Our PD Center for Mathematics and Adult Numeracy continues to create and adjust our offerings to meet your needs. One way that we do this is by developing blended courses. For example, when piloters told us that it was important for teachers to be able to talk together—in the moment—about topics such as student work, we changed Analyzing Student Work from a totally online course to a blended course to kindle discussions. (By the way, this course starts on January 12; be sure to sign up today!)
We have begun developing a Making Sense series, starting with two blended courses: one on fractions, and one on proportional reasoning. Both courses start with a face-to-face session so participants can get to know one another before meeting online. Read more about Making Sense of Fractions and Making Sense of Proportional Reasoning below.
Because doing math involves so much more than simply learning procedures, it is not sound practice to offer quick fixes, for teachers or for students—learning mathematical practices such as proportional reasoning, understanding of fractions, or algebraic thinking is a long process that develops over time. While we are brainstorming ideas for some shorter professional development (PD) offerings (such as using number lines from Level A to E), some topics will require longer PD because we want to provide the same depth of understanding that our students need. We know that multi-session PD can feel like an investment of time that you or your teachers can’t afford, but time is necessary to develop real understanding—and research shows that it pays off in the long run, as both teachers and learners become more effective at their jobs. As stated in a 2011 National Commission on Teaching and America’s Future report on STEM teacher development, "learning is no longer preparation for the job, it is the job" (p. 4).
Therefore, you won’t see any brief videos on our website claiming to teach these deeper concepts in "Three Quick Minutes"! Instead, we’ll try to continue to develop ways to meet your needs without compromising our High Quality Professional Development standards: “a set of coherent learning experiences that is systematic, purposeful, and structured over a sustained period of time.”
Proportional Reasoning: What’s the Big Deal?
I suspect that many directors spent the previous months trying to figure out their program’s cost per student, teacher cost per hour, space allocation, etc. I also suspect many of you spent at least part of the holiday season trying to determine which item was the “best buy.” Without consciously realizing it, all of you were engaging in proportional reasoning.
How many of you actually set up your proportions so that you had to perform a cross product (setting up two ratios, making them equal to each other, and cross-multiplying)? I suspect very few. If that’s the case, then why is it that almost every adult education math workbook only refers to that one strategy? And why is it that so many teachers teach only that strategy—one that is effective in some situations but not others?
In other words, teachers teach how to do the procedure for figuring out a missing value in a proportion, but we don’t actually teach how to reason with proportions.
According to the National Research Council, “The concept of ratio is much more difficult than many people realize” (p. 417). Proportional reasoning is considered the capstone of elementary school arithmetic and the gateway to higher mathematics (p. 242), so it is critical that we teach our students how to reason with ratios and proportions. Because of this need to understand how to reason proportionately, our Center decided to develop a course for teachers to deepen their own understanding in order to help students develop proportional reasoning over time, beginning at the most basic level.
Last month we piloted a blended course—Making Sense of Proportional Reasoning—which we are now ready to roll out, beginning January 31 in the Boston area. If you’re not near Boston, we would love to offer it in your region. Just give us a holler (email Donna Curry [firstname.lastname@example.org]), and we'll make arrangements with you!
Spiraling Into Control, Not Out of Control
Usually when we hear that someone or something is spiraling out of control, this is a negative thing. But at our Center, we focus on spiraling in order to get better control.
In most adult education workbooks, you see a chapter on fractions, a chapter on algebra, etc. Each topic is covered separately; students are shown a procedure and then given opportunities to practice that procedure on a bunch of similar problems. After the practice (during which students often seem to be able to remember the procedure . . . for the moment), the topic changes to something new.
Based on test scores and success rates (or lack thereof) in developmental education classes, this method for teaching doesn’t work. Instead, we need to teach a topic, then come back to it again, always in a little more depth than the last time it was discussed—a method that the College and Career Readiness Standards for Adult Education (CCRSAE) calls coherence. The CCRSAE often use verbs such as “apply and extend” students’ previous understanding, which suggests that there is a developmental progression in teaching math concepts. (Check out our leveled progressions for fractions and number lines for more concrete examples.)
This spiraling approach also helps when our students do not show up regularly for class. Even if students miss the first introduction to a topic, we (and the students) know that the content will be covered again.
Our Center is modeling the spiraling approach in a number of ways. Our offerings are designed to help you understand how to introduce concepts at a beginning level and then revisit them again and again. In our blended courses, the virtual sessions review and deepen the material in the face-to-face sessions. For example, our new Making Sense of Fractions course isn’t just about showing students how to use operations with fractions: We begin at the most basic level and then move to higher levels of understanding so that students can handle a variety of algebraic situations involving fractions.
This course, like our other blended courses, models the spiraling approach whereby the virtual sessions review and deepen the material in the face-to-face sessions. If you are a director who is tired of having students complain that they hate fractions, or a teacher who is tired of teaching and reteaching fraction procedures, this course is for you. It will be piloted in 2018—stay tuned!