Strategies to Maximize Math Instruction

By Dr. Michele Douglass

Multiple researchers discuss the best practices to use to maximize student achievement in mathematics. The good news is that the authors share many of the same big ideas.

One thing that stands out is a strong focus on procedures that has existed for years in the elementary grades. This emphasis is linked to the language and testing of our state standards, but lacks problem-solving development and the foundations of knowledge needed for higher-level mathematics.

The strategies that exist across multiple authors for improving mathematics achievement include:

  1. Making number sense a part of everyday instruction
  2. Focusing all content around problem solving
  3. Using communication every day
  4. Supporting learning with the use of tools and representation for all levels of students

1. Make NUMBER SENSE a part of every day’s lesson.

Number sense doesn’t build in a single chapter or topic; it builds over time. By providing students opportunities every day to build number sense, we help them become better problem solvers. They also learn to think about the size of the number while they learn to work with numbers in flexible ways.

Building number sense also develops estimation skills. We all know that in the real world, we rarely as adults pull out a pen and paper. Rather, we think about the numbers to estimate solutions. Think about how you solve the problem 15 x 16 without a pencil or paper. What about 45% of 250? We must support students to think about numbers in multiple ways so they don’t have to rely on an algorithm or the calculator on their cell phone.

Making this happen isn’t as hard as it seems. You can incorporate number sense into your warm-ups by figuring out the number sense that scaffolds into your lesson. Use a timer, as it’s easy to make number sense an entire lesson. Set the timer for 10 minutes. If you are working on exponents, your number sense might be on multiplying repeated factors to see how students group the factors to find the product. If your lesson is on multiplication, your number sense might be on multiplying numbers by 10 or 100. You might estimate your age in seconds or the height of 1,000 or 1 million pennies. You know the set of number sense topics that are critical at your grade level. Use these specific topics as the basis for your number sense problems. Some days, you might do a single problem, and as students learn methods for thinking about numbers, they will be able to do more than one problem.

2. Integrate more PROBLEM SOLVING.

Students learn new skills through the process of solving problems such as learning facts. Problem solving is a great way of connecting conceptual knowledge with procedural knowledge. While we often think as adults that the problem-solving problems are the hardest, children often need the context of a problem to connect the meaning within a procedure.

To begin with problem solving, choose problems that are open-ended, allowing for multiple ways to arrive at a solution. Problems need to allow students to make and test conjectures. They should foster creativity while either using formulas or connecting procedures to concepts. Many times you can find a problem in your textbook that you can turn into an open-ended problem. For example, turn a simple area problem into a comparison of two sets of dimensions and add a context. Which has the greater area and why? Ask students to justify their answers in more than one way.

3. COMMUNICATE, Communicate, communicate.

The one thing to remember about communication is that you can’t communicate either in written or oral formats without having the knowledge to express the idea in a coherent manner. To support students in communicating, begin with a safe learning environment.

Through communication, students are orally processing what it is they think they know. However, communication also gives the instructor the opportunity to be aware of how students are thinking about a concept or set of symbols or even a definition. For example, if all you ask is the answer to 2 to the 4th, you might not realize that the student is simply multiplying 2 times 4. When you ask students to go beyond giving you an answer, you learn whether they grasped that 2 to the 4th is the same as 2 times 2 times 2 times 2.

Oral and written communication in mathematics also supports language learners and students who struggle with language development in a content area. By speaking and communicating, students are building language skills and specific math academic language.

One way to begin with communication is to ask the question “Why?” And don’t just ask this when a student provides an incorrect solution. Asking “Why?” all the time makes students rethink the solution and the steps they used to arrive at the solution. Asking “Why?” to a student whose solution is correct enables you to hear the student’s process. If the process is accurate, the student is providing teaching to the class. However, there are many times when a student is getting the correct answer for the wrong reasons. If we never ask “Why?” then the student is being set up for making continuous errors.


Tools and representations help students build relationships among numbers, construct knowledge and meaning of concepts and ideas, and make connections between concepts and connecting procedures. Tools and representations have also been found to help maintain a positive attitude about mathematics, as they support the sense making of mathematics. Mathematics is abstract even from the nature of the symbols. There isn’t anything concrete about the way you write the number 5 to know that it represents 5 things. This must be learned. Tools provide methods for solving problems by allowing space to organize, think, reason, and test ideas.

As you begin with tools and representations, refer to your text and capitalize on the representations used by the authors. Supplement to use a variety of items. As you use different manipulatives, help students transfer understanding by showing a representation of the manipulative on paper. For example, if you use place-value blocks, you might teach students to draw a square to represent the 100 block but to use a line for the 10 rod and a dot for the unit pieces.

It takes time upfront, but the time we spend reviewing could be minimized if we spend more time supporting students in building concepts and strategies through these methods.

Michele Douglass, Ph.D., is the president of MD School Solutions Inc., a company that contracts with school districts on content and pedagogy with teachers and leaders. Her experience ranges from math instructor to director of curriculum and instruction at Educational Testing Services. She has authored several math curricula, as well as professional development and technology programs.

About Michele Douglass

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