Foundations of Fishtank Math

Written by former classroom teachers and leaders, the Fishtank Math curriculum is designed with students and teachers in mind first and foremost. We believe that all students deserve access to high-quality curriculum and that students should not need to prove they can do rigorous, grade-level math in order to gain access to it. We see these beliefs as key components of supporting anti-racist school practice, and we share our curriculum as a trusted resource for educators in this work. As a curriculum team, we are continually listening, learning, and iterating on our curriculum and resources to get this work right. We strive to help all students see themselves as confident and competent mathematicians who are able to apply their math knowledge both in and out of the classroom as global citizens. 

With this in mind, we designed the curriculum to include a balance of student-directed and teacher-led learning. Students have ample opportunities to investigate, explore, and be the drivers of their own learning. At the same time, teachers have what they need to ensure students are adequately guided towards strong conclusions. The curriculum is also designed to be comprehensive yet flexible. Every Common Core State Standard for Mathematics is covered in the Fishtank curriculum, with a focus on the major work of each grade and a thoughtful vertical progression embedded from course to course. Teachers are provided with a flexible lesson structure that gives them the content and tools they need to make the decisions that are right for their students. Lastly, we designed the curriculum to be standards-based and content-rich. This means you can be sure students have access to and practice with standards-aligned problems that are engaging, accessible, and supportive of productive struggle. 

Read more about the guiding principles that are core to the design of the Fishtank Math curriculum.

In the Fishtank Math curriculum, the content is placed front and center. Each lesson is organized around a handful of Anchor Problems or Tasks, created by the Fishtank curriculum team as well as selectively curated from trusted resources. To support the best learning opportunities for students, these problems: 

• Are accessible to all students with low-floor/high-ceiling opportunities for students to engage in grade-level content
• Offer more than one solution pathway and encourage the use of a variety of strategies
• Support worthwhile mathematical discussions 
• Are intentional in building strong conceptual foundations before introducing procedural algorithms or processes
• Support productive struggle
• Make connections to mathematical concepts explored within and across grade levels

Tasks put in front of students should be tasks that are worth doing. While not every task will encompass all of these characteristics above, those found in the Fishtank Math curriculum aim to engage students as active, thoughtful participants. At times, this means students dive into deep, complex tasks that challenge them to grapple with a new concept; other times, this means students are solidifying generalizations and working towards fluency. As a standards-aligned curriculum, problems stay true to the standard language and the aspect of rigor that is called for in each standard. 

Discussion offers unique benefits to teachers and students that are hard to come by any other way. It can reveal understanding and misunderstanding - to the speaker themself, their classmates, and their teacher. Discussing ideas helps students retain understanding and information, both by articulating their ideas and hearing them repeated by others in their own words. It supports language development and the construction of logical arguments. It also increases social skills, helping students develop effective ways to communicate with others like patiently listening, understanding other people’s points of view, respectfully disagreeing, and being empathetic and kind to their classmates. Perhaps most importantly, discussion helps empower students to see themselves as sense-makers, building their own confidence in their abilities and increasing their motivation, and decentralizes the power dynamic in the classroom. Thus, by providing a curriculum full of rich, discussion-worthy tasks that provide students opportunities to describe and evaluate their ideas, share strategies, and make conjectures, Fishtank aims for students and teachers to develop a deeper understanding of the content and themselves. 

Different students learn in different ways, but what is constant is that learning is a process. There are many elements that can be part of this process: making sense of problems, trying out both familiar and new strategies, drawing on prior knowledge, discovering patterns or efficient algorithms, and discussing ideas with peers, to name a few. In many ways, these elements are deeply intertwined with the standards for mathematical practice, which outline "processes and proficiencies" that develop students' mathematical reasoning skills and habits of mind. Another part of valuing students’ learning process is using mistakes as positive opportunities to make meaning and build understanding. By normalizing mistakes as part of learning, students may be more likely to take risks, build confidence, and engage in productive struggle. The Fishtank Math curriculum aims to recognize and support students’ individual paths to learning content by encouraging multiple approaches to solving problems, activating prior knowledge, and making connections across ideas.

While "assessment" can often imply tests and quizzes for many, the term can be thought of much more broadly. Summative assessments, like post-unit assessments, have value in evaluating student understanding, but formative information, like that gleaned from pre-unit assessments, Target Tasks, and even Problem Sets, is an even more powerful tool to help teachers make in-the-moment instructional decisions that are best for their students. Students also play an important role in monitoring their understanding, which is why pre- and post-unit student self-assessments are included in the Fishtank Math curriculum offering. With these assessments, students can diagnose their own understanding and reflect on what actions they took to be successful to build metacognition and foster a growth mindset. With each of these resources, students can demonstrate their understanding in various ways, such as through computing, drawing, or explaining, none of which is inherently more valuable than another. By providing teachers and students with frequent feedback, they are able to learn from their mistakes and improve their understanding. 

The Fishtank Math curriculum was designed by former classroom teachers who believe and acknowledge that teachers have both knowledge of their subject and knowledge of their students. The curriculum provides teachers with autonomy and flexibility so they have the support and freedom to make informed instructional decisions. Teachers are encouraged to make adjustments based on the students in front of them, as a way to foster student choice and engagement. The curriculum is designed so that teachers feel empowered to use a variety of classroom structures, from pair and group work to student-led opportunities. Whether teachers are veterans or in their first years of teaching, the curriculum provides the content support needed to make both teachers and students feel successful without a completely scripted lesson. Lastly, teachers have the desire to continually build their own content knowledge and become greater experts in their fields. To support them in this work, the Fishtank Math curriculum provides resources to support intellectual preparation so teachers can plan smarter and more efficiently and effectively. 

Anchoring standards-aligned, content-rich tasks

• Smith, M. S. & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 14(9), 548--556. 
• The National Council of Teachers of Mathematics (2014). Principles to Actions. (pp. 17-24). NCTM. www.nctm.org/PtA/
• National Council of Teachers of Mathematics (2021). Problem Solving. The National Council of Teachers of Mathematics. www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Problem-Solving/

Communicating mathematical understanding

• Smith, M. S.,  Stein, M. K. (2018). 5 Practices for Orchestrating Productive Mathematics Discussions. The National Council of Teachers of Mathematics. www.nctm.org/Store/Products/5-Practices-for-Orchestrating-Productive-Mathematics-Discussions,-2nd-edition/
• Chapin, S. H., O'Connor, C.,  Anderson, N. C  (2013). Classroom Discussions in Math. Math Solutions.
• The National Council of Teachers of Mathematics. (2013). Benefits of Discussion. NCTM. www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Benefits-of-Discussion/

Valuing the process of learning

• National Research Council (2001). Adding It Up: Helping Children Learn Mathematics, (pp. 333-335, 422-423). Washington, DC: National Academy Press.
• Hiebert, J., Grouws, D. A. (2007). The Effects of Classroom Mathematics Teaching on Students' Learning.  In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning. (pp. 379-393). Charlotte, NC: Information Age.
• Cintron, S. M, Wadlington, D., ChenFeng, A. (2021). Stride 1: Dismantling Racism in Mathematics Instruction. (pp. 66-69).  A Pathway to Equitable Math Instruction. equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf

Monitoring student progress

• National Council of Teachers of Mathematics (2007). Benefits of Formative Assessment. NCTM. www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Benefits-of-Formative-Assessment/ 
• National Council of Teachers of Mathematics (2007). Strategies for Formative Assessment. NCTM. www.nctm.org/Research-and-Advocacy/Research-Brief-and-Clips/Strategies-for-Formative-Assessment/ 
• The National Council of Teachers of Mathematics (2014). Principles to Actions. (pp. 89-98). NCTM. www.nctm.org/PtA/

Honoring and building teacher expertise

• Remillard, J. T. (2005). Examining Key Concepts in Research on Teachers' Use of Mathematics Curricula. Review of Educational Research, 75(2), 211–246. doi.org/10.3102/00346543075002211. 
• Short, J.,  Hirsch, S. (2020). The Elements: Transforming Teaching through Curriculum-Based Professional Learning. Carnegie Corporation of New York. www.carnegie.org/publications/elements-transforming-teaching-through-curriculum-based-professional-learning/