in articles that address one or more of these strands: Teaching and Learning, Access and Equity, Tools and Technology, Professionalism, and Assessment.

# What is the Format for CMT Articles?

CMT articles should be between 800-1200 words, including titles, tables, figures, and references. Authors should write for a broad audience of people invested in mathematics education (in Colorado, and even beyond).

Wondering what a completed CMT article looks like? Here is an example:

# What is the Submission Process?

Submitting an article to the CMT starts with a proposal.

Send proposals to this email address: cmt (at) cctmath (dot) org

In your proposal, include the following:

1. Subject line: CMT: Title of Your Proposed Article

2. A short paragraph summarizing the main points of your article.

3. An outline of main sections of your article.

5. References (or links) for up to three recent publications (or presentations). [We welcome first time writers. If you haven’t published or presented yet, do not let that keep you from submitting.]

6. A statement disclosing any commercial interests that you have in products described in the article proposal.

7. A statement describing any portions of the planned article that appear elsewhere. (Or a statement indicating that no portions of the planned article appear elsewhere.)

# What Happens after Submission?

The CMT editorial team will review your proposal. After your proposal is reviewed, a member of the CMT editorial team will contact you. The review process typically takes a few weeks, sometimes longer.

If your proposal is accepted, the CMT editorial team will ask for you to send a complete draft of your article. After submitting your draft, there likely will be one or more rounds of required revisions. If revisions are required, a member of the editorial team will work with you along the way.

If your proposal is rejected, know that the CMT editorial team carefully reviewed your proposal. The CMT editorial team cannot provide in depth feedback for all proposals received. If your proposal is not accepted, the CMT editorial team encourages you to send another proposal.

Inspiration for the CMT submission process came from Edutopia.org.

# What Are Authors’ Ethical Responsibilities?

The CMT editorial team expects that authors uphold the integrity of the CMT journal. Authors should submit only new contributions, which have not been published elsewhere. If authors report data, they should not misrepresent, fabricate, or manipulate data for their own purposes. Authors should not plagiarize others’ work. When authors draw on others’ research or ideas, they should provide references and/or acknowledgments to give appropriate credit.