Mathematics Their Way Activities Notes

Source: Lorton, Mary Baratta - , (1976). An Activity - Centered Mathematics Program for Early Childhood Education . Meno Park, CA: Addison - Wesley Publishing Company.

According to Mathematics Their Way children can participate in activities at three different levels: the concept level, the connecting level, and the symbolic level. These levels should be sequential for each concept being presented. In other words whenever a new concept is presented, the children should start at the concept level, move to the connecting level and eventually be able to function at the symbolic level. The concept level is the most basic level. Children should have hands on activities. They should be manipulating objects to develop an understanding of the concept. They should not be asked to associate the objects with the symbols yet. Examples of this level in a fractions unit would be cutting up pizzas, exploring with Cuisenaire rods how many in different combinations, and comparing pattern blocks. The connecting level is where the symbols are introduced in conjunction with the manipulatives. Children still manipulate the objects but then record their activities using the symbols. As the children progress, they may use pictures to represent the objects and then record using the symbols. Examples of this level include dividing up a pizza and then labeling the pieces, and labeling a pictorial representation of fractional parts with the appropriate symbols. The symbolic level is where children can perform the mathematical task using the symbols only. Children should only be asked to perform at this level if the concept is already mastered. Examples of this would be matching the written name with the mathematical symbol (one half is 1/2), and being able to compute fractions.


Free exploration. If you start to have students do what you have in mind too early they will want to continue with their exploration.

Reproducing patterns and designs.

Rhythmic clapping. Clap snap... Dot chart (use lines horizontal, vertical, curved, straight... Unifix cubes for patterns. People patterns, stand sit, ... People card patterns. Pattern blocks. Geoboard unifix patterns. Glue patterns on paper. Border patterns. Necklace patterns. Macaroni patterns.

Sample Patterns






Left to right progression

Patterns can be represented symbolically

Similarities and differences

Patterns can be changed to different forms

Patterns can be extended

Reason deductively

Analyze patterns

Sequence of size, length, area, volume

Numerals: Put numerals in order. Trace on board, in pudding, shaving cream, in air, glue on paper, trace on paper with pencil, chalk...

• Dot to dot

• Rulers

Sorting and classifying. Each can be sorted in a variety of ways and when students communicate about their collections they build vocabulary. Color, texture, shape...

Sort on an overhead projector.

Sort pattern blocks.

Sort people.

People pieces.

Sort and have students watch and wait to guess the categories.

Take a sort and classify walk (sort shapes found in objects...).

Sort words according to syllables

Sort words according to sounds, letters...

Sort shapes on geoboard (squares, rectangles, triangles, zigzags, faces, houses, ...

Geoboard paths. Put a unifix cube on each side of the board. Make a path to connect them. Describe the properties of the path. Can other paths be found? How many? Can they be classified?

How many ways can three lines be connected on a geoboard?

Make colored water with food coloring. Sort baby food jars with colored water.


Sort collections with Venn diagrams.


Count and turn

Counting off line

Counting off in chairs

Counting in the circle game

Objects in the box game, piggy bank game

Spill the beans game

Pendulum game

Jump rope count

Ball bounce count

Number lines

Counting on game put out 5 blocks cover up two, count on.

Silent count to rhythm, _ _ 2 3 4 5 _ _ 2 3 4 5

Counting backwards. (Use variations of all above)

Put pictures in sequence (baking cookies,...)

Snap and clap

Stand up sit down count...

Double circle, walk in opposite directions, count with hand slaps


Make a more or less book. More pencils less windows.

Comparing length of names on graph paper (first, last).

Record height, compare each child, use cord arrange in height

Compare mass with teeter totter (one to another)

Jar lids with nail hole in bottom, How long does it take to sink? Record time.

Compare length of time. Ice cubes take to melt, lifesavers to be sucked, birthday candle to burn out, wind - up toy to wind down, tie shoe, sip zipper, paper plane to fly, ...

How big is a handful?

Stack tell spin and win. (more or less spinner)

Play tic tac toe and keep score with unifix cubes

Play squares and keep score with unifix cubes

The hole in the box game. (unifix cubes in a box so one will slide into a corner at a time, guess how many cubes are in the box.)

Measuring with jars.

Massing and comparing

Guess when the time is up game

Mark the jar with the scoops of rice


Graphs comparing two with the real objects

Graphs comparing three with the real objects

Graphs with pictures to represent objects

Symbolic graphs with tallies ...

Four objects

Number at the concept level

Arrange a given quantity of toothpicks in different patterns to explore conservation of number.

Arrange a given quantity of pattern blocks in different patterns to explore conservation of number.

Arrange a given quantity of tiles in different patterns to explore conservation of number.

Arrange a given quantity of jewels in different patterns to explore conservation of number.

Arrange a given quantity of wooden blocks in different patterns to explore conservation of number.

Arrange a given quantity of beans in different patterns to explore conservation of number.

Arrange a given quantity of objects from a junk box in different patterns to explore conservation of number.

Arrange a given quantity of toothpicks in different patterns to explore conservation of number.

Arrange a given quantity of unifix cubes in different patterns to explore conservation of number.

Arrange a given quantity of squares on a geoboard in different patterns to explore conservation of number.

Record quantities from above activities by gluing squares, toothpicks into learning log.

The hand game. Take an amount of beans shake in two hands then open one and say how many beans then open the other while saying and, and how many beans in the second hand.

The peek through the wall game. Put a specific number of beans in a line. Then take a piece of cardboard cut like a picture frame and insert it somewhere in the line and say how many are on either side of the frame. 3 and 2.

The lift the bowl game. Put a specific number of beans under a bowl. Take some out and put them on top. Say how many are on top, open the bowl and say and, and then how many beans which are under the bowl.

Have the student count a specific number of beans into your hand. Hide some in the other hand, show the student how many are left and ask how many am I hiding?

The whale game. Use fish crackers. Pick a certain amount, have the student eat some, and ask how many are left. The whale was still hungry and ate some more, eat the next one, and ask how many are left? Then say three more swam up, how many are there? And continue until all the fish are gone.

Listen and count. Ring a bell a certain amount of times. Then turn over a card with a + or - sign.. If the sign is + add a number of objects that the bell rang to the card, if the card is - , then take that many objects from the card.

Concentration. Take some boxes of the same size and put pairs of different amounts of objects under all the boxes. Have the students turn a pair of boxes over, if the pairs match they can remove the boxes and the objects. If the pairs do not match they cover the objects with the boxes and it is the next person's turn.

The cave game. Make a row of objects. Cover an amount of objects and point to the objects outside of your hand and say the amount, then point to hour hand and say the amount under your hand. Then lift your hand and repeat the combination. Repeat with the same total amount of objects the same, but different amounts in and out of the hand. The teacher can vary the game by saying how many to hide ...

Put a row of objects on a card. Say the total of objects on the card and take away a number. The teacher then covers that number with another card and asks, How many are left?

Move them off the grid. Have students stand on a grid. Have them take steps from one square to the next by telling them how many squares to move relative to two points marked at two different points on the grid 3 from the x, 2 from the pok a - dot. Until all are off the grid.

Presto change O. Put two stacks of three, Presto change O, to three stacks of two. Presto change O to six stacks of one...

Line them up, group them up, cup them up. Put five groups of three, line them up, cup them up... How many cups should we use? Continue to line them up one group at a time, if there is one lined up for each cup then put them in the cup and repeat until all are used and there are none or some left over. If there are some left over ask if it worked.

Unifix trains. Make four groups of three. Put all your cubes into a train. What size of trains can we break or big train into? Twos? Did it break into trains of two evenly? Yes. Put your train back together. What other way? Did it break into groups of five evenly? No.

Word problems. The teacher reads word problems and the students act them out. Sue had a birthday how many candles on the cake. There were four chairs at the table, Juan put two more how many where there? ... Cardinal, ordinal (Jan, Meg, Molly lined up, who is second...), addition , subtraction, multiplication (Jon made six birthday cakes, each had two layers how many layers?) division (Jill had six pencils, she put two into three stacks, how many were in each stack?).

Number at the connecting level

Interpreting symbols: what d you picture in your mind when I write this symbol? (numeral). How could you picture this? (3 + 4) (5 - 2)

Make number books. Records of toothpicks, tiles, pattern blocks

One student makes up a problem and another records it. Use the hand game, peek through the wall, lift the bowl, cave game, move off the grid game, concentration, whale game.

Object cost grid. Make a grid with numerals from 1 - 50. Challenge students to bring in objects that cost the amount as each numeral on the grid. Students will bring objects from home.

Subtraction race. Build a train of unifix cubes. Roll dice and may remove the number of cubes as rolled on the dice or pass. Must remove the exact number of cubes to go out. This adds the idea of probability.

Crazy mixed up numbers. Students write random numbers on a strip of paper. Another student makes a set of objects for the first number and proceeds to tell what needs to be done to the first set to get a set equal to the second number on the list. This is continued for the rest of the numbers on the list.

Add or pop balloons game. Students pick a number and put that many balloons on a bulletin board. The next student draws a number and either adds balloons to get the next number or pops balloons to get the equivalent number of balloons.

Bell and student in the box. A student gets in a box with a bell. Rings the bell a number of times and other students tell the number of times the bell has rung.

My turn your turn. Select two teams and one student to stand in the middle of a row of squares. Challenge students to find the middle square of a row (must have an odd number of squares.) The first team rolls a die, the child in the middle faces that team, the team calls out their number, and counts aloud while the child in the middle walks toward them the number of steps indicated by the roll of the die. The object of the game is to have the child in the middle step off the game board at your team's end.

Order by volume. Put progressive amounts of rice in a collection of jars and mark each jar with a marker.

Number at the symbolic level

Use the activities and write all the predictions and possible results.

Numbered squares. Write numbers (1,2; 1,2,3; 1,2,3,4; 1,2,3,4,5) on different sized square matrices (2x2, 3x3, 4x4, 5x5 ...). Look for patterns of numbers.

Grow plants

Toss dice and record results on chart

Displace water with rocks and record the data.

Grab penny, nickel, dime bag. Grab a coin. Spin the greater or less than spin. Try to grab the next coin according to what the spinner reads.

Time trials. Students sit in a circle and pass an object around the circle, predict the time, have a counter count the number of times the bell rings (tape a bell ringing once every second) or the number of times a pendulum swings, before the object completes the circle. Change events and do another time trial.

Roll six dice and arrange them in order.

The magic box. Select a card, look at the red side, make the number of circles for the red number on your paper, put the card into the box with the red side up, when it flips over and comes out it has a blue six on the other side, put the number of X’s with the circles to make 6. Read the problem.

To do subtraction have the students turn the cards over with the blue side up, make that number of circles on their paper, put the card in the box so it flips over, when the red number comes out of the box have the students cross out enough circles so they will have that number of circles left on their paper. Read the problem.

Pattern two

Students observe patterns on fruits and vegetables and creatively describe and draw the patterns.

When students tire looking at the inside have them predict inside. Ask if seeds. How many? Cut open look at pattern, count seeds...

Look at leaves, patterns, number sequence leaves up branch...

Clothing patterns. Have students copy the pattern. Collect the pattern copies and have student match the patterns with the materials.

Matrix patterns. Have students arrange chairs in a grid. Have them arrange students on the matrix in a pattern.

Routes on maps. Have students make a map of a familiar route.

Geoboard patterns. Count on board from left to right, drop a unifix cube over a nail every time they say the number they are exploring.

Students construct a pattern, then they are asked to make a picture to explaining that pattern.

Surrounding patterns. Use tiles or cubes to make patterns that surround. Snowflakes, quilts...

Row, columns, and diagonal patterns with unifix cubes. Draw diagrams to represent each pattern.

Write names on square matrices to observe patterns.

Place value

Zorken’s game. Tell students you l have a magic number or message on slip of paper. Then ask them to think of a nonsense name, like zorken. Then tell them that name will be used instead of the word on the slip of paper. Open the paper to reveal a number (4 or 5) work well for the first time. Tell them that from now on the word zorken will be used instead of five. Five is outlawed. Then use a place value mat and show them that zorkens are allowed on the left side and single cubes are allowed on the right. No other combinations are allowed on the mat. It is outlawed. It is also helpful to have the two sides different colors. Green and white. Clear board. How many zorkens? 0 How many loose cubes? 0 When the bell rings add a loose cube. Ring bell. Zero zorkens and 1 single or loose cube. ... When the bell rings five times. Yell zorken and see if the students snap all five together and move the loose cubes to the zorken side. Then say one zorken and zero loose cubes. When students get the idea (about three zorkens and three) then tell them you are going to change the game. When the bell rings they are to take one away from the loose side. When they do not have any more zorkens left on the single side let them think what should be done. May need to remind the students the rule that only loose cubes can be on the white side. When students know the game play again. When they really know it play the next game.

Frump game. Played like the zorken game only use a different outlawed number, let students think of a new name, use beans, group them in nut cups.

Make up games for 3,4,5,6,7,8,9, and when get to ten don't make up a name use ten. Post the names

Have two flip rings with cards 1 - number of game. Have students flip each chart to a number. Then have them make the number on their place value mats. Then when the bell rings they can increase or decrease.

Counting game connecting the symbol to the concept. Have two flip rings with cards 1 - number of game. Have students flip each chart to zero and place the rings so the number zero is showing above the green and blue regions. Then when the bell rings have them place a cube and flip a card on the ring to match.

Giant number line. Have students play the preceding game. As they go through the chart have them record all numbers on a class grid with the number of unifix cubes beside it. Look for patterns. Repeat with other name games. Look for patterns on each and among the others.

Repeat the above and put the numbers on a matrix (4 - 4X4, 5 - 5X5 ...). Look for patterns. Color patterns with markers, chalk, crayon...

When students use base ten use the same terminology use ten one instead of eleven, ten two, two tens and two .. transition slowly. Make a matrix with pictures of cubes, numerals, word names as used by teacher, and add traditional words later.

Measure. Have students make measuring device (cord) and measure unifix cords in place value...

Estimate and count. Bring in bags of objects. Beans, M&M's, peanuts, safety pins, Q - tips, sugar cubes, crackers, cookies, macaroni, ... Students estimate, out number on slip of paper, count items by putting them into cubs of ten.

Count jars of objects. Use the same kinds of objects above, place them in jars, have the students guess and check. Switch jars, objects, and repeat.

Peas in a pod.

Cans of food. Mark the can with the price and name of food. Real filled cans is more real.

Unifix patterns for multiples. Use groups of blocks snapped together (4) place on paper draw count and record number, put next group, count up record, continue...

Perimeter. Make a pattern of unifix cubes, draw the outline on unifix graph paper, place one centicube on each graph line, count centicubes, write numbers from one ... last on each unit of the perimeter.

Geoboard designs and bases. Repeat some of above activities and place cubes on board, observe pattern

Store. Use cubes.... to find totals and change.

Movie theater. Arrange chairs in grids, have students find seats. Use different bases for seating arrangement.

Estimating and graphing. Have students estimate how many pendulum swings there will be will you stack 8 cubes, how much water drips from facet in one hour, then larger numbers? collect graph...

Unifix stacks. Roll dice stack cubes, turn card, add, subtract, roll dice add or subtract from stack... Turn cards on ring to match number.

Pattern activities

Pose questions explore, record, draw conclusions.

How many eyes do .. number of people have. Past pictures of people on a page and record the numbers beside each 2,4,6,8...

How many lines on stars? Show on paper. Draw label 1 - 5, 2 - 10, 3 - 15, ...

How many sticks on a popsicle?


Shapes: triangle, squares...

Jump ropes

Three leaf clovers

Join hand problem


Chain reaction. Give a student a piece of paper, cut in half, give a half away, cut in half give away, continue... Repeat next day and record what happened. Try with more sheets of paper....

Play London bridges. Each time a person is caught, they select a person from the line to start a new bridge, continue until all bridges, play next day record next, or if students demand to record today.

Apartment house interior designer, architect... Build an apartment complex from milk cartons, put one cube for each sofa, or chairs, or sinks... count chart...

Two handed take away... even numbers

Chip chop. Chop a board into x pieces how many chops?

Milk carton clothes pin game. If have milk carton and use clothes pins to clip to another, how many clothes pins? How many more are needed if add another milk carton? another....

Pattern block puzzles. How many triangles to make a hexagon? Stack them on top record stack another layer...

Make a tessellation. What is the pattern? write it...

Geoboard nail patterns...

City planner..


Intersecting roads. How many intersections with one road? two? ...

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