Math Exchanges Blog Tour Stop - October 4th

I'd like to thank the Academy Kassia Omohundro Wedekind and Zsofia (from Stenhouse Publishers) for including me on this blog tour! I'm thrilled to be a part of it! Make sure you check out the whole Blog Tour!

Monday, October 3rd at Catching Readers with Katie Keier and Pat Johnson
Tuesday, October 4th at Camp Read-A-Lot with me (Laura Komos)
Wednesday, October 5th at Reflect and Refine with Cathy Mere
Thursday, October 6th at Elementary, My Dear, or Far From it with Jenny Orr
Math instruction has been weighing heavily on my mind in recent years, so when Kassia tweeted about her new book, I knew I just had to have it! I can honestly say it's been years since I've read a book about math instruction. I've basically been following our Everyday Math curriculum with little regard to individual student needs. Yes, I'd scaffold my instruction and questioning, but I knew that wasn't enough. Forty pages into Math Exchanges, and I was hooked! I was excited about math instruction for the first time in... well, ever!

"Community, rigor, and joy are at the heart of the workshop approach to teaching and learning." (p. 2) What struck me right from the beginning was the connections Kassia makes between literacy instruction and math instruction. She references many Literacy Leaders who have deeply impacted my learning (and teaching.) She also explains the similarities between reading/writing workshops and her Math Exchanges. This is what made it all start to click for me! I love the framework Kassia suggests for Math Workshop, but I also love how she gives many examples and possibilities! She explains how there is no one "right way," which is what I already believe to be true with Literacy Workshops.

"We can make a shift from mathematics as something we simply "do" to a way in which we live our lives, relate to each other, and wonder about our world." (p.3) Starting with statements about what mathematicians do, Kassia describes her math exchanges and how it brings math to life for her students. As the teacher becomes more of a facilitator and coach, students learn to express their ideas, share strategies, and use their mathematical thinking skills to problem solve. Kassia goes on to explain CGI (Cognitively Guided Instruction) and how she groups students so their thinking will change and grow over time. There's a handy-dandy cheat sheet in the Appendix which I'm sure I'll use often! :) I know I'll go back to her "Try it out" sections over and over as well!

Two packages of Post-it flags later, I finished reading Math Exchanges! I know that I will keep it close at hand as I begin implementing Math Workshop and Math Exchanges with my first graders. As any good reader does, I asked myself lots of questions before, during, and after I read Kassia's book. Lucky for me, I also got to ask Kassia! Here's our Q&A session!

Me: Over the past 5 years, we've found children in our building are lacking in basic math fact knowledge. What do you think of math fact practice such as Mad Minutes or Rocket Math?

Kassia: I think it is important to question the message that these kind facts programs send to our children about math. Math is only about spitting out quick answers. Math is about how quick you are at facts. Math is a race. This simply is not true. Some of the greatest mathematicians of our time (and past) do not excel at fact recall. They exceed at problem-solving, innovation, creativity. This is where I think we should be spending the majority of our focus in our math workshop.

Additionally, there is actually some interesting research on this kind of fact practice program that shows that it has no effect (positive or negative) on math skills, but has a very significant negative effect on how children view themselves as competent mathematicians.

That being said, automaticity (not memorization) of facts is important. But I don’t think skill and drill leads to the accomplishment of this goal. Facts memorization practices treat each fact as separate and unconnected from the next fact—a sure success for failure for many people. Fact automaticity, on the other hand, is a longer process that occurs as children have work towards more and more efficient strategies for solving problems (one of the most efficient being use of facts). As children start to think about the relationships between numbers, they start to become more automatic with fact knowledge. They realize that if they know 5+5=10 that they can use this knowledge to help them with near-double facts like 5+6=11 or 4+5=9. They use what they already know to become automatic with more and more facts as they expand their understanding of numerical relationships.   

Me: I'm going to be the only one in my building doing Math Exchanges (right now, anyway.) What kinds of changes/difference will it make for my first graders?

Kassia: First of all congratulations on starting this journey! It takes a lot of courage to swim against the current or be the first one to try something different. I admire you for this.

I think one major shift you will see is in your first graders is how they view themselves, not just as do-ers of the work their teacher assigns them, but as mathematicians. In a math workshop kids feel ownership over their thinking and work. They feel a sense of pride when talking about the strategies they used to solve problems. They take on challenges and see themselves doing the real, authentic work of mathematicians.

I think you’ll notice that your students become more process focus, rather than simply looking for the correct answer. When you, the teacher, value not just the correct answer, but how students solved it, how they chose their strategy, how their strategies have changed over time, children will also take on this value in themselves and others. 

Also, I think you’ll notice that when a math workshop is based on problem-solving, students’ number sense and understanding of numerical relationships grows exponentially.

Me: My BOE and district is big into "data, data, data." What kind of data can I gather to help explain why Math Exchanges will make a difference for my students?

Kassia: I think a lot of people out there are struggling with how to quantify problem solving data, which is more difficult than say, math facts. Sometimes the most important understandings are the most difficult to quantify. Data is a big point of discussion in my school and district as well. The kindergarten team at my school decided to come up with a rubric for problem solving. At certain points in the year we’ve decided to score different kinds of problems on a 0-4 rubric. Here’s how we broke it down:

0—No attempt, or plays with materials.

1—Incorrect strategy (a strategy that doesn’t make sense for problem type), incorrect answer.

2—Uses a strategy that could result in correct answer, but either 1) doesn’t get correct answer or 2) cannot explain answer

3—Uses a modeling strategy that results in a correct answer and can explain what he/she did.

4—Uses multiple strategies or uses a more sophisticated strategy (counting, facts, derived facts, invented) to solve the problem.

We’re still finessing the rubric and how we want to use the data, but having a conversation about a way to quantify problem solving data that you might collect from an assessment or your anecdotal notes after a math exchange is another way of informing your instruction.

Me: I've already established some routines for math time (free exploring, games, etc.) We're constructing our statements for what mathematicians do. I love that you don't have a "scripted" moment-by-moment way of conducting Math Exchanges but do offer the great chart on page 44. That being said, I tend to jump in with both feet and get into deep water! I have a few ideas (from what I read in the book,) but how would you recommend starting Math Exchanges in first grade?

Kassia: Last year as a math coach I had the opportunity to work with an amazing teacher, Christy Hermann (she’s in the book!) and really deepen my understanding of first graders’ mathematical thinking. We began our year by offering students counting collections. These collections were things we had around school or our houses (shells, rocks, pencils, marbles, etc). We kids collections to count (choosing collections with numbers of items we thought we appropriate for the student) and record in a journal how they counted them. At first students counted by ones and often lost track of their counting. However, some students grouped items into fives or ten and ten counted the total. We used math exchanges and whole group conversations to highlight some more efficient counting and recording strategies. Counting collections is about counting, but it’s also about so much more. As children began to group by tens, we took a look at the collections. “62 toy monkeys. So, you counted that 10, 20, 30, 40, 50, 60, 61, 62. How many groups of ten did you count? How many extras? Where do you think that six groups of ten is in the written number 62? Where are the extras?” You are introducing the foundation for place value with counting collections without imposing a system they don’t yet fully understand yet. My book offers some more information on counting collections, the original idea for which came from an article by Angela Chan and Julie Kern Schwerdtfeger.

I would also suggest starting with some of the problem-types suggested in the book. Choose one with which you feel comfortable. In first grade, students usually are using a combination of modeling and counting strategies. Write one problem like “Marie has ___ shells and her friend, Sam gives her ___ more. How many shells does Marie have now?” Choose some numbers you might give to different groups. Let’s say you choose the numbers five and ten for one group. Some students might make a pile of five cubes and a pile of ten cubes and count them all from one. Some students might count on from five because it is the first number in the problem. Some students might count on from ten because they understand informally that a + b = b + a. Some students may know that how ten combines with other numbers and see 10 + 5=15 as a fact. Explore these strategies together. Highlight some of the more efficient strategies with your math exchanges.  

Me: If the goal is to focus on the problem type, you suggest using easier numbers. If the goal is to focus on numerical relationships, you'd use a problem type students will understand. (Did I say that right?! Yes, I think that's right most of the time! J) Would you ever include time, money, or fractions in the numbers part of the problems?

Kassia: Absolutely! I think you can apply these problem-types to most of the kinds of math you focus on in the primary grades.

Actually fractions are some of my favorite types of problem solving. It can start as simply as a problem like “Four friends have a brownie. How can they share the brownie equally between them?” Or get into answers that involve an answer that is greater than one. Two friends have three brownies. How can they share the brownies equally between them?” There’s a PHENOMENAL new book out, Extending Children's Mathematics: Fractions & Decimals: Innovations In Cognitively Guided Instruction by Susan Empson and Linda Levi. They offer problem types for fractional understanding and what this would look like in the primary grades.

Me: What do you think of the traditional daily "calendar routine" being used in many first grade classes? (date, weather, counting days of school, reviewing days of the week/months, etc.)

Kassia: I used to do a traditional calendar routine, but in more recent years I have really been asking myself “what’s the purpose?” of each part of my practice. Sometimes my only answer for parts of my calendar routine were “because I’ve always done it that way.” I don’t think that’s enough of a justification. This summer I read Jessica Shumway’s (my former colleague) book Number Sense Routines. She really offered some great thoughts on calendar and this year, with my  kindergarten class I’ve implemented some of the routines she talks about in her book. This is what I’m doing this year:

1)   Using a desk calendar to look at the date every day. We use the calendar just as adults do—we mark important dates (birthdays, field trips, the day our monarch caterpillars will emerge from chrysalises). I also will tear off each page as we go so they can see the whole year as it unfolds instead of just thinking of a single month at a time.

2)   Counting tape. Borrowing a routine from the Everyday Counts Calendar Math program, we’re making a post-it for every day we’re in school. The color of post-it is different for each ten days and then it repeats. Some kids are starting to notice this and talk about which color they predict will be next, so it’s getting interesting now after 18 days in school.

3)   Rock jar. We add one rock for each day of school to a jar. I ask questions like, “How full do you think our jar will be on the 100^th day of school?” How many rocks do you think will fill the jar?” How many more rocks do you think we’ll need to cover the whole bottom of the jar?”

4)   Unifix cube. We’ve been adding one cube each day to a long stick of Unifix cubes. We’ve been in school 18 days now, so it’s starting to get long.  The kids are starting to wonder, “What will we do when the stick of cubes gets very long?” I’ll be following up on this line of thinking asking them how we could organize the cubes (when we have a bunch of them) in a way that would be easy to count.

I’ll be reassessing these routines as the school year goes on and asking myself, “Is this something I should continue to do with my whole class? Or is this something that a child could do in as a job in the morning?” I’ll ask myself “Are there other important routines I can incorporate.

Jessica’s book is also available from Stenhouse and you can preview it (just like mine) in its entirety over on the Stenhouse website.

If you're interested in hearing Kassia talk about Math Exchanges, you should check out this great video posted by Stenhouse! :)

I highly recommend Kassia's book if you want to be inspired to change your views (and your students' views) on math! I look forward to continuing this conversation with all of you. Please feel free to comment and/or ask questions! One lucky reader from each blog will receive a copy of Math Exchanges (or another Stenhouse title if you already have it) at the conclusion of the book tour. Thanks for visiting!

~Komos :)