Having students find errors in presented solutions allows them to demonstrate their understanding of the mathematical concepts involved. Using CueThink enables teachers to see and hear students’ metacognition in this process.
How can 1 + 3 = 5?
Mathematically proficient students, "check their answers to problems using a different method, and they continually ask themselves, Does this make sense?" and “construct viable arguments and critique the reasoning of others,”. These statement come directly from the Standards for Mathematical Practices 1 and 3 (MP1 and MP3). With error analysis activities, students engage in these practices and as result detect “possible errors by strategically using estimation and other mathematical knowledge” which is mathematical practice 5 (MP5) - that is how 1 + 3 = 5 or more precisely MP1 + MP3 = MP5*. Inspired by Andrew Stadel’s blog post on error analysis, I tried this (and blogged about my experiences here). At CueThink, we looked at Jon Orr’s blog post about using technology (Explain Everything) for his error analysis activity. My first thought - “CueThink is perfect for this!”, so I decided to revisit error analysis with my students using the CueThink app.
*borrowing from Grace Kelemanik NCSM 2015 ignite talk, “When does 2+7+8=1?"
My Error Analysis Task
We were working on applications of the Pythagorean Theorem so I created and submitted three thinklets to my class gallery with common errors in them. Students were given the following directions:
- View all three of my thinklets, knowing they contained an error.
Critique each one and provide feedback using the app ANNOTATION feature.
Choose any one of the problems and create their own thinklet with a correct process and solution.
MP3 + MP5 = MP6
According to MP6 (Mathematical Practice 6), students must “attend to precision”. They must learn to carefully check [their] work, and I was pleased to see that most of my students spotted the errors fairly easily.
Not surprising, there were a few who thought I had solved the problem correctly. This was a perfect opportunity to suggest a different approach to critiquing - I had them take a step back and create their own thinklet as if it were a totally new problem. Once they did, they were better prepared to compare their answer to mine and determine the possible error I had presented. To quote one student, “[At first] I didn’t think you had done anything wrong - but then I did it for myself and now I know what you did wrong!”
My students have a difficult time with checking their work. When they take a quiz or test and they hand it in, I always ask them if they checked their work. Of course they say “yes”, but I know better. I believe in many cases they just don’t know how to proof-read in mathematics, and it looks good to them because they just finished solving the problems. By purposefully creating errors for students to find, they practice being critical, even if that means starting over and checking their answers to problems using a different method (MP1). My hope is that this translates to better self-reflection and that’s what I liked about this assignment -- they also had to record their own thinklet, which for some students was the only way they could uncover my errors.
Perfect for the Task
While I love Jon Orr’s lesson with Explain Everything, I feel CueThink is better suited for this task because it scaffolds students through the entire problem solving process, allowing them to slow down and take the time to notice, wonder, create a plan using one or more strategies, and carefully explain how and why they solved the problem the way they did. Then they can replay their thinklet to see if it really does make sense. I give Jon kudos for some stellar “app smashing” his lesson, but CueThink allowed me and my students to have everything we needed in one App.
Share Your Math Error Stories!
Let us know if you try this with your students! Email us, blog, or tweet using the hashtag #1+3=5. If you are looking for ways to get started with your own math mistakes lesson, check out mathmistakes.org. We’re always looking to share user stories on our blog, so please email firstname.lastname@example.org for a #makemathsocial blog or interview slot!