SMP 1. Make sense of problems and persevere in solving them.

Making Sense of Math

Calling all math educators! Have you heard of the Standards for Mathematical Practice? If not, here is the low down. The Standards for Mathematical Practice (SMP’s) are supplemental to content standards, but equally as important. They describe skills that mathematics educators should try to develop in their students. The standards stand independent of grade and course. Teachers of all courses at all levels should incorporate the SMP’s into their routine. These standards focus on process and proficiencies such as problem solving, communication, and reasoning.

The first standard is: Make sense of problems and persevere in solving them. This standard really has two components to it so let’s talk about the first one, making sense of problems. Remember that time you gave a math task to your students above something real…a real world problem. Let’s say this problem was about gardening.

When students are trained with more of an emphasis on procedural understanding, you may run into a problem with robots. Yes, computation robots! See the work of this student below. They trust their procedural understanding even though it is incorrect. "I got 42 trees because I did 6 x 8 which is 42."

The next student used their conceptual understanding to draw out the image and make sense of the array.

Students need the balance of procedural fluency and conceptual understanding so that they understand what they are doing when they use algorithms and steps. This is for the student that can memorize multiplication tables but doesn’t know how to apply it. Tricks and mnemonics are handy, but they don’t replace the understanding needed to endure.

Teacher Tip #1

In the classroom, have a balance of procedural fluency and conceptual understanding. Both are needed! As a student of No Child Left Behind, I have strong feelings about this topic. Throughout my early years as a math student, I was able to memorize algorithms and procedures quickly. I was a top contender for completing a 12x12 multiplication table in 2 minutes. However, as math became harder, I had a harder time keeping up. In high school, I struggled to know how to begin a problem and how to solve anything that was written in a non-traditional way, especially word problems. Why was this the case?

Let’s do a deep dive! Conceptual understanding and procedural fluency have been at the center of the pendulum swing in mathematics and I was caught in a hard swing to a focus on procedural fluency. Procedural fluency includes both knowledge of symbols and conventions for their use and the knowledge necessary to apply rules, procedures, and/or algorithms (Rittle-Johnson, Siegler, & Alibali, 2001). I lacked conceptual understanding. Conceptual understanding is the knowledge of mathematical concepts and understanding of the relationships/connections between concepts (Rittle-Johnson, Siegler, & Alibali, 2001). I couldn’t make sense of the problems. I didn’t know that multiplication was repeated addition. It is sad to say, but I did not fully develop a balance of conceptual understanding and procedural fluency until I was a teacher using the newly developed common core standards. The pendulum had begun to swing back.

To wrap up this tip, students need both! Conceptual understanding allows for a student to recognize how to solve a problem and then procedural fluency gives them the skills to solve it. Conceptual understanding acts as the guardrails for students to make sense of their procedures. I am not here to say how much of each or which comes first. It depends on the concept and your learners. But students need both to have a full and complete understanding of mathematics.

Take the time with your students to make sense of problems before solving them! What is the problem asking and what type of answer should they get? This will give struggling students an entry point and step in the right direction.

From CoreStandards.org: “Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals.” At the younger grades, that might include manipulatives, concrete objects, and drawings. At older grades, we add critical thinking about math relationships and using tools. These are skills that need to be developed in students.

A strategy is to do a close reading of the text with students. Break out the highlighters, pencils, and colors! Guide students through this initial process. Read through the problem and break it down into parts. Underline the quantities. Highlight what the problem is asking students to do. Make this a group process.

Next, find an entry point for the solution. Ask students to analyze the given information and suggest a way to start the problem. Take one student's plan and implement it! Work through the process outloud and demonstrate your thinking. Commit to their suggestion until something goes wrong or the problem is solved. Monitor progress towards their solution and change their plan if necessary.

If students don’t make explicit connections between mathematical ideas, teachers should help them clarify their thinking or explanations. For example, when a student highlights a strategy that would be helpful in another context, it’s important for the teacher to help the other students to see those connections.

Students need to build a tool kit for solving problems. Modeling the problem solving process gives students an opportunity to see your thought process. Non-routine problems such as word problems are especially challenging to students because they are not told what to do.

When deciding on a plan to solve the problem, students should rely on their prior knowledge and consider similar problems they may have solved in the past. They may also find it helpful to use simpler forms of the original problem, for instance using smaller numbers or a simpler situation. It is important that students construct a plan for solving the problem, rather than just attempting to solve it. For more ideas, read this previous MANGO math blog on problem solving!

Always ask the question: “Does this make sense?”

Preserve in Solving

Now, let's focus on the end of SMP 1: Make sense of problems and persevere in solving them. How often do students throw up their hands and say they can’t solve a problem? Why would a student say they can’t…because the problem doesn’t make sense to them! That is why the first SMP is a joint sentence. Students have to first make some sense to begin and then must persevere through a difficult task.

Teacher Tip #3:

Perseverance must be developed. Most humans aren’t born with the desire to complete difficult tasks. In math, perseverance is developed through rigorous problem solving. Sue O’Connell of the book, “Putting the Practices into Action,” says to “Pose problems with multiple steps or with multiple answers. Discuss ways students may have gotten ‘stuck’ when solving a problem and how they got ‘unstuck.’ Praise patience, reflection, and perseverance.”

You may have noticed throughout this blog that everything focuses on task-based problems. That is because math is a means to an end. Math is our tool to solve real life problems. These types of problems are more rigorous, but they help answer the big “why are we doing this” question.

I love Ted Talks. They are short and impactful. One of the most inspiring talks that I have seen is The Power of Passion and Perseverance by Angela Lee Duckworth. This is a little bit of a spoiler (still worth the watch), but let me tell you what Angela discovered. Leaving a high-flying job in consulting, Angela Lee Duckworth took a job teaching math to seventh graders in a New York public school. She quickly realized that IQ wasn't the only thing separating the successful students from those who struggled. In the talk she explains her theory of "grit" as a predictor of success. And guess what, grit IS a predictor of success. While some students appear more inclined to understand math, it was the students with grit that persevered to the end of a tough problem. They had the stamina and drive to overcome obstacles while students without it did not. Developing math perseverance takes students beyond “their ability” in math. The Ted Talk is six minutes and worth the watch whether you are in education or not.

Take the time to tell students when they will be given a difficult task and encourage them to persevere! Teach them problem solving strategies such as working backwards, looking for patterns, drawing, and even guess and check. Give them tools in their tool belts so they don’t give up on math problems, but try again.

Teacher Tip #4

Another way to develop perseverance is through Growth Mindset with Carol Dweck from Stanford University. As Dweck describes it in her book:

In a fixed mindset, people believe their basic qualities, like their intelligence or talent, are simply fixed traits. They spend their time documenting their intelligence or talent instead of developing them. They also believe that talent alone creates success——without effort. (They’re wrong.)

In a growth mindset, people believe that their most basic abilities can be developed through dedication and hard work——brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment. Virtually all great people have had these qualities.

Dweck, Carol S. Mindset the New Psychology of Success. Ballantine, 2008.

She teaches that the ability to learn is not fixed. She discovered that when people read and learned about the brain, they were more likely to persevere because they learned that intelligence wasn’t fixed. Failure wasn’t the end! This view of life is called having a growth mindset.

Teaching Growth Mindset is a fantastic way to start the school year or summer program. While Dweck’s work is not specific to a single subject, or even the field of education, it is powerful. Someone that took this research and applied it directly to the math classroom is Jo Boaler from Standard University (I wonder if they know each other!) Boaler has taken the idea of Growth Mindset and worked it into a curriculum called the Week of Inspiration Math! On the Youcubed website, Boaler has over 50 videos, resources, and tasks designed to build up student mindset and classroom community in all grades levels. The idea is that you pick five resources to build a Week of Inspirational Math to share with your students. I do this each and every school year and since the pandemic, I include a mid-year refresher.

Next time a student says they “can’t” solve a problem, say “you can’t solve it yet!”