Mathematical Practice 2 states that children should be able to “Reason abstractly and quantitatively.” What does that even mean??
This is one of the math practices that really stumps many of the teachers I work with, and rightly so! The practice itself is a little more abstract than we’re used to when it comes to math! But what it really means is that we’re looking at numbers connected to words and words connected to numbers, which we do when we apply mathematics into real world scenarios, or when we do word problems.
Just like fractions (which we explored in the previous blog series), word problems tend to strike fear into the hearts of teachers and students alike. In this blog series, we’re going to demystify the dreaded “story problem” and explore real life math so we can help our students learn to “reason abstractly and quantitatively.” We’ll look at a process to help students with reading comprehension skills using all kinds of different problems, from part/whole addition to part/whole subtraction, part/whole missing addend, part/whole multi step, part/whole multiplication and division, additive comparison, multiplicative comparison, fractions and more.
The Problem with Story Problems
Story problems can help students learn the basic arithmetic of math – the mechanical adding/subtracting/dividing/multiplying of numbers. But when students are asked to apply that arithmetic to real world situations or to think more deeply about those mathematical concepts, they get confused because they don’t have a deeper understanding of the mathematical processes or what the problem was actually asking.
We have the simplistic story problems like, “3 frogs jumped in the pond, and then two more joined them. How many frogs are in the pond?” But we also need to have more complex problems such as, “I baked four dozen cookies last night. 1/3 of the cookies that I made were chocolate chip, 3/4 of the remainder were gingersnaps, and the rest are peanut butter. How many peanut butter cookies did I make?” Answering a question about frogs is much different than answering a fraction multistep comparison problem.
The complexity of that kind of problem can stop a student dead in their tracks. Their first response is usually a high-pitched questioning voice wanting to know Do we just add (or subtract or multiply or divide)? They watch the response to their questions very carefully – if the teacher doesn’t seem happy with one suggestion for an operation to use, they’ll quickly suggest another until they figure out what they think their teacher wants. When it comes to story problems, students just don’t seem to have the perseverance they need and they aren’t interested in “productive struggle.” They just want a teacher or adult to tell them what to do and how to get the answer and save them from the story problem.
However, that approach isn’t really going to help kids in the long run. Everything that we’re doing with 21st century mathematics is really about applying it to situations in the real world. I’ve talked about this many times – I can just pick up my phone and ask Siri What’s 25 times 35? And Siri will regurgitate that answer without a problem. But if I can’t apply it in a real world situation, that’s where the struggle begins with the application of math.
Early Problem Solving
We’ll start this journey through the development of word problem skills at the beginning: with our littles. I always talk about peeling the layers back to look at where a child is coming from in the early years of problem solving. How do we do that?
Ask almost any teacher where we start with reading, and they’ll say something about oral language, phonemic awareness, all those great things that happen to allow a child to put sounds together in order to read words on a page. Decoding in reading is a lot like calculation in math. As a student progresses with the mechanics of reading, we start to think about reading comprehension, which is another skill altogether. The same thing follows suit with math. As students become more competent with math calculation, we move into what’s called math reasoning.
If you’ve learned with us before, either in an on-campus workshop, or through our M³ coaching, or even just reading our blog, you’ll know that we really emphasize peeling back the layers in math calculation with Math Mights and concrete, pictorial, abstract (CPA). Clearly, math calculation gets a lot of attention because it’s so foundational for the rest of mathematics. But, if you’re like many of the teachers we work with, you might feel a little less confident helping students with that next layer of math reasoning. The good news is that we can start building math reasoning skills very early on!
As we know, the first part of this involves real objects in the physical world. We want students to have thousands and thousands of experiences with real objects in the physical world to give them the opportunity to explore real life mathematically. This doesn’t have to be a formal math lesson with counters and flashcards for your 2-year-old! It can be picking up two objects and talking about which one is heavy and which one is light. It can be making a tower out of red blocks and building a tower of equal size out of blue blocks. It can be setting the table for a family dinner and knowing how many plates and forks will be needed. This is where real life math begins!
Creating a Math-Friendly Home
We just finished an amazing #sis4students Virtual Math Series (check out the archive here!), where we partnered with Making Math Make Sense to help parents, teachers and families work with their students to deepen their understanding of math. One of the things that we created for that Virtual Math Series is something called Math4Littles, since littles is where this all starts!
Creating a math-friendly home is the topic of the first Math4Littles series of videos. It is a room-by-room guide to helping math come alive and become concrete. Each video has five specific ideas or activities you can use to have math-focused conversations with your little as you go about your daily routine, and its all written out on a one-page printable so you don’t have to remember everything!
The kitchen is a great place to begin to bring math into reality! Next time you’re eating breakfast with your little, count together to see how many tablespoons of Cheerios it takes to fill up your cereal bowl.
When you head outside to explore, let your little pick up different types of leaves and rocks and sticks to carry home. Then, work together to sort them by an attribute like shape, color, or texture to help them see mathematical concepts with real things in their life.
After lunch, play “I Spy” in the family room, but describe the target object with mathematical vocabulary – shape, quantities, etc.
Then, after dinner, before your little jumps into the bathtub, take a second to observe the waterline, maybe even mark it with a bathtub crayon. Then, when your little is in the tub, help them notice what happened to the water line. You don’t need to go into a complicated explanation of volume and displacement, but simply calling attention to the change plants the seed for understanding volume.
Keep it Up!
It might seem insignificant, but strategically incorporating math-focused conversation will lay the foundation of early problem solving, which feeds mathematical reasoning skills later on! These concepts are crucial to our littles’ development of oral language with real objects in the physical world. Even if you don’t have a math degree, even if math “isn’t your thing,” all it takes is a little guidance and some intentionality in your conversation to help your little develop early problem solving skills in the comfort of your own home! Check out the tutorial videos and let us know how it goes!