Mathematical mistakes are made unconsciously. That is, when students are made aware of a mistake they can adjust their own work because they have the conceptual or procedural understanding necessary to do so. Our students are brilliant and experience the world in innumerable ways; accordingly, they uniquely receive, process and express information and we cannot expect that they all commit mistakes in similarly unique ways. In response, we model self-monitoring when encountering mistakes rather than adjusting instruction with new instructional methods and strategies.
Proving cues to students when we discover mistakes models self-monitoring and, ultimately, self-awareness of the habits of mind that lead to them. Mistakes often occur when computing, modeling and communicating abstract mathematical representations (e.g. numbers and symbols, expressions, equations, algorithms) and we have a range of cues to support students in developing this awareness.
| Cue | Example |
| Playback |
A student says, “I then combined 2x and 7x to get 10x and then used that to…”. You respond during the next appropriate moment, “We heard you say 2x combined with 7x is 10x.” The student realizes that isn’t correct, and adjusts their work accordingly. You move on with instruction as planned. |
| Example |
A student struggles to recall the definition of a prime number. You say, “1 is an example.” After a few seconds of silence, you add, “2 is an example.” The student then says, “A prime number is a number whose only factors are 1 and itself.” You continue with instruction as planned. |
| Pinpoint |
A student multiplies 23 and 98 using the standard algorithm. When multiplying the partial product of 90 and 3, they write 230. After they finish, you point to the 3 in the tens place and make a pensive face. The student says, “Oh, I mean 270.” They complete it again and arrive at the correct product. You continue with instruction. |
| Comparison |
A student multiplies 23 and 98 using the standard algorithm. When multiplying the partial product of 90 and 3, they write 230. After they finish, you set their work side-by-side with a correct response and ask, “How are they different?” The student says, “The 3 in the tens place is different from the 7 in the tens place there.” You wait. They then say, “I see, I multiplied incorrectly. The partial product should be 270 and the product would change to 2,254.” You continue with instruction. |