Activities that let students get immediate feedback on how their are doing are extremely beneficial. Activities that allow students to self-check their own work allow for this immediate feedback and correction. These types of activities can allow the teacher to focus their time with students who are having conceptual misunderstandings and not get bogged down helping students find and correct computational errors. While students are engaged in self-checking activities, the teacher can also be working with small groups of students on mini-lessons or conversations/conferences. Below are a few of my favourite activities and routines that allow for students to check their own work:
Team Post-Its - I recently saw this activity described in a post by Julie Morgan. This activity is very easy to set up and does not require much front loaded time to create. The teacher posts a list of questions for small groups of students to work on. These might even be questions from the textbook. Each group solves the question, writes their answer on a sticky note and posts it on the whiteboard. As other groups complete the questions, they can compare their answers to those from other groups to see if they agree. If they don't agree, they double check their work. I would suggest that each group of students starts with a different question.
Add ‘Em Up - In this activity, students do a set of problems, either on their own or as a group. These problems typically have numerical answers. The answers to the set of problems are added up and compared to the sum provided. If the sum is not the same, then the student knows that one or more of the problems in the set was done incorrectly and works to find the error. I first saw this activity described in a blog post from Kate Nowak. I later saw a blog post from Amy Gruen describing a simple and quick way to do this same activity that I used occasionally. There are many descriptions of how to organize this activity including one in a detailed blog post from Sara VanDerWerf.
Row Game - I also first saw this activity described in a blog post from Kate Nowak. Typically, a row game is a worksheet of problems organized in two columns. The worksheet is completed by a pair of students, one doing the problems in column A and the other doing the problems in column B. The problems in each row have the same answer so if the students' answers don't match, they can work together to check their solutions to find the error. To make row games a bit easier to create, you can create an additional column with the sum of the solutions from column A and B (similar to the Add 'Em Up activity from above). This allows you to use any two problems and not have to create two problems with the same solution. Kate Nowak has a shared google folder with a large selection of crowd-sourced row games.
Added 03Oct017 - I recently saw a great idea from Heidi Neufeld. She asked students who finished quickly to make a new row for the row game and create two different problems with the same answer.
Mathematical Circuit Training / Around the World / Star Chain / Question Stack - There are lots of different names for and variations of this activity. The essential part is that there is a series of problems and the answer for each problem leads you to the next question to ask. The answer to the final question leads you back to the starting question. This activity can be organized as a simple worksheet, a stack of cards to turn over, a set of cards to chain together or questions posted on signs around the classroom or hallway. If you make a mistake, you won't be able to find the next question and you know to try again. This can be done individually or in small groups.
Added 26Sep2017 - Thanks Alicia!
Invisible Ink - The description of this activity is from a blog post from David Petro. Students solve a set of problems on a card. When ready, they can check their solutions using an answer card. This card has the correct answers written with "invisible" ink that can only be seen by shining a small UV light on it. Once the student has solve the questions correctly, they move on to the next card containing more complex questions. David says, "Students really seem to like this style of activity as they feel empowered to move from card to card when they are ready and the added feature of checking the answers with the UV pen gives a sense of novelty."
If you know of any other self-checking activities that I've missed, please let me know and I'll add them here.
Have you ever had a student look at you like you're talking in a foreign language during a math lesson? It happens. Sometimes there are gaps in students' prior knowledge. Sometimes students need more time to process a new concept in order to construct and develop their understanding. Sometimes I haven't explained something clearly and I need to think of a new approach to tackle this topic. When I encounter those blank stares, I think, "That's Numberwang."
Numberwang is a skit from That Mitchell and Webb Look, a British sketch comedy show from 2006. If you've never seen it, take a two minutes to watch a video of this skit. The premise of the skit is that while the presenter and contestants seem to understand the rules perfectly, they are completely inscrutable to the viewer. We're left scratching our heads in confusion just like our students sometimes do in class.
So what do you do when you sense that students in your class are not getting it? I suggest that you invest some time to uncover your students' thinking. As Guildenstern implores in Tom Stoppard's play Rosencrantz and Guildenstern Are Dead, "Delve. Probe the background, establish the situation." Take the opportunity check in with students to determine their level of understanding. Here are some strategies you might use:
Reflect and Respond
Once you have a better picture of the misunderstandings and misconceptions that may be present in your class, you can plan your next steps. Was there really a misunderstanding or did you make assumptions about prior knowledge that weren't true? Were just a few students struggling or was it a commonly held misconception? Tracy Zager, in her book Becoming the Math Teacher You Wish You'd Had, writes, "If just a few students were confused, she could work with them individually. If there was a really interesting mistake, or patterns among the misunderstandings she saw, she could use those examples as her next teaching opportunity."
When I see those "Numberwang" looks I am reminded that even a well planned lesson can sometimes miss the mark. Reflecting on how a lesson went and how I can improve it helps me refine my teaching practice and be more responsive to students' needs. Don't let those "Numberwang" moments go by ignored. Matt Larson, in his August NCTM president's message, wrote "Making mistakes, getting feedback from our colleagues, and making iterative improvement are part of the natural process of continual growth. We should never forget that perseverance isn't just for students—perseverance also applies to us as professionals."