Self-checking activities allow students to have immediate feedback on how they are doing. When these activities are completed in small groups, it gives students an opportunity for meaningful mathematics discussions. Students determine if they have the correct answers and if they don't, they can work together to determine where their mistake is. This allows the teacher to focus on groups that have misconceptions or misunderstandings as students will often find and correct their own computational errors.
A self-checking activity that I've recently been using is called "Odd One Out." I was inspired by a couple of activities that I found on TES. The first example was a page showing a number of expressions to evaluate using the order of operations. In the center of the page was a bank of possible solutions. There were 15 expressions and 16 solutions. The solution left over when all the expressions were evaluated was the "odd one out." The other example showed four sets of five linear equations to solve. All of the equations in each set had the same solution except for one. The goal was to identify the equation with the "odd one out" solution.
Below are two "Odd One Out" problem sets that I created. The first one is to practice solving systems of equations and the second is to practice solving precentage problems.
You could also use this activity as a quick warm-up with fewer numbers. Jo Morgan (@mathsjem) shared an activity from MathsPad on her Math Gems #74 that had a sets of 9 radicals (surds): 4 simplified, 4 unsimplified and 1 odd one out. You could also use the same format as above with fewer questions. A couple of short examples below.
Craig Barton (@mrbartonmaths) highlighted Odd One Out activities in a "Maths Resource of the Week" in late 2016. He describes a few different varieties of this activity and also points to a variety of different examples of this resource.
Have you used an odd one out activity with your students? How did you use it?
I recently had the opportunity to try out a digital breakout with a Precalculus 12 math class. The classroom teacher and I wanted to create an opportunity for students to have some interleaved practice as a cumulative review for the course. We liked the idea of a breakout game but we wanted to make sure that all of the students got a chance to do a wide variety of problems. Our solution was to do a digital breakout in small groups of 2-3 students.
This was the first time that I had created a digital breakout game so I went hunting online for some examples that might spark some ideas. I found Tom Mullaney's (@TomEMullaney) Digital Breakout template page to be very helpful in figuring out what I was going to do. It gave me lots of ideas and inspiration. I also found I found a post from Meagan Kelly (@meagan_e_kelly) showing an example of a math digital breakout that I was just what I was looking to do. I learned how to create a google site and conquered a number of new technical challenges. While creating the site took some effort, the classroom setup was easy and there were no materials required.
I thought the breakout went well. The students were very engaged and they reviewed lots of different concepts from throughout the year. They liked working in groups and having a variety of different types of puzzles to solve. Many students were consulting their notes and examples from the textbook to find solution strategies. They were also using online tools like https://www.desmos.com/ to help them graph and visualize mathematical relationships. All the problems were self-checking. If the combination for a lock didn't work, they knew that they had made a mistake and had to work together to find and solve it. They also all got to work at their own pace. To add a bit of additional flair, we added a final physical lock and box for students to unlock with a small treat inside.
If you'd like to give this breakout a try, check it out. The link is: bit.ly/PC12Breakout.
Some colleagues recently told me about an activity they had used in class called "Math Market". I'm not sure who originally created it. The teacher who shared it with me learned it at a math conference several years ago. I decided to give it a try with a Calculus class that was just finishing up a unit on integration.
Here is how the activity is run. Students work in small groups (we had groups of three). Each group starts with $5 and selects a captain who can buy questions of different levels of difficulty from the market. Easier questions cost less and have a smaller profit. More difficult questions cost more and have a higher profit. The captain takes the purchased question back to their group to solve. Once they all agree on a correct solution, the captain returns to the market to sell the solution for a profit. The card is added back to the bottom of the market pile and some other group will have an opportunity to buy it. If their solution is correct, they buy a new question and continue working. If the solution is incorrect, they have to buy the question again to attempt a revised solution (or they can purchase a new question at a different level of difficulty). We decided to purchase the solution at a reduced price ($1 less) if they forgot to include the "+C" at the end for the constant of integration. The easiest questions were free so that if groups went bankrupt with an incorrect solution, they would still be able to "buy" another a problem.
I printed the questions on coloured card stock and cut them out. Each question was marked with its level of difficulty. I also added a letter to the card so that it would be easy to find its solution to check the answers.
How it Went
I like that students got immediate feedback on their work. If it was wrong, they had to work with their group to correct their mistake. This was a test review for the class so there were lots of different types of problems mixed together and students had to determine what strategy would be best to solve each problem. It is a nice way to introduce some interleaved practice. This activity could be done with nearly any topic but it worked really well for integration as the questions were challenging and took them some time to solve. This made the market area less crowded.
I'm sure there are lots of variations of this activity. If you have some suggestions, I'd love to hear about them.
Being on Twitter and following hashtags like #MTBoS and #ITeachMath allows me to see classroom mathematics well beyond my physical horizons. I get to glimpse creative and engaging mathematics education around the globe. Recently I saw a couple of different ideas that I've tried to adapt and apply for myself.
Since the Nova Scotia grade 8 classes are working on integer multiplication and division, I decided to create a math mystery of my own. Another nice source of math mysteries is the book Mathematical Team Games: Enjoyable Activities to Enhance the Curriculum by Vivien Lucas.
I liked this idea because it is relatively easy to create; just a find a series of questions with unique answers. Also, students get instant feedback. If their answer isn't on the map, they know they've made a mistake. I would call this purposeful practice as there is a goal to achieve at the end of the activity. There is a reason to persevere. Once students are familiar with the activity, you could give them a blank template (or they could hand draw their own version) and they could work in small groups to make their own treasure hunt activity (and answer key) and share it with each other.
The Role of Practice
I recently read Mark Chubb's (@MarkChubb3) blog post on the role of practice in math class. He discussed the differences between "rote practice" and "dynamic practice". Rote practice involves following procedures, drill and repetition while dynamic practice involves active student thinking, playful experiences and puzzles. I think that the Mystery activity is a more "dynamic" activity than doing the Treasure Hunt activity. However, I think that creating your own Treasure Hunt activity does involve additional characteristics of dynamic practice.
I'm a fan of self-checking math activities. These activities give students immediate feedback and help them to find and correct errors. Many students will be able to correct their own computational errors, especially if students are working in pairs or small groups. When students are unable to fix their errors due to more serious misconceptions, the teacher can step in to help develop understanding. This helps the teacher use their time efficiently and focus on students facing challenges.
I've recently seen one math activity used in a number of classrooms in a variety of forms. I'll call this activity a "question chain" although I've seen it referred to using lots of different names. This activity starts with a set of questions and associated answers. Students start by solving one of the questions. The answer to this first question leads the student to the next question. This process is repeated until the student arrives back at the starting question. The answers form a "solution bank." If the student can't find their answer, they know that they've made a mistake and need to find and correct their error. Below are three different ways that I've seen this activity implemented in classrooms.
Questions on Cards
Questions Posted on the Wall
Questions on a Worksheet
Selecting a Method
During a recent professional development session with math teachers, we tried this activity using all three methods. Participants were split into three groups and each group was given a different method. All three versions of the activity included the same ten questions (see the files below).
After completing the activity we had a discussion to compare the three methods. All of them took about the same amount of preparation and could be quickly created using questions from a textbook or other problem bank. How would students record their work in each method (on paper, mini-whiteboard, etc.)? How would the teacher assess students work in each method? Would each method work better individually, in pairs or in small groups? How might this activity be used in a combined grade classroom? Which method might be most culturally relevant for your students and how does your knowledge of your students inform your selection of a method? Which method is the most engaging for your students? We had a very productive and rich discussion.
Have you used this type of activity in your classroom? Another variation of this method is the "I have/ who has?" oral classroom activity. Have you used a different variation of any of these methods in your class? Do you have a favourite method? Why is it your favourite?
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.
I was introduced to row games while reading Kate Nowak's blog several years ago. A row game is an activity for a pair of students to work on together. Problems are organized in two columns. The first student completes all of the problems in column A and the second student completes all of the problems in column B. The questions in each column are different but the answers are the same. Students collaborate to verify that their answers match. If they do, they move on to the next question. If the answers do not match, the students work together to find out where the error was made and how to fix it. This allows students to have immediate feedback on their work. It also generates great discussions between students as they check each other's work. Another benefit is that students can correct each other's computational errors and the teacher's time can be focused on helping students with more serious comprehension errors.
Row Game Links
There are a couple of great resources for row games online. Kate Nowak has a shared Google drive folder packed with mathematical row games for a variety of grade levels and topics. Another row game collection is available on John Scammell's Orchestrated Experiences for High School Math website.
Nova Scotia Mathematics 10 Cumulative Review Row Game
Below is a row game that I created as a cumulative review for Mathematics 10. I created about half of the questions myself and appropriated the rest from row games created by Kate Nowak, John Scammell, and David McGuinness.