In this monthly column, we speak with a notable member of the mathematics education community about their work and their perspectives on the teaching and learning of mathematics. This month, we had the pleasure of speaking with Dr. Christopher Danielson.
Christopher Danielson has worked with math learners of all ages—12-year-olds in his former middle school classroom, Calculus students, teachers, and young children and their families at Math On-A-Stick at the Minnesota State Fair. He designs curriculum at Desmos. He is the author of Common Core Math For Parents For Dummies, the shapes book Which One Doesn’t Belong?, and the forthcoming counting book How Many? He blogs about teaching on Overthinking My Teaching, and for parents at Talking Math with Your Kids.
First things first, thank you for taking the time for this conversation!
Besides teaching mathematics and curriculum development (at Normandale Community College and, most recently, at Desmos), one of your main interests is helping parents support their children’s mathematical development, as the title of your website Talking Math with Your Kids suggests.
As you write on the website, parents know that they should read with their children every day to support their literacy development, but are typically less familiar with strategies to cultivate numeracy. Why might this be the case?
I feel I should make clear that my responses here are based on my knowledge of culture and schooling in the United States. I expect that much of this also applies in Canada, but I really don’t have extensive experience to make those kinds of claims.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]We understand that people read for pleasure, but we don’t really understand that people figure things out for pleasure.[/perfectpullquote]Math is tied up with school in our culture. We understand that people read for pleasure, but we don’t really understand that people figure things out for pleasure. Probably our education system—and especially our mathematics education system—has a lot to do with that, but whatever the origins, this perception has consequences. If reading is pleasurable, then it’s natural to curl up together on the couch with a book. If math isn’t pleasurable, then we don’t go looking for more of it in our daily lives.We have clear and simple messages about supporting literacy—read out loud with children from a very young age, and surround them with books and words. But we haven’t helped parents make the same kinds of connections to math. Building with blocks, doing puzzles, putting away dishes, sorting socks…these are all examples of activities that support children’s math learning but we haven’t pitched math this way.
So parents only know to look for things that look like school math—flash cards and their electronic equivalents.
The main strategy you offer to support children’s budding numeracy skills is conversation—in particular, you recommend that parents talk (often) about math with their children “as they encounter numbers and shapes in [their] everyday lives” (Danielson, n.d.). On your website, you offer a wide variety of examples of math talks with your own children taken from everyday life, including conversations about weighing onions (2015, May 22), bedtime (2014, January 20), underpants and armholes (2013, August 17), and many more.
How do you recognize authentic opportunities for math talks, and how might parents with less experience with mathematics (i.e., without a PhD in mathematics education) develop this skill?
Yeah. That’s one of the big questions at the heart of my work right now. Bedtime Math is a project that has made some headway with parents—especially math-anxious parents. They’ve teamed up with the University of Chicago to research the effect of non-school-based math interventions and found that introducing even small amounts of math talk into math-anxious households has meaningful positive effects on kids’ achievement in elementary school math. The Bedtime Math approach to introducing this math talk is an app with simple story problems about some silly or interesting news item. Parents can open the app at bedtime (or anytime), read that day’s problem, discuss with their child, and be done with it. It’s a simple and elegant solution.
But it’s also a limited one from my perspective. It makes progress, but only so much. One key to its success is that it looks like school math. Everybody recognizes what it is, and so it’s clear what to do with it.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]I’m greedy. I want more from kids than solving a silly story problem that someone else thought up.[/perfectpullquote]I’m greedy. I want more from kids than solving a silly story problem that someone else thought up. I want the ideas children have—that all children have—to be the starting place for conversations and wondering and learning.How do you help parents learn to do this? I’m figuring that out. At the outset I needed examples of the thing, and that’s the origin of the Talking Math with Your Kids blog. Over time, I have started to think that a more effective strategy than writing about these conversations is putting things in the hands of teachers and students that can help to create the conversations. That’s where Which One Doesn’t Belong? and Math On-A-Stick come from.
Which One Doesn’t Belong? is a shapes book that invites parents and children to have a conversation. The opening is a tutorial that makes clear everyone’s ideas are valued, and that there are many ways to think about shapes. Then I turn them loose with increasingly complex sets of shapes to consider. The book closes with an invitation to look for similarities and differences in the world, and to design your own sets.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Having the parents and the kids in the same place—while the kids are using their math brains with joy—is a really powerful tool.[/perfectpullquote]Math On-A-Stick is a large-scale family math event that runs all 12 days of the Minnesota State Fair each summer. There, I get to help parents notice the mathematically smart things their kids are doing, and to help them have a fun time together noticing, discussing, and playing with shapes, patterns, and numbers. Having the parents and the kids in the same place—while the kids are using their math brains with joy—is a really powerful tool. It’s a limited one, though. Thousands of families come through, but we can only reach them 12 days a year, and we can only reach the families with the time and resources to attend the fair. So I’m still looking, still considering opportunities. I have ideas for parent nights that are more like Math On-A-Stick than like a traditional “How to help your kid with homework” night.What advice do you have to offer to parents who are interested in supporting their children’s numeracy development, but who harbor anxiety towards the subject—perhaps due to less-than-positive experiences in their youth?
This is important work. Often, such parents will avoid math talk out of fear that they’ll harm their children. If math is about right answers, and if I’m not sure I have the right answers, then I may feel I need to steer clear of math with my kids. Plus, math anxiety is a real thing that produces a real stress response, and people tend to avoid things that make them anxious.
The University of Chicago research suggests that introducing even small amounts of math talk into such homes has a significant positive impact. I have three goals with math-anxious parents.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]You don’t harm your child when you misread a word in a book, or drop the ball she throws, or when you don’t know the answer to a math question.[/perfectpullquote]The first goal is to alleviate their fears that they’ll harm their children by not knowing right answers. You don’t harm your child when you misread a word in a book, or drop the ball she throws, or when you don’t know the answer to a math question. In all of these scenarios, participating in the activity supports children’s learning.My second goal to help them notice the things they’re already doing that support their children’s math learning. If they’re counting, or reading shapes books, or taking the kids to the grocery store, I’ll make sure they know that these things are important to continue doing.
My third goal is give them one or two strategies for increasing the opportunities for learning. The simplest of these is asking How do you know? When a child mentions numbers, ask how she knows. Then listen to the response and compare it to how you know this. You don’t need to focus on whether she is right or wrong. Instead, focus on understanding her thinking.
Presumably, as children enter their teenage years, they become less interested in conversations about weighing onions and armholes… How are the math talks you have with your son (12 years old) different from the talks you have with your daughter (9 years old)?
I have to be a lot more strategic with the 12-year-old. With him, it’s more about finding ways to exploit his motivations. He loves to argue, and he relishes being right. He’ll find mistakes in the world, or I’ll find them and ask what he thinks. I’ll make him convince me, sometimes taking up a contrary position that I don’t really believe. Also he is interested in money. He loves the independence of having his own money. This leads to opportunities for making him think. When there’s a job I’m paying him for, I’ll make it by the hour or the pound—no flat rates. Then it’s up to him to figure out what I owe him for the work, and to convince me that it’s the right amount.
The 9-year-old is still a kid and finds nearly anything fun to think about or imagine. We can still talk armholes and underpants. All the examples on the blog, where the target audience is four- to ten-year-olds, still pertain to her.
Weighing onions, by the way, is still good stuff with the older child too. He likes the competitive aspect of making the best guess. Another thing we do that involves the whole family is guessing the total bill at a restaurant. I know this can contravene certain norms of polite society, but I’m telling you it’s a good time and kids get good at it quickly.
You have often written that in engaging in mathematical conversations with their kids, parents shouldn’t necessarily worry about whether the child can get right answers (Gahan, 2013), and should not be afraid to discuss concepts with which they are unfamiliar. As you write: “DO NOT let the idea of being wrong get in the way of your math conversations. DO NOT be afraid to play around with ideas you know little about” (Danielson, 2013, August 17).
While play and math talk may cultivate budding mathematicians’ curiosity and creativity, can they—and should they—also foster precision and rigor?
Short answer: Yes!
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Precision in play and conversation comes when there’s a need for it. Often that need comes from questions that arise naturally.[/perfectpullquote]Longer answer: Precision in play and conversation comes when there’s a need for it. Often that need comes from questions that arise naturally. My daughter has been playing with stairs. She loves taking them two at a time, but is dissatisfied when there is one stair left over at the end. She has learned all of the major staircases in her life (home, school, bus, grandmother’s house, etc.) and knows whether they are even or uneven (her word). We talk about this from time to time, and for a while we reached an impasse when trying to specify the number of steps that any particular staircase has. Just the other day, we focused just on that and resolved that—for us—steps are different from stairs. When she goes eight steps (in four sets of two), she counts this as 9 stairs. I only count it as 7, because I don’t want to count the floors where you start and end as stairs. But we now have precise language, and the need for that precision came from needing to communicate clearly.But initially, the play didn’t require precision and so we didn’t have it. Precision without a need for it is pointless. I think this applies to school math too, and I think we all too often proceed from a place of precision and rigor without helping students to arrive at this place by routes that make sense to them. We lose a lot of potential mathematicians—and potential mathematics—that way.
Recently, you have also started offering mathematical playthings at your Talking Math With Your Kids online store as another way to support parents and children in math activities and conversations. These include tiling turtles, spiraling pentagons (and other tiling pentagons), pattern machines, and more [head to http://talkingmathwithkids.squarespace.com/ for details].
What do these playthings—which, on the surface, do not look particularly “mathematical”—have to do with mathematics in general, and “school” mathematics in particular?
I design things that foster kids’ play with numbers, patterns, and shapes. The turtles were designed by two mathematicians—Kevin Lee, a colleague of mine at Normandale Community College, adapted them from Jos Leys’ original art. Jos is a Belgian mathematician and artist. For Kevin and Jos, the turtles are artwork to be admired. I want kids to get their hands on the turtles. I don’t want kids to notice what someone else has made; I want kids to make their own things.
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]Children aren’t doing mathematics when they’re admiring someone else’s creation. They are doing mathematics when they are building relationships for themselves.[/perfectpullquote]Each has its place. I love to look at and admire mathematical art. It inspires me. But children aren’t doing mathematics when they’re admiring someone else’s creation. They are doing mathematics when they are building relationships for themselves. Figuring out how to tile the turtles is the entry level; they imagine what the turtles will look like when rotated or flipped. They notice that each turtle put into the tiling creates a space for another turtle, and so they begin to consider an unending process: infinity. Children begin to use the two colors of turtles to make patterns—alternating light and dark, or using the colors to highlight structures they notice in the tiling.When they play with Pattern Machines, children notice and use rows and columns. Seeing and using such groups is an essential foundation for multiplication and for place value.
But as a general rule, I don’t really talk a lot about the work I do in relation to school math. Mostly this is because I have critiques of school math. My daughter is a whiz with place value. She sees and understands groupings—especially groupings of ten—and is able to exploit them in clever and useful ways for thinking about things and solving problems. But it doesn’t show in the ways her teachers talk about her mathematics. In school, she is supposed to say that there are 3 tens in the number 435, when she knows that there are really 43 tens there. In school, she is supposed to answer the question, What is the value of the 4 in 435? by writing 400 not hundreds. In school, she is supposed to figure out what answer the teacher wants when she asks Which One Doesn’t Belong? So I really don’t want the success of my work to be judged by the ways it prepares kids for school math.
But it does support kids’ math learning in a wide range of environments, including school. My daughter may object to there being only one right answer to a Which One Doesn’t Belong? prompt on a fourth-grade assessment, but you’d better believe she knows at least one way that each shape in the prompt is different from the others.
Lastly… why mathematics? In other words, besides the potential to increase success in academic (school) mathematics, why should parents actively work to foster their children’s ability to attune themselves to the mathematics in their daily lives?
[perfectpullquote align=”right” cite=”” link=”” color=”” class=”” size=””]My work really isn’t about making my son or his sister—or any other kid—love math. The point of the work is for math to become a tool that they can use to do whatever they want.[/perfectpullquote]I was just talking with my 12-year-old son about this the other day. He gives me a hard time for being nerdy. As an early adolescent, it’s part of his job to reject things the people around him hold dear, so I can take it. But I told him that my work really isn’t about making him or his sister—or any other kid—love math. That’d be nice, but it’s not the point of the work. The point of the work is for math to become a tool that they can use to do whatever they want. I know many people for whom mathematics was an obstacle that cut off possibilities. While my son gives me grief about my love of math, I see him using proportionality and probability and patterns and shapes in the thing he loves dearly—arguing (especially about politics).My daughter and I figured out together that the turntable in our microwave takes about 20 seconds to go around. She isn’t tall enough to reach the back, so she always puts things in for 20 or 40 or 60 seconds. If she needs to do 10 seconds, she’ll push the plate as far back as she can so it comes around to the front in those 10 seconds. That’s math being used for making her life a little better, not for school math. More, it’s empowering. She will develop an expectation that a little bit of mathematical analysis can help her do other things better, too.
Finally, mathematics as a purely intellectual endeavor is one of the beautiful cultural legacies we hand down to our children. Just like fine art or classical music doesn’t need to have a practical application to be valuable, mathematics needn’t be applied in order to be part of the culture we pass down.
Thank you, Dr. Danielson, for taking the time for this conversation. We look forward to your upcoming work and to continuing the discussion in the future.
Ilona Vashchyshyn
References
Danielson, C. (n.d.). What is talking math with your kids? Retrieved from https://talkingmathwithkids.com/
Danielson, C. (2013, August 17). Armholes [Blog post]. Retrieved from https://talkingmathwithkids.com/2013/08/17/armholes/
Danielson, C. (2015, January 20). Bedtime [Blog post]. Retrieved from https://talkingmathwithkids.com/2014/01/20/bedtime/
Danielson, C. (2015, May 22). Weighing onions [Blog post]. Retrieved from https://talkingmathwithkids.com/2015/05/22/weighing-onions/
Gahan, Z. (2013, November 27). Real dad: Christopher Danielson. Minnesota Parent. Retrieved from http://www.minnesotaparent.com/real-dad-christopher-danielson