Teaching for Thinking, Not Just Right Answers

Guest blog post by Susan Smith. Susan is a retired math teacher with 34 years of classroom experience. Passionate about mathematizing, she shares practical tips and strategies to make math meaningful for students. 

“It is very important that your child knows their multiplication facts. In 4th grade they will be multiplying by multi-digit numbers and will be bogged down in the process if they don’t have fact mastery.” 

This is what I always told parents each year at Curriculum Night. I defined fact mastery as recalling single digit multiplication facts within 3 seconds. To me, this was a critical foundation for math success. This was when I believed math was about rote-memorization and mimicking step-by-step procedures. I “differentiated” for my students by providing multiplication charts for those who still struggled with fact recall to not hold them back from learning the steps of multiplying place value partial product  and ultimately the U.S. traditional algorithm. I thought I was ensuring students’ success with math in 4th grade and beyond. When in reality, I was only ensuring correct answers. Does any of this sound familiar? 

When "Right Answers" Was My Only Goal

In episode 298: Who Gets to Math? One thing that Opens or Closes the Gate, Kim and Pam present the problem 39 x 25 to start the podcast. In the past, solving this for me would have meant using an algorithm. But now having rewired my brain to view math as figure-out-able and having worked on my own numeracy, I am empowered to think about the relationships and the most efficient way to solve such a problem. And I agree with Kim, this one “screams” Over strategy. I immediately thought: (40 x 25) - 25, which is 1000 - 25 = 975. BAM!

Reflecting on my math journey, I am struck by the stark contrast between my class when answer-getting was my primary goal to when it shifted to a goal of building mathematical reasoning in my students. 

What Literacy Taught Me About Math

Before becoming a math teacher, I was a reading specialist and literacy coach for 22 years. In that role, I was clear that I wanted students to not just learn to read fluently with comprehension, but to become lifelong readers. When I moved into math, for some reason my goal was a narrower one; for students to know how to accurately solve the types of problems outlined in the 4th grade state standards. 

My goal has since grown and now mirrors my work in literacy. I want to build lifelong math-ers. That does not happen with a focus on answer-getting and a belief that kids must learn some prescribed list of “basics” before being able to truly “math”.  Lifelong math-ers know the joy and power of math-ing and that happens when we build kids’ brains to be thinkers, to use what they know to reason about problems and own mathematical content and relationships.

The Problem With Easy Answers

In the episode, Kim makes the point that when we provide cheap and easy ways to get answers, students don’t want to think and then become stuck.  I will never forget when one of my students, who resisted using reasoning to solve any subtraction problem, said to me, “ can do the algorithm without really thinking at all and still get the right answer — so why would I do it any other way?” Is that what we want in math class?  Students who can get answers without thinking? Really?! Kim draws an analogy to moving a barbell in a gym. If there is a quick and effortless way to move the barbell, it will get moved, but you won’t gain any muscle. The goal of math is not to move the barbell, the goal is to strengthen the student so that they are capable of moving the barbell. (WOW, well said, Kim!)

Pam and Kim also contrast the mental actions between applying an algorithm and using thinking and reasoning. The algorithm is monotonous, single-digit oriented, and demands automated recall. The other requires understanding of relationships, connections, and allows choice, curiosity, wondering. Solving a problem using those mental actions is empowering and exciting. When kids view math as boring or drudgery, I wonder if it is rote-memorizable, repeated practice, mimicking steps, fake math they are actually experiencing. Because real math-ing is an intriguing endeavor.

The Day My Class Went Silent

My 4th grade curriculum has a standard that states, “Students can fluently use the standard algorithm to subtract multi-digit whole numbers (up to 1,000,000)” Last year, I used Problem Strings to teach relationships that lead to strategies for solving these large subtraction problems. During these lessons, there was always a buzz of energy in my class as students grappled and solved problems with choice, cleverness, using what they know to think and reason. I also felt obligated to teach the standard algorithm — it is, after all, one of my required standards. So, one day I said, “Today, math class will be a little different. I have to teach you a process for solving subtraction problems called the standard algorithm.” I began with manipulatives to build conceptual understanding first and then connected it to the steps of the algorithm. It was an “I do, We do, You do” structured lesson. The sharp, significant dip in energy and thinking in my class was palpable that day. During the “You do” practice portion of class, one student called out, “This is really boring.” I agreed and assured everyone that the next day we would be back to thinking and reasoning and using what we know to solve problems! There was a marked sigh of relief from the entire class.

As discussed in the podcast, I often regularly hear the argument from other educators and parents, “We have to teach the basics first.” By 'basics,' they mean facts and algorithms — the building blocks of answer-getting. This episode helped me clarify that I do agree with teaching the “basics”, we just need to re-define what those “basics” are in math education.

Redefining the Basics

I agree with Pam and Kim that the “basics” are those underlying properties, generalizations, and relationships that students need to own to be able to math using thinking and reasoning. Through Problem Strings we can high-dose students with patterns such as doubling and halving, the relationships of numbers to each other in space in time, scaling by 10, the five meanings of fractions, and connections between problems so that more kids can do more real math. We are making a false assumption if we think kids must first trudge through the basics of rote-memorization and mimicking before they can be given the opportunity to engage in thinking and reasoning to solve problems. It is an assumption that holds many kids back from the joy of truly math-ing and experiencing math success and confidence.

I am so grateful to the Math is FigureOutAble movement for showing me how to invite ALL students to math and become lifelong math-ers. 

An Invitation to All Students

If this resonates with you and you're ready to shift your own math teaching toward reasoning and teaching math as figure-out-able, a great place to start is Developing Mathematical Reasoning: Avoiding the Trap of Algorithms by Pam Harris. This book gives you the strategies, Problem Strings, and models you need to move your students away from rote procedures and toward genuine mathematical thinking, the kind that sticks, transfers, and builds confidence across grade levels.

Whether you're just beginning to question the algorithm-first approach or are already on your journey, Pam’s book meets you where you are and gives you concrete tools to bring into your classroom right away.

You can find all of Pam Harris’ books on her website. Once you start, there's no going back to boring.