When I was learning math as a student, I had no idea that math was a beautiful and interconnected story. Sure, I could do mathematics. I even graduated from UCLA with a Bachelor of Science in Mathematics. But I didn’t deeply understand mathematics. For example, I had never noticed how simple single digit multiplication progresses…
If you’ve used one of my problem-based lessons or seen me present at a conference, there’s a pretty good chance that you’ve used my Problem Solving Framework with your students. I first started working on it in 2012 after seeing something similar on Geodff Krall’s website. It’s now on version 8.1 and I wanted to…
Recently I was looking at data that Google shares about who visits your website including what pages they go to and how long they stay on your site. I had some hunches about which lessons would be the most popular and I was curious to see if I was right. SPOILER: I was very wrong.…
You’ve probably heard that there are tournaments for board games such as Scrabble. These winners are often held in high esteem for their intelligence, knowledge, and strategic thinking which they use to propel themselves to victory. They are thought to be experts of the languages they use… but I wonder if that is always the…
The 2017 College Football Championship game came down to the final play with six seconds left. Watch the first 14 seconds of this clip so you understand what happened. Next, I want you to read the excerpt below from a CNN article, and pay special attention to quarterback Deshaun Watson’s quote at the end. Think about…
If you’ve ever taught students how to find the area or perimeter of a shape, you won’t be surprised to read that students commonly confuse the two measurements. For example, if you ask a student to find the perimeter of a rectangle, they will often give you the rectangle’s area. Based on my experiences, this seems to be a pretty typical outcome for all math educators.
I realize that we’ve come to accept this as normal, but have you ever thought about why it happens? Does this also happen in real life? Could this possibly be a problem of our own creation? After all, when a person is buying grass turf and fencing for their home, does that person ever get confused as to which measurement is which? I can’t imagine that happening often.
This makes me wonder about whether it’s possible that the reason students confuse area and perimeter is because we often present problems with fake contexts. As a result, the terms “area” and “perimeter” remain abstract labels rather than something attached to a relatable meaning.
Don’t you hate when your lessons are derailed and don’t go as you planned? Wouldn’t it be nice if you could better anticipate these issues and come up with ideas to lessen or eliminate their impact? If so, here’s an idea to consider that has great potential to help: a pre-mortem.
Here’s a question to think about: if you can completely remove a problem’s context and still solve it, can it really be mathematical modeling?
Consider the math problem below from a middle school math textbook. It uses the context of a baseball diamond to discuss rational numbers. It is listed as a “Real-World Link” and demonstrating Math Practice 4 (notice the box in the upper right).
I was watching students work on this page and wondered how important the context was to solving these problems. Then it made me wonder, “What if there was no context? What would change?” This is what the problem might look like it the context was completely removed.
Math educators are on a never ending quest to help students improve their problem solving skills. To succeed on this quest, they must choose between a variety of tools and it can be challenging to determine which are better. So, I wanted to share my take on two main categories I see:
problem solving tools designed for complicated situations
problem solving tools designed for complex situations
If you are frustrated that your students will try a math problem once or twice and then give up, this blog post is for you. Let me (re)introduce you to one of my favorite tools for helping students persevere when problem solving: the Open Middle Worksheet.
To explain why many educators find it invaluable, I’ll need to tangent and briefly explain a central theme of one of my favorite books, Freakonomics. Life is all about incentives. There are positive incentives (I go to the gym because I want to stay healthy) and negative incentives (I don’t steal because I don’t want to go to jail). Without these incentives (or if the incentives changed) what you do would change.
My favorite tool for helping students solve problems is the Problem Solving Framework. I love using it because it scaffolds students often underdeveloped critical thinking skills and helps them develop their own problem solving techniques.
One of the most common concerns about implementing 3-Act Tasks (also known as problem-based lessons, application problems, or math modeling problems) is “How do you assess them?” Here are the four ways I use most often.
Here are two ways to integrate a problem-based lesson into your unit as well as a third that should be avoided. I use an 8th grade standard on the volume of spheres, cones, and cylinders as an example.
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…
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…
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…
This search engine searches all of the sites below to quickly help you find a problem-based lesson (also called 3-Act Task, mathematical modeling, or application problem): The links below are the pages that are being searched by the search engine: 101 Questions Andrew Gael Andrew Stadel Catherine Castillo Dan Meyer Dane Ehlert Emergent Math’s Problem…
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…
Several years ago I had a profound moment that led me to completely rethink what it meant to understand mathematics. I was still in the classroom and had been working with 6th graders on adding and subtracting mixed numbers. My formative assessments and observations showed that most students were proficient, and I felt pleased. To…
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…