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The Math Debate?
John Carroll
Tuesday, January 08, 2019

Recently someone made a reference to the nation-wide “math debate” and advocated for students to practice math facts. She drew a parallel to music education and noted how competent musicians spend a lot of time practicing scales. Her analogy reminded me of teaching young marital arts students who wanted to know the “secrets” of Karate. The most obvious “secret” is to practice basic skills… a lot… until they become routine (boring) and then practice some more. 

So I agree that mathematical competence requires practice and rote learning much the same way that music or Karate does.  In fact, becoming good at almost anything requires a lot of basic practice.

The problem with traditional math instruction is not that it demands rote memorization and practice. The problem is that it creates the perception that rote memorization and practice ismath, that “real math” is about knowing how to solve equations and get correct answers quickly. Tragically, students who don’t have a natural knack for speed, accuracy and memorization (which aren’t math) often come away from school thinking that they are bad at mathand do their best to avoid it later in school. Many carry this belief into their adult lives. We have lost countless creative and innovative math practitioners simply because we sent them the wrong message in 4thgrade. 

We don’t judge musicians by their ability to play scales with speed and accuracy and we don’t award a Black Belt in Karate to a student because she can demonstrate basic punches and kicks. We also shouldn’t judge mathematicians by their ability to do math facts and apply set formulas with speed and accuracy. 

The drill-and-practiceadvocate at the Board meeting said that kids need good “foundational” skills: math facts and formulas.  I agree that kids should know facts and formulas but I strongly disagree that those things are foundational.  

That is the heart of the current “debate” about math.  

In order for us to improve math learning among our students, we have to undo a long history of belief that was based on a misunderstanding of what is foundational and what is not: I see many older teachers and parents always trying to attach problem solving or practical application to non-foundational math facts and algorithms. (If I can’t tie a complex problem neatly to an algorithm, it must not be worth doing!)  Even the adults who understand that math is not memorization sometimes find themselves defending engaging and relevant math applications, puzzles and games on the grounds that they help kids learn their facts and formulas. It is difficult to move beyond that backward misconception; I continue to be surprised at how intractable it is.  

But we need to get over it; too much is at stake. We need to get behind the idea that algorithms and facts (though important) do not form the foundation of math.  They rely on the real foundation: deep conceptual understanding. Facts and formulas are tools that can be applied only by those who have a strong foundation.  

To keep it simple, let’s look at addition:  The foundationof addition is the understanding of the abstract principle that one or more variables combined into a set with other variables can be expressed as a number that is dependent on what the preceding variables were. 

Without that foundationalunderstanding, math facts (such as 2+2=4) have no meaning. They are shorthand expressions that can be useful for calculations… but only for people who understand the underlying principles. Same with subtraction, multiplication, and so forth. You might train a parrot to repeat the multiplication table from one through twelve but the parrot will not be doing math. (Sadly, the parrot would get a better grade than some 4thgraders who actually understand the concept and can apply it but are not as fast and accurate on a timed quiz.)

The concept of addition is actually more complex than we might recall. As adults, we understand it, so we are comfortable going quickly to (non-foundational) calculations to apply it. We get so accustomed to our favorite method of calculation (usually an algorithm) that we begin to think of the short cut as real math.  

But for a young child, it takes time and consideration to master the concept.  For example. If you add two apples to three oranges how many do you have?  Well, the answer depends on what you are categorizing. If it’s apples, you can add oranges all day and still have 2.  If it’s fruit, you have 5.  What if you are adding total weight?  2+2=4 only applies when you know when to use it. 

The point of learning math facts and algorithms is to get kids to use them as tools when appropriate and not to view them as the final goal for their learning.  Traditional programs focus nearly 100% on calculations and offer “application”, word problems and puzzles as though they are extras that might help some students understand why they are doing the abstract calculation… if they are still awake.  It’s totally backward.  No wonder so many people forget what they learned in math class so quickly after graduation.  

Try this little experiment that I got at a recent training at Stanford University: 

1 ¾ divided by ½ 

  1. Calculate it and give the correct answer. (algorithm)
  2. Draw a diagram or picture showing what the equation means- don’t use numerals.
  3. Explain in writing what the equation means and how to solve it.
  4. Describe a real-world application.

Students with deep conceptual understanding can respond competently to those 4 prompts. However, many students who are identified as “good at math” get the first one fast but have serious trouble from there on.  Among advanced high school math students, #2 can be very frustrating.  One honors Trigonometry student I observed could not liberate herself from numerals in her diagram because the standard equation was so deeply ingrained in her thinking. In response to #4 she suggested that there probably wasn’t a real-world application. 

Even as an adult you may struggle with all 4 prompts because you have forgotten the algorithm you learned so long ago. (Who can blame you? If the problem were ever to come up in the real world, you would ask your phone! - a technique I would contend is every bit as “foundational” as doing it with pencil and paper... faster and more accurate too.) Does this mean that you are bad at math?  Of course not, it just means that you may have had a math education that stressed the importance of the tool over the importance of the task. 

The debate about math instruction (Traditional vs. Conceptual) is a false choice.  The advocates of traditional programs almost always mischaracterize the other side, saying that we are trying to throw out practice and memorization in favor of concepts, puzzle and games. (I have yet to meet a progressive math teacher that advocates the abandonment of practicing facts and learning formulas.)  The truth is we can have it both ways.  Students need a deep conceptual understanding of math and that forms the foundation for their education. With that solid foundation in place, they will be better able to approach problems from a variety of perspectives and not quit when they don’t get the “right” answer fast enough. Along the way we should teach them the shortcuts and tools (facts and formulas) to make their work more efficient so that they can continue to learn and not see math as so many of us did, as something to get through and be done with.