Category Archives: Mathematics

My February Clicks



This covers ALL math!  This is by far the best math app that I have seen.


Truly this has to be the coolest yet.  I have looked at several apps and found this one ideal for history and science taught the classical way.  Full of activities for students, including quizes.


I have been looking for ways to meet core standards.  Despite the problems and current environment surrounding this rather hot issue, I have found that many of the standards, particularly in history, are emphasizing things that we, as classical home-schooling families, already consider important to the logic and rhetoric stages: using document-based tools and lessons.

Internet History Sourcebooks

Reading Like a Historian

Library of Congres:  Constitution  Includes the Teacher’s Guide and Resources.  I am using these resources for a week’s study.

Ben’s Guide to U.S. Government An older site, but it displays well on a tablet.

The US Government’s Official Portal for Kids A great place for the grammar-stage learners. (BETA) (Free for schools and up to 100 books can be downloaded each month for the membership fee.)

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Filed under General, Mathematics, Resources, Social Studies

Multiplication Copywork

I will confess that I did not focus on multiplication memorization last  year.  As a matter of fact, I handed my student a multiplication chart.

Oh yes, I did!  Our public education had already failed us miserably in this area of learning.

I was not going to “wait” for the facts, and I didn’t.  However, I use and used a variety of tools, and I require copy-work …. FOR MATH!

Download Multiplication Copywork (pdf) Zip File

You will find the zipped folder contains the multiplication fact copy-work from the 2’s to the 12’s. I do not see any reason to complete 0’s or 1’s. I do not think that you need any explanation for why. It should be rather evident.  However, I will be adding addition, subtraction, and division soon.  (I gathered multiplication early for the Hive!)

A few things more …

If you use a memory book organizer, I would encourage you to make a list of each set of facts on index cards or insert flashcards for each set.  Have your student read through them as you decide to schedule.  I add fact cards to Friday.  I change the set once I feel that most of the list is retained in memory.  I move them to another day for review.  (Being that we are fastly approaching the concept of inverse operations, I am using the triangle flashcards that are available at for review and practice.)

Memory:  Triangle Flashcards

In addition to the above, I created a multiplication and division practice book.

Multiplication Practice and Drill

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Finger Counting

I learned to add and subtract in the elementary grades by understanding how to count.  I repeated my numbers with due diligence when prompted by my mother.  I counted forwards.  I counted backwards.

Addition was counting up.  Subtraction was counting back down.  This was an easily understood concept later demonstrated by macaroni noodles, buttons, and blocks.  It was simple counting, which is why I also encourage teaching math facts as soon as counting is accomplished.

Let’s digress. We teach our little ones to hold up their fingers and count to ten.  How many times did you hold up a finger and run through the digits until you reached 10?

Many math resources use an abacus, number line, or special blocks.  If we use physical objects to demonstrate or enable problem solving, why can we not use our fingers?

Have you used a manipulative of some sort?  Regardless of what you use, you are still using a visual representation.  I want to note that I will always have my fingers with me, or at least under normal circumstances I will.  If not, I can use my toes!  (I will only need to commit to wearing flip-flops.)

Is there anything wrong with using your fingers?  In ancient history, it is shown that Greece and Persia used a method of finger counting.  We subscribe to classical educational techniques, but we recoil when children use their fingers?  (The classical finger-counting method used by these ancient civilizations is explained by Estellvenia Sanders.)

If you can complete the same problem like another who does not use fingers, why balk?  Are we so intent on focusing on speed that we forget with math that it is the accuracy that matters?

This, of course, means that I must make an off topic statement.  I detest and discourage timed tests for math!  I see no reason to fail a student for taking too long to get the right answer. 

Consider that you may very well have a visual learner.  As you have more than likely experienced over time, learning math facts has become or is near impossible.  The numbers need to be represented in order for them to understand and to master the arithmetic.  Do not be discouraged.  Through repetition the finger solutions will inevitably become memorized.  How many times do you solve one plus two with your fingers until you have committed the answer of three to your memory?  We learn by doing and by repeating.

At this point, we could argue against finger counting.  Speed is sacrificed.  You could also argue that poor memory is the fault of the child not putting the time into the task.  So, memorization is a part of math.  I concede, and I agree completely with both arguments.

However, if the concepts of the arithmetic are understood, then you have learned the theory, or number sense.  You just have not successfully learned the math facts of the four basic operations.  Yes, it would make the arithmetic easier, but does understanding follow memorization too?  I would rather have accomplished number sense versus math fact.  With the understanding, you can progress to higher arithmetic or math concepts.  There is no need to hold a child back because they use their fingers and fail to memorize facts.

So, how do you move on without math fact memorization?  Concept and theory do not go hand in hand with fact memorization.  You must weight carefully where your own balance and focus will be as well as the time to push forward regardless of memorization.

Memorization is a key component to academic studies.  You cannot disregard the need in math either, but it should not be criteria for moving forward with learning.  As well, consider not teaching the mental tricks until the child understands the numeric context of why the tricks work.  Stick with strict memorization in early math instruction.  For those that are visual learners, create pocket charts.  While you will need to develop tools, like charts, in order to keep from pulling your hair out while your child adds nine times eight, you do not have to stop the learning.  Not memorizing the facts does not stop them from understanding or prohibit them from moving forward, although you will need to make concessions through chart use or extra time for work.

In reality, this will depend greatly on the type of learner that you have.  I have a fantastic memory, but I still often use my fingers!  At the same time, I am teaching addition math facts 1-10 to the little “terrorist” who is four years old.

Finally, ask yourself if it is so terrible that your child still or has begun using their fingers (which they often start doing out of the blue).  Is it that much of a problem?  Is the balking by the current math fads truly something to pay attention too?  Math U See says go back a book.  Should you really? No. Can it really hurt number sense?  No.  Will it stop at some point?  Maybe.

Finger Binary

How to Count to 99 on Your Fingers

Counting with Fingers – Is it Bad?

No Abacus Handy?  Use your hands.


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Rote Memorization of Math Facts

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

Trends in Math Achievement

Mathematics and Science Iniative Concept Paper

Why Memorize Math Facts?

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.


Filed under Mathematics