Practice is important. Whether it is playing the piano, shooting free-throws, popping an ollie or solving a quadratic equation you need to practice to improve. Some practice routines are more effective than others at helping students solidify their understanding. Practice can often seem tedious and it can be difficult to maintain the motivation to practice.
In mathematics, students practice skills in a variety of ways. One style of practice that can help students stay motivated and engaged is purposeful practice. Instead of a page full of repetitive problems, students focus on an activity that has a mathematical goal to achieve. Dan Meyer wrote a blog post a few years ago titled "Purposeful Practice & Dandy Candies" that started me thinking about how to make activities in my classroom more purposeful.
One of my favourite sources of problems with purposeful practice is Open Middle. There is a large selection of questions organized by topic and grade level. Each question has an "open middle" meaning there are many ways to explore and solve the problem. Below is a question submitted to the Open Middle site by Robert Kaplinsky.
In this question, students try to find the arrangement of digits that yields the product closest to 50. Students will try numerous different arrangements of digits and get lots of practice multiplying decimal numbers without it seeming tedious. The question can also be quickly modified to give additional practice. For example, just add a hundredths place onto one of the factors and use 5 different digits.
Games can be a great way to encourage students to practice. There are lots of examples of but I'm going to mention just two. The first goes by several names. Joe Schwartz wrote a great post about Factor Captor. A similar game is described on the NCTM Illuminations site called the Factor Game. Students alternate turns playing on board filled with numbers. The first player selects a number to cover and adds that number to their score. The second player finds all the factors of that number, covers them and totals those number to add to their score. The roles are then reversed and play continues until there are no uncovered numbers remaining. There is a lot of math in this game and it is fun to play.
The second game is Horseshoes from Math4Love. This game is played with a deck of cards numbered 1-9. Two cards are drawn to form a two-digit target number. Then four more cards are drawn. Players use these four digits to create an equation using addition and subtraction that is as close as possible to the target number. For example, let's say that the target number is 25 and the four digits given are 1, 3, 6 and 9. A student might create the equation 39-16 = 23. Another student might make the equation 13+6+9 = 28. There are lots of way to tweak this game for different levels of complexity.
Both of these games allow for lots of numerical practice in a format that engages students. There are many excellent sources of ideas for mathematical games. My current 'go-to' resource is a book titled Well Played, 6-8.
There are several mathematical puzzles that include lots of practice with numerical computations in pursuit of a solution. KenKen puzzles and Maze 100 from NRICH are two such puzzles that I've used. I also think that Yohaku puzzles are great. They are numerical puzzles where you need to determine the number that is in each square in order to make the column and row sum/products.
For classrooms with the available technology, activities from Desmos.com are another way to practice with purpose. An activity that incorporates a lot of meaningful practice is Marbleslides. In this activity, student try to capture stars on a Cartesian grid by creating a path using functions that marbles roll down. Students work to refine their functions to capture as many marbles as possible. Another activity that generates lots of practice is Transformation Golf. Students use a series of rigid transformations to move a shape to specified location. They have to find an efficient path around several obstacles. Each successive challenge increases in complexity.
There are also activities that have a non-math goal for students to achieve. Lots of online math games have incentives for students such as badges to earn, experience points to accumulate or virtual prizes to win. There are also worksheets and activities with non-math goals. Worksheets such as "Algebra with Pizzazz" and "Punchline Algebra" have a riddle to be solved once all of the questions are completed.
In my classroom practice, I used a number of activities with these types of incentives and I think that many students find them exciting and fun. If students are excited to do math, I consider that a win. However, I think that these types of activities should be used with caution. We don't want to inadvertently send a message to our students that math isn't fun by itself so we have to disguise it (like sneaking vegetables into their favourite foods so kids will eat them).
If you have a favourite math activity, game or puzzle that gets students practicing math with a purpose, I'd love to hear about it. Please leave me a comment.
My son and I recently spent a lovely fall afternoon exploring the carnival games and amusement rides at a local fair. My son is quite adventurous when it comes to amusement park rides and is eager to try just about any ride that he meets the height requirement for. While we were walking through the midway, I spotted a carnival game called "Roll Down" that appeared to have a bit of mathematics involved.
The object of this "game of skill" is to roll six balls down and inclined ramp to land in one of six numbered bins. If the sum of the six rolls is under 10 or over 31, you win. Is this game worth the $5 price to play? What are my chances of winning? Should I go for under 10 or over 31?
The bins are just wide enough for a ball to fit so it is very difficult to aim a ball with accuracy. You also have to question if the balls roll straight and if the board is smooth and level. Lets just assume that the balls fall into a random bin (you could then play an equivalent game at home by rolling 6, six-sided dice). With six balls, the smallest sum possible is 6 (all 1's) and the largest is 36 (all 6's). How many ways are there to get each possible value?
There are only 31 possible sums (6-36) that you can score. To roll a sum of under 10, you can score 6, 7, 8 or 9. To roll a sum of over 31, you can score 32, 33, 34, 35, or 36. At first glance, it looks like you have a 9/31 chance of winning but this is not correct.
This reminds me of a bet in the casino game craps that looks good, but on further inspection is really bad. The field bet is a bet on the sum of the next roll of two six-sided dice. If the sum of the two dice is 2, 3, 4, 9, 10, 11 or 12 you win. If the sum is 5, 6, 7 or 8 then you lose. It has the illusion that there are more ways to win than lose, but you are much more likely to roll one of the losing numbers.
With practice at Roll Down, you might be able to achieve better than the random results that I detailed above. Instead of this practice, I decided to spend my $5 at the concession stand to buy a hand-battered, deep fried corn dog. A midway concession stand can also be considered a bit of a gamble, but in this case it was a delicious win!
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 recently wrote an article about podcasts for the Global Math Department newsletter. I had more to write about podcasts, so I'm continuing my thoughts here. There are lots of really well done podcasts focused on mathematics and education and I wanted to share the best of what I've been listening to on my daily commute to and from work. I have just enough of a commute to get through about a podcast each day. I really enjoy how a good podcast make a dull commute alone in a car feel like a road trip with friends.
One new podcast that I started listening to is called Infinite Insights. Duane Habecker and Maggie Peters started this podcast to discuss articles and research regarding mathematics education and how it applies to your classroom. I really enjoyed listening to their first podcast about using math for your Day 1 routines.
Five Fabulous Podcasts Episodes
Instead of just recommending a bunch of podcasts, I wanted to hone in on specific episodes that I think you'll enjoy (assuming you're into mathematics). Check these out:
So what makes a good podcast? I used four criteria for selecting the podcasts above: Length (I find that 30-40 minutes is about right for me), Mathematical Interest (I like to learn new things about math or education), Content Density (I like a podcast to get to its point and not linger on idle chit chat), and Production Value (a good microphone and a sound booth help).
If you're interested in a longer list of math podcast recommendations, David Petro has a nice collection listed on his Ontario Math Links website.
I'd really like to hear about your favourite mathematics or education podcasts. Even better, I'd like to hear about a great episode of your favourte podcast. Please let me know what I've been missing out on.
I recently saw a photo posted on twitter by John Golden. The photo showed the question prompts "What makes a good question?" and "What makes a question good?" on a white board. John said that this was an intro activity to a discussion on questions. These questions really got me thinking and reflecting on my own practice.
There seems to be a lot of discussion about "what makes a good question." There are entire books filled with examples of rich question and engaging problems. Two of my favourites are Good Questions: Great Ways to Differentiate Mathematics Instruction by Marian Small and Good Questions for Math Teaching: Why Ask Them and What to Ask, Grades 5-8 by Lainie Schuster and Nancy L. Anderson. I've seen much less discussion about "what makes a question good." I think it's harder to define the effective teaching practices and routines for asking questions in a manner that makes them effective.
What are the characteristics of a good question? I recently read a post from Geoff Krall in which he wrote about teachers adapting questions they had found to make them even better. He said that to do this, "You start to turn from 'I like this task' to 'What do you like about it?'" I think that this is an important question to ask yourself. Teachers need to think about their students and the context in which they teach to determine what questions are going to best facilitate effective student learning in their classroom. Here are some possible characteristics of good questions that you might consider:
Jennifer Piggott in an article on NRICH wrote, "In essence, rich tasks encourage children to think creatively, work logically, communicate ideas, synthesise their results, analyse different viewpoints, look for commonalities and evaluate findings. However, what we really need are rich classrooms: communities of enquiry and collaboration, promoting communication and imagination." This really resonated with me. It is not enough to have a great problem. As teachers we need to know how to effectively present and lead the exploration of a problem in order to reap its benefits.
There are a number of strategies you might consider when exploring a question:
You Need Both to Succeed
My grandmother was a big fan of the card game pinochle and she taught me to play at an early age. It's a trick taking card game where players work in pairs to score points. I learned early that even the best hand of cards can be beaten by someone who knows how to play well. My grandmother played to win and she would often remind us to "mind your p's and q's" while playing to pay attention and play carefully. I learned a lot of math counting up tricks and keeping score. The important lesson here is that to be successful at pinochle, you have to have both a good hand of cards to meld and play those cards correctly to maximize your score. Just like pinochle, when planning a lesson, you need to consider two important and complementary components. Both finding good questions and using them effectively are equally important to the success of your problem solving lesson.
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."
My nephew visited me this summer during a family vacation. He will soon be starting grade 11 (junior) year. He is a hard working student and would like to study mathematics in university. He understands that this is the year when he needs to start thinking about university and scholarship applications. We started chatting about his post-secondary plans one evening and he asked me two questions. The first question was, "What can I do to make sure I have competitive university and scholarship applications?" The second question was, "What sorts of careers are possible with a mathematics degree?" Here is the advice that I offered him.
Break Out from the Pack
As a high school teacher, I have written a lot of reference letters. The hardest reference letters to write are for talented students who come to class every day and work hard but have nothing to set them apart from the other 30 or so students in class. Ask yourself, "What sets me apart or makes me different and special?" Here are some things I suggested to my nephew:
Careers for Mathematians
There are some great careers for mathematicians that don't get a lot of press. Here are some careers that I find really interesting:
So what advice would you give to a grade 11 student who is interested in a career in mathematics? I'd love to know what you think.
A couple of months ago, I posted a list of six exceptional math Ignite Talks. For those unfamiliar with the format, an ignite talk includes presentation slides that automatically advance every 15 seconds. Exactly 20 of these slides result in a 5 minute talk. Since I posted my first list, Suzanne Alejandre (@SuMACzanne) at the Math Forum has been busy posting videos from older Ignite sessions. These playlists are a gold mine of mathematical thought and reflection. Additionally, the Ontario Association for Mathematics Education (OAME) had their 2017 conference ignite talks. There are so many great talks posted recently that I felt the need to recommend an additional six talks. It will take just 30 minutes to watch all six... time well spent.
I've been working on my own ignite talk and have discovered how difficult it is to craft one. There is a lot of reflection and thought involved in refining what you are really passionate about as an educator, determining how to explain it clearly and figuring out how to make it entertaining. At this point, my talk is a still just an organized collection of notes, ideas and images. I don't have any plans to actually present this ignite talk, but I feel that the process of creation and reflection is very worthwhile.
June is here and the end of the school year in Nova Scotia is rapidly drawing near. In grade 7 math classes, it is time for students to break out their geometry sets and practice geometric constructions. The outcome for this unit (7G01) states, "Students will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors."
The intention of this outcome is for students to be able to describe and demonstrate these constructions using a straight edge and compass. In developing the understanding of these constructions, students are exposed to a variety of methods and tools such as paper folding, Mira, and rulers. I've written about geometric constructions in the past, so I won't get into a discussion of the merits of Euclidean constructions. My concern is that this unit could turn into a series of steps to memorize for a short list of basic geometric constructions that lacks coherence and context. If you really need to construct a heptadecagon using only a straight edge and compass, I'd expect you to look up the steps, not have them memorized. Additionally, in my opinion, the exercises in the student textbook are routine and dull. I think that this is a unit that has a lot of potential for student engagement but could easily become tedious.
Dan Meyer gave an inspiring ignite talk titled "Teaching the Boring Bits" at the 2013 CMC-North conference. In this talk, Dan encourages teachers to engage students by creating an intellectual need for new knowledge. Teachers should strive to provide students with a reason to want to know new mathematical skills and methods.
A grade 7 teacher that I know thought that incorporating Islamic geometric designs into this unit would give a purpose and context for doing geometric constructions. Another factor in selecting this context was that she has a number students who are recent immigrants from the Middle East in her class. We brainstormed some ideas and developed several activities to infuse into this unit that might help give this outcome some coherence and allow students to be creative and artistic. The teacher started by using a template and pattern from Eric Broug's School of Islamic Geometric Design. Students used the template and followed the instructions to create and colour their designs which were then tessellated in a grid to make a group composition on the bulletin board.
Later in the unit, students were challenged to construct eight-pointed stars using geometric constructions without a template (although a template could be used for students that needed additional supports). Creating this design using a straight edge and compass required students to perform the majority of the constructions required by this outcome. Students also had the opportunity to use their creativity to personalize their design and make it unique.
A number of students were very interested in creating designs of increasing complexity. They were able to pursue this to apply their geometric skills to create some very impressive designs. Have you used any creative or artistic activities to teach students geometric constructions? If so, I'd appreciate hearing about it!
A few colleagues and I recently initiated a bit of self directed professional development. We decided to watch a selection of math ignite talks as a focus for discussion regarding mathematics instruction and educational practices. In preparation for this, I reached out the the #MTBoS on Twitter to ask for suggestions. I was pleased to get some valuable feedback.
We took the feedback we got from our Personalized Learning Network (PLN) as well as sharing our favourites with each other to create a playlist of math ignite talks. A big thanks goes out to The Math Forum at NCTM for having such a well organized and easy to access YouTube channel containing math ignite talks from numerous conferences. What are your favourites?
Has watching these videos inspired you to create your own mathematics ignite talk? If it has, and your looking for advice, look no father than Robert Kaplinsky's web page. He has a post titled, "How I Prepare for an Ignite Talk" that will provide you will lots of tips and ideas.
Update: I've added a part 2 playlist of six additional math ignite talks here.