The reason we have problems with the multiplication table is that we are too good at pattern matching. Start with a 9x9 grid. 45 of these are duplicates. 1,2, and 5 are trivial to remember. That leaves 21 things to remember. Of these 7x8 is probably the most difficult for me, and it is probably that the pattern seems broken. 21 things to remember is not a lot, and you could argue that 3, 4, 6 are fairly easy too. That would cut it down to 6. Perhaps some of the problem is that we teach these difficult ones the same way we teach 2, 4, 6, 8, and expect our considerable pattern matching abilities to see a similar pattern in 7,14,21,28,35,42,49,56,63 but there is not a recognizable pattern.

3 4 6 7 8 9

3 9

4 12 16

6 18 24 36

7 21 28 42 49

8 24 32 48 56 64

9 27 36 54 63 72 81

I think we are in a transitionary phase in mathematics, like the one that occured when the printing press was invented. When the printing press was invented, and the long journey to literacy started, I suspect there was loud opposition to reading because it was ruining our memory. I see a lot of this same issue in Mathematics. Many believe memorizing a spelling word, or a formula is somehow related to creative ability. When I was a student, 45 years ago, it was necessary to learn and memorize the multiplication table, but in 1975 when calculators went below $10, the need for this ended. We subject children to years of memorization, calling it mathematics, and wonder why most people hate Mathematics. We make students do long division, something that they will never use. We totally complicate the teaching of fractions. and many high school students do not know how to add 1/5 and 2/5, because we complicated it too soon. Calculation is a separate skill than pattern matching. While I understand the arguments which teachers use to justify that these skills are important, they all seem to revolve around the fact that someday they may not have a calculator, and then where will they be, or they will fail a test that does not allow calculators. These are not rational arguments. I do not know the answer to this problem, but I suspect it will disappear when the people like me who had to memorize the multiplication table are gone.

When teaching multiplication to my students, I want them to understand the
concept before memorizing facts. I have them use arrays, numberlines,
skip counting, draw pictures, etc. while I'm teaching multiplication.
While we are working on these strategies in class, I will have the students
memorize using flashcards at home. You can have the students use index
cards to make flashcards if they don't have any.

Another motivation for students is "Banana Split Multiplication." As you
are studying multiplication, have the students take quizzes to show mastery of
their facts. When they show mastery of a group of facts, they get a
bowl. As they master more groups, they can earn a spoon, banana, scoops
of ice cream, toppings, etc. At the end of the year, the students get to
eat whatever they have earned. I have seen a teacher display the
student's progress through laminated cutouts of each thing that the student
earns. After they master the group of facts, the student gets to put the
next thing on their display.

It's good for students to learn that multiplication is simply a faster form of addition. Once they see that relationship, multiplication will make more sense to them. Specifically, you can use a skip-counting game, where each person adds the factor to the previous answer and you go around the room to some specified product.

For Algebra teachers whose students are struggling with multiplication, you can add a variation that will connect this to linear equations: If you are working on 3's, the first time around, the first student starts with 0, the next student 3, then 6, 9, etc. until the first round ends. That's your times table for a factor of 3. The next round, the first student starts with 1, the next student will add 3 to get 4, 7, 10, etc. Third round starts with 2. Then, you can show them how this represented y=3x, y=3x+1, and y=3x+2.

I had once a girl who was horrible on multiplication, but crazy about horses. So if we have 4 horses, how many horses legs are there? If there's another paddock with 5 horses, how many legs then? If food requirements are 2 hay bales and 8 gallons per day per horse, how much do you need for both paddocks? If it costs $3 a bale, how much do you spend each week in hay? etc.

She knew her tables cold within a week.

Of course, this means tailoring the curriculum to a student's interest. Which means finding out what the student is interested in, which can be difficult, if not impossible if there's 30 kids in the class. Maybe assign similarly interested kids to a few different groups, one for horses, one for vampyres, etc? If 4 vampyres each need 5 quarts of blood per day, how many victims......:)

Music helps with memorizing rote facts. There are things out there like "Multiplication Rap". You can also teach them some of the tricks for certain numbers (like 9s, with the digits of the answers adding to 9) If they aren't good with skip-counting, that's another thing to reinforce to help them remember. And sometimes, if they get to a certain age and still don't know those math facts, they really aren't likely to learn. Then some accomodations need to be made so that they are allowed to use tables or calculators, or they won't be able to go further in math. A lot of kids can understand concepts for higher math but just can't remember all the facts, and shouldn't be penalized for that.....but that's just my opinion.

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