Waldorf 2nd Grade Math

Well, I’ve happily returned from my summer of professional development and taken a little bit of time off to relax at home (kinda, not really), but now I am just as happily hitting the books to prepare for 2nd grade.

The Waldorf 2nd grade curriculum is a nice little, gentle leap from the dreamy fairy tales of first grade and into the stories of saints and fables. Someone once told me that the 2nd grader is essential a cartoon character with the devil on one shoulder and the angel on the other — so the saintly stories appeal to their angelic side, while the stories of the animals in the fables appeal to their more devious tendencies. It’s all just perfect. But this year I’m trying something I’ve never done before — I’m skipping the story theme of the year entirely at the beginning and starting the year with a math block.

Starting with a Math Block

I was surprised when one of the teachers in our summer training recommended beginning this way, because I’ve always started the year with the story content that is the theme of the year. This was recommended in my teacher training, years ago, and I’ve always found it satisfying and successful. But, there are a number of reasons why it makes sense this year.

• I like pairing our class play practice with a math block (to avoid the competing story content). It doesn’t always work out that way, but it worked better if I started with a math block.
• I wanted to spend the first block reviewing some phonics rules from first grade before we hit the ground running with phonics in our second block of the year. I’m taking it quite to heart that most of my students will begin to read this year and I want to set ourselves up for success.
• I knew that the story content of the first block was going to be knights and dragons, which feels like a good segue from first grade fairy tales, and a good introduction to second grade (more of that angel/devil thing). It feels better to pair our first language arts block with the simple stories of the fables, rather than complex stories about knights, dragons and adventure.

So, for the past few days I’ve been sitting down with my math curriculum books and putting together a plan for the entire year, as well as specifically the first block. And, as usual, along the way I’ve developed some opinions.

Waldorf 2nd Grade Math

So, the primary content of the 2nd grade math curriculum is place value. In our summer Waldorf training, we learned that this means you’re primarily doing the following:

• Teaching the idea of “groups of 10” and that in a double digit number the first number represents the 10’s while the second number represents the 1’s.
• Addition and subtraction with double-digit numbers, in columns.
• Carrying and borrowing (or regrouping — which isn’t a much better term)
• Reviewing the other processes, keeping the times tables 12 and under

In our summer training, our instructor gave us a nice little image for how we can present the idea of the 1’s, 10’s, 100’s and so on. Teachers use all different images, but basically you use some object to represent the 1’s place — let’s say apples, for example. When you have 10 apples, you can no longer carry them in your arms, so you need to put them in a basket. So a basket represents 10 apples. Then, when you have 10 baskets, you need a box (which represents 100), and so on. I completely agree in the value of using an image in a story to bring the basic idea. And then it’s important to practice. So, using gems and pouches, for example, you can ask students to represent various numbers. There are many, many other ways to practice the idea and, according to my most recent research, practicing is really important. It is VERY easy for children to lose sight of the value of the numbers as they work with them. Practicing with manipulatives helps them remember that 29 is 2 tens and 9 ones. So . . .

29 = 20 + 9

I plan to spend a good week and a half with activities to help solidify this idea.

But this is where I get opinionated.

So, once you’ve practiced this, the logical next step is to teach addition and subtraction with these larger numbers. The way that you and I and most people, at some point, were taught to do this is to stack the numbers and add the columns. Simple enough. But here’s the thing . . . As soon as you stack the numbers and add the columns, the numbers lose their value. Now, bear with me here, I can’t stack numbers on the screen so I’ll talk you through it. Take the problem 21 + 34. If we stack the numbers and solve the way we were all taught, we do 4+1=5, 2+3=5 and we get 55. But notice that in working that problem through I did not add 20+30. I added 2+3, completely losing sight of the value of the numbers. Also, because I stacked them, I totally missed the fact that it was actually a really easy problem I could do in my head. If I’d been presented with them vertically, I probably would have just taken the 1 from the 21 and added it to the 34 to make 35. Then added the 20 to 35 to get 55. This is the kind of creative solution that keeps thinking flexible, which we want to encourage in our students.

The primary resource I’ve been using for this preparation is Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2 by John Van de Walle and others. It was recommended to me in my teacher training (years ago) and I’m a huge fan of this book. This book really shows the value of these alternative solutions as it outlines the progression that students follow when they learn computation. It goes like this.

Instructional Sequence for Computation

1. Direct Modeling. This is the most basic approach. Students compute by counting by ones. This is where almost all of my students are right now. To do 5+4, they put up 5 fingers on one hand, 4 fingers on the other and begin counting on one hand, starting at 1 and then move to the second hand. Some of my students will do this kind of problem by remembering the 5 and starting their counting at 6 to get to 9. Once they feel confident in this, and they’ve learned about the value of 10, they use base-ten models (either physical or mental.) For example, 8+5 is easier when you take 2 from the 5 and add it to the 8 to make 10, then add the leftover 3.
2. Invented Strategies. This is where students find their own ways of solving problems, like in the 21+34 example above.
3. Standard Algorithms. This is the technique or “trick” we all learned to solve the problem. Addition and subtraction with double-digits is the simplest, but long division and long multiplication are other examples of algorithms. They require lots of guided instruction and should be followed by an intuitive check. At best, they should be seen as just another strategy — rather than the “right” way of solving the problem. They should also only be used if the student truly understands why they work. Teaching students a rule that “just works” does not encourage them to think creatively and confidently.

So, ideally, we want to encourage them to stay in the “invented strategies” phase for awhile. If they stay there long enough, they may even “discover” the algorithm themselves. And if they do, great! Then they can use it, because they understand why it works!

“Invented Strategies”

Student-Centered Mathematics outlines the benefits of those invented strategies for supporting future learning.

• Children make fewer errors. When students figure out their own methods, they understand them better, so mistakes are less frequent. When they make mistakes with an algorithm, it is usually an error with implementing a step in the algorithm. This kind of systemic error can be really difficult to correct, once it has become a habit, because it doesn’t come out of the child’s number sense.
• Less reteaching is required. It takes some extra time to allow students to explore other strategies, but that extra time results in a “meaningful and well-integrated network of ideas that is robust and long-lasting.”
• Children develop number sense. This is the biggest benefit, as I see it. They don’t forget that they are adding 20-something and 30-something. These strategies are number-oriented and they keep the value of the number intact.
• Invented strategies are the basis for mental computation and estimation. These strategies come in handy whenever the students are working with numbers and computation does not then need to be taught as a separate skill. As they become more skilled with these other strategies, they can do them in their heads, without writing down steps. (Or pulling out the calculator on their iPhones.)
• Flexible methods are faster. To see this, just try doing 100-98 using the algorithm. You’ll have to “borrow” twice (with zeroes, which is even more complicated). Whereas, if you just think about the numbers, you’ll see it’s a pretty easy problem.
• Strategy invention is itself an important process of “doing math.” When they invent strategies they are making meaning. When they implement an algorithm, they are carrying out a process, without much meaning.
• Children who use invented strategies perform similarly or outperform their counter-parts who are taught only standard algorithms. Of course they do. They understand what they are doing.

Encourage Invented Strategies

So, according to this book, the best way to encourage students to use invented strategies is to delay the teaching of those standard algorithms. As I see it, students are much more likely to find other ways of solving when they are presented with problems that are written horizontally. Also, there are types of problems that can encourage one or another alternative strategy. I’ve gone on long enough in this post (only the true teacher-geeks or math nerds are probably still with me), so I’ll write another post soon that outlines examples of those different strategies. There are specific ways of thinking about these strategies and I’ll point out how the problems you choose can help your students discover different strategies. Suffice to say, we’ll be working with place-value practice and horizontal problem-solving right through our first math block! And if you’re one of the nerds who made it to the end of this post, leave me a comment.