Introduction to Mathematical Mindset Algebra

This 4-week curriculum unit we have developed is one that can be used to introduce algebraic concepts at any grade level. It draws upon algebraic research showing that it is more helpful for students to learn algebra through studying pattern growth where a variable represents a case number, and can vary, before learning about “solving for x.” When students start learning algebra by solving for x they come to believe that a variable stands for a single number and does not vary. Later when they need to understand that variables can vary, they meet a conceptual barrier, and many do not ever get past that barrier. We recommend that students learn first about pattern growth and see that algebra can be useful for describing growth. Later, when they encounter situations when the variable stands for one missing number, they see this as a subset of their broader learning about variables and there is no confusion. For a detailed review of research on algebra learning, see Kieran, (2013).

A second goal of our curriculum is for students to learn that algebra is a problem-solving tool. Students will learn to examine different functions that they explore visually, numerically, graphically, physically constructed, and algebraically. Students will be generalizing, representing, modeling, describing, and interpreting the relationships between two quantities. They will also be distinguishing between linear, quadratic, cubic, and exponential growth within multiple representations.

A third goal of our curriculum is to help students develop stronger number sense, as many students fail algebra, not because the algebra is difficult but because they lack a strong foundation in number sense (Gray & Tall, 1994; Boaler, 2016). In our number sense activities students will learn ways to adapt numbers and to use grouping symbols that will help them understand and use algebraic expressions.

Throughout the 4-week unit students will receive opportunities to make important brain connections, as they experience algebra in different ways, forms and representations.

References & Further Reading.

Boaler, J (2016) Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching. Jossey-Bass/Wiley: Chappaqua, NY.

Gray, E. and D. Tall (1994). “Duality, Ambiguity, and Flexibility: A “Proceptual” View of Simple Arithmetic.” Journal for Research in Mathematics Education 25(2): 116-140.

Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 390-419). New York, NY, England: Macmillan Publishing Co, Inc.

Mason J. (1996) Expressing Generality and Roots of Algebra. In: Bernarz N., Kieran C., Lee L. (eds) Approaches to Algebra. Mathematics Education Library, vol 18. Springer, Dordrecht.

Schoenfeld, A. & Arcavi, A. (1988) On the Meaning of Variable. The Mathematics Teacher, 420-427.

Thompson, P. W., McCallum, W., Harel, G., Blaire, R., Dance, R., & Nolan, E. (2007). Intermediate algebra. In Algebra: Gateway to a technological future.