- "Exponents represent repeated multiplication." This is very much embodied for students who are comfortable with mutliplication of numbers. This embodied form is limited to positive exponents only.
- Exponent properties listed in 1.3.2. These are symbolic properties derived from our embodied understanding of positive exponents. Memorizing the properties without regular reference to embodied examples regularly leads to incorrect properties.
- Negative, zero, and fractional exponents are formally derived from the properties of positive exponents. Letting students know this (in more informal language) can let them know that this is difficult and may need more attention.
Subsection 2 Tips for Teachers
Subsubsection 2.1 The Ideas Behind the Text
This text is written for a first-year college audience who will be going on to take a Precalculus (with trigonometry) course followed by the Calculus sequence (mostly STEM majors). This will not be the first exposure to Algebra for most of the students, but it might be for some of them. It is not intended to be a terminal mathematics course. It is assumed that subsequent courses will expose students to the idea of functions as the primary object of study. In this course our goal is for students to gain literacy with symbols in mathematics. However, we don’t just mean the ability to follow rules to get correct answers. We mean for students to attach concrete (possibly geometric) meaning to the symbols they write and understand mathematical operations as real actions.
How do we help our students both build the ability to make meaning, while also making sure they obtain manipulative fluency? The approach of this text is to blend the so-called three worlds of mathematics as described by David Tall in [3] (his descriptions in italics):
-
Conceptual Embodiment builds on human perceptions and actions developing mental images that are verbalized in increasingly sophisiticated ways and become perfect mental entities in our imaginations.In this text, this approach to mathematics often takes form in geometric understanding. For instance, we think of the variable \(x\) as a length and manipulate geometric objects to make algebraic manipulations more concrete. For instance, we do this in 2.2.5 and 2.2.31 to give extra meaning to things most students have previously encountered.It is the belief of this author that as mathematical ideas are understood, they all become somehow embodied.
-
Operational symbolism grows out of physical actions into mathematical procedures. Whereas some learners may remain at the proceduaral level, others may concieve the symbols flexibly as operations to perform and also to be operated on through calculation and manipulation.It is often the case in Algebra courses that we jump straight to symbolism, and then are satisfied with students only gaining procedural competence. This is the "rules without reason" type of learing described in 1. By making sure we only proceed when students understand the meaning of the symbols we can have students work with them more flexibly.
-
Axiomatic formalism builds formal knowledge in axiomatic systems specified by set-theoretic definitions, whose properties are deduced my mathematical proof.It may appear that formalism is limited in an introductory algebra course. This isn’t really true. Indeed, many of the things students have the most difficulty with are due to formality that is unexplained.
Example 2.1.2.
It is worthwhile to note as you teach from this text whether you are teaching something from an embodied, symbolic, or formal viewpoint. If you make a sudden swithch, make some sort of distinction of it for your students. As teachers of mathematics, we often move subconsciously between these three worlds. This can be jarring for students. If something is initially described in a way that is meant to be intuitive, but then moves to formal, students may think they missed something and are "just not a math person".
Subsubsection 2.2 Text Specifics (for Faculty)
This text is meant for students to read and answer all the questions in it. Be sure to let students know where you are and what they should be reading. There are two exercise types in the text that are meant as breakpoints in the exposition.
- Checkpoints: Checkpoints are online exercises that you can do to help make sure everyone following what is being done. They are Webwork problems that are mostly included in set definition files for the course (available on request). Working through some of them in class will help students gain confidence without having the class bog down into a sequence of sample homework problems.
- Questions: These are questions that often have essay type answers. They are often difficult questions that are meant to be thought provoking and/or start a conversation with your instructor or a tutor. They are meant as formative assessment. Some subset of these should be collected and graded (perhaps with a generous rubric) weekly.
