Chapin, S. H., & Johnson, A. (2006). Addition and Subtraction. In Math Matters: Understanding the Math You Teach, Grades K-8 (2nd ed., pp. 66-67). Sausalito, CA: Math Solutions. 
Chapin and Johnson describe the skills that students need to move from modeling strategies to number sense strategies. Students must be well versed in combinations to 10, doubles facts, the commutative property, and the inverse relationship between addition and subtraction. Students will move back and forth between modeling strategies and number sense strategies when encountered with new challenges. The opportunity to apply skills to as many problems as possible seems to support the use of mental strategies.

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Bruce, C. D. (2007, January). Student Interaction in the Math Classroom:Stealing Ideas or Building Understanding. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/bruce.pdf
In this article, Dr. Catherine D. Bruce, an assistant professor at Trent University researching mathematics education and teacher professional development, lists five strategies to support student interactions in math discussion. The article first outlines difficulties teachers may encounter with facilitating this kind of community engagement, like time, experience, math skills, and opportunities for professional development. The five practices to support facilitation of math discussions are “rich math tasks,” encouragement of strategy justification, student questioning prompts, wait time, and sentence stems for interacting with peer strategies. 
 
Campbell, P. F., Rowan, T. E., & Suarez, A. R. (1998). What criteria for student-invented algorithms? Yearbook (National Council of Teachers of Mathematics), 1998, 49–55. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&db=eue&AN=507609750&site=ehost-live
 
I read an article about criteria for invented algorithms by Campbell, Rowan, and Swarez. The article began by talking about classroom features that support the mathematical exploration and how teachers encourage inventive strategies. This work also needs questions to make the thinking transparent and test the limits of the algorithms, pushing for conjectures.The article then goes on to describe three criteria for which an invented algorithm can be assessed. The article talks about using these criteria to determine if the algorithm is “acceptable.” They are efficiency, validity, and generalizability. This article was very helpful and I feel like it can inform our work with moving students from direct modeling to invented algorithms. 
 
 
(D. Fuentes, personal communication, September 30, 2019).
 
We talked about ways to support development of academic language through paraphrasing. She suggested having students paraphrase each other, using this as a scaffold and assessment. Some major takeaways were around scaffolds to support engagement with each others thinking. She suggested assigning students “thinking jobs,” (Fuentes, 2019) in which they had to look for similarities and differences between their strategies and the presented ones. She suggested building silent gestures into the discourse so they can be subtly recognized for their work with their thinking jobs. Debra suggested having students come up with a clear take way away, even recording something they will try. 
 
 
Daly, I., Bourgaize, J., & Vernitski, A. (2019). Mathematical mindsets increase student motivation: Evidence from the EEG. Trends in Neuroscience and Education, 15, 18–28. doi: 10.1016/j.tine.2019.02.005
This article was about a study of motivation with exposure to math problems designed to be open through recommendations from mathematical mindset theory. Motivation was examined through self-reported reflection and through examining the neural patterns believed to show motivation, recorded with an EEG. This study was interested in the mathematical mindset theory and the observation of increased motivation with students are given more open ended math questions. The study provides empirical evidence for the commonly accepted correlation between implemented motivational mindset theory and increased motivation in students. 
This work was used to support my understanding of the benefits of the CGI problem structure. Students are more motivated to solve these problem types in their own inventive ways because they know there are multiple ways to solve it. 
 
 
Yeager, D. S., & Walton, G. M. (2011). Social-Psychological Interventions in Education: They’re Not Magic. Review of Educational Research, 81(2), 267–301.https://doi.org/10.3102/0034654311405999
In this review of educational research, Yeager and Walton present the perspectives of social-psychological interventions in education, challenge perspectives about causality and effectiveness, review various case studies that support the argument, suggest ways to implement these interventions on a large scale basis, and identify challenges and opportunities of scaling up interventions. 
Scaling up social-psychological interventions will increase the data we have to continue refining the development of such interventions. The benefits are not only there for students, but schools and psychologists as well. Yeager also makes a point that SPI are not meant to replace other forms of school reform, as he began, these are a cure-all. The SPI are made to help students “take better advantage of learning opportunities that are present in schools.” (Yeager, 2011, p. 293)
This work was used to reflect on ways to build a growth mindset with regards to math, status and identity.