NCTM Denver 2013: Lester's Whatever Happened to Problem Solving in the Math Curriculum?

Annual Meeting - Saturday, April 20, 9:30 am

Frank K. Lester - Indiana University

Frank Lester was the editor of the 1234-page Second Handbook of Research on Mathematics Teaching and Learning. This was an interesting talk, not so much for any specific content but for how it was put together. Lester began the talk by demoing Algebra Touch, an iOS app that promotes fluency with symbol manipulations in solving equations. He asked, "What will the math classroom of the future be like?"

Frank K. Lester

Then Lester went into problem solving, something he feels has slowly slipped out of most mathematics curricula. Problem solving, says Lester, is "What you do when you don't know (or aren't sure) what to do." That leads to the teacher's role in the classroom:

Lester: "The teacher's job is to make students better at being productive when they don't know what to do." #NCTMDenver
— Raymond Johnson (@MathEdnet) April 20, 2013

From here, Lester showed some of his favorite problem solving tasks. The first is probably familiar to most of you:

A snail is at the bottom of a well that is 10 meters deep and it wants to get out. Every day it climbs up 4 meters. It then slides back 2 meters when it rests at night. If it does this day after day, how many days will it take the snail to reach the top of the well?

As part of the discussion, Lester related his problem solving heuristics, harkening back to Polya's How to Solve It. Suddenly a talk that began with a discussion of technology and the future was using a (good?) problem that felt like it was from the 1980s and (good) strategies that were from the 1940s. Lester's choice of heuristics to apply here were "Draw a picture/diagram" and "Be skeptical of your solutions," since many initially reason that the snail reaches the top of the well in 5 days.

Lester then looked at finding the square root of 12,345,678,987,654,321. His heuristic -- one he called a "super heuristic" -- was to look for a pattern. I couldn't help but feel like this was a trivial problem with a trivial answer.

Things got better with the next problem: "On a European river cruise, 2/3rds of men are married to 3/5ths of the women. How many men and how many women are on the cruise?"

Lester joked that this problem predated talk of same-sex marriage, and I found it to be a bit out of touch. Lester said the problem could be adapted to involve pairings of animals or objects. Another heuristic here was "make reasonable guesses, not as final answers, but to get you started." After discussion of this problem, we moved on to one more: "A club has 500 members. At the Spring dance, tickets for new members were $14 but $20 for longtime members. All of the new members attended but only 70% of the longtime members attended. How much ticket revenue was collected?" It seems like there isn't enough information, but solving this plays off the fact that $14 is 70% of $20. That might elicit some interesting reasoning, but again I think this trivializes the problem.

Lester returned to technology at the end, mentioning strategy games like Math Dice and Rush Hour. He advised that teachers have an important role to play when students play games. Prior to the play, teachers need to help students be clear about the rules for playing, model how to play, and discuss special situations. I think that depending on the game and the goals, this could turn into too much guidance. During student game play, teachers need to watch students play, attend to their thinking, help and point out misunderstandings. It's important, says Lester, to not suggest strategies for playing. Kids should be left to figure those out for themselves. After gameplay, reflection is important, just as with other classroom activities.

Lester's slides can be found at the conference planner.

NCTM Denver 2013: Danielson's They'll Need it for Calculus

Annual Meeting - Saturday, April 20, 8:00 am

Christopher Danielson - Normandale Community College, Bloomington, Minnesota

As Steve Leinwand noted in his Thursday talk, math teachers are a relatively conservative, risk-averse bunch. Perhaps our conservatism comes from the perceived slow but steady progress of math over millennia where it's easy to take comfort in the old because the new can seem so difficult to obtain. Some of this rubs off in the way we teach, the activities we choose for students, and our judgement about what's important for students to know.

Chris Danielson's session kicked off by calling out some mathematics that gets taught in the name of "needing it for calculus," despite no widespread need for it anymore. Simplifying radicals. Rationalizing the denominator. Simplifying rational expressions. Factoring quadratics. Composition of functions. The binomial theorem. It's not that someone, somewhere doesn't have a use for these things, but what is increasingly becoming the exception should not prove the curriculum rule. Mediocre proficiency with these topics is not what leads students to be successful in calculus. What students really need for calculus is a deep understanding of slope as a rate of change and accumulation.

Christopher Danielson

This is a familiar story for some of us. We cringe when we ask students "What's slope?" and they parrot back, "rise over run" without knowing much beyond that. Yes, that might be one way to describe slope, but there are other, and arguably more important ways to describe slope. Danielson's focus on slope as a rate of change not only is most fundamental for calculus, but it is in alignment with the research on teaching slope (Lobato & Thanheiser, 2002; Peck & Matassa, 2012; Stump, 1999, 2001).

Danielson led the well-attended workshop through a number of middle-school appropriate tasks involving rates of change. Because the tasks were set in informal contexts, students would be most likely to work in terms of "dollars per bicycle rental" or "enjoyment per piece of candy," depending on the context of the problem. Time was spent not just looking at rates, but doing simple calculations to compare changes in rates over time, a fundamental conception needed for calculus. The problems in the workshop were adapted from tasks found in Connected Mathematics, a popular NSF-funded curriculum for the middle grades.

For Christopher's presentation, related tweets, and participant notes, see his post at http://christopherdanielson.wordpress.com/2013/04/21/the-goods-nctmdenver/.

References

Lobato, J., & Thanheiser, E. (2002). Developing understanding of ratio-as-measure as a foundation for slope. In B. H. Litwiller (Ed.), Making sense of fractions, ratios, and proportions (pp. 162–175). Reston, VA: NCTM.

Peck, F., & Matassa, M. (2012). Beyond “rise over run”. RME in the classroom. Workshop at ICME-12, Seoul, South Korea. Retrieved from http://rmeintheclassroom.blogspot.com/2012/07/icme-12-workshop-and-sharing-group-on.html

Stump, S. L. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124–144. Retrieved from http://www.springerlink.com/index/R422558466765681.pdf

Stump, S. L. (2001). High school precalculus students’ understanding of slope as measure. School Science and Mathematics, 101(2), 81–89. doi:10.1111/j.1949-8594.2001.tb18009.x

NCTM Denver 2013: Fennell and Wray's Math Specialists Get Ready Now: Common Core Assessments Are Coming

Annual Meeting - Friday, April 19, 3:30 pm

Francis (Skip) Fennell - NCTM Past President; McDaniel College, Westminster, Maryland
Jon Wray - NCTM Board of Directors; Howard County Public Schools, Ellicott City, Maryland

Skip Fennell, Jon Wray, and Beth Kobett (who was absent for this presentation) are the leads on ems&tl, the Elementary Mathematics Specialists & Teacher Leaders Project. As the name implies, the focus here is on supporting math specialists, such as district-level curriculum directors, instructional coaches, and anyone who is in a position to support mathematics teachers.

For this presentation, Fennell and Wray looked at the upcoming Common Core assessments, PARCC and Smarter Balanced (SB), and suggested ways math specialists can help teachers prepare for the tests.

Francis (Skip) Fennell

The challenge Fennell and Wray presented was essentially to focus on the upcoming assessments and respect the influence they will have on curriculum and instruction, without focusing too narrowly on the assessments and cause instruction and learning to suffer. This means, for example, not turning classroom practice into test prep, and using sample items from both PARCC and SB wisely.

Fennell and Wray used the concept of assessment literacy to describe the ability for teachers and specialists to understand a testing program. Many teachers have no formal training in assessment, so math specialists must be able to help them build their assessment literacy. Part of this is simply becoming more familiar with the schedules and formats of the upcoming PARCC and SB assessments. Both consortia offer more than just an end-of-year test, and teachers are going to need to help students interpret new kinds of technology-enabled assessment tasks.

Jon Wray

Fennell sees great potential in the CCSSM, but said, "If the Common Core becomes political, it's dead." Teachers and specialists need to work with the standards in ways that doesn't reduce them to a checklist of vaguely connected ideas. Using a number of items and task prototypes, Fennell and Wray showed examples of sample items from PARCC and SB and showed the many ways these could be used richely in lessons if the teacher provides the right support and instruction. "There are a lot of ways sample items can be used as instructional gems, " said Fennell. A list of potential questions and strategies for various tasks can be found in their slides.

The presentation wrapped up with an urging to better understand the role of formative assessment around these sample tasks. Also, encouragement was made to use materials from both PARCC and SB, regardless of the test your state has adopted. More task resources were linked to, including Illustrative Mathematics, the Institute for Mathematics and Education (especially the progressions documents), The Mathematics Common Core Toolbox, the PARCC Educator Leader Cadre Portal, and the Smarter Balanced Scientific Sample Pilot Test Portal.

The slides for this presentation are available here.

NCTM Denver 2013: Building Mathematics Learning Communities with NCTM Reflection Guides

Annual Meeting - Friday, April 19, 2:45 pm

NCTM Professional Development Services Committee
Chonda Long, Director of Professional Development at NCTM

The take-away from this session is pretty simple. To help facilitate professional development, NCTM has produced a series of free, online reflection guides that leverage lessons in NCTM journals.

Chonda Long

The idea is that teachers in professional learning communities can use these reflection guides to help focus their efforts around particular lessons. As an example, we worked through a problem presented in a 2005 issue of Mathematics Teaching in the Middle School. I left to attend another session before the ending, but the reflection guide and a link to the lesson can be found at http://www.nctm.org/profdev/content.aspx?id=23531.

To see all the reflection guides, visit http://nctm.org/reflectionguides.

NCTM Denver 2013: Hirsch's Mathematical Modeling: The Core of the Common Core State Standards

Annual Meeting - Friday, April 19, 2:00 pm

Christian R. Hirsch - Western Michigan University

Hirsch might claim that modeling is at the core of the Common Core, but at a glance it looks like a standard without standards. Yes, the fourth Standard for Mathematical Practice is "Model with mathematics," but the high school content standards chooses to mark standards in other domains as related to modeling instead of grouping the modeling standards together. This makes it more difficult to see the modeling connections across the high school standards, but that shouldn't reduce their importance.

Christian Hirsch has been at Western Michigan for 40 years and he is probably best recognized as the principal investigator for the Core-Plus Mathematics Project. Along with IMP, Core-Plus is one of the most recognized secondary, integrated, NSF-funded curricula to come out of the post-Standards curriculum development period in the 1990s.

Christian Hirsch

Hirsch opened his talk by detailing how all of the mathematical practices can be addressed with a modeling-focused framework of curriculum and instruction. "Real world problems, if even solvable, take a lot of time and perseverance." To Hirsch, Standards for Mathematical Practice 1 and 4 are the focal points of the entire process, at least in classrooms with good instruction. "I'm talking about classrooms where classes begin with problems. I'm not talking about classrooms where the problems are saved until the end."

The key to modeling and making mathematics problematic, says Hirsch, is to identify problems in context, study those problems through active engagement, and reach conclusions as the problems are at least partially solved. The learning lies not only in the solutions to the problems, but the new mathematical relationships that are discovered along the way.

Hirsch used several examples of problems involving modeling in this presentation. The first dealt with the business prospects of a climbing gym. Assuming a survey had been conducted that found the number of expected climbers is related to price \(x\) by the equation \(n(x) = 100 - 4x\), how many daily climbing wall customers should the gym expect? I didn't catch all the details of this problem, but the next question involved finding the optimal and break-even revenue points for the gym, which is nicely modeled by a quadratic. Hirsch advocated using a computer algebra system to assist with the calculations, and advised to help students realize that rounding to the nearest cent, if necessary, also slightly moves answers away from their true zeroes or maximums.

Hirsch's next problem dealt with finding the optimal location for an oil refinery with wells 5 km and 9 km from shore. I sense that this and the previous problem are in Core-Plus, but unfortunately that wasn't made clear and no handouts or downloads for this talk have been provided. While I don't like leaving presentations early, at this point I had a pretty good sense for this one and left to catch an overlapping presentation starting at 2:45. The problems Hirsch chose and the approaches to solve them were pretty solid 30 or more years ago and are still pretty solid today, and I wasn't feeling like the presentation was suddenly going to break new ground. (For me, at least. I totally understand that problems and approaches like this might be new ground in many classrooms.)