George Lucas Educational Foundation

4 Ways to Have Your Best Year Ever in Math

4 Ways to Have Your Best Year Ever in Math

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At the end of last year, I met with a group of K-5 teachers to talk about a math training they had just completed. As we wrapped up a discussion about new strategies they had learned and put to work in their classrooms, I asked if they had any regrets. The most common answer was, “I just wish I had known all of this on the first day of school!”

With that in mind, I want to share four “big ideas” that came out of the training.  Enjoy!

A Little Confusion Goes a Long Way

The concept of “productive struggle” has gained a lot of momentum in education in recent years. The basic idea is that students learn most when they are engaged, and one of the best ways to engage them is to present them with a problem that they might be able to solve, then let them struggle with it for a few unscaffolded minutes. Sounds scary? It is! But relinquishing control for a few minutes each day and letting the students wrestle with a challenge offers a huge payoff in the end: (1) your students learn what it feels like to sweat, to stick it out, and to overcome a challenge; (2) when it comes time to solve, they are more invested in the solution; (3) the strategies they come up with activate prior knowledge and set the foundation for new learning — and tip you off about how they think and what they know and don’t know.  

Try this: Select a story problem your students might be able to solve. With only minimal instructions to get them started, set a timer for 5-10 minutes while they grapple with the problem. See where the productive struggle leads!  (Note: Teachers have warned that it can take 5-6 attempts to find the right challenge level and get the timing down — but it’s worth it when you hit your stride!)

“So you’re saying…  Is that right?”

“Academic conversation” has become a popular buzzword in education circles. It’s well-documented (and self-evident in any elementary classroom) that students learn a lot by hearing their classmates’ ideas and discussing amongst themselves. But a student can also learn a lot just from hearing his or her own words restated. You don’t have to bring a handheld recording device to every discussion — it’s enough to parrot their words right back to them and ask what they think!

Try this: When a child answers a “how” or “why” question, repeat the answer back to them without editing or cleaning it up.  “So you’re saying… Is that right?”  You’ll know you’re doing it well when a student politely disagrees… with herself!

Conjecture Boards

Math is all about identifying patterns. It’s critical for students recognize patterns, see connections, and make generalizations about their learning. Conjectures are simple, unproven observations about shapes and numbers (eg: “all triangles have three sides” or “the product of a whole number and a proper fraction is less than the whole number”). They can be put to excellent use in the elementary math classroom.

Try this: Every time a student makes a conjecture, CELEBRATE! Restate the conjecture, write it down, name it after the student (“Donovan’s Conjecture”), and post it in a visible location, so you can return to it in the future. You will be amazed how often students refer back to their classmates’ ideas — and how quickly they internalize the importance of seeing the big picture!

Give Students Hand Signals to Show Their Thinking

Many teachers use hand signals (some adapted directly from American Sign Language) to help students express themselves without interrupting the flow of a class. You may already use signals for going to the bathroom or agreeing with another student. Hand signals can also be used in academic discussions. Without a word, students can show that they want to paraphrase or build on an idea, make a conjecture, share a “point of interest” (confusion about another student’s idea), and even encourage classmates to “keep going” when they’re explaining their thinking. Not only do the signals provide new entry points into conversation for all students (including those who are shy or not proficient in English), but it gives the teacher insight into what their students are thinking throughout a discussion.

Try this: Introduce academic hand signals one at a time. Encourage students to incorporate them into math discourse.  Hand signals will quickly become a part of daily classroom practice!

This post was created by a member of Edutopia's community. If you have your own #eduawesome tips, strategies, and ideas for improving education, share them with us.

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Vi Sampaio's picture
Vi Sampaio
English Teacher in Brazil. In a constant learning process.

This is a quite interesting article. I've been working with Math Challenges in Middle Years groups, but never tried them in my 4th Grade classroom. Definitely giving it a try, even though the school year is almost over here in Brazil.

One more thing: what kinds of hand signals do you suggest? Is there a list of most common/effective ones you could share with us?


Ellie Cowen's picture
Ellie Cowen
Educator and math specialist

Hi Vi,

I'm glad you found the article interesting! I've been lucky to work with some wonderfully creative educators on these ideas.

The hand signals I recommend most to get kids thinking and reasoning are "agree" (we use the ASL sign for "me too") and "disagree" or "point of interest" (we hold up the index finger, which is also the ASL sign for the letter D).

Check out for more ideas. I don't have a list ready to go, but if I find a good one I'll share it! And of course... your students may come up with some original signs if you put the same question to them. :)

Good luck with the end of your year! Best,

Monica Phillips's picture

I find that students really do thrive when given problems that allow for productive struggle. I have found that if my students can't solve the problem in the amount of time I have given them, we put the problem to the "side" for the time being and revisit it later--it just depends on the connection that particular problem might have to what I am currently covering in my curriculum.

One you have a particular website where you can retrieve rich problems for middle school?

I teach 6th grade and am always looking for great websites to provide my students (and myself) with engaging real world problems.

Ellie Cowen's picture
Ellie Cowen
Educator and math specialist

Hi Monica,

Great question: start with There are a variety of challenging questions there for all grade levels. I also recommend "Three Act Math," Dan Meyer's strategy for teaching with problems, which can be found at

Thanks so much for sharing your experience and tips!

JPowell's picture

This is a really an interested article. I tried the "productive struggle activity" this morning and my students were so engrossed that they begged for more time to solve the problem. It was amazing how they responded as they are normally students who gets frustrated easily.

See Xiong's picture

I really enjoy reading your article. I also do the same thing with my students. When introducing a new math concept, I start with the easy problems; and then once students grasp the concept, I move on to the difficult ones. Very interesting article :)

Dolly C's picture

Your math strategies are amazing.
The math game that would like to share is. For example, there are 18 kids in Kindergarten-Prep; they are age of 4 - 5 years old. Here is the game: first, everyone should know there are five fingers in each hand, second, show them my five fingers on my left hand, (every one said, "5"), and another five-fingers on my right hand, (every one said, "5"). Now I showed them ten fingers at the same time, (everyone said, "10".) To make sure that everyone knows this game and have played for several days. Third, show them "5 fingers" on the left and "1 finger" on the right. (Everyone said, "6".), then let them count from the group of higher number to the lower number, so here they counted, "5 fingers first, 1-2-3-4-5, then continue to another one finger - 6.) (Note: they will learn 5+5+10, 5+4= 9, .....5+0=5, and which number is greater and lesser, fewer, or smaller. On the same lesson, some kids will learn counting numbers, while some kids will learn writing number sentences.)

Ellie Cowen's picture
Ellie Cowen
Educator and math specialist

Little makes me happier than hearing that students are clamoring for challenge and growth. Thank you for sharing. :)

MsHayleyPate's picture

I'm going to school to be a teacher, so I find this article very helpful!
I struggled all throughout school with math, it was my worst subject. Because of that I have a fear of teaching lessons on it! I want to make sure I'm a good math teacher who can set the foundation for kids!
I like the idea of kids using hand signals to show what they are thinking. I first heard of and saw something like this last year, while I was shadowing a fifth grade math teacher. She had students put their heads down after the lesson and working out some practice problems. Then the teacher told them to put a thumbs up if they knew how to do the math problems, a thumbs in the middle to show that they are starting to understand, but need more practice, and a thumbs down if they are lost and need help.

Ellie Cowen's picture
Ellie Cowen
Educator and math specialist

Hi Hayley,
Sometimes I think those teachers who struggled in math at elementary school are the greatest gifts to students who struggle today. You probably understand better than most the need to give your students time and plenty of opportunities to make sense of everything that they're learning!
Thanks for sharing.

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