Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Teaching Strategies:

Pre-Teaching

I will pre-teach the use of x or another “variable” as a placeholder for an unknown value (The word variable appears in quotations, as the letters used for unknown variables in the word problems given throughout the unit are fixed). This will further prepare students to translate word problems into expressions and equations, as they will recognize an unknown quantity in the word problems to be x. Vocabulary terms that I will pre-teach students include unit, term, expression, and equation, so that students can successfully complete the problem sets which use this terminology.

Inquiry-Based Learning

This unit plan relies heavily on inquiry-based learning, which is learning through formulating new ideas and solutions to given problems. This learning will be conducted individually, in pairs, or in small groups- depending on the complexity of the given word problems- so that students can deliberate and help one another understand the proper use of a variable and its units in writing expressions and equations from word problems.

Kikan-jyunshi/ Kikan-shido

The Japanese concept of Kikan-jyunshi or Kikan-shido is the “purposeful scanning or monitoring” of the class to “elicit [individual] students’ ideas and bring them forward for whole-class discussion.”⁷ Japanese teachers very much prefer to question and elicit response rather than to lecture. In this unit, it is important to use Kikan-jyunshi or Kikan-shido to identify student work that promotes algebraic, arithmetic, and illustrative means of solving given word problems; this may facilitate transfer of knowledge from images or arithmetic to algebra for some learners. As Yale mathematics professor Roger Howe suggests, “…teaching symbolic algebra, and its use in solving word problems, might profit from the linguistic perspective that students might benefit from seeing, studying, discussing and working through the translation into algebra of many examples of verbally formulated situations, including work of solving these problems both with and without algebra, and comparing the algebraic and the arithmetic solutions to these problems.”⁸ The practices of Kikan-jyunshi and Kikan-shido encourage pathways of learning that not only follow Howe’s suggestion but provide personal relevance to the students and give learning a deeper meaning.

Culturally Responsive Teaching

I plan to incorporate word problems that are relevant to students’ day-to-day lives. Too often, word problems are irrelevant to my students; they convey scenarios that my students either don’t care about or have never experienced. Therefore, I plan to create word problems that not only are relevant but that are actual problems students may have interest in solving.

Cooperative Learning (Group Work)

Students will work together in pairs and small groups to support each other’s learning. Studies have shown that “Use of co-operative learning almost always improves affective outcomes.”⁹ Yet, “group goals” and “individual accountability” are necessary for this improvement to occur.¹⁰ Therefore, I will explicitly show students how group work is intended to operate; this way, students will have a better understanding of their individual roles in a group setting.