## Coffee and Mathematical Understanding

I spent the majority of this past week visiting with friends and family.  My destination ended up being in snowy and icy northern Michigan.   The house that I was staying in lost power for five straight days.  Thankfully the house had an efficient fireplace and a small gas-powered generator that ignited a few space heaters to keep one of the rooms fairly warm as outside temperatures hovered around 20 degrees.

In normal circumstances, the first person that wakes up in the house starts to brew the coffee for the family.  Since we had no power the coffee machine wasn’t an option.  You see, my family definitely enjoys their coffee. Being official coffee addicts my family has a decent understanding of the integral parts of the coffee-making process: hot water, filter, coffee grounds and cup. The one missing ingredient in this process was the hot water.  One of the family members found a pan and began to boil water on top of the fireplace.

The water was then used to complete the coffee making process.  Success!  All of the family was able to sit around the fire and drink our coffee.

As you can imagine or already know, they’re many ways to make coffee.  My family knows this and that understanding led us to a solution that was adequate.  We substituted a different process in the coffee making flowchart and arrived at decent tasting coffee. Regardless of the process used, the user ended up with the same solution.  In the end some type of hot coffee was served.  Understanding the key components of any process allows opportunities to substitute yet arrive at the same solution. Because my family knew the process in-depth we we’re able to substitute the missing item and still have the same result.  I feel like this type of thinking applies to the classroom.

Having a limited understanding of place value and number sense can limit opportunities for students.  Students need to be exposed to an array of methods to complete problems, not just shortcuts.  Only understanding the formula/shortcut doesn’t necessarily show mastery of a particular concept.

At the upper elementary level students are expected to find the product of  3+ digit numbers.  If students have been exposed to using only the traditional method to solve these types of problems they know how to multiply large numbers using one method.  Although that process might be effective for them, it doesn’t cement a deep understanding of multiplication.  Students often have problems when decimals are introduced when finding the product of these types of problems.  Having a more in-depth understanding of place value and multiplication can give students the tools to solve more complex math problems.

On the other hand, if a student has been given opportunities to use repeated addition, partial-products, lattice and traditional methods, students might have a better understanding of the role that place value has in the multiplication process.  Having that understanding of place value will help students when they approach decimal computation and throughout their academic career.  Having multiple tools/strategies also encourages students to be independent and choose the correct method to find a solution.  Even more important, students that are then able to apply their mathematical understanding to practical situations (beyond the test) can often immediately see the benefits.

In the end my family used the tools that we had (fireplace, metal pot, water) along with other materials (coffee filter, ground coffee, cup) to create coffee that we all could enjoy.  If we were fully reliant on just the coffee maker and electrical power, we wouldn’t have had coffee to drink.  Understanding the details of the process and that there are multiple ways to find a solution is an important skill to have as adults and as students in the classroom.

The power came on and I believe we all found a newly acquired appreciation for the electrical grid in Michigan.  Our coffee story is unique and yet I feel as though it’s mathematically relevant as teachers will be back in the classroom to start the second half of the school year.  Enjoy the rest of 2013 and I look forward to a successful 2014.

## Math Shortcut or Conceptual Understanding?

Image by:  Renjith

There are “tricks” or “shortcut” techniques that many teachers have in their tool belt when it comes to teaching mathematics.  These techniques are often memorized by students to be used later on some type of assignment.  I’m not saying that these types of techniques are good or bad, but often students come away with little conceptual mathematical understanding of the concept being taught via the shortcut. Click on the picture below for more information on conceptual understanding.

Without that understanding, students are not necessarily being prepared to apply the concepts later in middle school/high school.  A very small sample of the short cut techniques I’m referring to are below.

PEMDAS – order of operations

King Henry Died Drinking Chocolate Milk – Metric System

FOIL Method – Algebra

Negative Multiplied by a Negative = Positive

Gallon Guy/Gal = Capacity

Students generally remember the shortcuts and utilize them on assignments/tests.  Is that a bad thing?  I can almost hear math teachers around the world grumble.   If the students truly have a conceptual understanding of the concept then why not use these techniques?  Many of these types of shortcuts are used at the late elementary level. When students understand the technique – such as King Henry .. but don’t understand the concept (differences between the units of measurement and in what context they can be applied) then students/teachers run into problems.  Students are expected to be able to apply the concepts in multiple situations. Middle school teachers are then are held responsible to deepen the mathematical understanding of the concepts behind the techniques that were briefly utilized at the elementary level.  This topic has been on my mind lately, and finally made it’s way into this post based on this post.  Elementary teachers, as most teachers do,  attempt to use innovative and engaging methods to produce excitement related to learning and school. That motivation is often contagious and beneficial.  Whether teachers use these techniques or not (obviously, it’s up to you!) students should understand the concepts before memorizing nifty sayings that don’t really relate to the concept itself.  I’m not blaming teachers for using these techniques to engage students, but ensuring that students have a mastery of the concepts should be near the top of the priority list.