When It Comes Out of the Attic…

1. When It Comes Out of the Attic… Thomas Adamson Bigfork, MN Scott Adamson, Ph.D. Chandler, AZ

2. Austin, 2011 and Jacksonville, 2012 What belongs in the attic? • Square root algorithm • Rationalizing denominators • Synthetic division • Factoring quadratic polynomials • Finding zeros • Etc.

3. Why the Attic?

4. NCTM The vision of school mathematics in Principles and Standards is based on students' learning mathematics with understanding. Unfortunately, learning mathematics without understanding has long been a common outcome of school mathematics instruction. In fact, learning without understanding has been a persistent problem since at least the 1930s, and it has been the subject of much discussion and research by psychologists and educators over the years (e.g., Brownell [1947]; Skemp [1976]; Hiebert and Carpenter [1992]). In the twenty-first century, all students should be expected to understand and be able to apply mathematics.

5. MAA The goals of College Algebra should be to get students to internalize mathematics and come to understand certain ideas conceptually. Bob Moses and most others who have been successful with all students know that these goals can be obtained only if the mathematics is connected to everyday life. What does that mean for College Algebra? In my judgment it means starting with the applications, not the techniques. What problems should all college graduates - those from two-year and four-year colleges who do not major in a math-intensive subject - be able to solve? It is not how to factor a polynomial or complete a square; it is how to put together a budget and schedule for a proposal or building project or sales campaign. It is not how to solve a quadratic equation; it is how to use a statistical process control or model a production or health system.

6. AMATYC Becoming an efficient, independent problem solver should be a goal of every mathematics student. But for many students, mathematics is viewed as a “string of procedures to be memorized, where right answers count more than right thinking.”Authentic problem solving does not necessarily involve memorizing procedures and usually involves being motivated to solve the problems. “…good problem-solving behavior usually is not fostered by having students imitate how teachers solve problems. Because teachers typically demonstrate only correct moves, students often come to view problem solving as that of delving into a mysterious bag of tricks to which only a select few are privy.” To build problem-solving skills, faculty need to engage students actively in the learning process, create opportunities for exploration, and help them recognize that there may not be a rule to memorize or algorithm to follow for a given problem.

7. Common Core Standards Standards for Mathematical Practices • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics.

8. Common Core Standards Standards for Mathematical Practices • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

9. Want to or Have to? What if I have to or want to teach_____________? • An opportunity for students to make sense, problem solve, reason, etc. • Approach from an historical perspective • What are the learning goals? To be accomplished at doing a procedure or understanding mathematical structure? • The history of mathematics can be rich and interesting!

10. African Logarithms • Due to size constraints, the pictures from this slide are not included in the PPT

11. African Logarithms • Due to size constraints, the picture of an African textbook is not shown on this slide.

12. When It Comes Out of the Attic…Computing with Logs Seeing there is nothing…that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square, and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors. --John Napier

13. When It Comes Out of the Attic…Computing with Logs Until the arrival of affordable calculators in the 1970s, every student of math or science had to be expert at logarithms and they are still used in many African countries. --The Little Book of Mathematical Principles, Theories, and Things by Robert Solomon

14. When It Comes Out of the Attic…Computing with Logs 46.3 X 0.0796 • Look up each logarithm in tables • Add the two logarithms • Look up antilogarithm

15. Log Tables • Google Log Tables to find a log table…

16. When It Comes Out of the Attic…Binomial Expansion • Dad – what do you want to do here?

17. When It Comes Out of the Attic…Polynomial Division • I have some ideas here…this is still often taught…but without any attempt for understanding…just follow the procedure.

18. When It Comes Out of the Attic…Cramer’s Rule • I have some material that didn’t make it into our textbook…maybe I can use it here?