*Alexandra's article below reprinted with permission from Dr. Alicia Holland-Johnson, where it originally appeared as a guest blog posting on her Helping Tutors Become Their Best blog.
Introducing Multiplication to a Student with Different Learning Needs
by Alexandra Berube, bostontutoringservices.com
The basis of my math instruction is always to move from one concept that the student firmly understands and then apply that concept to the next level of mathematic reasoning. In working with my favorite third grade student, I found the opportunity to introduce multiplication to him. This student was born with Agenesis of the Corpus Callosum (a complete or partial absence of the corpus callosum, the band connecting the two hemispheres in the brain), and I wrote about him in a previous blog post. He learns concepts in a completely different order than most people would expect. He is still solidifying his addition and subtraction facts, and adds and subtracts any amount, including one plus one, on his fingers. And yet I know that he is often ready for more advanced concepts, and that these advanced concepts will actually help solidify previous concepts that he is still working on mastering.
In his third grade class, the student is working on geometry, including perimeter of squares. This was the perfect opportunity to introduce multiplication. In a square, all the sides are the same size. If he has a square with the side length of two, then he will do 2×4. I worked with him on this geometry concept for a while with different shapes such as pentagons, hexagons, and triangles: any shape that has sides of equal length. He quickly grasped this concept and then we moved on. I like to use a dry erase board in my instruction, because it's another form of media (‘media’ used loosely, I suppose) than pen and paper, and it allows the student to draw shapes and manipulate the written material in a new way. I had the student make shapes of his own, and we would see how many sides that shape had. We would give each side a length, and then see what the multiplication problem would be as a result.
We then worked on the worksheet I've included a link to here, which shows pictures of groups of objects (for example, four triangles with three stars in each triangle). It asks the students to write an addition problem (so, in this case, three stars four times, so 3+3+3+3) and then the multiplication problem (3×4). He picked this up very quickly, and so we moved on to the last game of the session.
Using a pair of dice, we played a game to visually show the amounts to be multiplied. First we rolled one die, and then drew the dots shown on the die on a piece of paper. We wrote the number value above the dots. Then we wrote a multiplication symbol, and then we rolled the other die, which would act as the 'multiplier.' The second die dictated how many times the first die would be multiplied by. So if the first die was a four, we drew four dots, put a four over it and then a multiplication symbol next to that. Then if the second die was a three, we wrote the number three next to the multiplication symbol, and then drew four dots two more times for a total of three sets of four dots. This way he could see why we were multiplying--we were adding the same number a multiple of times.
He then added all of the dots on the dice and found out that 4X3=12. Below all the dots we wrote 4+4+4 (the addition problem like on the worksheet I just described), to further enforce that multiplication is an extension of addition. He'd already mastered adding groups of numbers, so this was the next logical step. He smoothly transitioned into a student who understands the basis of multiplication. Of course, he's not going to be memorizing his multiplication tables in the near future, but he understands what multiplication is now, and he grasps that it is an extension of addition that applies in real life.
About Alexandra Berube
Alexandra is the Managing Director of Boston Tutoring Services, a tutoring company that offers one-to-one in-home tutoring in Massachusetts. She is also a former Kindergarten teacher who also tutors students in grades
K-8, in all subject areas, including test preparation.
Would you like to print a copy of this article?
Alexandra has given me permission to convert her article to a pdf format for printing, which you can access by clicking on the link below (to share with your child's teacher):
Introducing Multiplication to a Student with Different Learning Needs - printable version
Multiplication Worksheet
Watch for a new guest blog post here next week from Teacher/Tutor, Alexandra Berube.
As a teacher and professional tutor, Alexandra plans to share more future guest blog posts here--where she will reveal additional insight into her student who has Agenesis of the Corpus Callosum and how it affects the student's education, and she will also be sharing teaching strategies that have helped her student.**
ADDITIONAL INPUT FROM PARENTS OF KIDS WITH ACC:
1. "My son learned the [multiplication] tables through Touch Math songs. He still struggles but I see him sort of singing the songs when stuck.
Calculators are truly a kid's best friend!
Now about place value....."
Parent of child with ACC, 5th grade
Touch Math multiplication songs video - on Teacher Tube
http://teachertube.com/viewVideo.php?video_id=210943
********************************************************************************
2. "My son, 10 years old, PACC (and other things), has been learning his multiplication tables by making up songs to remember them by. He still struggles if it has been a while since he practiced, but the songs seem to help. He is in 4th grade, and is repeating the year mostly because he has struggled with the times tables, and other math, and logical thinking. We find that anything Kinetic that we can do to help him seems to help."
Note: Kinetic or Kinesthetic Learners - learn through a tactile [touch] hands on, feeling, touching, manipulating objects, moving, doing, experiencing things approach.
Parent of child with partial ACC, 4th grade
********************************************************************************
3. "Concerning the use of songs, using familiar tunes and following the pattern of repetition of the words seemed to work well. We used the tune of Frere Jacques to memorize the 9 times table and it worked great. Each part of the verse is repeated twice so you get the following (original song words appear beneath the multiplication version so you can see how they match up):
9 times 1, 9 times 1; 1 times 9, 1 times 9
(“Frere Jacques, Frere Jacques; dorme vou, dorme vou?”)
9 is the answer; 9 is the answer
(“Sonnez la matines; sonnez la matines”)
9 times 1; 1 times 9
(“Ding, ding, dong; ding, ding, dong”)
But once used for the 9s, we couldn’t use it for another multiplication table because the words for the 9s were already tied to this tune in my child's head. Trying to associate the same tune to another multiplication table likely would have resulted in
incorrect remembrances.
And believe me, we needed to be able to illustrate that it doesn’t matter in what order the numbers appear, the answer will still be the same.
8 X 2 is 16 and 2 X 8 is 16.
She learned the multiplication tables for 2, 5 and 10 fairly quickly. Most kids can count by twos, fives and tens. But she didn’t get the relationship that the answers in the 4 times tables were twice the answers in the 2 times table. So there was a slight disconnect there in terms of relating one set of multiples to another. She did eventually learn all of the multiplication tables (through 10 X 10), but it took a good deal of repetition...and I believe it was truly a rote learning experience. I think she would probably have to have to think pretty hard to explain that what is meant by 6 X 8 is 48 is that there are six groups which each contain 8 items and represent a total of 48 items. But she can pretty reliably give you the correct answer whenever you ask her “What is ‘this number’ times ‘that number’?”
Parent of child with ACC, (in 8th grade now)
********************************************************************************
4.
"Multiplication and division are tricky. As they progress from learning the basic concept, it becomes necessary to memorize. Kids can "count on" on their fingers to do addition and subtraction, if need be, forever. Honestly, my child [who has partial ACC] was doing it still in algebra, but it was so subtle most people would not notice. It's quick and can be pretty efficient and accurate as well.
Not so for multiplication (and its inverse, division). Counting on becomes tedious, and is far more likely to be erroneous for these operations. It needs lots of "drill and kill" for some kids, and not just those with learning problems. I found several things that worked well in my classroom; I used them for all my students, not just the ones who I suspected would have trouble. In that way, all of them had the same background, and nobody who needed them longer was seen as having to use a crutch. Here are some things that worked for me.
Jewels
Many programs that emphasize hands-on math use jewels. These are strings of plastic beads (they are attached to the strings) that can be cut apart into sets of ones, twos, threes, fours, and fives, each a different color. Kids can easily use them to visually represent multiples of each when beginning to multiply. strings of them are available at teacher stores and in math catalogs. The drawback to me is that there are only five colors, so you can only go to 5 X 5. This may be OK for first graders, and even some in second, but by third you need more.
I solved this problem with Christmas garlands of plastic colored beads. If you can find four more colors besides the ones from the store (gold, silver, pink, purple, perhaps), you are set up to 9 X 9. Kids do have some problems losing place when they count them, but they are physically there so a recount is possible. These jewels are cheap, easy to store, and they reinforce the concept of multiplication as repeated addition. They are great for division as well.
Using the margins
When kids get older, they are expected to do multi-digit multiplication before some of them have their facts memorized. This often results in very slow work, with repeated, excruciating counting on. And it doesn't seem to stick from one time to another. So I taught them to utilize the margins of their papers to write vertical lists of the multiple they are working on, like this:
7
14
21
28
35
etc.
Then, the next time they need a multiple of 7, it's either there, or they can easily extend the list to find it, not having to repeat the same feat again and again. There is usually enough room for 4 lists on each side of the page.
I learned this just by chance, trying to help kids who had a terrible time multiplying. One by one, I noticed over time that fewer and fewer kids needed this technique, as they were realizing it was easier to just memorize facts than go to the trouble of listing. But in the meantime, they were not spending all their class time counting on their fingers. It had the added benefit of showing me instantly when someone had made an error, so I could point it out and help that student get it right. Some people use multiplication charts, but I prefer that the student at least do the computation him/herself at least once, and it seems to stick better when they do. (They can even do it on state tests, when manipulatives are not available.)
Computer Software
This is one area where a computer can really help. There were games available, even some really inexpensive ones, that filled the bill of "drill and kill" without any pain. I'm sure there are many more now. Some will even adapt to a child's level as s/he plays--I highly recommend those. I have been out of it for several years, but look online for some. I'm sure other teachers and other parents are also good sources. If you have the chance to play it before buying, do try it out. You want to make sure that a child can't "game" the system, by using something other than knowledge of facts to succeed at the game--like guessing.
Math replacement units
Marilyn Burns has a great set of math replacement units. These are books that give plans for units of study that cover one topic, such as money or probability, that teachers can use to replace (or supplement) the ones in their texts. I have used several of these at different grade levels, and I highly recommend them.
I used the one on multiplication fairly often, especially when I had average or low third graders and high-achieving second graders, because it was easy to use with them altogether. It's intended for 3rd grade, but it can be used also as enrichment for younger kids and remediation or reinforcement for older ones. She explains a number of activities set up as one- and two-person games (an adult can play with a child) that give practice in the meaning of multiplication. Kids can go back to them again and again for reinforcement. Kind of pricey, I think around $30, but you can likely go to a teacher store and peruse its pages to see if you think it would help you.
Tip: you can find the older out-of-print used version online titled: "Math by All Means: Multiplication, Grade 3" by Marilyn Burns through Amazon and other online sites. This is the version I used while teaching students.
The newer updated version is titled: "Teaching Arithmetic: Lessons for Introducing Multiplication, Grade 3" and can be found at
Math Solutions, a company founded by Marilyn Burns. It is also available (both new and used) through Amazon.
My daughter (who has partial ACC) did have some trouble learning her multiplication facts, but we just practiced them relentlessly. At school she used jewels, and we used the software at home. I hadn't discovered Marilyn Burns or using the margins at that time.
As I recall, memorizing multiplication facts was not as hard for her as addition was. Maybe because she was older. She did have problems remembering how to do multi-digit multiplying and long division, though. I decided she needed more practice. So I had her dad (she was a daddy's girl) write 3 or 4 problems, just for practice on the white board easel we kept in the kitchen, and she'd work on them while we made dinner. She just really needed a lot of practice, over and over, doing the same algorithm. And when she learned it, she needed lots of reinforcement for it to permanently stick."
(Teacher) and Parent of daughter, 24, with partial ACC and septopreoptic holoprosencephaly
********************************************************************************
If you would like to share input about multiplication and your child who has ACC, please send me an e-mail and I will be glad to include it here.
********************************************************************************
RESOURCES - (for your consideration):
For many kids who have ACC, learning through music is often times a helpful and successful method of teaching.
Multiplication Mountain CD, by Hap Palmer - listen to sound clips of songs
Singing Multiplication Tables CD, by Hap Palmer - listen to sound clips of songs
Tip: check your local library (and online library system) to see if you might be able to borrow the multiplication music CDs by Hap Palmer to try with your child.
**Please Note: Agenesis of the Corpus Callosum has a very wide range of effects--ranging anywhere from no symptoms--or mild learning disabilities--to severe mental and/or physical challenges. It can sometimes also be seen with other medical conditions, genetic syndromes, chromosomal anomalies and more. Every person with ACC can present differently in terms of their development, cognitive abilities and educational needs.
Each child who has ACC is a unique individual with their own abilities, weaknesses, challenges, motivations, strengths, as well as their own style of learning.
