## Two exceptional pieces from NCTM journal on technology and modeling

I just got my copy of the National Council of Teachers of Mathematics (NCTM) December 2015/January 2016 issue of* MATHEMATICS TEACHER*. Two wonderful articles grabbed my attention and I kept bouncing back and forth between them. I commend them both to you (and give a full citation below) and suggest you get to your institution's library to read them for yourself or open your own copy of the journal if you are an NCTM member and sit and enjoy.

In "Teaching in a World with PhotoMath" the authors, Corey Webel and Samuel Otten, describe PhotoMath, an app for your smart phone or tablet through which you hold your device up to a mathematics problem in a textbook and the app shows you the answer to the problem as well as the steps in solution - that simple (and yet pedagogically complex as the article goes on to describe). A highlighted text says, "If we ask students to do only what PhotoMath can do then maybe our learning goals are misplaced." That IS the point for me in the article.

Imagine mathematics innovators in the Middle Ages saying, "Well if this place value stuff can keep track of things so easily then why are we still using the Roman numeral system in commerce?" Duh!!! The current authors give some of the options from banning PhotoMath to incorporating its use in the classroom. Needless to say, the latter is considered under the thoughtful guidance that the use of technology should enhance and promote mathematical reasoning. Bottom Line Up Front is that "Technological advances allow us to refocus learning goals." To which I might add a proper refocus, for I can remember years ago when I got access to a computer algebra system (Maple on a VAX!) one colleague said, "Good, now we can have students check all their by-hand integration techniques." Was that a refocusing or just a pounding of students into the ground? My refocus was to take students beyond the by-hand world into the world of applying the mathematics in context, to mathematical modeling, which leads me to the second article in the same issue.

In their piece, "Integrating Modeling Steps int the High School Curriculum," in the same issue of *Mathematics Teacher *authors Ted Wendt and Kevin Murphy demonstrate how, "Textbook word problems can be modified to change the way the questions are posed without changing the prerequisite content." They give crisp examples or reworking problems and in so doing, they suggest, "Like many skills, modeling is best developed through practice of smaller components gradually accumulated over time." The idea is that we should introduce modeling opportunities in small doses and like Chicago voting of yore, do it "early and often." They suggest, outline, and give examples of an integrated, incremental process of bringing modeling into the classroom for students. Through several restatements (reworking) of one problem the authors provide us a window into how we can shape our classroom teaching based on traditional mathematics books, by simply reformulating a questions and asking additional questions about the process and results.

We here at SIMIODE are encouraging colleagues to do both, use technology and perhaps reshape problem formulation into a modeling scenario to take advantage of technology, as well as do more and more modeling to motivate the mathematics, before, during, and after studying the concept of the moment. So we say "Write on!" to both sets of authors and a big "Thank you." For if high school teachers take these ideas to heart imagine how rich our college level mathematics coursework will be

Full citations for these pieces are given and we commend them to your reading list:

Webel, Corey and Samuel Otten. 2015. Teaching in a World with PhotoMath. *Mathematics Teache*r. 109(5): 368-373.

Wendt, Ted and Kevin Murphy. 2015. Integrating Modeling Steps int the High School Curriculum. *Mathematics Teacher*. 109(5): 374-379.