Valid numbers

This title describes the logical math learners.  Math is logic.  It is an automatic strength for these students.  They are wired to think in a linear way.  They need organization , step by step processes and orderly surroundings.  Usually engineers, scientists, doctors, computer programmers, accountants and researchers are strong logical learners.  Two of the above noted are Albert Einstein and Bill Gates.

The very young valid number learner asks a  lot of questions.  They like to count.  They like to figure things out.  Teachers should model this reasoning ability in multiple ways.  A problem solving strategy could take the form of a story that ends in a question that is acted out.  This action and way of thinking as we retell what we did and how the answer was discovered can be practiced over time before it is committed to writing numerals on paper.

Math vocabulary and language can be incorporated as you solve the problem if you are taking something away (e.g. subtraction, difference ) or putting it altogether (e.g. addition, sum ).  this student will learn better with visual materials and hands on projects.  Computer games and learning activities with accelerated challenges will be very attractive to this student.  Repeating patterns are natural for this student.  Growing patterns is a challenge an early primary student will enjoy.  Counting comes easy.  Number sense is a pattern so this sequence is interesting and appealing to the logical learner.  We always start with the hundreds board!

The older primary logical learner enjoy valid number puzzles, brainteasers and games of strategy.  Chess could be introduced to a seven or eight year old.  These students have a good sense of how numbers affect and relate to each other.  They are interested in gathering information about a problem and then brainstorming.  When given an assignment, these students need a well-defined goal based on math reasoning in order to access their ordered, categorized thinking.  Just as we pretend play in early primary, the older students can access the same skills by playing "detective".  This time the results will be recorded using valid numerals and probably record findings with graphs or charts to categorize results.

Early Primary Logic Activity

Since we know that these learners love patterns and sorting, here is an inexpensive and colorful way to get this started.
-  Fall and sometimes yearly, drug stores and $1.00 stores sell large round plastic platters that are sectioned.
    There is usually a round area in the middle and maybe four sections.  These are typically used for outdoor
    eating.  They are excellent for sorting.  They usually come in brilliant colors. One platter could work with
    two or three small children.
-  Also at the same store you can find bags of foam shapes, bags of just about everything.  For example,
    look for the colorful and relevant items for the season.  It may be dinosaurs, Fruit, vegetables,          
    bingo counters come in different colors and are wonderful for many activities.
-  You may already have manipulives that will work.(numbers or number blocks)
    These same manipulitives can be used to create a repeating pattern.  Make a pattern of four different
     things and say them out loud.  Ask, "What will come next?" Continue this on then make the pattern
  longer. After this is mastered, you can graduate to "growing" patterns.

Older Primary Logic Activity 

We know the valid number learner needs to be precise, and use mental calculation.  This can be accomplished by practicing step by step math processes with logical numbers to increase critical thinking.

After the math problems are in language, give the students tips for understanding.  This is how we give the steps for solving.  It can be committed to a math journal with the second example, numbered steps for solution, illustrated and explained in their own words.  Read a question from the board.  Read again and highlight or underline the Key words that tell the operation ( information the child needs in order to solve the problem ).  Discuss or point out that there may be words in the problem that are not needed to solve the problem.  Cross or line those words out. Point out that the words or math vocabulary will tell you what operation is needed  The problem may be using one or more steps.  Like with younger students, this may be acted out or aided by the use of manipulatives.

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The logical learner is doing most of the work in his head.  Repeat this process with a new example that will require the students to work with partners or in small groups. They will use the same steps.  This time they will record their steps and learning in their math journal.

These students who collect and organize data enjoy board games.  Goals are defined - there are rules.  Logical learners are quick to find what numbers or combinations which are equivalent.


 

I'm All That I Need / We Are All Needed

The intrapersonal learner has been described as a loner.  A person with high self esteem that enjoys talking about themselves, This is a strong-willed person that needs space and probably feels that they have all that they need.  Equip them with an independent study or self directed activity with self checking materials after you know their learning level and they are fine to work alone ( preferably with space in a quiet area)

This learner enjoys setting his own goals and reaching them. He is very thoughtful and caring to others, while enjoying expressing personal feelings.  One strong teaching strategy is to connect learning to his personal life.  Allow him to have choices within the activity.  He would enjoy a hobby of genealogy

All primary math teachers assess their students at the beginning of the year. I read of a primary teacher who used the following strategy in math.   Those top performing students who work beyond the core grade level are given a file of self-directed word problems and puzzles with self-checking materials.  These files are monitored closely to see if progress is made and that is all he needs.  Pacing with the student is the key.  Students working alone are given a math journal to communicate problems, frustrations or successes with the work.  Interviews with the students to talk about their learning material and this style is very important for him to express his feelings.  He needs to know that he taking part in directing his own instruction.

- When teaching math, demonstrate how geometry is relevant to how their house was built:  straight lines, rectangles, triangles, circles, squares, etc.

- Have a learning journal to express how it is related to them, express opinions and feelings about what they have accomplished.  What can they do with this knowledge.

- Teach shapes with their own body.  Use different body parts to draw shapes in the air or use the body and transform themselves into a shape. The intrapersonal learner has all that they need to accomplish this feat themselves.

- Work alone to talk about a learning event, or about themselves into an audio tape

- They need a camera

- Keep scrapbooks, or albums of accomplishments

- Gather materials, choice of products, create self-made products

- Self-directed treasure hunts

- Rock collections - self directed, then sorted and look for patterns

- During math, have multiple centers already explained, that give choices for successful experiences

- Have a "safe space"  this is an area that is safe psychologically and emotionally

- Without pushing, continue to encourage the intrapersonal learner to socialize in group games 

In conclusion, the intrapersonal learner has a rich inner life.  He really would think I'm all that I need.  He is going to benefit from his own research through computer software programs or the internet.  With monitoring, this medium will unlock his self-directed style of learning.

-Some famous names associated with the intrapersonal learning intelligence are, Ghandi, Helen Keller, General George Patton, Aristotle, and Martin Luther King, Jr.

We Are All Needed

At the other end of the spectrum of the quiet corner is the social leader, the Interpersonal Learner.  This person can establish relationships easily and make others feel at ease.  They can innately read reactions and empathize. Their barometer can sense conflict.  The interpersonal learner will be the first to help mediate the situation.  

"Be The Teacher"

These students may struggle with independent seat work, anything abstract or out of context.  Their math learning needs to apply with everyday life.  These students learn through conversation with personal relationships and associations.  Self-expressive learners tends to visualize and create images, then  pursue multiple strategies with others.  This is a collaborative style of learning. We are all needed for learning to take place.
The interpersonal learner is comprehending highest when he is sharing, relating, cooperating, comparing, interviewing others----"Be The Teacher".  When teaching math as a resource instructor, I did this.  As students accomplished a math principle, I would allow them to come up to the overhead and illustrate/explain their learning.  This not only reinforced their knowledge, but encouraged other students to want to demonstrate a different method to arrive at the same answer.  We learned multiple ways to solve a problem.  In other words, they taught that principle.  I found that even in a diverse resource group of under achievers, with positive guidance, at least one interpersonal learner at each table was all that we needed.  He talked out and shared how he solved problems without any judgement, benefiting all in that setting.

Classroom Suggestions For the Interpersonal Learner

- Peer tutoring

- Interactive math centers with activities already practiced - work with 2 or 3 students:

   a.  Act out math word problems.  Place on cards with paper, pencils and manipulitives

   b.  Manipulitives with instructions on making different kinds of patterns with examples

This is a commercial model.  The coordinate map can also be made with masking tape on the floor or use tape on a clear plastic shower curtain that can be folded up and preserved.

   c.  Floor coordinate graph with bean bags and sticky notes.  Label graph 1-10 on one

        side and A-J on the other side.  Toss the bean bag, write the location on the sticky note,

        (e.g. d,7 ),  This can vary by placing toys or articles on the graph and having the student

        write the location of each of articles on the graph on the sticky note.  When done, 

        the items can be replaced at different places.

    d.  Assorted math puzzles

    e.  Number Line on board with location cards to record jumps.

Lyrical numbers

Research shows that when children spend time playing or listening to music, the cortex, which is responsible for highter thought and development is stimulated.  Children practicing a musical instrument are better disciplined and pay attention to detail.  These are valuable skills needed when learning math concepts.

Prue Whoo submitted on the website, Songs for Teaching, repeats that Plato said about music, "is a more potent instrument than any other for education". 

For the young child, listening to music and expressing themselves through dance or sway to the beat is a good beginning.  This lyrical learning encourages them to move differently to varying sounds or rhythms and motivates the brain to categorize sounds and recognize patterns within the art of music.

Read a rhyming story and allow children to clap out the beats after the first read.  This another strategy for the brain to think in a linear mathematical way.

Borrow stick or small instruments for the students to shake or tap in rhythm when listening to music at their own pace.  Finding the rhythm and keeping the beat is the first connection between music and math.  Discovering this connection will build confidence.  An older primary student hopefully will get the opportunity to learn to formally play an instrument either at school or at home.

Clap Rhythms
By later first and second grade this lyrical math activity will integrate music with math principles such as counting, fractions, parts of a clock and is a great mix for other styles of learning such as act out learning.
 Instructions:

Get 3 pieces of different color construction paper.
1st piece (brown) is a whole note - demonstrate with a clap
2nd piece (purple) is cut in half - demonstrate with faster 2 claps
3rd piece (red) is cut( first folded ) in four pieces - demonstrate with 4 very fast claps

Students already know how to clap their hands.  They clap for others for jobs well done.  Teachers may clap for attention.  Begin by modeling clapping for and with the students in a simple pattern. of 4 equal and fast beats.  Have the students repeat.  Do again. With the help of a familiar song have them vary the beat to (e.g. 2 (half) purple, and 2 (whole) brown beats) to match the music they are hearing.  The next goal is to enjoy and explore rhythm for its own sake.  Eventually you'll want to pose the question, "What is the difference between clapping the half notes ( faster clapping) and clapping the whole notes (steady beats).?  Can you see the relationship of the fast beats to the slower beats?  One whole note = 2 half notes.  This can be illustrated with the papers aligned on the board with the whole note on top.  The colors help imprint or clue the beats and can be called out. 

If the students say the colors as they tap or clap, it will aid them to keep the rhythm steady.  With practice, this lyrical activity will reveal that clapping 2 reds ( quarter notes ) equals the time for clapping one purple  (half note ).  Help the students see that the reds must be clapped twice as fast as the purple.  Also, 2 purple papers make 1 brown paper.  Demonstrate the beats for them.  Allow them to practice with partners and talk about how many beats make a whole beat. 

Experiment


In small groups or partners, allow students to move rhythm our of their own minds to others understanding.  One group can teach the other members of the class.  This can be a lyrical learning experience that continues to build for a class project, or take it out for school performances.

Math Integration

Although the math mind has been integrated in this activity, there are math principles that can be integrated with this activity.  As you stack the paper sections with the whole note on top, the two halves, under and the four fourths next, you are teaching fractions.  This can also be modeled on a number line.  Understanding time and the clock with half and quarter hours are also being taught at this time.  Here is also another extension.

some resources from SEDL.org

I have used Cd's for teaching Number Sense principles lyrically for many years.  These songs have catchy beats, different voice tones, variety of music styles that intrigue the listeners.  This is usually done at the end of the period for the last five minutes.  We usually write in the air to describe what we are learning.  The songs are learned very quickly and begged to be played again repeatedly.  There is knowledge in this way of learning and it imprints during the musical experience. 

Free Teaching Music Downloads with Graphics on Mathwire.com/music/music/html

Act Out Numbers

It may seem that these students are not listening.  They may be dropping their pencil, tapping their foot, making faces and unable to stay seated.  Is there a learning disability here?  Aren't they usually boys?  We are discussing young elementary students, however they have probably been in a classroom setting for several years.  They know the drill.  You are looking at a kinesthetic or act out learner.  This is a person that benefits from movement, touching you when they are talking to you,  and touching items as they learn.  They like to watch demos and then imitate the actions.  Commercial magnetic numbers, number puzzles to touch and move around is effective in acting out his learning.  Lessons should be kept short with breaks that allow movement.  

Brain Gym is a book about exercises that teachers should use at the beginning of the lesson  to engage both sides of the brain.  It basically has you and your students twisting and moving arms and legs crossing the trunk of the body and stretching.  This activity is beneficial for all types of learning styles but I'm sure a relief for the acting out learners. 

As a math resource teacher, I came up with an interactive math activity that seemed to delight kindergarten and first graders.  This can be translated to higher grades with other areas of the curriculum.  I used one blank foam die that had white board material on all six sides.  With a white board marker, I would put a number sentence, ordinal word or even math words being taught at the time, something different on all six sides.  I may have a group of four to five students.  I also used a small white board or let one of the students keep score.  I put each students' names on the white board.  I would make the students stand back and tell them I was to roll the die and they could not touch it.  As soon as they solved or recognized the answer to yell it out!  The first person to answer correctly received a tally mark beside their name.  If there was a tie, then both students got a tally mark.  This continued until we got to five tally marks with the cross over.  Many of the students were just learning tallies, so we had a double lesson going on.  This game created much movement and excitement with added learning of data on the die.  I shared this with the primary teachers and suggested that it be used with reinforcing high frequency word recognition.  I will be offering this idea and the materials on my site later.  Our students get lots of opportunity to act out.

A floor number line that you can make is ideal for the kinesthetic or act out learner.  We can add and subtract, always starting with the larger number.  We can get on a number and jump back three to subtract,  or jump forward to add.    This also helps with the organization of lining up.  When the act out learners are moving and touching, the brain is engaged.

Another interactive game is "Guess My Number".  I went to a tile store and got basic least expensive squares of tiles for several groups of students.  With permanent markers, I numbered them 0-9.  I would make up describing facts about my number. (e.g. I am an even number, two digit, less than 9 tens, greater than 8 tens and my ones rhyme with great)

There are books at Creative Mathematics, Kim Sutton with innumerable ideas.

"Pass the Pen" by, George Woodbury's Blogarithm is another interactive game involving movement for the act out learner.  Mr. Woodbury uses this activity as a closure to reinforce a concept he has just taught in math.  A practice problem already taught is put on the board.  He says "pass the pen".  Students wanting to solve the problem will raise their hands to participate.  He will call on a student that does a preliminary step.  That student turns around and says, "pass the pen".  This creates excitement and attention as the student decides who to choose next.  This action continues on until problem solved.  It sounds like higher math, but it could be used for growing patterns, geometric shapes, number skipping, etc.

In conclusion, the act out, or kinesthetic math learner needs space, ways to learn by touching, movement and we can't underestimate interactive computer math games.

Look for my post on Math Manipulatives.  There are too many to name now!

Optic Numbers

Outdoor Hundreds Board

This is another way of describing a visual way of learning.  These learners need information presented with images or pictures.  Illustrations that describe what words are telling.  Visual or optic learners prefer colors, maps, to organize information and communicated with others.  They are able to use their “mind’s eye” to visualize the objects plans and outcomes.

Mastering Number Facts: 

Despite great effort, visual or optic students have persistent trouble “memorizing” basic number facts in all four operations.  This is after great effort expended to do so.  4+5=9 or 7x3=21 facts can be labored over for years with an outcome of counting fingers, or pencil marks.  This could be an inefficient and unsuccessful memory strategy. 

A simple individual facts chart helps the students to proceed to more complex applications and problem solving.  If he is able to find the same answer in the same location each time, it helps recalling what it is.  Also, by blackening over each fact that has bee mastered you will prevent the child from over reliance on the chart.  This increases the motivation to learn the next one. 

These cards show that the students have already been exposed to thinking in combinations .

Tips for Teaching the Optic/Visual Mathmind

-There are commercial interactive, motivational games

(Child’s interest and keen attention is as important as time spent)

-Emphasize reverses  (6 + 3) (3 +6).  This can be done vertically or horizontally. 

-Use of FACT FAMILIES :  Give 3 numbers to the child.  Draw a simple house.  Using addition and subtraction, make four rooms with number sentences (e.g. 12, in the top of the house:  8,4,12.  Answer; 4 + 8 = 12,

8 + 4 = 12, 12 – 4 = 8, 12  - 4 = 8, This is a visual/optical way of using facts to demonstrate graphically.

- The Hundreds Board is an endless way to show patterns of numbers with every operation of math.  There are hundred’s board book ideas that can be purchased.  With counters, students can be taught to add ( e.g. put counter on 5, count up 3, what is the number, etc.)  Put counters on all even numbers up to 20.  This is done in kindergarten and first grade.  It is the beginning steps of multiplication.

-Students should make their own math journal with illustrations and key math words.

- With a partner make a poster about a key math operation with colorful markers, construction paper, and other art supplies.  Have materials available; just make sure the math principle is being portrayed.

- Flash cards – Limited time, fast and furious concentrated daily.

- DVDs or videos to demonstrate a math activity

- On-line interactive math games

- Watch someone perform or demonstrate a math discovery

- Watching themselves performing a math activity played back on video

- While using visual cues, also using motion or music interactively

- Piggy Bank, empty and recognizing coins and counting money

- Cooking – quantity, measurement tools

- Block building – counting blocks in a line or in a tower

- Play dough – cutting – making sets of objects, patterns/fractional parts

- Pictures with all kinds of paper medium and color, color, color

- Manipulatives, colorful counters, to add and make groups put some in:

   Take some out, Make up pretend world stories

- With maturity these sets can be drawn

- Legos, blocks, cubes, dice, counters, are all immeasurable tools to help

   the child see how math all fits together.

In conclusion,  you can already see that the visual or optic math learner is needing some of the same tools that the other learners will need.  The styles will interact, mix and hopefully further engage the brain more fully.  

How Do We Ignite The Math Brain?

The unborn child's brain begins to relate to  patterns and rhythms of mother's voice as she reads and music when she  dances before he is born.

After birth, the stimuli in voice and vision begin to build the brain cells.  The first two years of life is the period of time for the greatest growth of brain cells in vocabulary building for speech.  This is the time we must interactively ignite the brain with exposure to words of every variety.

  After the age of three, unused brain cells will diminish.   That's why feral children that have been abandoned or "raised" with animals ( in rare instances) may never gain full speech.
That's because they lost that early period of speech development with humans.


So, we have to make it count from the earliest days.  It's never too early to read to infants.  We can do it in a rhythmic way.  A rattle, and a bounce toy can be played interactively to music.  Babies up to two years should be showered with pictures and books of animals, places, things, numbers everything we can talk about, name and describe.  Attention span is very short but the material has to be surrounding the child.  Remember, brain cells are being ignited to build on new learning.

These are my top ten Prerequisites after excellent prenatal care:

1.  Safe and (physically and emotionally) secure home surroundings

2.  Excellent nutrition according to pediatrician

3.  Attention to all health matters (illness or Doctor's appointments

4.  Secure the child in loving, gentle soft touch and firm holding

5.  Show affection with touch, voice,  kissing, snuggling and physical contact

6.  Interactively play with child and expose to tactile materials (soft, sticky, stiff, rough fuzzy,bumpy, etc.  and tell the describing names.)

7.  Read to your child early in a rhythmic way and describe the pictures.

8.  Have at your disposal stacks of pictures of any and everything from  people, animals,  outdoors scenes, mountains, exotic places, buildings, numbers.  Cards can be purchased, magazine pictures, or free downloads, etc.

9.  Dance with your child.  Hold your child. Put on music and dance to beats. Count the beats.

      As the child gets older he will count with you and get the patterns. This is building the math mind. 

10.  Interactive play with toddler toys to incorporate math principles such as counting blocks.  Take one away, cover one up,  put two more in, practice counting songs.  "Let's stack 4 blocks, take 2 away, how many blocks are in the tower?  Hide and Seek!  Count to 5...
Count to 10.


Now we are moving into NUMBER SENSE.  This is how we know Quantity.  The base ten numeration system is the foundation of understanding number sense.  The place value system is based on ten symbols.  It is the ultimate recycling plan.  After nine there are no more symbols, so we go to ten.



Let's make a Number Map!


On a large chart paper, write, or use commercial numbers, 1-10.   Find inexpensive stickers to count out that number.  Stick that many under the number.  Write the ordinal number above and show the child the word for that number.  A toddler can only keep their attention for a few numbers at a time.....but that's OK!   We can go back to it later with different stickers that maybe more relevant to where his interest lies.  This ignites counting and number sense.

Introduction: Hello!

     My name is Kay Allen. I am an elementary school tutor after serving as a classroom teacher. grades kindergarten through third,  support math advisor, gifted and talented teacher,  grades 1-5.  I have also been a math resource teacher for students with learning differences.

I formally retired from the system in North Carolina in June 2010.  I am married with two grown sons with the delightful blessing of a grandson aged three and one-half.  Can you imagine the interactive strategies I have been using with him since I could get eye contact?  Yes, I have bought fancy manipulitives and games, but we also make up math games and activities using sets of cars, trucks, dinosaurs or however I can produce a math principle.  Of course we move quickly because the attention span of his mind and imagination is fast paced and we are off to the next pretend place and event. 

Learning to make patterns with colored Goldfish

I have continued to work with children from ages one to ten years of age in math.  Out of 21 years of experience and devotion to teaching, there are obvious truths I have discovered.  Math is a naturally occurring phenomenon that when directed to children's attention becomes an exciting, mind expanding fact that bears repeating by them over and over again.  For example, patterns in nature observed while going for a walk, man-made products;  make it fun to make patterns with about anything (small objects, toys or items at home).  

We've made a great pattern, but now it is not complete without our tractor trying it our as a bumpy road.