You write quite a bit about the importance of the mathematical foundation students are getting in the early grades. Do you have any thoughts on common misconceptions or common areas of weakness in our instruction that need to be addressed?
As a math coach, one of the most interesting parts of my job was the wide perspective I got as a result of working with many more students than I saw in a single class. I really got the chance to delve into why kids were struggling. And the most interesting realization that came out of this unraveling of understandings and misunderstandings was, that in almost all cases, when I really analyzed what students understood, partially understood, and did not understand at all, most of their misconceptions were very similar to each other. The good news of what I learned though is that I think that we can fix, or rather prevent, these misconceptions from occurring—which is really a lot easier to do than trying to fix them after the fact.
One common misconception that I talk a lot about in my book is the long and complicated journey to the understanding of place value. (See Chapter 7: Building Number Sense through an Understanding of Ten) I think we’re not giving kids enough time and opportunities to truly construct an understanding of place value. In some senses we talk a lot about place value—we talk about the tens place and the ones place, we build numbers with base ten blocks, we make groups and regroup). And yet, the time we give to really consider why we group, why ten is so important in our number system and really think about how numbers are composed and decomposed is really very limited.
So, here’s where the good news comes in. I think there are some very simple ways we can prevent misconceptions and build stronger place value. I think we can change some of the tools we use and the kinds of problems we give to students. In the research I did for this book, these small changes had a significant impact on children’s understanding of the number system and their construction of place value concepts.
It is clear throughout your book that you work in a wonderful, supportive, collaborative atmosphere. Do you have any advice for teachers attempting to begin a math workshop on their own, without such support in their building?
That’s a great question. Math workshop is a much newer, less well-established practice than reading and writing workshop. Here are a couple of ideas:
1) Find at least one other person who is interested in teaching math through a workshop model. It’s so important to have a partner (even if it’s someone at another school you’re emailing with) to discuss what is working and what is not.
2) Think about what you value in reading and writing workshop. How can you translate that to a math workshop? Play on your strengths. If you know how to get rich conversation going in your reading workshop, think about how you could establish this in math workshop.
3) Start small. Start simply. Math workshop does not have to be complex or a complete change from what you are already doing. Perhaps you’re just adding one small group math exchange to your structure during the time your kids are playing a game or working on a problem you've taught them from your current math program. Perhaps you’re adding a short counting routine at the beginning of your math time. Use the resources you already have and then add new components.
Creating an environment in which children can and will talk about their thinking, in mathematics and other areas, is an important piece of math workshop. Can you share what you think are the most important things for a teacher to remember in working towards this goal?
Whenever you can make math meaningful and relevant to your community of learners, you’re going to change how your students think about what math is. You help students make a shift from thinking that math is something that is static, to something that is very much alive and evolving. That’s powerful! So, when my class worked on problems about how many potatoes we could grow in a raised bed at our school, how much cat food my mom needs to buy for her six cats, or how many more stops we’d go on the metro to get into Washington, D.C., these problems mattered. It mattered if the answer was six or sixty. They were acting as real mathematicians solving real problems.
No matter what structure you use for your math instruction, you can include these kinds of problems. Here’s one way I have found to be powerful: Have students work in pairs and solve problems on chart paper. Have them use markers to write (for ease in seeing what they wrote and also because then you can see all the steps they took with no erasing.) Talk about the strategies. Compare them. Talk about the strategies and the math. Also talk about how they worked with their partner. How did they negotiate the solving of the problem? What did their partner teach them? What happened when they disagreed?
Just as in reading and writing, a critical shift occurs when students take ownership over their mathematical lives.
Establishing routines in the primary grades is a huge piece of the start of the year. What sort of a timeline did you have for getting math workshop started? What did you introduce first? How did you add pieces? How long did it take to get the full workshop up and rolling?
It is always tempting to move too fast at the beginning of the year! Every year I have to remind myself to slow down and really make sure routines and structures are strong before taking the next step forward. In September I focus on three aspects of the workshop: 1) the warm-up routines, 2) the independent practice part of the workshop, and 3) the reflection at the end of the workshop.
Our warm up routines (counting around the circle, dot cards, math read aloud) are a short, but powerful number-sense focused part of our math workshop. (See Jessica Shumway’s new book, Number Sense Routines, which is all about this topic). In September we focus on a different routine each week and we really delve into the expectations for the routine.
I also focus on making sure the structure of the independent practice is in place (be it centers or partner tasks). I teach them how to take care of materials, how to talk to their partners when playing a game, how to switch centers, etc. I want to make sure that there is meaningful talk and play going on during the independent practice portion of the workshop. Only then can I feel comfortable removing myself from this part of the workshop and working with small groups.
I also focus on the reflection at the end of the workshop because this is a place where we share our strategies, discoveries, investigations. I teach children how to talk to each other, agree and disagree respectfully with one another, add on to each other’s thoughts and connect to each other’s ideas. We learn how to look out to the group when we’re talking and not just at the teacher. I teach them to value this part of the workshop as a place where we learn from one another, and not just from me, the teacher.
In mid to late October I start thinking about how I want to work with small groups. I start by working with just one group per day. I get my kids up and going in the independent practice, work with one group, and then return to check in on the independent practice. When this is going well, I’ll start working with two groups per day.
Think about what you most value in your workshop. Take the time to teach the expectations for this. It’s ok to go slow (I’m writing this to remind myself too! This is hard to do!). Start simple. Take on one part of the workshop at a time, if that works for you.
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