The term open-ended is frequently applied to problems that don’t actually have an open end, but rather an open middle. Here are two problem-based lessons I’ll discuss to make the difference clearer: Do Hybrid Cars Pay For Themselves? How Much Does A 100×100 In-N-Out Cheeseburger Cost? First, let’s consider how each problem ends. With the hybrid car…
I recently attended a full day workshop with Jo Boaler where she shared a different way of thinking about convincing someone from Thinking Mathematically by J. Mason, L. Burton, and K. Stacey. In the book, she identified three levels of convincing: Convince yourself Convince a friend Convince a skeptic I immediately realized that this…
Recently I’ve become aware of the reality that as math teachers, we often talk about what problems we will be using and even how we will implement them but frequently we miss opportunities to discuss why we’re using them. Consider Graham Fletcher’s amazing sugar cube task on volume of a rectangular prism. I can think of three different…
Only a few months ago, I used the words “complex” and “complicated” interchangeably as synonyms for “difficult.” Recently I learned about the differences between their meanings and I now see their significance everywhere, especially in math education. Specifically, I am concerned about Depth of Knowledge (DOK) being viewed as a complicated issue when it is really…
In listing the Common Core State Standards’ three shifts, the authors define “Rigor” by stating that: “rigor refers to deep, authentic command of mathematical concepts, not making math harder or introducing topics at earlier grades. To help students meet the standards, educators will need to pursue, with equal intensity, three aspects of rigor in the…
It’s challenging to have a conversation about depth of knowledge (DOK) when everyone has different ideas of what it is. So, I’ve decided to refine how I distinguish between DOK levels and turn it into a tool that can be used to facilitate a conversation. The two-page tool covers topics from 1st grade through high school as well as topics ranging from number sense, fractions, geometry, probability, and functions. The pictures below give you a preview of the tool and you can download it by clicking the button at the bottom of the page.
When teachers begin to implement problem-based learning, several questions consistently come up and I have a slide to discuss them during my presentations. At the 2014 California Math Council (CMC) North conference in Asilomar, Michael Fenton mentioned that these questions would be worth sharing in a blog post. Anyone else think @robertkaplinsky should write a…
I wanted to share a resource from Tina Cheuk out of the Understanding Language department at Stanford University who created a very useful graphic organizer to show that the Common Core State Standards (CCSS) for Mathematics, English Language Arts (ELA), and Next Generation Science Standards (NGSS) have significant overlap and can be collectively implemented for…
I recently shared how I came to realize Webb’s Depth of Knowledge’s (DOK) importance and why I believe that it is critical to integrate these types of problems into our curriculum. I also expanded into how to distinguish between the DOK levels using specific examples that covered fractions on a number line, perimeter and area, and…
The most straightforward method I have found for distinguishing between the first three levels of Webb’s Depth of Knowledge (DOK) is to take a single mathematical concept and provide a problem at each DOK level. Here are three examples that illustrate the differences well using: fractions on a number line perimeter and area surface area and…
When the Common Core State Standards (CCSS) for Mathematics were written, the authors defined three shifts (coherence, focus, and rigor) to articulate “how the standards differ from previous standards—and the necessary shifts they call for.” The shifts are useful in articulating the big picture as to how things have changed. This blog post delves solely into…
I am no longer using these posters. If you are interested in a tool to help math teachers make sense of the depth of knowledge in mathematics, please check out this post with a reproducible handout for educators.
Today we found this problem on the Smarter Balanced Practice Test (Grade 8 – Question 15): It provided the basis for a worthwhile conversation on the importance of students having conceptual understanding. The problem is based on these 8th grade standards: There are two main ways to approach this problem: conceptually or procedurally. If you…
I’ll be honest with you: I remember first learning about Bloom’s Taxonomy when I was getting my teaching credential but I never really thought it was that big of a deal. I had a decent understanding of what it was and how you could implement it. What I didn’t get was why it mattered. As…
Articulating why the Common Core State Standards are such a big change for mathematics can be challenging, especially if your audience is unfamiliar with them, like parents and other non-educators. After all, isn’t it just changing the standards? Fellow educators also feel the stress, but may have trouble stating exactly why that is. How do…
It is often helpful to have a map to your destination when taking off on a journey. Without one it is easy to lose your way. This holds true with groups of teachers collaborating as part of a Professional Learning Community (PLC). Sometimes teachers find themselves in situations where they have time to meet as…
How do you think 32 eighth grade students would respond to this nonsensical question: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?”* Take a guess as to what percentage of them would give a numerical answer and then watch the video below: Of the 32 students I…
Have you looked at a mathematics problem-based lesson and thought that while it would be fun to do one with your classes, you weren’t sure if it would work with your students? Maybe you’ve already tried out a few lessons but keep running into the same issues? This Tuesday at 9 PM EST I will…
About two months ago I had the opportunity to work with a class of struggling 8th graders in a support class. I chose to do the ticket option lesson with them and wanted to share my experiences and some student work samples. From my previous experiences working with these students, I knew they struggled mathematically…
Many teachers reading the Standards for Mathematical Practice are confused by what “Model with Mathematics” (Math Practice 4) means. I define mathematical modeling as the process of taking an often real-world context, turning it into something you can manipulate with mathematics, and then returning to the context with the new knowledge. For example if you want…