15 Solutions to 1 Problem
I spent the first 7 years of my teaching career teaching 4th grade in 3 very different schools. By fourth grade, most of the students that joined my math class fell in two categories:
They were on track or possibly ready to soar: These students typically liked math and enjoyed tackling tough problems and situations.
They lacked foundational skills: These students were missing key understandings of their development of the four operations. These students typically disliked math and found it to be something that you either knew how to do or you didn't.
As a fourth grade teacher, I did the best I could. I knew the basics of math. I knew the way I learned, and I knew what my textbook told me to teach. But I hadn't LIVED IT. I hadn't seen the way operations are developed. I began to study and learn and even step outside my comfort zone to the primary grades.
Once I did...WOW! If you haven't hugged a primary teacher lately, make sure you find one. While upper grade teachers help students use operations to solve complicated problems, primary teachers lay the foundation this solid house is built upon. And it must be SOLID in order to not topple.
So, I want us to take a minute to check out my latest research project in thinking about how students develop addition and subtraction understanding, especially when it comes to word problems. Most students understand addition as getting some more and subtraction as taking away. No biggie. They can act it out, use cubes and even solve with an equation in Kindergarten.
But then, by Grade 1, we decide to throw some kinks in the mix. We are still going to represent addition and subtraction action stories, but sometimes you don't know how many are being added or subtracted. Sometimes you don't even know how many you had in the beginning. Sometimes the problem clearly shows addition, but you really need to subtract! Yes...all in Grade 1.
In Kindergarten and Grade 1, students also learn that sometimes addition and subtraction don't describe a physical action. Sometimes it's merely a way to compose or decompose a set. These same problems can be done with addition and subtraction but aren't really the same thing.
There's a reason students in Grade 1 spend so much time on numbers within 120. They are working on so many complexities within numbers to 120. This is not just adding or subtracting. This is truly understanding the part-part-whole relationship of addition and subtraction and the way operations can be used inversely.
I feel like my Grade 1 students understand these with some confidence. And then I throw them the next curve: comparison problems. Teaching students to find the difference using the language of more and less or fewer. For some reason, students always understand more, but the word fewer....complete blank stare! I can literally hold up 5 red cubes and 7 yellow cubes and ask: How many more yellow cubes are there? There are 2 more yellow cubes. How many fewer red cubes are there? Nothing. It's the same digit as the answer so it must not be the right answer.
As students develop this understanding, many of them begin to think that when there's more, we should add and when there's fewer, we should subtract. This makes complete sense when you consider the meaning related to a quantity. But sometimes we might know information about the lesser quantity to find the greater quantity which would require us to add. Or we may even know how much more one quantity is than another but need to subtract to find the lesser quantity.
This concept takes concepts that students may have considered as absolutes and turns them around. Those key words don't work anymore. It's all about understanding what those words are describing!
If you've kept track, that is 15 total addition and subtraction scenarios that students are expected to understand by the end of Grade 2. They will use these understandings to solve multi-step problems in the upper grades and perform the same problem types involving fractions and decimals. These 15 problems lay a solid foundation for our student's mathematical understanding.
So what do we do? We know this is a challenge and can be problematic, but there is a solution. First of all, use models to help students visualize operations. Use number bonds and bar models to help students see where parts are being combined or separated instead of relying on key words. You can find several blog posts and even some great FREEBIES about number bonds and bar models in the blog feed.
Second of all, start from the ground up. There's a reason students in Kindergarten begin with numbers within 5, then within 10 and move to numbers within 120 in Grade 1. We begin with numbers that are easy to manage before trying to tackle numbers that are too great to represent. That's why Jerome Bruner's research on the Concrete-Pictorial-Abstract theory is so powerful.
We can teach students to read through word problems using numbers within 10 to first comprehend the problem type. We can help them draw models that make sense to show whether to add or subtract. Then, we can provide them with the understanding of algorithms to solve it with greater numbers. Once we do that, it's amazing to watch them soar!
I've been fortunate to expand my teaching experience from Kindergarten to 8th Grade in the past 8 years, and I've loved teaching a virtual course to some awesome students right now on this very concept. Watching these students see the subtle differences in problems and helping them feel success has been so much fun.
If you are thinking this is something you want to try, I want to encourage you to check out my new student workbook where I break down these 15 problem types. Each problem type has 3 videos, 1 basic problem, 1 on-level, 1 challenge and 4 practice problems to complete. Students who work through this book with the matching teaching videos will have these concepts down within a few weeks! (Imagine if all our K-2 students had this foundation!) It's a steal right now in my shop as a digital download, print book and even a digital teaching bundle with teaching Jamboard lessons. (Email me for bulk discounts...we will make it happen!)
If you want to check out some of the great ways to use models to help students make sense of problems, check out this FREE student Jamboard I'm sharing with you today. I used this Jamboard to help students look at the different forms of the part-whole relationship and how it can be used to solve other addition and subtraction situations. It's one of the Jamboards included in the digital bundle. You can grab the download for free by clicking below and try it out with your students.
I hope you find this daunting task not quite so intimidating after checking these resources out. As I mentioned before, if you haven't hugged a primary teacher lately, make sure you show them some love! They are tackling some tough concepts.
Also, if you like this workbook, stay tuned for the sequel: The Problem Solving Handbook: Multiplication and Division!