Using Manipluatives to Understand Mathematical Patterns
Since mathematics is a language through which we can express the patterns and rhythms in creation, we have tried to find ways to demonstrate this to the children. Sometimes it is the children who show us. Several years ago, one of the second grade students was “experimenting” with some colorful manipulatives. With his teacher’s permission, he put a sign on his classroom door which read “Arithmetic Laboratory, Do Not Enter.” He was experimenting with various ways of arriving at a series of numbers through addition. The simple fact that a number, 7 for example, can result from adding many different combinations of numbers (0+7, 1+6, 2+5, etc.) is fascinating to a child if it is presented in a concrete, rather than abstract way. The pattern which emerged was very clear and beautiful. Now we use his discovery as a means of showing “number families” which are analogous to words families in language.
Without doubt, children need to know their addition, multiplication and other basic facts by heart, but we have found abstract memorization to be much less effective and satisfying. Having objects which provide visual and spatial experience and can be moved around by the children greatly enhances their understanding of simple arithmetic facts and makes learning much more meaningful. It is the concrete approach which opens the door to these “discoveries”.
One very kinesthetic way we use to give our students the feel of multiplication is by jumping the times tables on a long number line. Jumping one number at a time, then two numbers at once, then three numbers, then four and five numbers at once gives them a very concrete experience of the difference between addition and multiplication.
When we tried to incorporate this technique into our math stories involving Paddy Plus (addition) and Peter T. Times (multiplication), we made some interesting discoveries. The whole story is too long to tell here, but once Peter T. Times showed Paddy Plus his a special “dance floor”. Jumping the times tables on this “dance floor”, he makes some wonderful patterns. Making this “dance floor” - a specially designed string board - and drawing these patterns has become a regular part of our teaching beginning multiplication. Instructions for making a string board are at timespatterns.html
Once the children have been introduced to the four processes and have mastered to some degree simple addition and subtraction, we present the idea of place value. The concept of place value may seem elementary to adults but to many young children, it makes no sense. To an adult mind the symbolism of place is significant, but to a child’s mind (at least at the beginning) if the number looks the same, it should be the same. When you tell them there is a difference, they look at you as if to say, “If a 3 is means 3 here, why does it mean 30 there?” The idea that the value of a number is dependent upon its location is a very abstract construct. Where else in a child’s life does the value of something change with its location? To help the children understand place value we have employed the aid of Chatty, the Squirrel
Chatty was once a disorderly squirrel named Scatty (as in scattered) until she was given an important responsibility by Earnest Equal, King of the Numbers. Rather than leaving nuts all over the forest floor, she now keeps order by storing the nuts in various size trees. Chatty’s smallest tree can store only nine nuts, so she stores the rest in various colored bags and puts them in larger trees. Her second tree is larger and can store nine blue bags of ten nuts; her next tree, larger still, can store nine red bags each of which contains ten of the blue bags. You probably get the idea. Each time she fills a bag she says, “Look here now I have ten, I’ll bag them up and begin again.” Practice with borrowing and carrying (re-grouping) is done through many story problems, sometimes acted out in costume.
At St. Michael’s the children generally memorize their multiplication tables by playing Up the Hill. This game uses dice and specially dyed wooden base ten blocks. The units are small cubes dyed green, tens are bars dyed blue, hundreds are flats dyed red and thousands are large cubes dyed green. The object of the game is to be the first player to reach a predetermined total of blocks. A child working on his two-times tables would roll the dice, add them and multiply the sum by two. He would then take from supply of manipulatives the appropriate number of blocks. The rules of the game are such that children working on various times-tables can play at the same time and have an equal opportunity to win. Initially the children use the “number family” blocks to count by twos, threes, fours, etc. Variations of the games allow for practice with adding, subtracting, re-grouping and more complicated multiplication.
The same manipulatives are used to teach column, or multi-place, multiplication, e.g 45 x 38. The manipulatives are very helpful becuase the children can actually see the reason for all the steps required. We have a limited supply of these manipulative available in The School Store
Children need to know the multiplication tables not only forward, (5 x 4 =20), but also backward (how different ways are there to get to 48?), and “inside-out” (easily recognizing common factors). We have a couple games available for free download in the Curriculum Ideas
section of the website.
We have found that children not only need to be able to translate a story problem into one of the arithmetic processes, but also be able to develop an appropriate story based on the number problem. One technique we have use to solidify understanding of story problems involves having the students make up a problem represented by a number problem written on the board.
When it came time to introduce the middle grade students to geometry, we wanted to follow the same principle of “from the concrete to the abstract”. All the geometry books we had seen started the study of geometry with the definition of a point, a line, a ray, a triangle, etc., then proceeded to various geometric proofs. We wanted something with a more intuitive approach. Then we discovered a publisher called Key Curriculum. Although they have workbooks on several aspects of mathematics, we were interested in their approach to geometry because the entire course is done with a compass and a straight edge - just like the ancient cultures. The students learn about all the forms and principles of geometry in an experimental way rather than through memorization. The course does not provide very good tests, so made our own. These are available for anyone interested.