How my Dog taught me about Math

Over the summer, my wife and I adopted a puppy found in a local dumpster.  After reading about 3 or 4 books I decided I was going to teach my dog how to sit, stay, roll over, bark and……mathematics.  

How do you teach a dog math?

Very easy, but first you have to teach your dog how to bark.  Once this is done, I have trained my dog to bark twice every time I say "One plus one".  She barks three times when I say "Two plus one".  Lastly, I have trained her to remain silent when I say "Four times zero".

She understands math correct?

Before an argument is started, I do not believe she truly understand math, but only has memorized mathematical commands.  I wonder how many students go through math class with the knowledge similar to my dog; memorized facts, but has very little understanding.  

Years ago, my class was set up in a way that I was training dogs, not teaching students.  I would give students questions out of context, assign redundant homework, and lastly reward speed and repetition with marks.  My dog has taught me a more valuable lesson than I could ever provide to her; there is a large difference between memorization and knowledge.

BEDMAS last along with Lottery Winners.

I would like to address a convention in mathematics that needs to change; BEDMAS.  For those who are unfamiliar with this it stands for: Brackets, Exponents, Division, Multiplication, Addition, Subtraction.  This is how I was taught order of operations.  However, division and multiplication actually are completed from left to right, and the same goes for addition and subtraction.  Therefore, we really need students to memorize also BEDMAS, BEMDAS, BEDMSA, depending on the order of the operations.

Also there could be operations inside brackets, so in actuality all that must be memorized is B(BEDMAS)EDMAS, B(BEMDAS)EDMAS, B(BEDMSA)EDMAS, …

I hope we are all confused as I am.

I have had a conversation recently where I was told, “If we don’t focus on order of operations then 3 + 2 * 10 could be 23 or 50, and there can only be one right answer”.   Now without a context 3+2*10 is 23.  However, students need to know WHY to put brackets, and when to add sometimes before we multiply. As for 2 examples:

If you are taking a cab which has a base charge of $3, and $2/km, then it would cost you $23 to take the cab 10 km.

If you are planning a party for 10 people and want to supply cookies, which costs $2/person, and cake, which costs $3/person, the total cost would be $50.

I understand that we need a procedure to solve questions that have no meaning, but for students who are just learning a new concept there should always be meaning in their learning.

This idea of memorization through acronyms can been in many subjects:

HOMES – Great Lakes

Mrs. Vasquez Eats Many Juniper Seeds Until Nurished – Planets.

I could go on and on.  I am not saying knowing the order of things is unimportant but should not be more important than the meaning and application of the knowledge memorized.  For my own students, I would rather they understand why the order of the planets are important for the equilibrium of our solar system than just the order of the planets.The sad truth is that the only time students are going to see a question, as 3+2*10, is when they need to answer a skill testing question for a prize.  Are we teaching critical thinkers, or lottery winners?

Understanding first, memorization second

NY times posted the following article.  The article claims that,

“Taking a test is not just a passive mechanism for assessing how much people know, according to new research. It actually helps people learn, and it works better than a number of other studying techniques. “

After laughing I continued to read on.  The above statement is supported by

“students who read a passage, then took a test asking them to recall what they had read, retained about 50 percent more of the information a week later than students who used two other methods. “

Alas, the research opens the reality doors.  Of course, if you are asking students to only RECALL information, it makes sense that they more they see a certain topic the more they will understand.  I don’t want my students to recall information I want my students to understand concepts.

According to Jeffery Karpicke, “I think that learning is all about retrieving, all about reconstructing our knowledge”.  This is the idea that understanding is the outcome of memorization.

I disagree greatly!  Memorization should be the outcome of understanding. Recalling information will become easier as the students use the knowledge more and not because of force or requirement.  If I was to tell you a fact over and over again, would you understand why the fact is true?  Students should be told the why part, and after understanding it and applying the “why”, the memorization will occur.

Understanding will always last longer in a mind than memorization.