Category Archives: mathematics

Mathematical Thinking at Play (5 year olds are amazing thinkers)

A few years ago I filmed a couple of the Fletcher gang playing  a game I love called “Set”.  Set shows us that challenging means fun.  The game that requires some pretty sophisticated thinking.   Initially, I didn’t know if  my 5 year-old could handle a full game of set, but I thought I would try it anyway.  And… she surprised me.  Even though she just turned 5 a week, earlier, she was capable of so much.  Now,  I’m not saying that she is a genius.  Rather, I make the point that average intelligence is still one of the most powerful forces the universe has ever produced.  We would do well to accept it.  When it comes to intelligence, however one defines it, average means powerful.

Check out the video to see how I worked with the Doo Dah.  The toy company markets Set as “the game of visual perception” but there is a lot more to it.  I use it primarily to encourage a) Planning/Executive Functioning  b) Simultaneous/holistic processing c) control of attention d) fun.  In the video you will notice that I use very little direct instruction.  Instead, I ask questions and encourage the Doo Dah to think for herself.  I could have given her the answer at one point, but that would have stopped her from thinking.  Instead, I  gave her just enough information to allow her to solve the problem for herself.  Watch this video with an eye for the process.  Maybe you do some of this yourself.  What you will see is: 1) I encourage the Doo Dah to survey the task/tell me her plan 2) I prompt her by asking questions/guiding her attention 3) I model for her by talking about my thinking 4) I ask her to summarize her own thinking.

Can Intelligence Be Taught?

How we think is very much a product of our culture and our relationships.  Play often leads to meaningful interactions,  creativity in learning, and new thinking strategies.  Even attention is developmental and  therefore trainable.  If you like, you could try meditation sometime.  You may find that the beginning  sitting practice is tough.  In fact, Buddhists say we have  “monkey mind” because our thoughts and attention bounce around like a monkey.  With practice, however, one learns to sit for longer periods.  One learns to select the breath as the object of attention.  Meditation is the art of resisting distracting thoughts, at least that is what I have learned.

On another note,  I can’t for the life of me understand how “experts” continue to ignore the important role that emotions play in learning.  Any reform that doesn’t place emotions at the center is likely to fail or have limited effects.  In response, several years ago I decided that I would develop video games as learning tools.  Few would disagree that it  is the child’s nature to play.  I go further.  I believe it is the adult’s nature to play, as well.  You might be aware that the average age of the most frequent video game customer is 40 http://bit.ly/brmQA .  If we care to bring the emotions into learning, we would do well to pay attention to video games.  I include play in learning precisely because play brings emotional experiences.  When a person feels intense feelings, he/she often performs better.  Boredom, conversely,  interferes with learning.  Anxiety interferes with learning.  Play, on the other hand, reduces harmful stress.  Video games are play. Video games can teach.

In a short while, my team and I will release what I think will be a very important iPad app.  I hope you will stay tuned so that you might try it.  I will give updates as the release date draws nearer.

For more on the role of emotions in learning, see http://bit.ly/bVRYul , http://bit.ly/aEQeFi and http://bit.ly/b8BYmE .

Can children learn the entire elementary school math curriculum in 30 hours?  For evidence that we might be underestimating what kids can do, see http://bit.ly/c7QETj

I recommend Todd Kashden’s excellent work on curiosity.  Please check it out:  http://www.toddkashdan.com/

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The Psychologist’s Role in Teaching Mathematics

Historically, psychologists have studied math but have been noticeably absent when it comes to doing direct mathematics work with children.   Notable exceptions include Diennes, Bruner, and Vygotsky, but to this day Psychologists tend to assess and make recommendations for treatment.  This surprises me, given the cognitive processing involved and the anxiety that so often appears in achieving and struggling math students.  I have designed and delivered treatment programs for struggling mathematics students for nearly 15 years, and I believe that other Psychologists could really contribute.  Psychologists have a unique and relevant skill set that would add value to a treatment team.  I have used mathematics as a vehicle in promoting a variety of therapeutic outcomes.    The hours I spent working on math may be my most favored memories of my practice.   To learn math for many of these youngsters is to conquer fear.  I offer a few thoughts on the role of psychologists in teaching math.

Psychologists can:

1) Discover the structure of the child’s thought through constructing mathematical models and problems.  Given space, children can do more than simply apply the sometimes confusing rules that might work on tests only to later fail to appear when encountering real world problems.  Psychologists aim to enhance thinking performance which leads to confidence and resiliency.

2) Psychologists might help make sure that math enjoyment survives past 2nd grade.

3) Psychologists get training in helping children use the imagination.  These techniques are well suited to mentally manipulate mathematical relationships, calculation problems and decoding symbols as mental objects.  Psychologists teach visualization, for instance, which goes far in enhancing performance by using visuospatial skills.

4) Help children understanding  concepts,  meanings, and numbers. Math is a language developed for measuring and describing the natural world.  Symbols appear, in part, to prove  underlying ideas, relationships, and concepts.  Physical objects are the reality, the symbols allow us to manipulate it.  (There are exceptions in higher math, of course).

5) Help children use discovery to make math meaningful. Rote Memorization=boredom.  When we play and build math problems, the inherent creative strategies transfer to real life problem solving.

6) Psychologists can make math fun. What if Math were fun? What if Math became an aesthetic experience?  I often repeat “fun is solving a problem mentally” which is how   Raph Koster defines it in his book A Theory of Fun).

7) We feel better by doing better.  The Psychologist might help  through a combination of play, motivational psychology, mindfulness, assessment, talk, and other interactive skills.  The psychologist trades in  building resiliency and helping patients solve problems.

Chime in and let me know what you think.  Have I made a case for it?

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Working memory matters!

For over 20 years I’ve been thinking about working memory in primarily two ways.  We accept  working memory as functioning as a visual-spatial sketch pad and as a  phonological loop.  Put simply, visualizing and rehearsing information by repeating it were the ways in which most of us recognized working memory.  By working memory, we mean that special type of short-term memory that allows us to hold information in mind long enough to solve a task.  For example, think about how you might remember the correct sequence to a new combination lock fiddle with the dial to open your locker.

The Visual-Spatial Sketchpad (VSS) allows us to work with our imaginations in a visual way.  We may remember the shape, color and location of an object as an imperfect picture.  If you are imagining where you need to go while walking down a hallway, then you recognize the  VSS.   The Phonological Loop (PL) works on information which requires a specific order to have relevance or meaning.  Imagine thinking about an unfamiliar  phone number or a series of steps shown to you in an unfamiliar math problem.   We often use the PL as a strategy  when processing language or other auditory information.

In 2000, scientists began to discuss a third type of working memory. This would be known as  the Episodic Buffer (EB).  The EB  helps to  contextualize information in a meaningful way. EB is a short-term working memory function that appears to take place primarily in the frontal lobes.  We use it to suddenly  pull information together in a meaningful way to recall events or complete a task.  The EB can bring information together from many sources to create a single episode or unified memory.

Three ways to enhance your working memory include:

1. Make use of your VSS, practice visualizing things. Hold math problems in your head.  See the quantities, not only the symbols.  You may actually take your finger and draw problems in space and then hold them in your head as execute the operation.

2. SET is a terrific game for enhancing working memory, categorization and strategizing.  I demonstrate the game in a earlier blog post.

3. Poker, bridge, and chess are other ways to practice holding information in your brain and manipulating it for successful output.

4. Physical exercise, especially aerobic activity, has a strong and measurable effect on these types of thinking.

So much learning depends on working memory strategies.  We might consider attention, problem solving and conceptual thinking as depending on working memory . Playing games and practicing strategies can really help. Sudoko to you!

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Coping With Math Anxiety Can Be About Perspective

Many children experience discomfort around math, we generally call this Math Anxiety, a fear that math is going to be uncomfortable and full of failure.

Here’s an interesting post via Elizabeth Stevens on Math Anxiety, I wished to share with you all.  Math performance is so important today.  Math is actually fun.  But, we have to approach it in an engaging way.  We can do this and we are doing it.  Thanks to colleagues like Elizabeth.

Let’s remember:

1. Math aptitude is not inborn, math skills can be learned.

2. Math is super creative, there are many different ways to get an answer.

3. Boys and girls can both be great at math, it depends on what we tell children about their experiences with math.

If you wish to delve deeper check out Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Thomas Carpenter, et al.

Breathe… and enjoy Dr. Marty

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5 Year Old Discovering Equivalent Fractions

This is the Doo Dah playing math.  Fractions. This is where kids often learn to hate math.  Guess what?  We can change it.  I’ll describe what is happening in this short video.  The Doo Dah is a) seeing what a fraction is b) playing w/ the proper toys/tools to help her discover its meaning c) going from concrete (seeing/touching, etc.) to abstraction (writing symbols, abandoning the manipulatives and doing the math in her head).  I hope you enjoy the short video.  I mainly want you to see how much fun we have doing math.  Math is a puzzle.  Math is a game. But, it is very hard to learn anything if you are not curious/interested in what you are doing.  So, let’s make it a game and play!

Anxiety is first the anticipation of an upcoming event. From that, there is an irrational fear that the outcome will be terrible, horrible, awful, etc.  Math Anxiety is the irrational fear that math is “too hard” and that “I’m going to fail” and “I’ll never get it b/c I’m not smart enough”,  etc.  Nonsense.  Just watch the Doo Dah.  Have fun and Game On.  Also, if you like the blog, please subscribe.  I blog on parenting and education issues mostly.  Thanks, Dr. Marty

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5 Year Old Doo Dah Likes #Algebra

The Doo Dah and I “play math” before she goes to Kindergarten.  We keep it fun, but do some pretty abstract things.  You can really see her thinking as she factors a polynomial.

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Algebra w/ Doo Dah (she’s five) #math

People have asked to hear what the Doo Dah is saying, so I ducked the music down.  I know that this may disappoint all of those Herb Alpert fans out there.  Anyway, the purpose of the video is to show that even Algebra (especially Algebra) seems no different from most other games and puzzles.  Just pay attention to Doo Dah’s thinking and her emotion.

I started focusing on Math with two of my children.  Now all of them are asking to play math, including my 12-year-old.  The summer can be a great time to get way ahead in math before September.  We are going to make this one of the fun things we do this summer.  Pretty cool.

 

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