AMERIC AN MAT HEMAT IC AL ASSOCIAT ION OF OPENING DOORS THROUG H MATHEMATICS

THE

DEVELOPMENTAL COMMITTEE OF

T WO -YE AR

COLLEGES

MATHEMATICS AMATYC

April 2012

The views expressed do not necessarily reflect the views of AMATYC.

INSIDE ISSUE:

THIS

Chair’s Report

1

New Life Project

2

Next Generation at

3

the Dana Center Reclaiming the Mathematical Lives of Community College Students

6

Texas State Center for Mathematics Readiness Hosts FOCUS on the

7

Accelerating Students through Developmental Mathematics: A Success Story

8

Future!

A Helpful Approach to 9 Determine a Teaching Entry Point Extending the Inverted Developmental Math Classroom

10

Tips for Algebra Class

11

The Real Perspective on Word Problems

12

DM Mission Statement

13

Newsletter Editor: Jessica Craig [email protected]

CHAIR’S REPORT BY LINDA ZIENTEK I am looking forward to serving as the AMATYC Developmental Mathematics Committee (DMC) Chair for the next two years and continuing the work established by the previous DMC chairs. I am happy to inform you that the committee membership has continued to increase. Currently, there are approximately 185 DMC members. Please encourage your colleagues to become a member of AMATYC and the AMATYC DMC. AMATYC Conferences In Austin, there was a number of excellent sessions devoted to developmental education. We look forward to more great sessions in Jacksonville, Florida. At the November conference, we will have an hour and a half session scheduled for the DMC committee meeting. During this time, we will discuss relevant DMC topics, which include the intermediate algebra and teaching position statements. The DMC committee submitted a themed session enti-

tled ―Evidence-based Developmental Math Redesigns.‖ Google Groups and DMC Website One reason for joining the DMC is to share information with and learn information from your colleagues about developmental mathematics. There are several venues that DMC members can use to communicate with their colleagues, which include the newsletter, Google Groups and the DMC Website. The newsletter will be printed twice a year so please consider providing submissions to the newsletter. Members of the DMC can send a message to the membership through Google Groups. In addition, we have the DMC website. The website is intended to be a living and evolving website. If you have information that you believe will be of interest to the membership, please send the information to me and I will include the information on the website.

Webinars We have two planned AMATYC webinars that are sponsored by the DMC. On Tuesday April 24 at 3 EDT/2 CDT, Kathleen Almy will present the following webinar: New Pathways for Developmental Math: A Look into Mathematical Literacy for College Students. On Wednesday June 6 at 4 EDT, Uri Treisman and Jack Rotman will present the following webinar: Issues in Implementing Reform in Developmental and Gateway Mathematics. Descriptions of the webinars are in the newsletter. Position Statements Currently, there are two position statements under consideration by the DMC. The committee is charged with revisiting the Teacher Qualifications for Developmental Mathematics position statement. We can recommend one of the following actions: no change, a minor revision, a major (Continued on page 2)

DMC

Page 2

CHAIR’S BY

LINDA

REPORT

NEWSLETTER

(CONTINUED)

ZIENTEK

change, or retirement. The committee will vote on this action at the DMC Committee meeting in Jacksonville Florida. The position statement is available at the following link. http://www.amatyc.org/documents/Guidelines-Position/other-statements.htm In addition to the Teacher Qualification position statement, the DMC is creating an Intermediate Algebra position statement. A committee of five DMC members is compiling the information from the November meeting and will be sending a draft position statement to the membership through Google Groups. We will hopefully be able to have a position statement available for a vote at the November DMC meeting. I hope that everyone has a great summer. Linda Zientek ([email protected])

THE BY

NEW JACK

LIFE

PROJECT:

NOTES

FROM

JACK

ROTMAN

ROTMAN

Quite a bit of activity has taken place, relative to the New Life Project. The online community continues to be active. The wiki site (http://dmlive.wikispaces.com) has a high level of use for this type of thing – consistently 20 to 40 different visitors per day. The companion blog (http://www.devmathrevival.net) averages about 40 visits per day. The New Life community has several pilots and implementations underway at this time; if your college is planning or doing one, it would be helpful to let me know (via email [email protected], or posting on the wiki site). Some of us have submitted proposals for the Jacksonville AMATYC conference this November on our New Life work. The AMATYC program committee is currently reviewing these – whatever the outcome, I appreciate our group‘s willingness to share (and the work involved in sharing). There is now enough activity on implementing New Life courses that textbook companies are developing products to support these courses.

I have had conversations with editors at Pearson and at Cengage. They would prefer that I not share details of their plans (since publishing plans are subject to changes for many reasons) … however, the companies are working hard to get materials ready. Based on what I know, some preliminary materials will be ready for class testing this coming year (2012-2013), with published materials later (late 2013, early 2014). I hope this information helps you plan as you consider whether you can implement a New Life model (MLCS and Transitions) at your college. Oh, in case you‘ve forgotten the ―New Life‖ course names – the first one (MLCS) is ―Mathematical Literacy for College Students‖, which provides the mathematics useful to all college students (including some algebra). The second course (Transitions) is ―Transitions to College Mathematics‖, which provides appropriate mathematics for students continuing on to college algebra, pre-calculus, and other courses at that level; Transitions is intended to be more symbolically-oriented than MLCS (which emphasizes numeric and graphical methods, with some symbolic methods).

Transitions is intended to be more symbolically-oriented than MLCS (which emphasizes numeric and graphical methods, with some symbolic methods). This activity related to the New Life model has also risen on the radar of companies that provide placement tests (especially Accuplacer and Compass). Accuplacer now has a process to develop ‗customized placement tests‘ in the computer adaptive model, based on a group identifying the content needed and a commitment from a set of colleges to use the customized placement test. Many of us are using ―placing into beginning algebra‖ as a proxy for ―placing into MLCS‖, and our pilot projects will provide some information whether this works in practice. There is less confidence about using a proxy (―placing into intermediate algebra‖) for the Transitions course, so this might be an area for us to consider in the near future. Such a placement test is important because we do not want to assume (or require) that all students need to complete both MLCS and Transitions; we want to provide a mechanism for many students to place directly into Transitions. (Continued on page 3)

Check our the DMC’s wonderful Webinars at http://www.amatyc.org/publications/webinars/index.html

Page 3

THE

DMC

NEW

LIFE

PROJECT:

NOTES

FROM

JACK

ROTMAN

NEWSLETTER

(CONTINUED)

For us to proceed with a customized placement test, we would need to form a work group to coordinate with the company (including the gathering of the commitments from colleges to use the customized test). We are probably not ready for this step, quite yet … though it might be appropriate to form the work group later this year. Just let me know if you have any questions about any of this … or if I can help you in your New Life work in any way. Jack Rotman Department of Mathematics & Computer Science; Leader of AMATYC‘s New Life Project for Developmental Mathematics (subcommittee of the DMC) Lansing Community College [email protected] 517.483.1079 http://dm-live.wikispaces.com http://www.devmathrevival.net

THE BY

NEXT AMY

K

GENERATION

AT

THE

DANA

CENTER:

THE

NEW

MATHWAYS

PROJECT

GETZ

Uri Treisman and the Higher Education team at the Charles A. Dana Center at the University of Texas recently announced details for the next phase of its work to reform developmental and gateway mathematics in community colleges. The New Mathways Project (NMP) is the Dana Center’s vision for a systemic approach to improving student success and completion through implementation of processes, strategies, and structures built around three mathematics pathways and a supporting student success course. This project builds on the Dana Center‘s earlier work in partnership with the Carnegie Foundation for the Advancement of Teaching to develop the initial versions of the courses called StatwayTM and QuantwayTM. The NMP is based on two fundamental principles from this early work. First, it seeks to offer students mathematics courses better aligned with the skills needed for their programs of study, their future jobs, and their lives as informed citizens and consumers. This is accomplished by providing three options with different mathematical content: the Statistics Pathway, the Quantitative Literacy Pathway, and the STEM Pathway. Second, the NMP is structured so that students are able to move from developmental math to and through a college-credit course in an accelerated timeline, as compared to traditional developmental math sequences. The Dana Center intends to develop curricular materials for the three pathways. These materials will be designed to build foundational mathematics skills through meaningful and challenging contexts using contemporary applications and real-life data. They will contain integrated routines and structures to support key success-based learning strategies such as self-regulated learning. The Dana Center will also develop a student success course to be taken in conjunction with the student‘s first math course in the pathway. The mathematics course and the success course will form an interconnected experience that not only enables students to be successful in mathematics but also builds the skills needed for the completion of a degree or certificate program in their chosen field of study. Course materials will be built into a comprehensive online platform that allows the Dana Center to keep the materials up-todate and respond to the rapidly changing needs of students and faculty. This platform will provide instructional aids for the classroom and a homework and learning support system for out-of-class work. In addition, the online platform will provide extensive instructional support. It will allow instructors to access this support at different levels of detail depending on their individual needs or interests. The platform will also include opportunities for faculty to participate in Communities of Practice facilitated by practitioners in the field around specific issues of interest. (continued on the next page)

Page 4

THE NEXT GENERATION (CONINUED)

DMC

AT

THE

DANA

CENTER:

THE

NEW

MATHWAYS

NEWSLETTER

PROJECT

The NMP Structure Accelerated Pathways Figure 1 below shows the structure of the NMP Accelerated Pathways, which are designed for students who have completed Arithmetic or are placed at a Beginning Algebra level. The term ―Accelerated‖ refers to the fact that these pathways allow students to complete their college credit courses more quickly than the traditional sequences. The three pathways have a common first course—a quantitative-literacy-based course that introduces and prepares students for college-level math. Students take this course along with the mandatory co-requisite student success course. Figure 1: The New Mathways Project: Accelerated Pathways

As a part of the student success course, students learn about the three pathways in relation to different careers and programs of study. Students declare a major, create a completion plan, and register for courses with the support of their instructors and advisors. Students end their first semester not only prepared for their next math course but with an understanding of how that course fits with their overall academic and career goals. Upon completion of the first-semester course, students in the Quantitative Literacy and Statistics Pathways move on to the specialized pathway course in which they earn transferable college credit. Students in the STEM Pathway enter into a two-course sequence that prepares them for Calculus. Students earn transferable college credit in the second STEM course. There is also an option for students who place at the Intermediate Algebra level to go directly into the STEM sequence without taking the common introductory course. In this case, students take the student success course at the same time as the first STEM course. Intensified Pathways The Intensified Pathways will serve students who are close to college-level but need support to be successful. These students will go directly into a transferable college-credit course in Statistics or Quantitative Literacy along with a mandatory co-requisite student success course that will include just-in-time support for the foundational math skills needed in the math course. The Dana Center recognizes that this new vision for college mathematics will require more nuanced placement methods based on multiple measures. The Dana Center anticipates the first implementation of NMP courses will take place in Texas in Fall 2013. The Center will work with national experts and state leaders to make recommendations for placement that will provide a model for other states. (continued on the next page)

Page 5

THE NEXT GENERATION (CONINUED)

DMC

AT

THE

DANA

CENTER:

THE

NEW

MATHWAYS

NEWSLETTER

PROJECT

Quantitative Literacy and the NMP The design of the NMP demonstrates the Dana Center‘s deep commitment to the importance of quantitative literacy. Because quantitative literacy is often interpreted in different ways, it is appropriate to briefly describe what the Dana Center means by the term. The Dana Center defines quantitative literacy as the skills and concepts necessary to understand, use, and communicate the quantitative information presented to citizens and consumers in today‘s society. This definition is based on the belief that all people need the skills to critique and evaluate quantitative information and make justifiable decisions based on that information. Because quantitative information is almost always presented in a form that incorporates text, quantitative literacy includes skills in reading and writing. The mathematical concepts of quantitative literacy include fluency with large numbers, estimation, proportional reasoning, descriptive statistics, and the basics of algebra (including modeling). Authentic uses of technology (i.e., uses that replicate the types of tasks performed in ―real life‖) should be incorporated whenever possible. A quantitative literacy course should include the analysis of data and statistics in the contexts of personal finance and the use of statistics, health information, and social and political issues. The Dana Center will publish an issue brief with more information on this topic on the NMP website in May. Faculty and the NMP The Dana Center strongly believes that the ultimate success of the NMP will depend on building upon the expertise and experience of faculty. No reform effort can be successful without faculty support and buy-in. The Dana Center is eager to communicate with the field and invite feedback. As the work progresses, there will be opportunities for faculty to contribute to the development of the materials and the tools and services that support the work. Faculty interested in learning more about the NMP may visit its website, www.utdanacenter.org/mathways. The following are some of the materials posted on the site:  April 17 webinar with Uri Treisman and Dana Center staff explaining the NMP, including information not in this article on the principles behind the project and its structure  Version 1.1 of the New Mathways Project Implementation Guide  Samples of Statway and Quantway materials  News and resources related to developmental and gateway math reform The Dana Center welcomes comments and questions at [email protected]. Getz, Amy K [[email protected]] 1

The original versions of the Statway™ and Quantway™ courses were created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching, and are copyright © 2011 by the Carnegie Foundation for the Advancement of Teaching and the Charles A. Dana Center at the University of Texas at Austin. STATWAYTM/StatwayTM and QuantwayTM are trademarks of the Carnegie Foundation for the Advancement of Teaching.

Page 6

DMC

RECLAIMING CARNEGIE

BY

GAY

THE

MATHEMATICAL

FOUNDATION

FOR

THE

LIVES

OF

ADVANCEMENT

COMMUNITY OF

COLLEGE

NEWSLETTER

STUDENTS:

TEACHING

CLYBURN

Two recent Community College Research Center reports revealed that too many community college students are placed into remedial classes, get frustrated and drop out, often ending their academic pursuits. While the issue of improper placement raised in those two reports is certainly a problem that has to be addressed, the other side to that coin is that there will still be large numbers of students who actually do need better preparation in mathematics or English, the areas identified now as those where students most likely need remediation. In an editorial response to the release of the reports, The New York Times suggested that colleges move quickly to find a new approach that gets help to students who really need it. While the Carnegie Foundation for the Advancement of Teaching is not yet working in the area of remedial English, we do think we have a new approach to reclaim the mathematical lives of community college students that will put them on a pathway of success, not just in college but in their lives and careers as well. Recognizing the grave consequences for individual opportunity and more generally for our economy and society if we do not accept our responsibility as educators to prepare mathematically literate citizens, Carnegie has engaged networks of faculty members, researchers, designers, and students in the creation of new pathways, one in statistics and the other in quantitative literacy. These Networked Improvement Communities (NICs), as we call them, have embraced an audacious goal—to dramatically increase from 5 percent to 50 percent the percentage of students who achieve col-

lege math credit within one year of continuous enrollment. We are well on our way to achieving our goal. The initiative is vital out in the field. Statway™ went live in classrooms this past August and Quantway™ achieved this milestone in January. We now have 30 colleges participating in our two NICs—22 in Statway™ and eight in Quantway™. Our collaborators—the faculty, students, Carnegie researchers, and developers—are testing and refining the materials and the faculty supports and student supports embedded in them. They are discovering what works and what doesn‘t and we are taking that information and quickly revising what goes back into the classrooms. This continuous improvement process, using the tools of improvement science that has worked in other ―industries‖, has gotten us to Version 2.0 and we‘ll keep going. The approach—working in networks—is different, but so are the pathways themselves. We have ample evidence that teaching students the same content over and over again in the same way just doesn‘t work. Students who failed to learn the basics of ratios/fractions and proportions in elementary school, and who did not benefit from attempts to reteach it the same way in middle and high schools, are very unlikely to learn this material if presented in the same fashion still one more time in community colleges. To provide a grounded comparison, for example, between Quantway™ and a typical developmental algebra course, you merely have to compare syllabi. The

lessons in the first module of our developmental Quantway ™ sequence have titles such as ―Seven Billion and Counting,‖ ―The Credit Crunch,‖ and ―Has the Minimum Wage Kept Up?‖ These are problems based in the mathematics of algebra but centered around the themes of citizenship and personal finance. Compare this with typical chapter titles in an intermediate algebra text such as ―Inequalities,‖ and ―Exponents and Polynomials.‖ Carnegie believes that what our students need is not more complex math content, but rather to learn how to use math well in the complex problems we confront at work, in our personal lives, and in exercising our civic responsibilities. Carnegie also knows that it will take more than changing the math that we teach to ensure success, so we are also attending to how students learn. Drawing on findings in cognitive science and learning theory research, classroom lessons are built around rich problems where students are asked to think, struggle with ideas and then in this context come to see how basic tools of algebra, statistics, data visualizations and analysis can help them understand better. Such instruction constantly seeks to make explicit connections about both the big ideas of mathematics as well as the specific tools that we might apply to different problems. We also now know a lot more about the design of effective homework. There is much more to this than rote repetition of the same algorithms over and over again to a similar set of problems. In addition, we know that we must see our students as persons, attend closely to engage them and ultimately educate them. (continued on the next page)

Page 7

DMC

RECLAIMING

THE

MATHEMATICAL

LIVES

OF

COMMUNITY

COLLEGE

NEWSLETTER

STUDENTS

(CONTINUED)

Students placed in developmental math often come to the classroom with what Jim Stigler at UCLA calls ―math scar tissue‖—the residue of years of failure in mathematics courses. Many students have experienced schooling that says to them ―you are not good at this, we don‘t expect you to achieve.‖ But we now know from extensive research that math ability is in fact malleable; with effort and deliberate practice, new skills and understandings can be acquired. These students can learn math, and even more importantly come to see its relevance in their lives. Most students work hard in developmental math classes—studying long hours, nights and weekends—yet do so using ineffective strategies. Some simply withdraw effort soon after the course begins. Student success requires both persistence in studying and attendance and efficient and effective use of time and energy. We call this productive persistence—a key driver for success in our pathways. A number of promising social psycho-

TEXAS STATE CENTER FOR “FOCUS ON THE FUTURE!” BY

ISAAC

J.

logical interventions are now being integrated into pathways instruction in seeking to make a difference here. And we know that there are language and literacy barriers that impede instruction and learning. So we are also attending to where the language of instruction, rather than the mathematics, is the real barrier to learning, and how this barrier may be manifest in textbooks, homework assignments, classroom lectures and discussion activities. While it is early on in the life of these networks, initial results are promising. From our fall surveys of Statway™ students, we know that that they are finding instruction more interesting and useful than the math they had before. They are less anxious and more likely now to believe that with hard work they can learn math. Student persistence rates in Statway appear strong and initial analyses from end of first semester assessments indicate comparable student learning to that seen among other community col-

MATHEMATICS

READINESS

lege and university students who have successfully completed a college-level statistics course. Faculty who may have been doubtful when we launched this program have stepped up as leaders. As one faculty member put it: ―All of my comfort zones are deeply threatened, but it is like the comfort zones were exactly what has been wrong. The whole system is set up to protect us from being challenged to improve…‖ Carnegie‘s work in these two pathways is about simultaneously advancing on what must be held as the triple aims of educational improvement— more engaging learning for diverse students, greater overall effectiveness, and accomplishing this with the most efficient use of resources. We are convinced it can be done. Gay Clyburn [email protected] The Carnegie Foundation Website is http://www.carnegiefoundation.org/

HOSTS

ORTEGA

The Center for Mathematics Readiness recently hosted the Fundamentals of Conceptual Understanding & Success (FOCUS) symposium, FOCUS on the Future. On Wednesday January 11, 2012, the Center for Mathematics Readiness welcomed guest speakers Dr. Dolores Perin, Dr. Claire Ellen Weinstein, and Dr. Philip (Uri) Treisman. Dr. Dolores Perin is a Professor of Psychology and Education, and Senior Research Associate at the Community College Research Center at Teachers College, Columbia University. Her research interests include instruction in developmental education and adult literacy programs. Dr. Claire Ellen Weinstein‘s research focuses on her Model of Strategic Learning and its validity and usefulness for developing educational programs and diagnostic assessments in schools, colleges, and business/government training settings. Senior advisor to the Aspen Institute‘s Urban Superintendents‘ Network, Dr. Philip ―Uri‖ Treisman is professor of mathematics and of public affairs at The University of Texas at Austin. Serving as the founder and director of the University's Charles A. Dana Center he also serves as senior fellow at the Carnegie Foundation for the Advancement of Teaching, and serves on the boards of the New Teacher Project, Education Resource Strategies, and the AFT Innovation Fund. For more information about the services provided by the Center for Mathematics Readiness, please contact the center‘s director, Dr. Selina V. Mireles, at [email protected], (512) 245-2338. Isaac J. Ortega [email protected]

Page 8

DMC

ACCELERATING STUDENTS A SUCCESS STORY BY

JOYCE

THROUGH

DEVELOPMENTAL

NEWSLETTER

MATHEMATICS:

WALBORN

It‘s difficult to pick up any higher education journal these days without finding at least one article about developmental education. The large number of students that enter college and require at least one developmental course is at the forefront of most administrators‘ minds. Organizations such as the National Center for Academic Transformation have urged colleges to redesign their developmental course sequence to give students an opportunity to accelerate through the remedial work. Results of research through the Achieving the Dream initiative suggest that students who completed all of their developmental education in the first year were more likely to persist to both the second term and to the second year.

cessful in an accelerated Prealgebra course. How these same students would fare in the accelerated Introductory Algebra portion of the course was cause for concern, but I proceeded nonetheless.

Even with research data in hand, many mathematics educators still question the wisdom of moving students through the curriculum at an accelerated pace. I, myself, was skeptical as to whether or not rushing developmental math students through the coursework was wise or effective. Still, I was willing to provide those students that desired to complete their math courses more quickly the opportunity to do so. With this in mind, I proceeded to develop an accelerated course that combined Prealgebra and Introductory Algebra into a one semester, 5 credit class. Our current developmental math offerings consist of three courses: Prealgebra, Introductory Algebra and Intermediate Algebra. This new combination course would then allow students to complete two courses in one semester, saving them both time and one credit of tuition. The target group for this course was students who tested on the high end of the Prealgebra scale. Those students, theoretically, would be suc-

The high persistence rate and average grade for students in this course were indeed unusual, and considering the fact that this course is a combination of the two lowest levels of remedial mathematics courses offered, the success of these students is certainly worth discussion. It is actually not surprising that these students fared well in Prealgebra, as their initial placement scores indicated that they almost qualified for Introductory Algebra. What is surprising is that they continued to score well into the Introductory Algebra course material. Midway through the course, I asked the students for their opinion on why they felt they were doing so well. Overwhelmingly they stated that having class every day, with no opportunity to procrastinate on the homework was the reason they did well. In their opinion, they were able to learn the material and take the test before they forgot everything. While I agree with their assessment, I would also add that developing a routine of working

The course met Monday through Friday and consisted of daily lecture and homework. Assessments consisted of chapter tests and a final exam. In addition to the lessons in mathematics, the students participated in several 10 to 15 minute Study and Life Skills Workshops throughout the semester. Overall, the average grade in the class was 93.7%. In terms of persistence 83% enrolled in the next course in the mathematics sequence.

on mathematics everyday contributed to their success. In addition to the daily routine of the class, I believe other factors contributed to the students‘ success. On six separate occasions during the semester the students were given a lesson on college success skills. The workshops were interactive and provided students with activities that they could do to increase their success at school. Workshop topics included dealing with test anxiety, improving memory skills, time management, developing positive habits, creating a support system and listening and taking notes. Students commented that they enjoyed the workshops, found them to be valuable and even asked to add an extra workshop at the end of the semester, which we did. Addressing students‘ concerns outside of the math classroom, I believe, was a factor in their success. Finally, whether or not they realized it, the students did form friendships in the class. Because the class met daily, I believe it was easier for students to initially get acquainted and to develop amicable relationships. Outside of class, I would notice students sitting together, and some told me that they studied together for exams. An informal assessment would suggest that regular attendance and the development of a connection to the class resulted in greater success. Finding a successful strategy in the classroom is always the goal of any educator. Finding success in (Continued on the next page)

Page 9

DMC

ACCELERATING STUDENTS THROUGH A SUCCESS STORY (CONTINUED)

DEVELOPMENTAL

NEWSLETTER

MATHEMATICS:

developmental coursework is difficult to obtain, but very rewarding when discovered. While I did not conduct a formal research experiment, I do believe that this class was successful for some very specific reasons. Regular and frequent class meetings, college success strategy workshops and the students‘ own connections to the learning community all played an important role in the high class average. The high persistence rate also provided evidence of the merits of accelerated developmental coursework. Perhaps the traditional requirement of three semesters to complete the developmental mathematics sequence should be revisited so that students can both accelerate and achieve high success. Joyce Walborn [email protected]

Opening Doors Through Mathematics

Check AMATYC’s webpage: www.amatyc.org

A

HELPFUL

BY:

APPROACH

NATALIA

TO

DETERMINE

A

TEACHING

ENTRY

POINT

DARLING

Too often I have found myself either assuming students need a review of some math rules and definitions, or glossing over some math facts because they are supposedly prerequisites. A technique that helps with pre-diagnostic assessment is the use of the ―KWL‖ graphic organizer. I spend time at the beginning of new chapters using the KWL as a review and as a way of modeling how to organize notes. For instance, in an introductory algebra course, an assignment sheet entitled ―Prerequisite recap‖ is given to students as a ―graded‖ exercise. The sheet has a table with 4 column headings: A) Description of prerequisite, B) K: What do you Know, C) W: What do you Want to know, and D) L: What have you Learned. The rows contain definitions or questions such as 1) given a geometric figure; state the formula for perimeter, area, and volume. If students know the answer they write the information under the ―K‖ column; if they are unsure, they write a guess or some questions in the ―W‖ column. This enables all students to write and prevents a feeling of discomfort if they don‘t have an answer. Credit is received regardless of what they know, so long as they participate. They then compare answers and as a group share their results. If the answer is correct I write it on the blackboard under ―K‖ and under ―L‖ I place a checkmark. If they give an incorrect answer I place it under ―W‖ and then we interactively discuss how to correct the information. This correction is clearly written on both the board and their sheets under ―L.‖ The playing field is now equalized after this discussion (all students will have the correct results based on peer-teaching & some instructor guidance.) Class time is spent on discussion that is needed versus guesswork and, as a perk, students learn to organize their notes. Natalia Darling, Assistant Professor of Mathematics, Math Physics & Computer Science Department at University of Cincinnati Blue Ash College. ([email protected])

Page 10

DMC

EXTENDING

THE

BY

JASTER

ROBERT

INVERTED

DEVELOPMENTAL

The idea of delivering course content over the internet to create time in class for more productive activities seems to be spreading across the country. A quick search at youtube.com on the keyword combination flip classroom yielded well over a dozen videos created by educators from various states promoting the use of flipped (i.e., inverted) classrooms. Dr. Scott McDaniel successfully used an inverted developmental math classroom to promote student learning. An article that he and Dr. David C. Caverly wrote for the Journal of Developmental Education (McDaniel & Caverly, 2010) encouraged me to extend this approach to college algebra at Texas State University in fall 2011. Based on the level of engagement in problem solving that I observed in the classroom, and the understanding demonstrated on quizzes and exams, I am confident that most of my students gained a greater understanding of college algebra in the inverted classroom than they would have gained in a lecture-based classroom. Moreover, an end-of-semester survey revealed that most of the students preferred the inverted classroom to the traditional lecture. The inverted classroom allows the instructor to work more closely with the students. As described in the cited article, prior to coming to McDaniel‘s class students were expected to watch videos to gain enough understanding to engage in the in-class activities. Inside the classroom students focused on basic concept worksheets, worked in groups, and presented problems at the board to the other students in the class.

MATH

NEWSLETTER

CLASSROOM

Inside my college algebra classroom students also worked on problems. I was able to circulate through the classroom and work individually with students. I gave a quiz at the end of each class to encourage students to work earnestly on the assigned problems during class. The quiz also provided an incentive to take notes while watching the videos and to write the solution for each assigned problem since I permitted students to use their notes and in-class work during the quiz. This semester I am again teaching my students in an inverted classroom. The interaction with individual students is mutually beneficial. The student is able to get help when it is needed the most, while solving problems. I am gaining a better understanding of the difficulty that students have in learning mathematics. This idea has also worked well in developmental mathematics classes. As part of his dissertation, Long (2010) created and implemented an action plan to increase success rates in developmental mathematics in community colleges. Long‘s study included 63 students in two section of an Intermediate Algebra course in California. The action plan contains two components: provide students with access to mathematics videos and proactively strive to create a sense of community with student participation groups. Long used an end-of-course survey to collect participants‘ perspectives on action plan components and the course experience. One of the openended survey items asked students to express their thoughts about mathematics videos. Over 96% of the responses

were positive. Another survey item asked students to express their thoughts about collaborative learning and community building. The vast majority of these responses (87%) were also positive. Long‘s dissertation contains numerous excerpts from student comments that he received in response to these and other survey items. Long reported that the implementation of the first iteration of the action plan was very successful, and anticipated enhanced success in its next iteration. Robert Jaster Texas State University [email protected] References McDaniel, S., & Caverly, D. C. (2010). Techtalk: The Community of Inquiry Model for an Inverted Developmental Math Classroom. Journal Of Developmental Education, 34(2), 40-41. Long, G. W. (2010). Community and videos: An action plan to increase success rates in California community college developmental mathematics (Doctoral dissertation). Retrieved from ProQuest Dissertations and Theses database. (3398690)

38th AMATYC Annual Conference Jacksonville, Florida November 8-11, 2012 Conference Theme: River-of-Knowledge, Ocean-of-Dreams

Page 11

TIPS BY

FOR

GEORGE

DMC

ALGEBRA

NEWSLETTER

CLASS

ALLAND

Some students in my introductory algebra class struggle with basic concepts. Based on my lessons learned with this diverse group of learners, following are some tips for teaching algebra to a general audience. 1. Solving Equations After explaining the basic concepts in solving equations, I show students a simple equation to solve such as 2x + 4 = 8. I tell students that the goal is to get x by itself on one side of the equation. So the first step is to subtract 4 from both sides, and we are left with 2x = 4. Most are able to follow this procedure. Then when I ask them what should be the next step, some students want to subtract the 2 from the 2x. So I explain that 2x means 2 multiplied by x, so to get x by itself we need to divide by 2. This satisfies some students but others remain confused. I have achieved good results by turning this equation into a word problem: Two cantaloupes = $4. I ask ―How much is one cantaloupe?‖ and they all answer $2. Then I ask ―How did you figure this out?‖ This helps them realize that they had to divide both sides by 2 in order to get the answer. 2.Collecting Like Terms I use a similar strategy when I am explaining how to add like terms. For example, if we have an expression such as 6x2 + 2x4 – 2x2 + 7 + 3x2 – 3 and the goal is to combine like terms, I first ask three students to name their favorite fruit. If I get answers such as apples, oranges and bananas, I first rewrite this expression as 6 apples + 2 oranges – 2 apples + 7 bananas + 3 oranges – 3 bananas. Then I ask them to evaluate how many fruits of each kind are left. Most can figure this out as 4 apples + 5 oranges + 4 bananas. Then I tell students that in math we also have different kinds of fruits but they have mathematical names such as x4, x2 and just plain numbers. They are then usually able to make the connection and get the right answer.

3.Factoring Polynomials Most algebra textbooks introduce factoring of polynomials before describing the purpose of factoring. I prefer to start by showing my students a graph of a quadratic equation which is a parabola, and explaining that the solutions of this quadratic are the x intercepts (they are already familiar with the concept of intercepts from linear equations). Then I indicate that there are three ways of solving this type of equation: 1) The quadratic formula, 2) completing the square and 3) factoring (when possible). Then the reason for factoring becomes more understandable in view of the ultimate goal. 4.Systems of Equations Before showing the two methods of solving a system of two equations (substitution and elimination), I first write this equation on the board: x + y = 4. Then I ask my students ―What is x and what is y?‖ After the common answers of 2 and 2, 3 and 1 and 4 and 0, most of them soon realize that an equation in two variables does not have a unique solution, because there are an infinite number of possible answers. Then I write two equations like this: x+y=4 x–y=2 I ask ―What is x and what is y?‖ Most are able to figure out that x is 3 and y is 1. This makes the explanation that we need two equations to solve for x and y more understandable. 5.LCM and GCF Students sometimes get confused between LCM and GCF. One of the reasons, I think, is that LCM has the word ―least‖ in it which can make you think that the LCM will be a smaller number than the ones given, and GCF has the word ―greatest‖, which makes you think that the number you are looking for is greater than the given numbers. What I tell students is to focus instead on the

words ―multiple‖ and “factor‖. A multiple will be greater and a factor smaller than the numbers given. 6.Finding the LCM (Least Common Multiple) of Rational Expressions This topic is often quite challenging for students. What I try to do is clearly demonstrate the analogy with finding the LCM of numbers, which they have studied in an earlier chapter and know fairly well by the time we get to rational expressions. In order to find the LCM of 12, 15 and 18, break each number down into its prime factors: 12: 3, 2 ,2 15: 3, 5 18: 3, 3, 2 So the LCM is 3 x 3 x 2 x 2 x 5 = 180 To find the LCM of x2 – 4, x2 + 4x + 4, 2x – 4, break each expression into its prime factors (after defining a prime factor for an algebraic expression): x2 – 4: (x + 2) (x – 2) x2 + 4x + 4: (x + 2) (x + 2) 2x – 4: 2(x – 2) Using the same procedure we used for numbers, we multiply each prime factor the maximum number of times it appears in any one of the three expressions: So the LCM is 2(x + 2) (x +2) (x – 2) However, as opposed to how we work with numbers, we don‘t usually multiply all the terms together. We leave it in factored form. 7.What is the Meaning of “=” (Equals)? In one lecture on the distributive property I wrote the following equation on the board: 2(x + 3) = 2x + 6 One student asked ―But what is the (continued on the next page)

Page 12

TIPS

FOR

DMC

ALGEBRA

CLASS

NEWSLETTER

(CONTINUED)

the answer?‖ I did not at first understand what the student was getting at, and I asked her to explain. She said she was told in her previous math classes in high school that equations always have an answer, so what is the answer here? Then I realized that ―equals‖ has actually two meanings. In one case, if I write x + 2 = 6 then the answer is x = 4. But in the equation above, the right side is a transformation of the left side—a transformation into an equivalent but differently formatted expression. Since then, I always make my students aware of the different uses of the term ―equals.‖ 8. I say “minus”, you say “negative” Some students prefer that I differentiate between ―minus‖ and ―negative‖. For example for this mathematical expression: 2(– 3) if I say ―two times minus three‖, some get confused because for them saying ―minus‖ means subtraction. So they insist that I say ―two times negative three‖. And for this: – (– 3) if I say "minus a minus three‖ that again breeds confusion. So I have been saying either ―the opposite of negative three‖, or a ―negative one times a negative three‖. There is a good letter to the editor about this in the December issue of Mathematics Teacher (NCTM). George Alland/ Mathemtiacs Instructor Rasmussen College - Brooklyn Park [email protected] www.Rasmussen.edu

THE BY

REAL LAUREN

PERSPECTIVE

ON

WORD

PROBLEMS

HICKMAN

According to Green and Emmerson (2010), the use of realistic problems should help students ―develop thinking skills necessary for analyzing the kinds of quantitative information they will encounter in their professional lives‖ (p. 114). However, students tend to approach classic word problems with minimal understanding of how to set up accurate representations of application problems (Hegarty, Mayer, & Monk, 1995, p. 19). Having the computational skills to solve problems is important, but without the ability to use contextual clues to correctly identify mathematical relationships, students‘ computational skills cannot be applied successfully in realistic situations (Hegarty, Mayer, & Monk, 1995). Miller (2010) argues the importance of incorporating other disciplines, such as writing and science, in the teaching of word problems. Miller (2010) believes that by forcing students to write about and discuss word problems and solutions,

students will gain the skills required to identify relationships and communicate results in real life. Students will learn that a numerical solution is meaningless unless the significance of that solution can be effectively connected back to the original problem (Miller, 2010). The AMATYC Standards for Introductory College Mathematics before Calculus include modeling as a Standard for Intellectual Development (Standard I-2, p. 10). The modeling process described includes ―(1) Identifying the problem…; (2) interpreting the problem mathematically; (3) employing the theories and tools of mathematics to obtain a solution…;(4) testing and interpreting the solution…; and (5) refining the solution techniques‖ (as cited in Crossroads in Mathematics, p. 10). Modeling realworld situations requires the incorporation of all five stages. In this way, realworld problems differ from problems without an application. Without addressing each stage in the teaching of word

problems, students may not develop the problem-solving skills required to be successful when encountering quantitative situations outside of the classroom. Teaching word problems is often ineffective at helping students develop the skills required to identify mathematical relationships, properly set up a mathematical model, and communicate the meaning of solutions (Miller, 2010). Currently, many developmental mathematics programs may approach classic word problems as applications of various topics, such as systems of equations. If the lessons with classic word problems can focus more heavily on the larger understanding of the situations described, as opposed to just ―setting up‖ the proper model, then the lessons could provide a greater opportunity for students to develop the thinking skills required to solve problems in their professional lives (Green, & Emmerson, 2010). (continued on the next page)

Page 13

THE

REAL

DMC

PERSPECTIVE

ON

WORD

NEWSLETTER

PROBLEMS

(CONTINUED)

References American Mathematical Association of Two Year Colleges. (1995). Crossroads in mathematics: Standards for introductory college mathematics before calculus. Green, K.H., & Emmerson, A. (2010). Mathematical reasoning in service courses: Why students need mathematical modeling problems. Montana Mathematics Enthusiast, 7(1), 113-140. Hegarty, M., Mayer, R.E., & Monk, C.A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers. Journal of Educational Psychology, 87(1), 18-32. Miller, J.E. (2010). Quantitative literacy across the curriculum: Integrating skills from English composition, mathematics, and the substantive disciplines. The Educational Forum, 74, 334-346. Lauren A Hickman, [email protected]

If you have not read the American Mathematical Association of Two-Year Colleges publication, Beyond Crossroads, or if you need a refresher, check it out!

Jack Rotman told you about the wiki site for the New Life Project in his article. Here it is again. (http://dm-live.wikispaces.com) Also check out the companion blog ―Developmental Mathematics Revival!‖ (http://www.devmathrevival.net) Check our the DMC’s wonderful Webinars at http://www.amatyc.org/ publications/webinars/index.html: Kathy Almy’s webinar “A look into Mathematical Literacy for College Students”, and soon Jack Rotman and Uri Treisman’s webinar “Reform in Developmental & Gateway Math Courses”.

DMC

Page Page 14 14

DMC OFFICERS DMC Chair Linda Zientek [email protected] Liaison to AMATYC Executive Board Margie Hobbs DMC Webmaster Chad T. Lower

DN ME CW S NL EE WT ST LE ER T T E R

The Developmental Mathematics Mission Statement Developmental mathematics programs exist in order to prepare students for collegiate mathematics courses, for other courses requiring a mathematical foundation, and for general academic success based partially on quantitative literacy. These developmental mathematics programs will allow flexibility for students and enable students to consider additional and higher academic goals.

Check out the Developmental Mathematics Committee Website! https://sites.google.com/site/amatycdmc/

DMC

REGIONAL

Northeast Mid-Atlantic Southwest Central Midwest South East

DMC

REPRESENTATIVES

Geoff Akst Bill Coye Mel Griffin Loye Henrikson Vasu Iyengar Richard Leedy

MEMBERSHIP

West Northwest

Eric Matsouka Carren Walker

At Large At Large At Large

Kathleen Almy Jack Rotman Sharon Sledge

FORM

If you know of anybody who might be interested in joining our committee (and if they belong to AMATYC), they can go to our web page to complete a membership form at http://www.devmath.amatyc.org/join.htm