The assigned reading for outside class from *Strength in Numbers* focused on creating tasks and lessons that align with the values expressed in the first few chapters of the book. This mirrors our shift into creating lessons for the high school. A particular part that stood out to me in *Strength in Numbers *is the section on mathematical fluency. *Strength in Numbers* states that mathematical fluency has five strands:

- Conceptual understanding
- Procedural fluency
- Strategic competence
- Adaptive reasoning
- Productive disposition

The productive disposition strand stood out to me this week, as I tutor students on the football team enrolled in MATH 095 and MATH 102 four hours a week. Productive disposition is defined as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.” What I have noticed most in my hours tutoring these students is their lack of productive disposition. While reading *Strength in Numbers* I began to think about why these students are missing a productive disposition. The chief complaints that I hear from students is that they are teaching themselves with videos, their work does not matter (and thus no partial credit is given), and the amount of homework is excessive. Keeping what I have learned in our class in mind, I began to look into the implications of having a productive disposition for mathematics courses. I was able to find one study that looked at mathematics in the middle school, and how mathematical teaching materials impacted students’ productive discipline, called “The Impact of Mathematical Models of Teaching Materials on Square and Rectangle Concepts to Improve Students’ Mathematical Connection Ability and Mathematical Disposition in Middle School” by Irfan Mufti Afrizal, and Jarnawi Afghani Dachlan. The study found that students who learned by using mathematical teaching materials through modeling had an improvement in their productive disposition.

Link: Mathematical Modeling

In the future, I hope to use this information in my own teaching practices. Not only has reading about this study been beneficial to me, but my consistent interactions with students who struggle in mathematics has been beneficial. It has helped me to realize the true implications of having a productive disposition, the importance of a teacher in the classroom, and to consider how I can work toward creating a productive disposition in students. The study found that productive disposition in students improved when they learned material with more modern methods of teaching mathematics, rather than traditional. It emphasizes the importance of mathematical modeling as an instructional strategy In my own classroom, I can utilize this information when constructing lessons and tasks for my students. In the past, I have written about how I hope to use mathematical modeling in my own classroom. Cultivating a productive disposition in students is at the core of mathematical fluency and helping students to be successful in mathematics. Much of what we have discussed previously in this class indirectly aids in creating productive dispositions for students. In the study it was shown that using modeling in instruction does help to bolster productive disposition, but I believe that tasks focused on reasoning and sense making, coherence, and other topics discussed can help as well. The emphasis of the process rather than the answer is inherent in each of these topics, can boost productive disposition because students themselves will begin to shift their focus to the process of finding the answer, and their ability to problem solve rather than finding the correct answer. Furthermore, there is frequently more than one approach to these problems so this also aids in building a productive disposition in students. These mostly aid in building the belief in one’s own efficacy. Regarding the belief of the usefulness of mathematics, tying in real world applications is imperative. How students view the mathematics will impact how they approach the problems that are presented to them. If they do not see it as useful, their productive disposition is hurt and they will not be able to sufficiently build their mathematical fluency. This particular strand is an essential part of building mathematical fluency, and a great consideration when developing tasks for students.