Teaching Students Sense-Making in Math Problems

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  • 0:05 Logic & Number Sense
  • 1:52 Estimation
  • 2:35 Techniques for…
  • 3:30 Techniques for Middle…
  • 4:30 Correct Answers
  • 5:15 Lesson Summary
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Instructor: Meredith Mikell
Teaching math strategies is one of the more challenging subjects teachers face. We will identify how and why to teach students to use logical reasoning in solving math problems and identify useful tools for doing so.

Logic & Number Sense

Imagine that you are teaching a sixth grade math lesson on inequalities. The students are working through a problem: 15 + 4 = 18 + x. Perplexed, some have both hands up as they count each side on their fingers, while others are writing down a subtraction problem to find the value of x. You tell them all to stop working and look up, because you have a way of solving this problem that is far simpler: using basic logic, or the use of sound reasoning, to solve a problem.

A more specific way of explaining logic in this way is identifying sense-making, using an intuitive understanding of mathematics and common sense to solve problems. You explain to them that you don't have to break out fingers or calculators to see that since 18 is 3 more than 15, so in order to make both sides equal, x must be 3 less than 4, therefore x = 1.

Teaching students sense-making not only saves time in solving a problem, but it also trains students to implement basic quantitative reasoning before plugging through a prescribed series of steps. Two key components of sense-making are number sense and estimation.

Number sense is the intuitive understanding of magnitudes, ranges, and estimates of numerical values based off of both logic and experience. Number sense is not only critical to the development of strong math skills, it is also a necessary element of basic reasoning through everyday situations.

For example, knowing that one kilometer is a shorter unit of distance than one mile would be helpful when traveling in a country that uses the metric system. On math problems, number sense allow students to first use logic to eliminate incorrect possibilities before calculating an answer.

Estimation

Estimation involves identifying the ballpark correct values to a mathematical problem, and can be a very important step in sense-making. This goes hand-in-hand with number sense. For younger students, preschool through elementary age, this can be best achieved through practice and frequent exposure to familiar mathematical values.

For example, with preschoolers, pointing out how many hands or feet each person has and how many fingers are on each hand are great basic exercises to develop number sense. A preschooler who grasps these basics of number sense would likely laugh at the absurd idea of someone having one hundred fingers!

Techniques for Elementary Students

Elementary-age students benefit greatly from using measurement tools to discover the length, height, or weight of various objects and compare them. Activities in which they sort objects they have measured in increasing magnitude help them further develop their sense of what an approximate value should be for a given set of objects.

For example, after measuring that an apple weighs about two pounds and their teacher weighs about one hundred and twenty pounds, a student can then be asked how much they think a watermelon could weigh: one pound, ten pounds, or one hundred pounds.

Another helpful and quick exercise in estimation would be to ask students to guess various values associated with their school day, such as: 'About how long does it take us to walk to the music room: two seconds, thirty seconds, or two-hundred seconds?' Students could then time the walk and see how close they came to the actual value.

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