In my combined first and second grade, I fully believed that understanding math does not replace memorization. It is a tool to facilitate carrying out math operations. Timed tests? Never, I thought, for these young and eager math students! Then I heard of a method and decided to try it. After children took a daily timed test, they kept a record of their scores on a personal bar graph. This focused their attention upon their own improvement. Much to my surprise, it didn’t take long for me to realize that they LOVED taking timed tests. And Anne S., as an adult writing about her memories, stated, “And I remember those one-minute timed tests. Somehow, you made them fun for us – so fun that my friend and I would practice them during free time, as if they were the newest game. You taught us in such a way that we didn’t know we were learning – we thought we were just having a blast!”

There were six tests. The first two tests were a plus test and then a minus one of 50 problems with sums to 10, for 1 ½ minutes. (Full sheets of 100 facts were cut in half to provide only 50 facts per test) Next, for three minutes, were an addition and then a subtraction test with 100 problems for sums to 20. And finally for especially capable students there was a multiplication and a division test also with 100 problems, each for three minutes.

Sometimes, however, after successfully completing the first three tests, a few children were progressing very slowly on the minus test with sums to 20. So, they were taught a new method which involved just knowing the sums to 10. When given an example, such as: 15 – 8 = (written vertically) they were reminded that 15 = 10 + 5, and then, looking at 15 – 8 =, told to first take 8 from the 10 resulting in 2 and then add the 2 to the 5 ones which = 7. Voila! The correct answer. It involved more than plain memorizing, but with a little practice it was simple for them to use, and they could do it very rapidly. Certain children loved this method and their scores quickly progressed to passing the test.

Correcting 24 or so tests each day was more than a little daunting. So that children could correct each others, the top row of numbers on a pair of tests, were the answers to the other. One child for each test would read the answers, and I would correct that child’s test. I would always correct any test that was passed. But when spot checking at times, the children seemed to be quite accurate. And if someone got mixed up, we’d hear a loud, “Wait a minute!”

Scores were recorded on one’s bar graph, filled in each day when the previous day’s test was returned. If, over time, I saw a child who wasn’t improving, the parents were informed. Usually they acknowledged that they hadn’t been studying the facts for a while. When studying resumed, the bar graph began climbing again, and the child was very pleased. Facts were to be memorized at home, but if that was a problem, some student or adult volunteers at school would help.

In order to reinforce their attention upon their own success, each day on the front blackboard, I wrote the names of about 5 or more who had improved the most. The name could be for a child on a plus test with sums to10 or one on a division test. This was the last activity for each afternoon. Children were so disappointed if for some reason we couldn’t have a daily timed test.

Read more about the math program, including the timed tests, in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**.* Included is a web site where the timed tests and graphs can be downloaded, in addition to downloading other materials for use in a classroom. Also, see 7 reviews on www.amazon.com

Materials supplied for constructing were: 5 small paint buckets, each one ½ full of Cuisenaire Rods; 1 inch wooden colored cubes; dominoes; Base Ten Blocks; and Unifix Cubes.

Building all kinds of structures were most popular – buildings, towns, roadways, and towers, supplemented with various other materials such as 1 inch cubes, dominoes, pegboards, etc.

There were times when too many structures were knocked down because someone carelessly walked by. So there was an agreed upon class rule that excluded anyone from the math center for a few days if they knocked anything down – even if it was a complete accident. The number of days agreed upon varied with each class, but usually it was two or three days. No blame was attached, and it was amazing how carefully children walked through or by the buildings after that. From then on, there were few accidents.

Constructing **allows opportunities for** developing concepts necessary for good achievement in **abstract math**. A good math student is able to conserve, recognizing that two amounts are equal even though one is a different shape or size than another – that five Cuisenaire Rods or five rulers are five. Building in the math center, children constantly see various arrangements of numbers of objects.

It is also an opportunity for **overcoming egocentrism**. An egocentric child cannot see or understand another viewpoint. An aid to overcoming this is continually looking at all sides and angles of each structure built. Success in math requires a good mental image of numbers of objects in many different configurations including the various shapes, sizes, and dimensions of these objects.

Children must have **concrete experiences** before they can abstract. While children were constructing, they were constantly sorting and classifying while **looking at balance, design, and structure.** These were opportunities for developing seriation plus spatial and numerical relationships eventually leading to paper and pencil math, no longer needing objects.

Read more about the math program and the Math Center in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**.* Included is a web site where programs and activities can be downloaded for use in a classroom. Also, see 7 reviews on www.amazon.com

In my combined first and second grade classes, a study of symmetry lasted at least half a year and more often all year. Library books were used for referral. Children noticed or were exposed to symmetry in their own world, inside and outside the classroom. They found symmetry in letters, people, animals, and kites and in nature with plants, flowers, leaves, and snowflakes.

Building in the math center produced many symmetrical designs using Cuisenaire Rods, 1” cubes, and many other manipulatives along with geoboards, pegboards, and pattern blocks.

Palindromes were quite a challenge such as “madam” and “aha”. Some classes found many three letter words such as “pop” and “eye”. They were fascinated with some phrases I found such as “a man, a plan, a canal – Panama” and “never odd or even”, etc.

There were many designs made using pieces of folded paper with paints and scissors. One, two, or three colors of paint were placed in a construction paper fold, then folded and pressed gently before opening and revealing the symmetry. A little more complicated was using thin paper folded once or multiple times before cutting a design, while retaining some folded parts so it stayed together after opening. Of course, making snowflakes was very popular.

Read more about teaching Math with other supplementary activities and games in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**,* beginning on page 113. Also, see 7 reviews on www.amazon.com

A child would roll the dice and pick up the number of beans indicated by the sum of the dice. When a player had 10 or more beans, he picked up a 10 bean tongue depressor and placed it on his game card. He then returned 10 of his beans to the pile and kept what was left over to use with the next 10. The winner, of course, was the first one to fill his card.

Read more about teaching Math with other supplementary activities and games in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**,* beginning on page 113. Also, see 7 reviews on www.amazon.com

Each pair has the following list: length of foot, hand span, head, neck, waist, wrist, arm span, height

An interesting discovery was made when children were asked to compare the measurement with outstretched arms, of the distance from finger tip to finger tip to that of their height. (It is about the same!)

Read more about teaching Math with supplementary activities and games in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**,* beginning on page 113. There is a web page available with a work sheet for this activity including a diagram of a person that can be printed for classroom use, in Appendix M. Also, see 7 reviews on www.amazon.com

**Sets: **within each of some sets, an equal number of objects

**Array: **an equal number of columns and rows.

**Cross Product:** every item in one set of objects combining with each item in another set.

**Measuring: **on a number line of objects, skipping an equal number of times across the line.

After explaining the four models, examples should be presented. It takes a while for children to realize that the numerical answer is not the answer – you want to hear which model it is. The following are some examples, first for a mother with a child or two and then for a teacher with a class:

Mother, for sets: there are 3 bags of candy. There are 6 pieces of candy in each bag. How many pieces of candy are there?

Mother, for array: on our cookie tray, we put cookies in 3 rows and 4 columns. How many cookies are we baking?

Mother, for cross product: if 4 butterflies landed on each of 3 flowers, how many landings were there?

Mother, for measuring: Mary took a walk. On every 3 blocks, she took a drink of water. She drank water 4 times. How far does she walk?

Teacher, for sets: we will listen to 5 children read 2 stories each. How many stories will we hear?

Teacher, for array: we need 3 rows of chairs with 4 chairs in each row. How many chairs do we need?

Teacher, for cross product: Jane has 4 skirts that can go with 3 different blouses. How many outfits does she have?

Teacher, for measuring: on a number line, Susan put a marker on every 3^{rd} number, 4 times. What number did she end with?

Again, it takes practice for children to answer which model it is. But when children are easily able to give the correct model, they will fully understand the use and purpose for multiplication.

Read more about teaching Math in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success*. Included is a web site where programs and activities can be downloaded for use in a classroom. Also, see 7 reviews on www.amazon.com

**Take Away: **taking a small set from a large set.

**Separating: **2 small sets within a larger set.

**Missing Set: **empty spaces in a large set.

**Matching Sets:** comparing 2 separate sets

If sets are not clearly understood, group/s can be substituted for set/s. After explaining the four models, examples should be presented. It takes a while for children to realize that the numerical answer is not the answer – you want to hear which model it is. The following are some examples, first for a mother with a child or two and then for a teacher with a class:

Mother, for take away: Katie has 5 brothers. Two brothers went outside to play. How many brothers are still in the house?

Mother, for separating: There are 6 red and blue balls. Two of them are red. How many are blue?

Mother, for missing set: The table is ready for 4 people for dinner. One person is already sitting down. How many more people can sit at the table?

Mother, matching sets: There are 3 chickadees and 2 blue jays at our feeder. How many more chickadees than blue jays are at our feeder?

Teacher, for take away: There are 20 children in our class. 5 children are going to the library. How many children are left in the class?

Teacher, for separating: There are 6 children in the library. 4 of them are girls. How many are boys?

Teacher, for missing set: There are 8 chairs at a table. If 5 chairs are occupied, how many more chairs are empty?

Teacher, for matching sets: There are 5 children reading and 4 children writing. How many more children are reading than writing?

Again, it takes practice for children to answer which model it is. But when children are easily able to give the correct model, they will fully understand the use and purpose for subtraction.

Read more about teaching Math in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**,* beginning on page 113. Also, see 7 reviews on www.amazon.com

Learning the three models of addition enables children to fully understand addition used in **problem solving**, various **operations**, and in **everyday life**.

** Putting Together**: two separate sets meet in one location.** **

**Adding On: **one set moves to join another set.

**Separating:** adding together 2 stationary sets.

If sets are not clearly understood, group/s can be substituted for set/s. After explaining the three models, examples should be presented. It takes a while for children to realize that the numerical answer is not the answer – you want to hear which model it is. The following are some examples, first for a mother with a child or two and then for a teacher with a class:

Mother, for putting together: Sam is in his bedroom and Julia is in the living room. Their mother calls both of them to meet her in the kitchen. When they arrive in the kitchen, how many people are there now?

Mother, for adding on: Lizzy is by herself in her house. Jane and Tom are playing outside. Lizzy asks them to come inside to play with her. How many children now are playing inside?

Mother, for separating: There are two green cars parked on the side of the street. Two red cars are parked in a parking lot. How many cars are parked?

Teacher, for putting together: There is a program in the assembly. Mrs. B’s class leaves their room and Mr. F’s class leaves their room and both go to sit in the assembly. How many children are in the assembly?

Teacher, for adding on: Our class is in our room and we invited Ms. Green’s class to come and see our play. How many are in our class now?

Teacher, for separating: Five children in Mrs. T’s class are wearing red today. Seven children in Ms. Smith’s class are also wearing red. How many children are wearing red?

Again, it takes practice for children to answer which model it is. But when children are easily able to give the correct model, they will fully understand the use and purpose for addition.

Read more about teaching Math in my book, *Early Childhood Programs: Opportunities for Academic, Cognitive, and Personal Success**,* beginning on page 113. Also, see 7 reviews on www.amazon.com