Logical Connections

• If you can understand your teacher's descriptions of what math is, how it works and how to make connections between concepts and the facts you are working on, you are developing an understanding of mathematics.
  But you have to understand more than these connections between fact and concept. You also have to be able to explain these concepts in a simpler way, as if you were explaining them to someone in an earlier grade. You also need to point out and identify principles in the math problem you are working on that help you to make the math work out correctly. When you meet up with a new concept and recognize it based on math you've already learned, you are beginning to understand mathematics.
  When you first begin learning math, you start with manipulatives such as coins or pieces of candy, and 1 + 1 = 2. As you progress through the grades, your math becomes progressively more difficult so that you have to develop an understanding of old and new concepts based on what your teachers taught you in earlier years and what your current teacher is having you work on. You begin from concrete steps (the candies) and progress to abstract concepts (logarithms and powers).
  
  

Mathematical Mindset

• Math works from rules that have been created to explain different concepts and how these concepts relate to the real world. In using these rules and making the connection to the real world, math has to make sense to students so a deeper understanding of the world can be gained.
  When you discuss the Pythagorean Theorem, for example, you think you're talking about triangles and numbers. In reality, you are discussing a theory that was developed centuries ago when a mathematician realized the connection between these triangles and numbers and how they related to the world. This Theorem (a2 + b2 = c2) is still widely used today, so students need to understand its origins and how it affects math and science today. Math majors have learned how to make this connection because they've learned to think mathematically. Math is a language, not just a jumble of numbers, letters and random symbols meant to confound and infuriate us. Learn to speak the language, and the land of mathematics won't seem so strange and alien any more--in fact, you'll feel right at home.