I want to share a question with you that has been clanking around my brain for a while. At first I wasn’t even sure if I wanted or was even “allowed” to share and write about this question, but it’s one I keep coming back to. It’s a question that makes me feel somewhat vulnerable. And it’s one I’d love to hear your thoughts about, so please feel free to leave a comment even if, like me, your ideas around this question aren’t fully developed either.

Is there room for math that isn’t hard?

One reason I wasn’t sure I wanted to write about this topic is because I really believe in teaching students to work hard to figure out mathematical ideas. I believe that math can be challenging and also enjoyable. I teach teachers and students that doing challenging work is what helps us construct new understanding. I really believe in the power of the “make sense of problems and persevere in solving them” (CCSS.Math.Practice.MP1) I believe that much of math learning will occur in this way. And yet, I keep coming back to this question. Should all math be rigorous? Should we persevere through all tasks? Or is there a room for something else too?

When imagining strong math communities, I like to consider analogies to literacy communities. In reading workshops, teachers spend time a good amount of time engaging children in read-alouds in which a primary purpose is learning to love books and enjoy reading. As a kindergarten teacher I often read Mo Willems’ Piggie and Elephant books not because they were rigorous, but because it’s fun to read the speech bubbles using the voices we imagine Piggie and Elephant would use. We lingered to laugh for a few moments on the page of David Shannon’s *No David* that shows David’s naked bottom as he runs down the street naked. In reading workshops book browsing is encouraged. Children squirrel away Lego encyclopedias in their book boxes to pore over with friends. They linger over photographs of animals and come up with their own wonderings like, “Why are reptiles bumpy?” “Why do pigs have curly tails?”

In reading workshop, children should not spend the majority of their independent reading time engaged in books that are difficult for them. Richard Allington’s research indicates that children should be spending the great majority of their reading time engaged in books at their independent level, books they understand and can read accurately. Children need to read and re-read. (This is one quick Allington article, but his work on the topic is much more extensive than this.)

So, I’m wondering, is there a math equivalent of this? If children subsist on a diet that consists only of difficult math, will they learn to enjoy it? Will they learn to pursue it beyond the walls of the classroom? I say that with the huge caveat that I believe challenging and difficult math can be engaging and fun. But I also believe there is space for math that is not terribly difficult, but is very enjoyable.

Strategy games (like Rush Hour, for example) can be extremely challenging. While playing Railroad Rush Hour recently, 5th grader Kevin told me, “I’m only on the first challenge. And it’s already hard. I can’t do it.” He stuck with it though, and figured it out. Kevin moved on to other Rush Hour challenge cards, but a few days later I noticed he was working on that first card again.

“Kevin, didn’t you do that one the other day”” I asked.

“Yes,” replied Kevin matter-of-factly. “But I like to go back to ones I’ve already done and think about it again. It’s fun.”

Was what Kevin was doing the problem solving equivalent of re-reading a favorite book?

This weekend I spent some time at the grocery store with my daughter watching an employee make a giant batch of guacamole. I watched with fascination. “I wonder how many avocados are in that bowl. How many containers of guacamole will that make?” I considered a few ideas in my mind. How much space would ten avocados take up? So, how many groups of ten avocados? I engaged lightly and playfully with the idea. The avocados offered an invitation, rather than a specific task to complete.

I held my almost-two-year-old up to see the giant batch of guacamole. “How many avocados are in there, Lulu?” (She loves avocados!)

“One and one and one and one!” (That’s what her counting sounds like now.) “Dos!” (That means any quantity more than one in her world.”)

It wasn’t hard. For me or for Lulu. But it was kind of fun to think about. I love looking for math in my life. I hope to inspire that in her and the teachers and students with whom I work. I hope to offer invitations to engage in mathematics. Sometimes in difficult mathematics and sometimes in something less serious.

So, is there space for this in our classrooms? Time for engaging, but not terribly difficult math? What other opportunities do children have to just wonder about math? To browse math? To engage in math play? To relax into math, the way one relaxes into a book. Is this important? Is this a thing in math? Is it something professional mathematicians do? Is it something everyone does as they wonder about their world?

I’d love to hear your thoughts.

~Kassia

Your blog post reminds me of the time we were ‘doing’ math during dinner – http://infinut.com/2014/03/09/anyone-can-do-math-2/ It was fascinating to my older daughter that the younger one could do division of strawberries equally, even though it wasn’t called that.

To engage in that kind of math play, kids need to relate math to the real world. Most of math taught in school is not that (not suggesting that’s what you do, its just what the curriculum is) – which is why the kids often find it boring. But, I built them games that explain math with something real. As a result, they have a deep interest in math – and they are solving problems that they haven’t even been taught. They make up the hard problems to challenge themselves. It’s not ‘hard’, its fun.

I agree that much of the math that students encounter in school is not playful or creative. We need to find more ways of bringing the kind of math talk some families do during dinner into the classroom space too, I think.

I love getting a post from you. Such good food for thought. Yesterday in our kindergarten the math was toooooo hard and i played the rigor card in my mind. Even though many kids did persevere it wasn’t that interesting, just hard. Today we played the build a city game/towers game from that old Kathy Richardson book and right away they were asking how many if they put both partners towers together and then did they have more than someone else and how many cubes were in the bag and all the bags and my favorite, “How many cubes do we have in our room?” Thank goodness my principal walked in today rather than yesterday. I want them to work hard and have productive struggle and I want them to love numbers and problems and ideas. I look to you for that stimulation. Thanks.

What a great question! I think it’s left me with more questions than answers for you: You use the word rigorous as a synonym for hard. What’s the difference? Your avocado example sounds like using “easy” math to think about a new problem, at least one you’ve never encountered before. Would that be like what I’ve heard called estimation problems (how many piano tuners work in the Chicago area?)? The math there is usually “just” arithmetic, but the assumptions used are where it gets cool. In your example there’s arithmetic and geometry. Would you only consider it easy if both of those came easily to students?

On another note, my son re-reads books all the time. He tells me that he loves the stories, so he loves going back to it. Now that he’s in high school it really doesn’t bother me (anything other than watching vids or arguing with me is considered a win) but when he was a beginning reader it did bother me. I’m glad I didn’t complain loudly enough to turn him off from reading!

It’s really interesting to think about the kinds of “pleasure” we–and kids–get from reading and the kind of “pleasure” we get from solving a problem. I wonder how the brain experiences relaxing while reading, vs relaxing while doing a math (or playing a game, or solving a puzzle). Same, yes, but different! Thanks for such a provocative post, Kassia!

This is probably not true of everyone but sometimes I like just doing arithmetic. In the middle of a hard problem where I could totally get out a calculator, or when I see some numbers near each other in the world, or when I’m bored and don’t have anything to read… sometimes I like doing mental math and sometimes I like just chugging through the standard algorithm. But it does remind me that I get pleasure out of rehearsing math that I’ve mastered, and using math that I’ve mastered to explore domains that are really interesting to me. Sort of like when reading becomes reading-to-learn and you might read an “easy” book because it’s about what you want to know about. Sometimes it’s fun to solve a math problem where you only use addition and counting, but the context and question engages you (see, for example, your guacamole experience!) Doing math with big numbers (last time I was in a 1st grade class and told them I liked to do math, they spent 10 minutes asking questions like, “What’s one thousand plus one million?”), and how many ___ are there all together, and how many avocados would it take to make guacamole for a giant, and how much will I have to grow to be as big as ____… where the math isn’t hard but is instead serving a kid’s sense of fun should totally get to be part of math time — and it makes me sad to realize I’ve never thought about that before!

What an interesting idea you bring up. I teach high school and too often my students have “learned” to dislike math. This is probably because they were more or less forced to do it on some level. Is there room for the type of math you discuss where students simply fall into it instead of being pushed and guided? That’s a great question you pose. I see more and more teachers going to this route by encouraging their students to “play” with math before engaging in a task, which a step in that direction, but still doesn’t really touch what you’re mentioning here. Thanks for an intriguing post.

Good write up. I think its an awesome point that you bring up. Allowing students to “fall” into math is a concept that many teachers strive for, but cannot always achieve. As a high school teacher, all too often my students have already learned how to dislike math over the years. I’m always fighting an uphill battle. I believe they have been forced, more or less, into learning mathematical concepts and have not had the ability to explore math for what it is. I’ve heard teachers discuss the idea of encouraging “play” in math (whatever the level) to help spur curiosity, and your post definitely reminds me of this. Thanks for an intriguing read!

It’s surprising that mathematics education doesn’t talk about this more. Young people need to practice easy math to develop the automaticity they need to keep basic knowledge in their working memory so that they can develop more complex concepts of numeracy as they progress. I teach high school math to students with learning disabilities in the area of mathematics. For them, math was always so hard in elementary school they never had this “play time” to develop automaticity and so the gap kept getting larger. I teach a number of 17 and 18 year old students who have found ways to cope enough to barely pass through “rigorous” Algebra classes but don’t truly “get” concepts as basic as multiplication as repeated addition. I wish we had time in our curriculum to slow down and play with ideas more.

One of the key considerations in the Reading Recovery program, before giving a student a task is “What makes it easy?” If there isn’t something that makes it easy, you shouldn’t do it. It’s pretty easy to teach students that they are failures by throwing incessant complicated and difficult tasks at them, with no chance to cement the nascent skills we are attempting to call forth.

Great post Kassia! I think this is such an interesting thing to consider and as I was reading the post, I was reflecting a lot on the Choice Time games we do in Investigations. I think that they allow student choice, can vary in difficulty through modifications, and are really just fun, however they do seem to be missing this “enjoyability” factor. Students enjoy them in the moment, but I don’t see many students at home choosing to play one of those games, as they do with reading books simply for pleasure. Like you mentioned in the avocado talk or Max mentioned above in the comments, math is something you are interested in or curious about and this curiosity can vary from “real world” situations to simply just playing around with arithmetic and numbers. I definitely think this is something we all need to think about making more time for in our daily routine. New Years Resolutions perhaps? I think I want to subtitle your post “…and how do we instill pure math enjoyment in our students?”

-Kristin

I think it’s a really good idea to compare with other subjects, I think of reading too, with my son. There were certain “quantum leaps” in his reading. Reading The Hobbit, for instance. And the Harry Potter books. But interestingly these leaps began with making it easy. I read the first few chapters. Then it time to sleep. Sam really wanted more, but my voice was tired. “Look, if you want to read a bit more before you sleep, you’ll have to do it on your own. But not for too long…”

So easiness can lead to hardness…

I was thinking about doing a series of blog posts on “how to make maths easy”. I’ve written one about how to make it hard:

http://followinglearning.blogspot.fr/search/label/hard%20learning

so I feel I’ve earned the right to write heresy now.

Yes, there should be a lot of room for math that isn’t hard. Several reasons:

(1) it is enjoyable

(2) it allows you/the student to groove the patterns and check to see that you really understood the idea. For example, when Kevin was playing with the first puzzle again, he was probably exploring how far ahead he could see the answer or how the different moves linked together. Not consciously/explicitly, perhaps.

(3) it allows you the opportunity to uncover hidden depths/connections. The 1+1 = 2 story is probably the most famous.

(4) it is easier to learn without a conscious strain.

Also, the definition of “hard/not hard” is very fuzzy. Some examples:

– Playing NIM vs finding a perfect strategy

– Folding hexominoes into cubes vs classifying all hexominoes (a recent activity for us)

– reading a proof for the main ideas and not reproducing every step or filling important gaps (which I recently did with this paper)

To borrow your reading comparison, these are like reading for pleasure rather than practicing literary criticism. In fact, there can be (at least) three levels:

– pure reading (equivalent to open math play)

– speculation (“maybe this book is actually about …” and “I wonder if this pattern continues like …”)

– focused analysis

I love this post!! Thank you for putting such an honest question on the table. I think you are completely right about the 5th grade boy replaying the first task… it IS similar to rereading a well-liked book. How fabulous to see that in math.

I’m an elementary math coach in Wisconsin, so like you, I am always toying with ways to make math more engaging, fun, and meaningful for students. Some of what you eluded to in your post about allowing natural curiosity to guide a lesson or math task reminds of this Ted Talk, which you may have already seen: http://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en

I think the more students are able to see how math relates to ideas, concepts, products, and procedures that they are genuinely interested in or intrigued by, the more they will want to continue to pursue math throughout their educational journey. Do all of those rich mathematical tasks need to be exceedingly difficult? That is hard to answer. I guess that those types of activities require a more in-depth thought process and trail of operations… but in the end the result is more satisfying and meaningful. I think it is satisfying and meaningful even for students who may have been a little lost during the journey. As long as their is an organized closure to the lesson, filled with great reasoning and discussion around strategies and efficient ways to arrive at the final answer, then maybe it doesn’t matter who was initially right or wrong, lost or leading the pack…

I also agree with the responses above! So much can be gained from revisiting “easy” math and taking a second look at what else you could “wonder” about it. Good stuff.

Anyway, very fun post to read! Thanks again 🙂

Feel free to check out my blog at http://www.rfsdmath.blogspot.com

On twitter @MrsBingenheimer

Play (the best way to learn) can be hard (like mountain climbing!) or easy (like KenKen). Sometimes we need the easy play.