Math Instruction

The National Council of Teachers of Mathematics (NCTM) recommends that teachers use tasks that:
  • Invite exploration of important mathematical concepts
  • Allow students the opportunity to solidify and extend knowledge
  • Encourage students to make connections and develop a coherent framework for mathematical ideas
  • Call for problem formulation, problem solving, and mathematical reasoning
  • Provide more than one solution path
  • Promote the development of all students’ dispositions to do math
Our expert math coaches work with your teachers to: 
  • Discover how mathematical tasks and questions differ with respect to the level of thinking required to solve them
  • Deepen understanding that learning mathematics involves students’ constructing ideas and systems
  • Recognize the role of productive discourse in students’ mathematical reasoning and thinking

Math Instruction 

Equations & Expressions are a major part of the 6, 7, 8 and Algebra Common Core Standards.  Up to 30% of the NY state math assessment (grades 6-8) consists of items aligned to these standards.  The core of algebraic thinking is to take a real-world scenario and condense it into a simple and eloquent mathematical equation or formula--a skill that goes well beyond the classroom walls.

​We know that students have trouble solving and writing equations. Often, the trouble lies in their conceptual understanding of a variable. Yet, we continue to use a traditional approach that emphasizes procedural steps.  In other words, we ask students to replicate procedures rather than understand concepts.  Take a look at the equations below. 
The most common approach to teaching how to solve equations is to perform the same operation to both sides of the equation. In the one-step multiplication problem above students are taught to divide both sides by 2 to find x. In the one-step division problem students are taught to multiply both sides by 4.  In the two-step multiplication problem students are taught to subtract 5 from both sides and then divide both sides by 2. Finally, in the two-step division problem, students are taught to subtract 3 from both sides and then multiply both sides by 2. What are students really learning? They are taught to mimic, to replicate, and to follow instructions. 

Alternative Approach

We offer an alternative instructional approach.  Instead of asking students to replicate a procedure, we ask students to think about the meaning of the equation. Students first put the equation into written and/or verbal words.  Translating mathematical concepts into students' natural language deepens their conceptual understanding of the concept. 
Once students  create a verbal or written statement, they then use the guess-and-check method to find a value for x that makes the equation true. Isn't the guess-and-check method a cumbersome and inefficient strategy? Yes, of course. But it builds understanding and promotes algebraic thinking.  Most importantly, given enough time, many students discover the procedural method on their own. A student's own mathematical discovery is more likely to stick. 

Try It! 

Spend a few lessons using this strategy! We've created these resources (with answer keys!) to support this alternative approach.  Each problem was carefully constructed to ensure integer solutions.  Give students a calculator, ensure they write/and or verbally express the equation and use the guess-and-check strategy. Keep count of how many students discover a more efficient strategy on their own. And let us know! 

Multiplication One Step
Multiplication Two Step
Division One Step
Division Two Step
Variables on Both Sides