What are the math facts? These are the standard set of “1+1=2” or “5-2=3” problems that we, as adults, have stored mindlessly in our brains. I include in this the multiplication and division facts, but I teach skip counting instead.

I abdicate that while our children are young that these facts should be stored in their spongy brains. I can hear you gasp now! Please close your mouth. We don’t want to encourage flies to travel that oral cavity or cause a choking hazard.

Before I counter the arguments that are forming as you read this post, let me present to you that this is no different from how the child eventually learns to read.

Mentally draw three connected lines on your mental canvas with your pencil of thought. This is now a symbol. It has no purpose. It does not mean anything.

Now, I want to teach you to call it by its name. We will name it “Mike”. Forever and ever, these three lines – drawn exactly as they are now – are known by this name.

Every time you see these three lines, I want you to say “zeph”. Forever and ever, these three lines known as “Mike” will prompt you to say “zeph”.

None of this makes much sense. You have no reason why those lines are called by this name or why you are not speaking the name of those three lines when you see them. All you know is that your mommy and daddy, or teacher, smile, clap and praise you when you say “zeph”.

This could have been any letter of the alphabet. Are you seeing things as your child does now?

Fast forward, eventually the connection to these symbols and their combinations will make sense. We will teach them to read! Until such time, these are random lines for which the child has a name and can respond with an oral sound. They do not make the connection that the sound will be text on a page. They don’t care. We don’t care. Together, we lay a foundation of indifferent memorization that will later have purpose and use.

Consider teaching the math facts. You can teach different orders of numbers and prompt responses for “1+1=”. No, they will not understand the foundation or concept behind all of the facts anymore than they did the letters that we taught them to so dutifully repeat and respond to from our prompts and questioning. Yes, this is abstract. However, it is simply counting up or down!

“I think it’s important to work by the principle of building understanding, rather than using rote memory only.” – Dr. Ruth Beechick

I have read many articles, and yes, I agree fully that children need a good foundation for understanding. However, I do not think that fact memorization is dependent upon understanding. Dr. Beechick’s defense of the inability to memorize multiplication facts is attributed to the lack of understanding multiplication. I beg to differ. Memory work is not associated with the skill. The memory work or memorization is a tool for the skill – much as learning the letters and their sounds to later learn to read.

Many of the points made by the popular home-school guru, Dr. Ruth Beechick, are not ones that I agree with entirely. She does not see value in drill or fact memorization. These are points that are not subject to dismissal in our learning. Facts and drills are a must.

At this point, you are still fussing and arguing with your monitor. However, consider the vast amount of mathematical knowledge that your child has acquired already. By four years of age, children work with cardinal numbers, sets, order, and general problem solving already. They have a foundation, even if it is not one formally handed from a text. They know the basics of addition and subtraction. As soon as this is demonstrated, even informally, there is no problem with teaching math facts without the manipulatives, number lines, or workbook pages. Just skip it. Learn them.

It is a fact that number lines, fingers, and counters (tools we use to teach the concepts by the way) will delay the memorization of math facts.

Are you still shouting at the monitor? This not nearly as strange as those three random lines!

Even the National Council of Teachers of Mathematics (NCTM) concedes and declares that the primal time to teach facts is between pre-school to the second grade. This is a narrow window. As well, the NCTM states that using math language early is every bit as important.

In defense of the math facts, the lack of fluency in math fact recall will no doubt hinder problem solving and hinder higher order math concepts. Consider solving word math. Does your child have a problem? It is probably related to fact mastery. Why would you wait until it is a must to have to learn the facts? Why make it an obstacle that has to be handled? If you treat it like the ABC’s, with as much casual acceptance too, you won’t have an obstacle.

Rote memorization was replaced in the early 1990’s in favor of conceptual understanding while rejecting memorization and declaring it unnecessary and unwarranted. This was also the point in which higher math or earlier instruction of formal math became the norm. Unfortunately, this is also the point at which computational skills became less of a focus and sacrificed. (Ask college math professors what it is like to teach trigonometry to these students.)

During this evolution, the use of manipulatives in place of numbers became the pop-culture fad. While I agree that the hands-on illustration has its place, what are we doing to math? The same thing that we did to phonics, but with manipulatives and number lines! Learn the math facts and master basic computation first. Do not hinder or muddy the math water with complex mathematical concepts or using manipulatives long after number sense should have been mastered. (Margaret Groves, M.Phil., M.Ed.)

Do not get caught in the “new” math era. We are striving to implement and benefit from classical techniques and educational methods.

Why would you not take advantage of this stage in which the brain is like a sponge that continues to clean up spill after spill of rambling fact?

With all of this pointing, I must say that I have experienced both sides of learning math facts. My oldest daughter is a product of learning facts early. She has no problem with math. My middle child is a product of the delayed fact memorization and “new” math teaching style. She went to public school for the first years. She struggles with math facts and learning more complex arithmetic as a product of not having the facts memorized. How much easier to have them fluent would it have been? I can report that the understanding would have been easy to teach, and it was for my oldest. Teaching and building upon those concepts has been difficult because memorization held no importance to my middle child. She was a product of needing to “understand” versus rote memory concentration early in her education.

Preschool Children’s Mathematical Knowledge

Mathematics and Science Iniative Concept Paper

Two Reasons to Memorize Math Facts

Another point entirely, but worth the comment, you don’t have to touch math. You can’t always touch math. Use caution with manipulatives and number lines! You may do more harm than good. Read Number Sense.

Brilliant. If it’s okay with you, I’d like to link to you in a new section on the http://www.Count10Read10.com site called “Why Count 10?”

Thanks for writing this!

Thank you for the link. I completely agree with the design. It is so important to both arithmetic and language arts to make habits. I look forward to further investigating this site. I was in agreement with many of the practices already!

This is always a good topic. I have spent much of time considering it, from both sides. There is such a rich interplay between the facts and the concepts. I am impressed by the best conceptual teaching (read VanDe Walle), but I am also impressed with the best direct instruction (when I was working on my mathematics education degree, my adviser was horrified by this). I know adults from all mathematical backgrounds ranging from “laissez-faire” to “nuns with sticks.” As adults, their ability and comfort with mathematics also seems to range the gamut, with no relation to their background.

I have more and more appreciation for the power of memory. As an adult, I have spent many hours memorizing the things I should have learned as a child.

You speak to the attitudes of most homeschooling parents. They, too, range between the extremes that you referenced. I appreciate your wording. It demonstrates descriptively how this is approached across all subject content too. Math is only one subject that I find a need to push for direct memorization, and it should not be the only one.

Note: I find the value of memorization to be an asset, and I recognize it. Perhaps this is because so much of my work is reliant upon memory for the math concept and logic analysis involved in working with technology, both hardware and software. I only wish I could find words adequate enough to prove that even day-to-day math would benefit and be rewared!

Thanks for the reply to my comment. I have a lifelong fascination with memory and memorization, perhaps because my own innate powers of memory are so poor!

Since elsewhere on your blog, you ask for blog links, I will give mine here:

http://www.urbanmythcafe.com/blog/

I am brand new to blogging, but my blog entirely concerns my own experiences with learning mathematics.

How would we speak, without a vast library of memorized words, thoughts, and expressions? Taking that isdea further, how can we think without that library? In math, the same thing is true, we need a library of fluent mathematical knowledge in order to think mathematically.

Nice Post,

It is extremely important to find out a working way for memorizing the maths formulas. I like all the points you mentioned here. We are running maths schools in Sunnyvale, Ca and we aim to teach mathematics in very effective ways. Our expert tutors have been using various strategies similar to ones you mentioned.

Regards.