Kickoff! #ElemMathChat

Tonight we kicked off a new weekly Twitter chat, #ElemMathChat. Hooray! As the name implies, the chat is designed for elementary school folks to talk about math.

I’ve been so excited to get this chat started! For the past two years, I’ve been a member of the MathTwitterBlogoSphere whose membership is primarily composed of middle and high school teachers. There are a few of us elementary-minded folks. We have appreciated all of the interactions we’ve had with the MTBoS. However, after meeting up at this year’s Twitter Math Camp, we decided our mission this year is to grow the elementary-side of the MTBoS.

And so it begins.

Tonight’s chat was a huge success! We had a great turnout with educators from around the US and Canada. (Thanks for catching my mistake @ChrisHunter36!) Everyone seemed excited about having a forum to discuss elementary math specifically. One person even commented that she was happy to have a place where she could be taken seriously. She said she’s tired of being considered “cute” for teaching first grade.

Our topic for the first chat was balancing problem solving with teaching/covering math skills. If you want to catch up on the conversation, you can check out the Storify put together by @davidwees. While the overall conversation was energetic and interesting, I was left a tiny bit disappointed.

I think it’s because I was the one who suggested this topic. Balancing problem solving and covering math skills is something I have struggled with myself as a teacher, and now as a district curriculum specialist, I am hearing from numerous teachers who are struggling to find the same balance themselves. So going in, I had some clear ideas of what I wanted to talk about and get out of the discussion.

The first question was “How do you define problem solving in the elementary math class?” This generated some interesting discussion. Some key points that rose to the surface for me were that problem solving involves thinking critically, collaborating, and using math as a tool. I especially like the “math as a tool” metaphor because it gives meaning to why we’re learning it in the first place. I think it’s often an implied message, but one educators need to try to make more explicit. I also liked how people described problem solving as a time to make kids get out of their comfort zones and make their brains sweat. I love the image that conjures in my mind.

The interesting thing that came out of this first question is that everyone seems to have different ideas about what problem solving is. Some people talked about it in a way that sounded like solving word problems, whereas others referred to rich and engaging tasks that focus more on the process than the endpoint. This is one area where Twitter chats can frustrate me. The conversation is happening so fast with so many people talking simultaneously that it can be challenging to pull the threads together into a coherent whole.

Maybe that’s what I need to learn how to do as a moderator. Instead of following my script of questions, I could have stopped and made question 2 be “So I’ve heard problem solving described as ___, ___, ___, and ___. What is one definition we can all agree on?” The conversation over the rest of the hour felt weaker because we didn’t necessarily have an agreed-upon definition to base our discussion on.

Question 2 also had some problems: “How do you define math skills?” This is where I had a clear idea of what I meant, but the majority of the group was on a different wavelength. Since we had just talked about problem solving, everyone seemed to think that I meant the Standards for Mathematical Practice or general thinking skills that are needed to solve problems. What I really meant, and I did try to clarify, are the nuts and bolts skills that teachers need to teach their kids: adding and subtracting whole numbers within 1,000, multiplying fractions with whole numbers, interpreting dot plots, and measuring angles, to name a few.

Here’s an example to illustrate the tension I was thinking about when suggesting this week’s topic. Learning a skill like long division takes time and effort. It is a very structured thing to do, but until students understand it, they are prone to making many errors. Can I do a few problem solving activities and have my kids somehow come away from the experiences as masters of long division?

You may be thinking right now, “But kids don’t actually have to know long division in order to solve problems. They just need a strategy that makes sense to them.”

I agree with you. However, in Texas and in Common Core, the standards do explicitly state that students learn to divide using the standard algorithm. So like it or not, it’s a skill that students are expected to learn.

Here’s where the tension comes in. Long division is just one skill. There are numerous other skills students are also expected to master in any given grade. How do you ensure the nuts and bolts mastery while at the same time providing ample opportunity for the types of activities that require critical thinking, collaboration, and brain sweating?

And please don’t take any of this the wrong way; I don’t fault anyone in the chat for not providing me a satisfying answer. To be honest, I don’t think a one hour Twitter chat is going to be the place to find concrete answers to big questions like this. It doesn’t mean I don’t want answers (hence the tiny bit of disappointment I felt), but I have learned over the past two years what Twitter can and can’t do.

What it can do is bring together like-minded people to fuel conversations and build relationships. The more I connect with people on Twitter, the more I get to know them. I can start chatting with them outside of our weekly chats. Perhaps I ask for help with a problem I’m having, or perhaps we set up a Google Hangout to have an actual conversation about a particular issue (good-bye 140 character limit!), or maybe we even collaborate on a proposal for a national conference.

Valuable professional relationships can grow from short, weekly conversations. It’s why I’m still here two years later, and it’s why I’m excited to get this specific chat launched. I’m eager to meet like-minded elementary folks and start forging some new professional relationships.

Adventure Time

One thing I’ve learned in my first month and a half at my new job is to be prepared for anything. A few weeks ago, my boss told me and the elementary ELA lead that we were going to be providing professional development to our district’s 107(!) interventionists. Not only that, but we would be providing them a total of 14(!) all-day PD sessions over the course of the school year.

On one hand, how cool is it that we get to work with every single one of our campus interventionists to create some shared vision around Response to Intervention and to help build their knowledge and skills of math and reading intervention?

On the other hand, holey moley! That is quite an undertaking on top of our other job responsibilities.

Just over a week ago, we held our first session. Thankfully over the course of planning for it (and getting feedback from interventionists who couldn’t believe they were going to be off campus 14 full days during the year) this adventure did come into better focus. For example, instead of having to plan 4 hour sessions each time, my partner and I only have to prepare 2-2.5 hour sessions on math intervention. In addition, instead of meeting 14 times during the school year, the higher ups knocked that number down to 9.

It’s still a big undertaking, no doubt about it, but it is feeling more and more manageable. On top of that, our first session went beautifully. Our primary goal, which I’m happy to report we achieved, was getting buy-in from the interventionists, many of whom had no idea this was even going to happen until a few days before the first meeting.  Now that the first session is over, I’m excited for the work ahead.

Since this is such a big project, I’m going to try to blog and reflect about it this year to see what I learn from it to apply to future endeavors. I also want to share my experiences in case anyone else out there can benefit from them.

The day started with all of the interventionists in a large group to hear from the higher ups about the purposes of these meetings and why they felt they are important enough to warrant so much time away from school. The district leads for RtI and dyslexia also went over some important changes that the interventionists needed to know about. When all of that was over, the interventionists broke up into groups to attend either a math session or ELA session.

Show time!

I opened the math session with everyone sitting in a community circle. My background is in a program called Tribes, and while I mostly used it with my students, the community building ideas apply to adults as well. The first thing I did was have everyone go around and introduce themselves, describe the types of intervention they provide on their campus, and then tell everyone a movie, book, or TV character that best represented how they felt at that moment.

I told them I felt like Lucy from The Lion, the Witch, and the Wardrobe, specifically the scene where she parts the coats in the wardrobe to reveal a newly discovered snowy landscape before her. I told them I felt like I knew what I was getting into with this job, but being asked to do these sessions this year opened me up to exciting new possibilities I hadn’t imagined. I added that I hoped nobody would be turned to stone or sacrificed on a stone table during the year.

As the interventionists went around the circle, I was impressed with how thoughtful, creative, and telling their responses were. Listening to them talk, I picked up on themes about wanting to be in control but feeling overwhelmed, about wanting to do the best job possible for their students, and about recognizing this opportunity to grow as a leader on their campus. While at first I had felt silly asking them to name a movie, book, or TV character, it ended up being a great way for them to share their feelings and realize that many people in the room felt the same way.

After introductions we moved into an activity called Talking Points. I learned about this activity in July from @cheesemonkeysf at Twitter Math Camp. I love Talking Points! They provide a way for people to improve exploratory talk, to dive deeper and have more meaningful conversation. You can download instructions and see examples of Talking Points on the Twitter Math Camp wiki.

The Talking Points that I had the interventionists do were all statements related to growth and fixed mindset. The statements didn’t use those exact words, but rather they got the groups talking about things like whether intelligence is something that can or cannot be changed. I did this on purpose because I wanted to know their current thinking on the matter. As interventionists, these people work with students who are having a difficult time in school. They not only need academic support, but they need someone who can help motivate them and encourage them to believe that they can learn. Like I told the interventionists, “If you didn’t believe that these kids can learn, then why would you bother showing up to work every day?”

Thankfully the interventionists seemed to believe in growth mindset by and large, so the groups tended to agree with each other, which I’m ultimately okay with. I was happy to see some dissenting opinions here and there though. Those groups were able to tease out some interesting ideas that the other groups missed out on.

After debriefing the Talking Points, I gave the interventionists a copy of our district’s math goals. I asked them to read the goals and do a quick write about how these goals are currently being met (or not) in their campus’ intervention program. Then they talked about their notes with the other folks at their table. This discussion was interesting because some of the interventionists didn’t even know we had district math goals. It also got some of them questioning whether their current intervention program was meeting the goals.

I used this discussion to segue into the foundation for our work this year, the What Works Clearinghouse guide on RtI: Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. The guide was put together by a panel including a research mathematician who is active in K-8, two professors of math education, several special educators, and a math coach. It provides 8 specific recommendations to help schools implement math intervention. The recommendations are based on the best available research evidence and the panel’s expertise in mathematics, special education, research, and practice.

Here are the 8 recommendations:

1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk.
2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee.
3. Instruction during intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
4. Interventions should include instruction on solving word problems that is based on common underlying structures.
5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.
6. Interventionists at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
7. Monitor the progress of students receiving supplemental instruction and other students who are at risk.
8. Include motivational strategies in tier 2 and tier 3 interventions.

All of the work we do this year is going to revolve around these 8 recommendations. To help kick it off, I had the interventionists look more closely at recommendations 2, 3, 4, and 5. I copied the sections about each of those four recommendations from the guide, and I had each interventionist read one of the sections. When they were done, they got together with the other people who read about the same recommendation. They collaborated to create two slides. The first slide contained up to 4 key points about their recommendation. The other slide contained noticings and wonderings their group had based on what they read. When the session was over, I put all of the slides together into one presentation that the interventionists could keep as a reference or that they could share with others on their campuses.

The interventionists were very receptive to what they read in the 4 recommendations we focused on in this session. They liked that the emphasis is supposed to be on number concepts rather than trying to keep up with what the teacher is doing in the classroom. Some of them said that by trying to fill gaps instead of focusing on key concepts they often feel like content mastery teachers rather than interventionists. They are hoping they can better define their role through our work this year.

We ended the session by revisiting our district math goals to see how they related to the 8 recommendations. It was very easy for the interventionists to see that these recommendations align very well with our district’s math goals. That’s not to say that their work with their students in meeting these goals will be any easier per se, but it is reassuring to know that the work they are doing with their students is going to be meaningful and supported by research.

All in all, the interventionists left excited about the adventure we’re embarking upon. They’re especially happy that they’ll have the opportunity to get to know each other better so they can utilize each other as resources. I couldn’t ask for a better outcome from our first time together. Part of me is even a little sad that I only get to plan 8 more sessions instead of 13.

Just a little. I still have plenty of other work to do.

Of

Yesterday on Twitter, I took part in an impromptu discussion of fraction multiplication. I’ll be honest. I often get frustrated diving deeply into meaty topics on Twitter because I’m limited to 140 characters per tweet (less when you take into account the fact that the handle of everyone tagged in the tweet is deducted from the total). However, this ended up being a very enjoyable conversation and reminded me of the power of connecting with folks from all over.

Tracey Zager was nice enough to Storify the conversation. If you’re interested to hear how a small group of elementary educators unpacked this topic, take a look. If you’re an elementary teacher yourself, especially an upper elementary teacher, you may appreciate it because you’re likely having similar conversations within your own district or campus.

One question that came up several times during the conversation was how and why the word “of” means multiplication:

As Math Minds’ tweet alludes to, there is a whole issue of teaching keywords and the damage they cause students, but that’s not what I’m focusing on here. Today I want to think through the idea of how the simple two-letter word “of” is related to the operation of multiplication in the first place.

This question made me think of the chapter I’m currently reading in Kathy Richardson’s book How Children Learn Number Concepts: A Guide to the Critical Learning Phases. It’s very timely that I’m reading a chapter titled “Understanding Multiplication and Division”. Richardson begins the chapter with the following quote from Keith Devlin (I like that I’m quoting a quote from a book.):

 “…in today’s world we are faced with a great many decisions that depend upon an understanding of quantity. Some of them are inherently additive, some multiplicative, and some exponential. The behavior of those three different kinds of arithmetical operations differs dramatically…”

Richardson goes on from there to discuss the need for elementary teachers to differentiate additive and multiplicative thinking.

“Central to understanding multiplying is the idea that the two numbers (factors) in a multiplication equation have two different meanings: one number describes how many equal groups there are and the other describes the size of each of the groups.”

And when we describe the relationship between the two numbers verbally, the word “of” can become an essential part of our description. Here’s an example from Ask Dr. Math:

Suppose items come 8 to a box.

If I have 2 of these 8’s, I multiply to find the total, 2 × 8 = 16.

(There are 2 equal groups and the size of each group is 8.)

If I have ½ of an 8, I multiply: ½ × 8 = 4.

(There is ½ of a group and the size of the group is 8.)

Going back to Richardson’s book, she goes on to describe the types of multiplication situations students should encounter in elementary school:

• Equal groups (equivalent sets)
• Rate / Price / Length
• Rectangular arrays
• Multiplicative comparison (scale)
• Combination problem (Cartesian product)

It’s the multiplicative comparison (scaling) situations that lend themselves best to understanding fraction multiplication. I found it very telling that in the CCSS grade 4 Operations & Algebraic Thinking domain, there is a standard that says students should interpret multiplication as a comparison. The standard uses whole numbers in its example, but the pump has been primed. Then in grade 5, this idea is embedded in the Number & Operations – Fractions domain in a standard that says students should interpret multiplication as scaling (resizing).

The trouble seems to be that up until fraction multiplication, the act of multiplying two whole numbers has always resulted in a larger number, and it has been easy for teachers and students to view it as repeatedly adding that quantity over and over. However, this idea of a quantity growing larger through repetition is only half of what’s going on. If quantities can grow larger, then they can also grow smaller, and our language to describe this needs to adjust accordingly. Instead of having 3 times as much or double the amount, we can now consider 2/3 of a quantity or half as much.

I don’t think the problem is that the word “of” doesn’t mean multiplication per se, but that as elementary educators, we haven’t opened ourselves up to needing different language to describe something new students are learning to do, namely using the operation of multiplication to decrease the size of a quantity.

Although, after writing all that, I want to revise my thinking. Richardson goes on to describe how children have difficulty grasping the word “times” when they first learn about multiplication. She recommends teachers use phrases such as “groups of”, “rows of”, “piles of”, “stacks of”, etc. There’s that word “of” again, and Richardson is advocating using it with children well before they learn about fraction multiplication.

If students can visualize and make sense of:

• 2 groups of 5
• 3 rows of 6
• 7 piles of 10
• 3 stacks of 9

Then we should be able to extend to this later on:

• 1/2 group of 5
• 2/3 row of 6
• 1/10 pile of 10
• 1/3 stack of 9

We come back to the idea that the two numbers in a multiplication situation have two different meanings. The first number in each example is the number of groups, whether it’s 2 groups or 1/3 of a group. The second number is the size of one whole group, whether the whole group is 4 pans of brownies or 1/3 pan of brownies.

So ultimately it seems that the word “of”, and phrases built around it, are mostly there to help students to visualize and make sense of a new kind of thinking. Up until grade 3, most students have been focused on additive thinking, so this is quite the paradigm shift for them, and they will grapple with it for several years. As a result teachers need to use familiar language and phrases, which include the word “of”, to help students expand their understanding of how we operate on quantities.