Pathways from the Pastis a pair of teacher’s manuals and two sets of activity sheets
for helping students master various topics in arithmetic, pre-algebra, and algebra.
Building on ideas from their award-winning book, Math through the Ages, Drs. Berlinghoff
and Gouvêa have crafted teacher-friendly tools for helping students understand basic
topics that often cause trouble. Their approach implements the spirit of the NCTM
document, Reasoning and Sense Making, as well as the “Standards for Mathematical
Practice” of the Common Core State Standards for Mathematics (CCSSM). The second
Pathways from the Past
II: Using History to Teach Algebra
This is a 72-page manual and a set of reproducible masters for 18 two-page activity
sheets that you can use with your students to strengthen their ability to think clearly
and reason effectively as they learn algebra.
First Thoughts for Teachers
1. Writing Algebra: Using Algebraic Symbols
Sheet 1-1: Symbols of Arithmetic
Sheet 1-2: Algebra in Italy, 1200 - 1550
Sheet 1-3: Germany and France, 1450 - 1600
Sheet 1-4: Letters for Numbers
2. Linear Thinking: Ratio, Proportion, and Slope
Sheet 2-1: The Rule of Three Direct Sheet 2-2: The Rule of Three Inverse
Sheet 2-3: False Position Sheet 2-4: Double False Position
Each of the four major chapters is organized like this:
Its mathematical focus is stated first.
“Historical Connections” lists the relevant regions and time frames.
A summary of the historical ideas comes next.
A section for each Activity Sheet starts with its main mathematical ideas and its
pedagogical purpose, followed by a detailed solution for each question.
The four chapters in this set progress from deciphering basic algebraic symbols to
tracing how the pursuit of solutions to cubic equations led to major changes in the
way we think about numbers. There’s something for every level of algebra instruction,
from the basic ideas of proportion and linearity to the equation-solving that led
to the complex number system. Connections to European and early American history
provide a fertile ground for interdisciplinary student activities.