Earlier this school year, I blogged about an exciting opportunity I had to offer PD to all 107 of our district’s interventionists spread across the entire school year. I anticipated blogging about the experience during the year, but life got in the way. Here we are at the end of the school year and I am about to offer my final PD of the year tomorrow morning.

At first I wasn’t sure what I would do with them tomorrow. The way the schedule worked, I haven’t had a session with them since February. And with it being the end of the school year, I wasn’t sure what theyd be interested in hearing about. More than anything, they’re probably counting down the days to summer like their students are!

I thought a lot about it yesterday and today, and I came up with a plan that I think does a great job of honoring the work we’ve done together this year, brings us full circle to where we began, and even hints at directions for next year. As with anything I plan, it won’t go perfectly tomorrow, so I want to capture my ideas tonight of how I’m imagining and hoping it will go.

I’m basically going to give the same session 3 times in a row for an hour and a half each. That’s the only way we can cycle through all of the interventionists and provide PD on math, ELA, and a third rotating topic. The trouble is that the first group in the morning is invariably late. The starting time is 8am, but I doubt I’ll have everyone until 8:15. The session post-lunch is just as bad. The only group that will maximize the hour and a half is my middle group from 10:10 to 11:40.

I have too much to cover, so I am planning to start with an activity that I can do with whomever is in the room when the session starts. I’m going to ask the folks in the room to brainstorm at their table all the different ways they can make 120 using base ten blocks. If you miss the activity, it’s okay because you might catch the sharing portion. Even if you miss that, you’ll actually see various representations in the reading we’ll move into after this activity.

All year long, the interventionists have been doing a book study of Kathy Richardson’s *How Children Learn Number Concepts*. It’s been a great book study because I didn’t require any reading outside of the sessions. Knowing how busy the interventionists are, I know many of them wouldn’t do the reading. Instead, I let them do all the reading during our sessions. Did it take a lot of time? Yes. However, it meant every single one of them got the chance to read and discuss the book. And this is a book you don’t want to miss!

Unfortunately because some of the planned sessions were axed from the calendar, we only got to fully read 4 of the 6 chapters. We don’t have time to read chapters 5 and 6, but I am having them read the introductions to each chapter which is 9 or so pages each. Chapter 5 is about understanding place value (hundreds, tens, and ones) as well as addition and subtraction. A key idea from this chapter is that students must understand how to decompose and compose numbers flexibly, which will serve the students when they add and subtract multi-digit numbers. The example from the chapter is composing 120 in a variety of ways using base ten blocks, just like in our opening activity.

As people finish reading, I want them to solve 397 + 205 in two different ways. We’ll chat about the reading for a few minutes to reinforce the importance of unitizing, and then we’ll do a brief number talk of the addition problem. I specifically want them to connect their strategies to the concepts in the reading.

Then we’re going to jump over to watching some videos of students solving 1,000 – 998 from the Math Reasoning Inventory website. The first two videos show students who quickly recognize that the difference is 2. The third video shows Ana, a student who uses her finger to draw and solve the problem on the desk using the standard algorithm. After comparing and constrating the students, I want to show them two more videos of Ana solving 99 + 17 and estimating 18 x 21. In every video, Ana gets the question correct, but she relies completely on writing the standard algorithm with her finger. She does not evaluate the problems or demonstrate any number sense.

I specifically want to focus on Ana because I want to talk about how an emphasis on skill-building over sense-making with intervention students can result in students like her who can get correct answers, but they only have a strong grasp of procedures, not necessarily mathematics. I want to ask them how they think Ana is going to perform when she moves up into middle and high school math. Is there a point at which her procedural skills aren’t going to be enough?

After this we’ll move into the pre-activity for the next chapter in Kathy RIchardson’s book. I’m giving the teacher’s this problem along with three sample solutions.

“There are three fish in an aquarium. The middle-sized fish eats 2 times what the first fish eats, and the big fish eats 3 times what the first or little fish eats. If the first fish eats 3 pieces of food, how many pieces of food would the other two fish eat?”

One of the solutions correctly shows how much food the other two fish will get, while two of the solutions show misunderstandings about the multiplicative relationships in the problem. I’m not going to call anybody out here, but I do want the teachers to sort of self-check before they read whether they are able to recognize the multiplicative relationships in the problem.

This problem is talked about in chapter 6 of Kathy RIchardson’s book, which is about understanding multiplication and division. I liked this problem a lot because she shared how questions like this were given to students, and the results showed that 45% of the second graders in the study were able to think multiplicatively, but only 49% of the 5th graders were able to do so with ease. This implies that despite three years of learning about multiplication in grades 3-5, the students in 5th grade did not demonstrate any greater understanding of multiplicative thinking than students in 2nd grade who have had little to no formal study of multiplication or division.

I love this chapter because it demonstrates how all of the work students should have done learning how to unitize tens and hundreds lays the foundation for unitizing in multiplication and division. Students must be able to think of counting groups of things and thinking of each group as a unit. She gives the following example to demonstrate the subtle shift in thinking students need to make to move from additive to multiplicative thinking:

Rick: I figured out 8 + 8, and that was 16. Then I added 8 + 8 again.

Emily: I had two 8s and that was 16, and two more 8s make another 16. That means four 8s makes 32.

From here we’ll briefly talk about the results from a survey given to over 1,000 teachers who were asked the following two questions:

- What reading skills do you most often teach to skilled readers?
- What reading skills do you most often teach to less skilled readers?

The results are, sadly, not surprising to me, but having the data helps drive home the point that we have to be mindful of what we’re teaching to differnent populations of students. You can read more about this survey and see graphs of the results here.

At this point, I want to share our district math goals, which I shared during our first PD session back in September:

The K-12 RRISD mathematics standards articulate five general goals for all students:

- That they learn to value mathematics,
- That they become confident in their ability to do mathematics,
- They they become mathematical problem solvers,
- They they learn to communicate mathematically, and
- That they learn to reason mathematically.

Just because a student is struggling in math and put in an intervention program does not mean that we should ignore these goals. If anything , we should strive that much harder to reach them with these students! Based on their struggles, these students are already at risk of not valuing mathematics and not being confident in their abilities. We should build up their skills, yes, but we should also help them find ways to value math, to learn how to use it for work *and* play.

To this end, I’m using a few imbalance puzzles I got from Sue VanHattums’ wonderful book Playing With Math, (they are originally from this blog post).

I’m giving the interventionists two to solve, though if they can figure out one, that might be all we have time for. I’m not advocating they give these to their kids, but I want them to experience something that requires some problem solving, but it’s also fun and gets you communicating with your peers. I want them to experience something that might inspire them to create some new interactions with their students that work toward our district goals.

I want them to feel like they are walking away with something they can actually use with their students, so I’m introducing them to open middle problems. I was already a fan of the website, but it was Michael Fenton’s recent Global Math Department session that put them back on my radar in a big way. If you’re unfamiliar with the open middle problem type, basically it’s a problem that has a lot of solution paths, though not necessarily multiple solutions. Here’s an example of one from the website:

This problem has one correct answer, but students can attempt to find it in a variety of ways. I’ll share a few more examples with them so they get a feel for why these can be powerful problem types to include at times in their work with their intervention students.

I’m going to wrap up the session by showing Megan Taylors Ignite Talk about replacing “teacher-proof” curricula with “curriculum-proof” teachers. I just saw this for the first time recently, and it resonated with me with regards to intervention. So often I hear the cry for “teacher-proof” curriculum materials for RtI that can be used with fidelity and “get results.” While I see why this appeals to some people, I don’t think it’s the right way to go.

When it comes to *response* to intervention, the key idea to me is that the interventionists should be constantly *responding* to their students’ needs. This means having a skillset that allows you to adapt and customize as needed to help the children grow mathematically, not to follow some prescription as though we’re trying to cure a cold. My goal this year has been to help build that skillset in the interventionists so they feel empowered to do that with their students, and I want to end our yearlong PD journey on that note so they can venture into their summer feeling like they are confident and ready to take on the challenges of next school year.