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

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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.

Associate Editor: Lisa Bejarano edited this article.



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.

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

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

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/

Enthought (2014). Canopy [Computer software]. Retrieved from https://www.enthought.com/downloads/.

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

Credits

alfinio_flores.jpg

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.

Associate Editor: Gulden Karakok edited this article.