Asking students to solve problems mentally, without paper and pencil, is always revealing and often surprising. I thought that asking students to solve 100 ÷ 3 would be sort of a slam dunk. My, was I wrong!

Tic-tac-toe is a game that has some advantages―it’s easy to learn, requires only paper and a pencil, and doesn’t take long to play. But the game has the disadvantage of getting boring pretty fast. Don’t give up on it. Try these variations, all of which give kids (even adults) a chance to think strategically in new ways.

I just learned about Factors and Multiples, a shelter-at-home game that’s engaging as solitaire and can be played as a two-person game either cooperatively or competitively. (I’ve played it both ways.) It’s intriguing for both adults and kids (as long as players know about factors and multiples of numbers up to 100). It’s a keeper.

Looking for an easy-to-play game that requires only the ability to count to 20, but has a real mathematical kick? Here it is. Teach it to your kids at home or to your students online to play with someone at home. Read on for the rules and some tips, including how to tweak the game to keep kids interested and challenged.

Riddles are usually a hit with kids, and with many at home and sheltering in place (as I am), diversions can be helpful. When rummaging through my book shelves, I found a book that I wrote in 1981―The Hink Pink Book. I wrote it shortly after I first learned about Hink Pink riddles, and also about Hinky Pinky and Hinkety Pinkety riddles. I think these riddles are good for some language play for kids at home, with a little math thrown in.

Good Questions for Math Teaching is a Math Solutions book that has long been one of my favorites. It’s a resource that I dip into when I feel the need for something fresh. And it speaks directly to our current shelter-in-place coronavirus crisis as many of us look for ways to mathematically engage students online, children at home, or both. Here are samples to get you started. I’ll continue to post more ideas on Twitter (@mburnsmath).

I’m often surprised by what I learn when I interview students. Watch this 46-second video clip of Jonah solving 100 ÷ 3. Then read how I used the clip in a lesson with a class of fifth graders, and also read the letters the students wrote to Jonah.

I’m a huge fan of math games, especially when they involve both strategic thinking and luck. And I’m always on the search for games that work with a span of grade levels. The Two-Dice Sums Game fits both. Learn about the game and read the letters of advice that 7th graders wrote to 2nd graders.

What am I doing on the floor? Teaching angles to fourth graders. Read about how instruction using Pattern Blocks and hinged mirrors, along with supporting number talks, can help students learn to understand and measure angles. Here I present a (sort of) photo essay to describe what actually occurred over the first three days of instruction. Ideas for continuing the instruction follow.

Fourth graders solve the problem 5 ÷ 4 in the context of sharing cookies, figuring out how to share five cookies equally with four people. The students came up with six different solutions―all of them correct! (Try and think of what they might be before continuing to read.)

Last year, I agreed to meet with a friend’s sixth-grade son. Oscar’s math teacher had raised an alarm for my friend and her husband about Oscar’s math progress. They were shocked. Oscar did his homework and was proficient with paper-and-pencil math. What was the problem?

Have you ever asked students to solve 12.6 x 10, and they respond that the answer is 12.60? I have, many times. Students who do this apply a pattern that works when they multiply whole numbers by 10—they tack on a zero to the end of the number they’re multiplying. But then they apply the same pattern when working with decimals. What can we do?

I thought I was on the right teaching track using real-world contexts to talk about fractions with a class of fourth and fifth graders. Then a surprise occurred! I’m still mulling over what I could have done. I’d love your thoughts.

When teaching students to add decimals, I wind up reminding students to “line up the decimal points.” This makes sense to some students while others follow the rule without understanding. How can we teach adding decimals to develop understanding and skill? Here’s a possible suggestion: Give the correct answer up front.

I believe strongly that mistakes are learning opportunities. At least that’s what I regularly tell students. But it sometimes feels different when the mistakes are mine . . . and especially when they are pedagogical mistakes that I make while teaching. That happened to me recently when teaching a lesson to fourth graders.

I love Twitter. On November 3, 2017, I saw this image in a Tweet posted by @MarkChubb3. The image stuck with me for several days. After talking about it over dinner with a teacher friend, and then again over lunch with another, I became curious to find out what students might think. I made arrangements to “borrow” three fifth-grade classes and made plans to teach the same lesson in each class.

A friend and I were talking recently about how much work we put into planning lessons. Even after all these years of teaching, I have to think through lessons as carefully as possible, both about the logistics and about the mathematical thinking I want to keep in mind and support. Here’s an example.

The card game Oh No! 99! is a keeper! It gives practice with mentally adding one- and two-digit numbers and with adding and subtracting 10 from two-digit numbers. The game encourages strategic thinking as students decide which cards to play and which to keep, and it’s also useful as an informal assessment. Read about how the game was used with second and fifth graders.

This blog post resulted from an email exchange I had with Jill Downing, a Title 1 Educator with the Helena Public Schools in Montana. My recent blog about using the children’s book 17 Kings and 42 Elephants included a link to an article I wrote, “Using Math Menus.” Jill read the article and was interested in more information. Her questions pushed me to reflect on some of the nuts and bolts I use when organizing math menus. Here I share what Jill wrote and how I responded.

Over a year ago, I blogged about The 1–10 Card Investigation. I didn't provide a solution to the problem and no one who commented asked for one. But a newly posted comment requested the solution. That pushed me into a conversation with myself about how I should respond, and about giving answers in general.

The children's book 17 Kings and 42 Elephants by Margaret Mahy is one of my long-time favorites. In this post I describe a division lesson that I’ve taught to third graders but recently revisited with fourth- and fifth-grade classes. With the older students, we tried extensions that differentiated the experience and put students in charge of deciding on problems for themselves. It was exciting to me to expand a lesson I've taught many times into a multi-day investigation.

Will Multiplication Bingo guarantee that students learn the multiplication facts? No. But it will help familiarize them with factors and multiples, engage them in a game that involves both luck and strategy, encourage them to make conjectures, and have them use data to guide decisions. Plus, the game provides a way to send home information to families about how their children are being asked to think and reason in math class.

In a recent email, a teacher friend commented on what she described as a struggle in her school district about math teaching. She wrote: "There is a bit of tug of war going on between student-centered teaching and traditional teaching." This isn’t the first time I’ve heard a comment like this. Here’s the response I wrote to her.

Lessons using beans and scoops have long been part of my teaching repertoire. I’ve used beans, scoops, and jars to engage students in all grade levels with a variety of mathematical ideas. In this post, I write about how I recently taught a lesson to give students experience with estimation, averages, multiplication, and more. Read about how I planned the lesson, how it unfolded, and suggestions for extensions and other lessons.

Have you ever thought about this numerical sequence—0, 1, 2, 3, 4, 5, 7, 8, 10, 12? What does the sequence have to do with unicycles, bicycles, and tricycles? And what's my mathematical and pedagogical quandary? Read more and find out.

When should a teacher resolve a question for students and when is it OK, or even a better instructional decision, to let confusion ride? I recently was confronted with this situation with fourth graders. Read about what happened and what I did.

Here's one of my favorite oldies. (No, not the photo—the border problem.) I was so pleased to see this math investigation included in Jo Boaler’s latest paper. This blog post presents a detailed lesson plan for using the border problem with students and also includes a five-minute video clip to give you a sense of how the instruction went with one class.

Teachers have always told me that paper-and-pencil subtraction when problems call for regrouping is hard to teach and hard for students to learn. Much harder than addition. So why subtract when you can always add? That’s what my friend Nicholas thought, and he taught me how.

I was recently planning to teach my friend Ruth Cossey’s elementary math methods class at Mills College in Oakland, California. Digging through my collection of student work, I found a paper from a third grader I had interviewed. When doing interviews, I typically ask students to figure out answers in their heads, but I agreed when Nomar asked for paper and pencil for some of the problems.

The fourth graders I’m working with on a regular basis are learning about fractions. During a class conversation, one student declared, “Fractions aren’t numbers.” Most of the others in the class agreed. I tried to help with the misunderstanding by teaching a lesson about placing fractions on a number line.

I like the multiplication game of Pathways. It engages students’ interest, helps develop their familiarity with the times table, and encourages them to think strategically. It's been a part of my teaching for a long time. Recently I came up with a way to introduce the game that made it easier for students to learn to play. It was a huge success. Read about what I did and how the students reacted.

Place value is one of the most important foundational concepts about our number system. Watch my assessment interviews of second graders and learn how you can find out what your students understand about place value.

At a math workshop, the presenter suggested that students have opportunities to be producers as well as consumers of their learning in the classroom. I put this advice into action with fifth graders, using the activity of Fix It to provide students additional experience with comparing and ordering fractions.

In a previous blog, I described a lesson I taught to second graders using the wonderful children’s book One Is a Snail, Ten Is a Crab. At John Muir Elementary School in San Francisco, I observed two other lessons using the same book, one in Kindergarten and the other in fourth grade. The lessons were a joy to observe, and I feel that my own teaching repertoire has now been enhanced.

Students begin learning about the equal sign in the early grades, and Quack and Count by Keith Baker is a terrific children’s book for helping with this in kindergarten and grade 1. It’s one of my favorite children’s books for teaching math. (Yes, yes, I know I have lots of favorites.) Here I describe the lesson I taught and what occurred.

Are you interested in a lesson that combines a wonderful children’s book with activities that engage students with organizing data and reasoning numerically? Read about how lessons using Chrysanthemum unfolded in two classes.

Read how 7th graders collected and analyzed data about the frequency of letters. They chose sentences, recorded the frequency of letters, and put their data on a class chart. Then we compared the class results to the actual frequencies of letters. Engaging the students in collecting their own data gave them an authentic math experience, not rigged by me in any way.

I’ve taught students in grade 2 through middle school how to solve KenKen puzzles. If you’ve never solved KenKen puzzles yourself, or haven't engaged your students with them, read about how I’ve introduced them in the classroom. But be warned: KenKen puzzles can be addictive.

A comment posted on my previous blog was a Yikes! experience for me. The comment was about one of the ways to make 11 that I included in the book I created for my grandson’s birthday. The comment was a wonderful reminder about how arithmetic, algebra, and geometry connect.

My grandson Jeffrey just turned 11 and I created a book to celebrate his birthday. Now I’m thinking that making books like this might be a good class project for students. Take a look.

In my early years of teaching, children’s books weren’t typically where I looked for help when planning math lessons. But that has changed. I now rely on children’s books regularly for engaging students with math. Here’s an example.

Bring an open mind is #1 on a poster of Sara Liebert’s expectations for her fourth grade class. Lynne Zolli used that expectation when introducing a math activity to the students.

Using games has long been standard in my teaching, and for several reasons. Games capture students’ interest and engages them in learning math. They’re ideal when students have extra time. And they’re effective options for the paper-and-pencil practice. Here’s one of my favorites.

Here’s an idea that I was first introduced to about ten years ago by Nicholas Branca, a math educator who contributed profoundly to my thinking about math and teaching. I’ve tried presenting it as a math-in-three-acts investigation.

In my June 2 post, I described how students solved 99 + 17. Actually I described only part of the lesson. Now, in response to a tweet, I explain how I also had students think about one of the important mathematical practices.

I began a back-to-school session for elementary teachers by asking everyone to write an opening sentence for Goldilocks and the Three Bears. The teachers were surprised by the request—the session was supposed to focus on teaching math. What was the connection?

I’ve been tweeting since September of 2014, and I’m hooked. I never would have predicted that I’d join Twitter, much less enjoy it and benefit from it professionally. In this post, I describe my initiation into Twitter and what I’ve learned.

The 1-10 Card Investigation has a big payoff with students. It engages their interest, involves them with making sense of a problem and persevering to solve it, and gives them experience with evaluating their progress and changing course as necessary. Plus it has a playful aspect that too often is lost in math class.

I asked a class of fourth graders to figure out the answer to 99 + 17 in their heads. In this post, I describe why I chose that problem, include a video of how the lesson unfolded, describe a teaching error I made in a subsequent lesson, and more.

A long-standing instructional practice has been to teach students how to multiply (or add, subtract, or divide) and then, after the students have learned to compute, give them word problems to solve. In this post I present a lesson with a different approach, where word problems become the lead and reason for learning to compute.

Students’ ideas often amaze me, and Lydia’s is one of the most suprising examples. She used 7 x 3 = 21 to figure out that 8 x 4 = 32. She reasoned that since the factors in 7 x 3 were each 1 less than the factors in 8 x 4, she’d just increase each digit in the answer, changing 21 to 32. She was correct! Read about Lydia's discovery, what I did, and what I learned.

After I told Steven, the man seated next to me on an airplane, that I was a math teacher, he described the Dealing in Horses problem that he was given at a corporate management training session. The problem has been one of my teaching staples ever since.

When a workshop participant raised his hand to offer how he was taught to subtract when he was in elementary school, I was surprised by the process he described. He called it the “trouble-coming method” of subtraction. I hadn’t ever seen it before, and I’ve never seen it since.

Here’s another post about subtraction. I’ve long been a fan of having students invent and use their own methods for computing, as long as they can explain why their strategies make sense. Here’s what I learned from Jesús, a fifth grader.

A word problem on a third-grade standardized math test didn’t call for a numerical answer, but instead asked students to decide if the problem should be solved by adding, subtracting, multiplying, or dividing. One third grader complained to his teacher, frustrated because he thought there was more than one correct possibility.

Word problems have long been difficult and frustrating for students to solve and for teachers to teach. A colleague recently forwarded an email from a woman looking for resources to help her fourth-grade granddaughter with word problems. I thought for several days about how to offer positive support to both the grandmother and her granddaughter.

This post is about subtraction, which is typically difficult for students to learn and for teachers to teach. Think about 503 – 398, for example. To estimate the answer, I can change the problem to 500 – 400 (rounding 503 to 500 and 398 to 400). That gives me an estimate of 100, which I know is close. But how can I know if the actual answer to 503 – 398 is greater or less than 100? I raised this question with third graders.

Several months ago I received an email message from my friend Sandra. She wrote, "If you want something new to distract you, try playing the new game 2048. I’m finding it addicting." I took Sandra’s advice and downloaded the free app. And, like Sandra, I found it addicting. But it also led me to think more about what I think is important when we teach math.

My 15-year-old granddaughter Charlotte is a diligent math student. When I checked recently to see how she was doing with her Algebra 2 math homework, I found her sitting on her bed with her computer, iPad, and phone, managing to text friends on and off as she worked. She asked me for help.

Reading may seem like an odd subject for my math blog, but here I describe how my love of reading and math connected (and my confusion as an emerging reader about hearing voices). This post was included in Open a World of Possible, an anthology from more than 100 contributors that you can access as a free e-book.

My friend Ann sent me an email about her unsettling experience at the supermarket deli counter. Ann has never felt particularly confident with her math ability, and I was pleased (and amused) that she asserted herself in this situation. Also, Ann’s comment to me about the work we face as math teachers rang true.

How much is 12.6 × 10? This is a question from the Math Reasoning Inventory (MRI) decimal assessment. What do you think were the most common incorrect answers given by the more than 7,800 students who figured out the answer in their heads? And what about the boy who answered, “One hundred twenty and thirty-fifths?”

When I entered college, I knew I would become a teacher. I’d always liked school and often played school at home with my sister and my friends. I enjoyed math, so I decided to major in mathematics and become a math teacher. But while math made sense to me in my earlier studies, my college experiences were very different. I struggled.