P E MD AS: aka Order of Operations

Please Excuse My Dear Aunt Sally.  There are a number of mnemonic ways to remember the order of operations, but I have never seen a hands-on demonstration that might help a child remember what to do first.  So here goes:

Quick review: if you don't recall, a math operation is the adding, subtracting, multiplying, and dividing that is applied to a set of numbers.  An expression is a combination of numbers and operations that can be simplified to a single value, in other words, whatever is on one side of the = in an equation.  When there are several operations to be done in an expression, mathematicians have agreed to do them in a certain order.   For example,  3 x 4 + (6+2) + 5 has a lot going on.  But just like in the emergency room, where the first patient in the door is not necessarily the one that gets seen first,  the numbers are not "operated on" from left to right. PEMDAS is the triage system that gets numbers in a long expression taken care of. 

So here is how to set it up:  The first thing you need are parentheses, and you'll make them out of paper.  Glue a pair of paper parentheses onto each of  few paper plates.  Now you need operations symbols, so glue some toothpicks or craft sticks together into addition (+) signs, which also double as multiplication (x) signs.  A dab of clay serves as the dot form (*) of the multiplication sign.  A single toothpick or stick serves as a subtraction (-) sign, and adding clay dabs above and below the stick turns (-) into division (÷).

Note:  the toothpicks are not glued onto the plates.

Either the clay dot or the toothpicks can be used as a multiplication sign, whichever your child is most familiar with.

Two clay dots turn a subtraction sign into a division sign.

The clay dots should be kept soft and sticky for a reason.  If you use clay to form numbers, it is best to use clay that can harden, in contrast to the "stickiness" of the multiplication and division symbols.



Now all you need are some 3-D numbers.  These can be magnetic numbers from the Dollar Store, foam puzzle numbers, numbers you form out of hardening clay or pipe cleaners (aka "chenille stems"), or in a pinch, you can cut numbers out of paper.

Use the numbers and operation signs to construct the expression.  If the numbers are in parentheses, they go on a plate.  For exponents, you can use a dab of clay-- aka the multiplication sign-- to stack numbers on top of each other, as many numbers as the exponent indicates.

So, for example, to indicate 4 to the third power, you would stack three 4's together with a dab of clay between each.   Use another piece of clay to form a small "3" to indicate the exponent.  For 5 squared, you would stack two 5's together with clay between them and a small clay "2" in the exponent position.  Note that the exponent itself is soft and sticky like the multiplication symbol, not hard like the numbers being multiplied. This model helps the student remember that 5 to the second power is actually 5x5 and not 5x2.

Now here is the explanation for the acronym "PEMDAS":

1. The P stands for "parentheses."  Or "plate." Either way, the parentheses form the outer shape of a plate, and you have to "clean your plate" first.  Or if you don't like that analogy, just stick to "do whatever is on the plate first."  So in this case, 6 + 2 is what you work out first.  6 + 2 = 8.

2. E is for Exponents.   Here we have 5^2, which is also know as 5x5, or 25.

Note: when typing exponents into a format that doesn't allow the little numbers in the exponent position, they are entered with a carat, like this:  5^2, which means "five to the power of two." Some computers have a calculator function that recognizes the carat also, and will calculate it for you.


Similarly, the asterisk * serves as a multiplication symbol, and the / forward slash as a division symbol.  If you want to see the ÷ symbol, try "alt" and /  or "option" and /.

3. Next, you look at the operation signs.  You will notice that both - and + have only a single horizontal line connecting the numbers on either side.  The vertical line on the + doesn't touch the numbers.  So these are kind of weak-looking links.  If you are using "x" as your multiplication sign, the "x" looks like a stronger link than addition, because it can touch both numbers in two places, top and bottom.  (Yes, I know that in real life the x doesn't actually touch the numbers, but we'll imagine.) The "dots" can be thought of as "sticky"-- like clay or glue dots or magnets or even planets with gravitational fields.  So they also make stronger ties than + or -.

So, after you take care of whatever is on the plates, and simplified the exponents, you go on to the strong/sticky operations.  That is the M and D, multiplication and division, of PEMDAS.  (Note that multiplication and division are done left to right, so it could just as easily be called PEDMAS.)

Last, you take care of the weak links, with addition and subtraction.  Again, these are done left to right, so if your - sign comes before your addition, you subtract first-- but PEDMSA is a lot harder to say.

So then,                       3 * 4 + (6+2) + 5^2:
Plate/parentheses:                       8
                                    3  * 4 +    8     + 5^2
Exponents                                               25
Sticky ops:                   3 * 4      
Weak links:                     12    +  8      +  25
Solution:                                              =  45


Easy, right?

Now,  if the exponent were on the outside of the parentheses/plate-- as in  3 x 4 + (6+2)^2 + 5^2, you would still do PEMDAS order.

In this case, you may need to help the student model the idea that since the whole "plate" is being squared, you can make two identical plates and stack them on top of each other.

Then it  obvious that you have to simplify the expression inside the parentheses-- add 6 +2 -- before you can square it.

So then,            3 * 4 + (6+2)^2 + 5^2:
Plate/parentheses:               8^2
                          3 * 4 +    8^2     + 5^2
Exponents                          64          25
Sticky ops:        3 * 4      
Weak links:         12    +  64      +  25
Solution:                                   =  101


If your student doesn't see the importance of "triaging" the numbers in the same order each time, you will of course want to take him through the operations the right way (PEMDAS) and the wrong way (straight across left-to-right or randomly) just to see the difference.  Note-- if the equation is already written so that multiplication or division happen first, it may not be as effective to compare the two results.  Try something like   3 + 4 * 6 + (4 - 2)^2  which if not done correctly, will not give the correct result of 31:

3 + 4 * 6 + (4 - 2)^2
3 + 4 * 6 + 2^2
3 + 4 * 6 + 4
3 + 24 + 4
31

Parents are Heroes: Nothing New Under the Sun

One of my all-time favorite reads in my adolescence was a book written the year I was born, called How to Be a Successful Mother.  This down-to-earth, no-nonsense book explained not only how to prepare for and care for a baby, but also how to cope with life as the mother of several preschool children while living in a fourth-floor walk-up apartment with basement laundry facilities, and using the bus for transportation.  (I take my easy life for granted!)  Later I discovered the same author had penned an earlier book, The Pregnancy Primer, in the 1950's, so I got a copy of that.  It was another amazing look into a past that I'd only glimpsed in episodes of I Love Lucy.

In both of these books, the author revealed a calm wisdom, a can-do attitude of working with what you've got, to do what you need to do, without complaints or hysteria.  I very much admired her attitude and pragmatism.  The fact that she recommended having your wine and ashtray ready before you sat down to nurse your baby-- well, that lent historical flavor.

So when I discovered Mrs. Featheringill had also written a book about math, I had to get it, too.  And again, she amazed me with her insight and intelligent prose.  (I have since learned that she also wrote poetry, and with her husband, owned a jazz record label  in the 1940's.  I would have loved to meet this woman, but she died in 1997 at the age of 81.)   Her basic point was, just because your child is being taught something you don't understand, doesn't mean you can't learn it.  You're plenty smart enough for the challenge.  Which was exactly her point in her earlier books-- don't sit at home crying because you've got three kids in diapers and no car to run errands--- just pack up what you'll need for the day and take the bus!   No excuses.

So, as parents and teachers now struggle with "Common Core" math,  railing against what appears to be poorly executed overhauls of a half-broken system, I again refer to the wisdom of Mrs. Featheringill and take a deep breath as I read her words from 1966:  "Math hasn't changed...The modern world has changed.  Getting ready to meet new kinds of problems, your child already uses "Higher Math" terms as he starts solving problems in kindergarten."

Granted, I don't recall solving any problems in kindergarten-- the only math I remember involved an annoying series of worksheets that had us circling sets of cows and pigs-- but I was in kindergarten four years after these words were published, so perhaps I did and just don't recall.  But the fact remains, she wrote this book in response to parents who were confused and frustrated with "the new math."  And today we again have parents who are frustrated and confused by another new math.

Will the confusion last?  Will our new math last?  Is it an improvement?  Was the old 'new math' an improvement over old old math?

Reading her book, I see striking similarities in the philosophy behind the change.  Proponents of the change were frustrated because students could run the algorithms but often lacked the understanding to apply them.  That's basically the impetus for Common Core (or in Arizona, College and Career Ready ) math standards now.  In theory, then, new math then and new math now are great-- and of course, getting students to understand concepts and apply them to real life is a worthy goal.  Unfortunately, the balance between understanding concepts, mastering algorithms and memorizing math facts is a delicate one.  Students need all of these skills to do math well.  And this truth is recognized by policy makers; there is actually an instructional model called Balanced Math that attempts to put all of these components together, so that students participate in math fact drill, algorithm practice, and concept exploration all in the same class period.

The problem is, it takes TIME to touch all the bases.  When I was homeschooling, I had the luxury of having each of my children working on two different math programs each day.  We would do a supplemental program together, such as Miquon or Key to Fractions,  and then they each did Singapore at their own level.  By middle school my daughter had caught up to her brother, but we still used one main program-- Math-U-See or Teaching Textbooks, and a supplemental curriculum, such as Key to Algebra.  I'm not even sure they noticed they were doing math twice a day; if they did, they didn't complain about it.  They had instructional time and independent work for each program, plus any math fact drill I assigned. We had all the time we needed, and if they needed additional explanation, I was right there.  They were probably spending a good 90-120 minutes on math each day.  In high school, their school used block scheduling, which allowed for longer periods of instruction as well.

Unfortunately, the elementary and middle school math teachers don't have the luxury of time.  In my years at the local middle school, the "fifty-five minute" class period I supposedly had for math was whittled away by the lack of a built-in passing period (the bell that rang to end one class was the start bell for the next, even if they were coming from across campus) and the housekeeping details of having students copy their daily objectives, stamping their agendas for homework completion, and attending to other business.  So the actual instructional time was much less than it looked like on paper.  Throw in the mandatory weekly, quarterly and end-of-year tests, and I calculated that the students had an average of three short instructional periods-- the equivalent of what my children used to do in a single day-- for each day of testing.

Kind of drove me crazy.  It wasn't so much that the objectives were unteachable, it was the frustration of needing more time to teach them.   And much of math is sequential, so that a student needs to have a good foundation in the basics before moving on to new material.  Unfortunately, the rush to have the students "solve problems" can sometimes mean they get short-changed on learning math facts and algorithms.

There is simply not enough classroom time to get everything done.  Just as the emphasis on "Whole Language" and "Six Traits Writing"  edged out phonics and handwriting, grammar and spelling rules in an attempt to make reading and writing more meaningful for the students, time that used to be focused on building a lowly, boring foundation of math skills is now spent constructing beautiful houses of math application.  But without a good foundation, these beautiful houses will fall.

So, as it was in the 1960's with "New Math," it falls to the parents to make up the difference. Which is why children come home from a long day at school with miserable amounts of homework. Which is why the academic gap between kids who show up to kindergarten never having been read to, and kids who have enjoyed regular read-alouds, only gets wider as the years go on.  Parents who may work two jobs to keep food on the table cannot (or make millions and will not) invest time in their toddler's intellectual development, and those same folks rarely feel capable of (or responsible for) helping with homework later on.   Happily, most parents I know are not like that at all.

How to Understand Your Child's New Math was written to another generation, but in its philosophy, it was written to parents struggling half a century later.  They are the same people.  They are the parents who are committed to helping their children succeed-- despite curriculum changes, despite time limitations, despite school funding shortages, teacher shortages, and everything else that threatens to interfere with the success of the next generation.

These parents are heroes.   Wine and ashtray optional.

The Homeschool Option: Choosing a curriculum to fit your child's needs

One of the most wonderful things about homeschooling is that you have the freedom to choose materials to fit your child's educational needs.  Ironically, one of the scariest things about homeschooling is that you have the responsibility of choosing the materials to fit your child's educational needs!

There are a lot of differences from one curriculum to the next.  Some differences won't matter to your child-- many students adapt well to a variety of approaches.  That's why "mass education" in a classroom setting can ever work at all!  However, you may be homeschooling your child (or considering homeschool) because what is offered at your local school is a poor fit for her.  That was our experience years ago-- by the time our oldest was in second grade, he and his younger sister both hated school and felt discouraged.  It just wasn't working for them!   So we set about finding a better fit for their education.  They're doing well in college now, so I guess we did okay!

If your child is not an average, "in-the-box" kind of kid, here are a few things to consider when looking at various materials:

Sensory Input / Learning Style:  Some children have a strong preference toward learning through a certain sensory pathway.  Visual learners remember best what they see; auditory learners remember what they hear. Tactile learners need to touch things, and kinesthetic learners need to move and do things.  For example, when learning how to get from one place to another, do you need to see a map, hear spoken directions, trace the map with your finger, or can you only remember after actually going there?  I can go somewhere multiple times, but until I've seen a map, I simply won't remember how to get there.  But some people are just the opposite!  Happily, there are materials for just about every subject that are designed for each type of learning style.

If you aren't sure about your child's preferred learning styles, here are a few sample tests online.

Processing Approach-- Analysis vs Synthesis:  Once the information gets into the brain, the learner has to process it.  And some people have a strong tendency toward processing by analysis-- breaking down the whole into its parts-- versus processing by synthesis-- building the whole from its parts. For instance, one child may remember a spelling rule by seeing many examples and figuring out the pattern. That's analysis, sometimes called the discovery approach.  Another child may need to be told the rule, and then use it to create examples.  That's synthesis, sometimes referred to as explicit instruction.

Focus-- Mastery vs Spiral:  Some students need a lot of variety and do well when a topic is introduced, practiced, and returned to often, while more topics continue to be introduced. Other students do better when they stick with a topic until they have mastered it.  While they may review it later, they have a definite closure before continuing on to the next topic.  In literature, some students may prefer a unit on poetry, a unit on short stories, a unit on novels, and a unit on essays, while others might prefer the spiral approach of studying all of these types of writing together over and over, organized by different themes.  Also, some math programs are spiral, while others provide completely separate studies for mastery of addition, subtraction, multiplication, fractions, decimals, and other topics.

Work pace and volume: Some students need a challenging pace, while others get overwhelmed easily if new topics are introduced too quickly or too often.  One student may need a lot of practice, while another might have a meltdown over too many math problems.  While either the pace or the volume of work can be modified if the curriculum is otherwise a good fit, it is something to consider.

Presentation--distraction and interest level:  Sometimes the page layout in a book can be too distracting for a student; another book might lack illustrations or color and not hold a student's interest.  The screen design or interactive features on a computer lesson can be an issue.  In a dvd lesson, the voice that encourages one child may put another to sleep.

Independence level:  If you need your child to be able to work on a certain subject independently, the instruction needs to meet her at her level.  This can be a concern whether she is reading a textbook or using an interactive computer program.

Consumability: If you have limited resources and multiple children, you may want a curriculum that is reusable for future years or able to be used by multiple users at the same time.

Price:  Few families have unlimited resources.  Whatever your budget is, there are ways to match your child's needs with an appropriate curriculum, but the lower the budget, obviously, the more resourceful you have to be.  Still, with a lot of help from the library and the internet, you can plan a good education for your child.  That would make a great future post topic...

Every child and every family's situation is unique.  Some of these variables will be more important to your selection than others.  Perhaps a few won't matter in the least!  But as you look over the choices for your child's education, I hope these ideas are helpful.

Does my child have a learning disability?

Every time I start a blog post, I end up with several other related topics that I feel compelled to write about.  My draft queue is full of post titles with one or two notes about what I want to discuss.  And then I get started on one and can't find a good stopping place!  This blog post is another one that has had me writing, adding more, then adding more, and then, oh wait-- there's this, too, and I ought to mention this, and...

I have worked with some learning disabilities from the parent side as well as the teacher side.  There are countless books and websites dedicated to individual diagnoses, and I am no expert on any of them, so I will try to limit my discussion. But it's hard.  Apparently I have a case of lexiarrhea.

Seriously, though, there are a lot of diagnoses out there:  Dyslexia.  Dysgraphia.  Dyscalculia.  Dyspraxia.  Scary words that mean a child is struggling with one or more foundational skills, interfering with her ability to do basic academic tasks, such as "reading, writing, and 'rithmetic."  A summary of each of these terms is here.  And that doesn't even complete the list of the possible barriers to a child's academic success.

But how do you know if your child has a learning disability?  And what do you do about it?

Many children have legitimate, brain-based learning disabilities.  Their brains actually receive and process information differently than other children.  This may be due to brain development issues that happened before birth, or even after.  Most of these issues do not affect actual intelligence, but they can frustrate his ability to absorb information or express himself, or both.

One child that I dearly love, for example, knew all her alphabet letters and their primary sounds at age two, and yet by third grade was still having trouble comprehending what she read.  Turns out she had auditory processing problems-- most likely caused by frequent ear infections that had resulted in functional deafness during her first year of life.  There was a domino effect:  her ears had not been correctly delivering sounds that she needed in order to process speech, so critical neurological connections were weak.  Although her hearing was restored before her first birthday, the damage was such that she still needed more time than her peers to think through what she heard in order to understand it. This affected how easily she could acquire new words from conversation.  By the time she started school, she lacked the wealth of vocabulary that most of her classmates had.  The same domino-effect processing damage can happen with children who have visual impairments.

Then there are those children whose hearing and vision are normal, with normal processing skills, but a lack of exposure to certain activities has made them functionally dyslexic, dysgraphic, or otherwise disabled.  They simply don't have the tools to learn the skills.  A common example is seen in children who show up to kindergarten without ever having been read to-- who don't even have books in their house.  They have no experience with letters or words in print, no nursery rhymes or Dr Seuss to form the basis of their phonemic awareness.  Similarly, some children arrive at school without being able to count, identify colors and shapes,  hold a pencil, or use scissors.  They are behind before they even start.   And because they are not ready for learning from the starting point that the school is prepared to teach them, they fall further behind each year.  (This is the motivation behind the government push for preschool for all.)

Years ago, lack of preparation was less of an issue, because kindergarten was all about learning these foundational skills.  Children sang rhyming songs and strung beads, kicked balls and colored and dug in the sand, all the while developing important brain connections.  Now, however, kindergarteners are expected to come with these skills already in place.  So those who come "unready" are at an extreme disadvantage and often never catch up, because there's no time allowed to go back and build the missing foundation.

Your child may struggle mightily with reading.  But before you decide that your child is dyslexic, make sure he has had the opportunity to develop phonemic awareness: your Dr Seuss and Mother Goose books are well worn.  You've played so many rhyming games, you could moonlight as a rapper.  Or, your child may be frustrated out of his mind by writing, avoiding it at all costs and having a meltdown over copying a single sentence.  Before you decide he is truly dysgraphic, consider whether he has had opportunities to develop his fine motor skills through tasks such as bead stringing, card sewing and button sorting.  If the struggles don't get easier,  seek help.

Does your child know his phone number and address, and can he remember where the matching cards are when he plays a memory game such as Concentration?  Depending on his age and what he has been exposed to,  not being able to do these things could be a red flag.  If your child is about seven years old and has developed phonemic awareness, but still struggles with putting the sounds together to make a word, or making sense of the words in a sentence,  there may be memory issues involved. Comprehension of sentences and paragraphs-- even reading longer words-- requires that the reader hold the first pieces of information long enough to put it with the last pieces, and process them together.

One quick screening test for short-term memory issues is the digit span test.  (With an explanation here.)  Research suggests that a child needs enough memory storage to be able to hear and repeat back (auditory memory), or see, look away and repeat (visual memory),  about six single-digit numbers in order to be most successful at reading.  This generally happens by age six or seven.  If he's not there yet, practicing digit spans can help.  If it doesn't improve, seek help.

Another indicator for reading problems is whether a child's vocabulary seems less developed than her peers'.  If your child often mispronounces words, doesn't seem to know what common words mean, becomes frustrated or angry or simply "tunes out" if you use too many words to explain something, or doesn't understand jokes that are appropriate to her age, she may have some "holes" in her language data bank.  Those were all clues for one child's auditory processing problems.  One day her parents tried turning on the subtitles in a comedy they were watching, and all of a sudden the girl was understanding the jokes in the dialog. All those words coming at her ears had been exhausting when she had to process them at the speed they were spoken. They realized that her low reading comprehension was tied in, too, because her vocabulary acquisition had been limited by the number of words she could actually process in a conversation.

ADHD is a common diagnosis.  There is a lot of debate over who has it, what causes it, and what should be done about it, but it's pretty clear that students who have trouble focusing on what they're being taught will have trouble learning.  I personally lean toward dietary/nutritional therapies-- undiagnosed chemical and food sensitivities can cause all kinds of behavior problems-- coupled with child-friendly, hands-on, active learning and ample physical activity, but sometimes medication is the only life-line a parent has.

If your child is in school, his teacher(s) may or may not suspect learning disabilities.  Don't wait for them to act first.  If you think your child's academic performance doesn't match his true intelligence, you may ask to have him tested.  That may take a while, but by law the school has a certain number of days to complete the test.  Then if they find evidence of a disability, they will begin to work with you to develop a 504 plan or an IEP (Individualized Education Plan).

A 504 plan outlines accommodations for a student whose disabilities interfere with his success in school. Accommodations can be specific changes that the teachers make in their requirements for the student, such as more time on tests, fewer items for homework, alternate testing methods,  or being allowed to turn in typed work instead of handwritten assignments.  They can also involve access to mechanical or electronic assistance, such as wheelchair ramps, computers, calculators, wiggle chairs, or amplified sound devices.  Special seating arrangements and permission to move around the room are also accommodations.

An IEP may include both accommodations and remediation.  It is designed for students whose needs are significantly different from the general student population, due to severe learning disabilities, cognitive or intellectual impairments, physical impairments, autism, or any number of other issues.  Remediation addresses the gap between what the student can do and what he needs to be able to do, and may include individual assistance, modified pace of instruction, alternate educational goals, physical therapy, and/or occupational therapy.

In either case, 504 or IEP, it is important to be a part of the planning process, and to advocate for your child if you think his needs aren't being met.

If you're a homeschooler, or your child is not old enough yet for school, you may still notice that things are not quite right.  In this case, you can still have him evaluated.  Knowing what the problem is makes it a lot easier to work with.

What's in a name? A great place to start reading!

You have Dr Seuss' ABC memorized.  You can recite The Cat in the Hat in your sleep.  If you read Richard Scarry's Best Mother Goose Ever out loud one more time, you're going to sprout goose feathers yourself.   But is your child ready to start actually reading?

(And by "reading," I refer to a child being able to decode not only the words he is familiar with, but also words he has never before seen in print that follow similar patterns.)

So---is he ready?  Maybe.  As long as you're not in a hurry. If you take it slow and provide plenty of hands-on practice, your child can probably begin to read, even before kindergarten.  But this is assuming his cognitive development is at the right spot.  Because after establishing good phonemic awareness, the next requirement to learning to read is being able to hang on to small pieces of information long enough to process them together.  For example, to read the word "dog," a child must not only know the sound represented by the letters, but also be able to string them together to form a word. A word like "documentary" takes a lot more memory to process.  And in order to comprehend whole sentences, the reader must remember the beginning words by the time she gets to the ones at the end.

Granted, there are folks who insist that you can Teach Your Baby to Read , but that is only possible by sight words, which trains the brain to "read" in very impractical way-- a whole word at a time, with no breakdown of the component sound parts.  A great explanation of the general outcome is explained in this dad's blog.  It's what I have seen most often in my students-- guessing at words, misreading one word for another ("diffidence" for "difference"), poor spelling, and a complete inability to break down unfamiliar words into syllables in order to read them.

In general, the older a child is, the longer sequence of information he can store in his memory, in order to make sense of it.  That's why you give a three-year old short instructions: "Come here, Joe."  "Stop hitting."  An older child can usually process more information, so that you can tell a ten-year old, "Take the hamper to your room and fill it with your dirty clothes."

It's the same with reading.  The younger the child, the fewer pieces of data he's going to be able to hold in his short-term memory long enough to make sense of it all.  Whether you're talking sounding out a word or comprehending a sentence, age tends to give the advantage.  That's why older children tend to pick up reading more quickly.  In fact,  research has shown that by the time students are in third grade, there's no noticeable difference between the reading ability of kids who started to read early vs kids who started "late," as long as they have developed phonemic awareness skills.

(Which drives me up the wall because schools insist on forcing *all* kindergarteners to read these days, or they can't pass on to first grade.  Forty years ago hardly anyone even began reading instruction until first grade, and then it was heavy on phonemic awareness.  Today some kids are so rushed, all they have time for-- and the cognitive ability for-- in kindergarten is memorizing the Dolch Sight Word list.  Then they just keep piling on sight words in the next grades and never get the full benefit of learning the phonograms. Are the kids now smarter by third grade, reading better, achieving more?  Um, no. Hopping off soapbox now.)

But let's say your child has excellent phonemic awareness.  She is already really good at making rhymes, and can distinguish the different sounds in a 3- or 4-letter word.  Maybe she even knows the names of the alphabet letters and some of their sounds.  What now?

The most important word to your child is her own name.  So why not use it to start teaching her to read and write?

Maybe your child can already write his name.  If not, even a toddler can learn to at least recognize his own name.  And if you make learning it a hands-on activity, it can be a lot of fun.  As mentioned in previous posts, using some type of moveable letters is the key. Starting off with 3-D letters, such as wooden or foam alphabet puzzles, is my recommendation.   (Using a puzzle with all capital letters is probably the easiest to begin with, although you might need two.  I have a puzzle with upper and lower case letters, so that is what I show here.)

Later on, letter tiles make a good transition from 3-D letters to 2-D print.   Some kids who have no trouble recognizing a stick-figure person or a simple line-drawing of a house will get all confused when presented with letters in print.  Why is that?  I suspect it is because they already know what the stick person or line drawing represents, because they understand what a person and a house are.  If the child has had real-life experience with the 3-D letters, then seeing and distinguishing the letters in print later may be much easier.

So if I were starting from square one to teach a child to read,  I would start by showing him his name in 3-D letters.  Even if the child's name is "Poindexter,"  he needs to see what his own name looks like.  If the child's name is Bob or Anna, so much the better!

(By the way:  if your son or daughter is at the age where s/he insists on being called "Batman" or "Elsa," feel free to go with that name.  One of my brothers used to sign his name "Lowly Worm.")

So I take the 3-D letters and place them in order on the table or floor.  I tell him that when we read, we use pictures of the sounds.  These sound pictures are called letters.  Sometimes a sound picture is one letter, sometimes a sound picture is more than one.  Each sound in his name has its own picture,  or letter.  I chunk the letters into syllables and say the sounds as I point to each phonogram:

For very young children, especially with long names, this is a good start.  Later I might take out the P and ask him to put it back where the /p/ sound goes.  Or have him point to the part of the word that is "dex."  But as mentioned before, that's about all his memory files can hold.


For a little older child (older 3, 4), I would later play with the letters a bit, move them around, and tell him,  "If we move these letters here, your name would be

   

(or   OBB or NANA or whatever you have).


Once I'm sure he knows what sound each phonogram represents, I might take some of the letters in his name to spell new words-- if that's not practical, you can add a new letter to make a nice rhyme. The main thing is to keep the sounds the same as they are pronounced in the child's name (instead of, for example, taking "Cristy" and spelling the word "city."  Try adding an M for "misty" instead.)

So our little Poindexter might see me make words like point, pet, den, Ted, text (adding an extra t) and net.  But at this point I would not separate the O and I, because in his name, they represent a single sound.

Of course, some names don't have enough sounds to get terribly creative:  Ed, Al, Bo.  Mae.  That's fine.  You don't want to focus on more than 3-4 sounds at a time, anyway.  You can add one or two letters just to give you something to play with.  Ed, plus a "b" and an "r" perhaps (bed, red), or Al with a "p" and "s" (pal, Sal).  Mae would be easier to rhyme with real words if you had spelled it May (though either way it still has only two phonograms-- M AE  vs M AY)  but you can always add f and r to spell Fae and Rae.

In the name at the top of this post, there are 6 phonograms:  B  R  AE  L  Y  N.  Unfortunately, there aren't a whole lot of combinations of these phonograms that will make actual words, so it might be easier instead to simply move the letters around:

"What would your name be if we took out the /b/ sound? " (Remove the B to spell  Raelyn.)
" What if we took out the /r/ instread? "  (Remove the R to spell  Baelyn.)
"Or what if we switched the /r/ and /l/?"  (Move the letters to spell  Blae-ryn.)

The important thing is that your student begins to understand that the letters represent sounds, and when you change the letters around or switch them out with other letters, you get a new word.  (Granted, it may be an imaginary word like "tob," but that's okay!)

You may want to encourage your child to try writing her name.  That's great-- most children are excited and proud to be able to write their own name-- but if she resists, please don't push it.  Fine motor skills that allow a child to control a pencil can take time to develop.

If your child is not yet ready to try printing her name,  she can continue to play with the 3-D letters, model some with play-dough, even stamp them onto paper using stamps or foam letters and paint.  Writing in a tray of sand or baking soda helps children get a feel for the the correct motions. But if your child is very young,  accuracy in letter formation is not really the point.

Understanding that the letters she is writing represent the sounds in a spoken word is the point.

At the end of each session, of course, you would want to have your child spell his own name correctly with the letters, helping him if necessary: "Which sound comes first?  Which letter shows that sound? (Note:  people do commonly speak of letters making sounds,  but to be most accurate, letters never actually make noise at all.  They just record in print the sounds that people make.)

After a few short sessions of playing with the sounds in her own name (depending on how many sounds you have to work with, it may keep you occupied for a while), keep adding in letters a few at a time. Eventually you will have the whole alphabet.  Stick to adding single-letter phonograms at first.  If the child's name has a multi-letter phonogram, of course, work with it "as is," using the same sound, for a while before you introduce the sounds for its individual letters:

You wouldn't go directly from learning Jean to spelling an, for example, because "a" does not stand by itself in the name.  Instead, you would first make rhymes like bean and mean.

A name like  H  ay  l  eigh  has 4 phonograms, but two are multi-letter, so you first might want to play around with the order of the sounds to make "Layheigh"  "Leigh-hay," and then make rhymes like Cayleigh and Tayleigh for a few sessions,  before eventually introducing the fact that the letter "a" by itself can be a picture of the sound /a/ in "cat."

As you add new letters, help your child be alert to the same sounds in other words.  Audrey and autumn share the same beginning sound, as do Casey and cat, Bentley and baseball.

So, a little bit at a time, you continue to play with the letters, adding new information gradually as the old is absorbed.  Before too long, your child will be reading on his own.

In a later post, I will explain how using colored pencils to mark words as he reads and writes can accelerate his mastery of spelling and reading.

Fractions, part 2: modeling mixed numbers, reducing, converting

If your student has not yet played with fraction manipulatives, please read Fractions, part 1.

Now that she familiar with the names of the fractions and what they represent, it is time for your child to start making those numbers do some work!

In this post, we will discuss 1) converting between mixed numbers and improper fractions, 2) adding fractions, and 3) subtracting fractions.  It is best to introduce the concepts in that order.

First, the definitions:  a mixed number is a number that includes a whole number part and a fraction part.  1 2/3,  3  4/5,  16 1/2  are all mixed numbers.  (Most children are already familiar with mixed numbers, because we tell their ages that way.  Preschoolers understand that someone who is three-and-a-half is older than a three, but younger than a four.)

An improper fraction is a fraction whose numerator (top number) is greater than the denominator (bottom number.)   5/3,  19/5, and 33/2 are improper fractions.

Once your child has learned that 3/3 = 1 and 5/5 = 5,  he can play with other equivalents.  Using fraction circles or strips, see how many he can match up exactly.  He should see that  1/2, 2/4 and 3/6 are the same length.  And 1/3 is the same as 2/6.  These are equivalents.  But 2/3, 2/4, and 2/5 are NOT equivalent.  Their denominators make them different sizes, and he can clearly see that two small things put together is smaller than two big things put together.

If your child is keeping track of fraction facts or fraction discoveries in a notebook, you can help him draw or glue in examples of equivalents and label them.

Older students can learn to reduce fractions.  Reducing means to write the fraction with the smallest accurate denominator.  In the photo here, 3/6 and 2/4 can both be reduced to 1/2.  2/6 can be reduced to 1/3.  But 2/3 and 2/5 cannot be reduced, because no fraction with a lower denominator fits.

Students who are familiar with multiplication facts may recognize that reducing a fraction on paper is a lot like dividing whole numbers.  The only difference is that she must divide the top and bottom by the same number.  It must be a number that goes into both the top and bottom evenly.  The best number to divide by is called the "Greatest Common Factor," or GCF.  The GCF is the biggest number that will go into the other two numbers evenly.

For example, the GCF of 10 and 15 is 5, because 5 is the biggest number that can go into both 10 and 15.  30 is the GCF of 30 and 90, because 30 x 1 = 30, and 30 x 3 = 90.  21 and 15 have a GCF of 3.  12 and 7, however, have a GCF of 1, because there's no other number that goes into both 7 and 12 evenly.

Dividing the top and bottom of a fraction by the GCF immediately reduces that fraction as far as it will go.

For example:

4     divided by 2    =    2
6     divided by 2          3

25   divided by 5    =   5
30   divided by 5         6

Of course, younger children who have not begun multiplication can just stick to using the strips and circles.

In the next step, you will need at least two sets of each fraction size: two circles or cut into halves, two into thirds, two into fourths, etc.  Or you can print out, color and cut a second set of fraction strips.

With the extra set(s) of fractions, the student can show that 3/2 (an improper fraction) is the same as 1 1/2 (a mixed number)  She can record her observations in "math code," writing 3/2 = 1 1/2.

Your student may note that while 7/5 is the same as 1 2/5,   8/6 can be reduced from 1 2/6 to 1 1/3.

For a bit of incentive, you can challenge her to find as many improper fraction/mixed number sets as she can in five minutes.  She'll need to write them down as she finds them so she can re-use the fraction pieces without losing her fraction discoveries.

Next, see how easily she can convert a mixed number to an improper fraction with and without using the fraction pieces:  can she show you that 1 1/6 is the same as 7/6?  Or that 10/5 is the same as 2?   See if she can tell you what the improper fraction will be before she shows you.

When your student can comfortably convert from a mixed number to an improper fraction and back again, she is ready to try addition and subtraction of fractions.

Phonograms? Phonemes? I just want my child to learn to read!

Now learn your ABC's right quick,
Or go get walloped with a stick.

This rather threatening verse in the classic book Heidi appears when the little Swiss girl begins to teach her goat herding friend Peter to read.  He is more than a little intimidated!  (Yes, I know, Heidi was not originally written in English, but it's the same idea.)  Learning to read and write in English can indeed seem a daunting task.  People point to sentences such as

 

and decide that the language is simply "not phonetic," therefore impossible.

It's not, really.  English is phonetic.  It uses letters to represent sounds, and these letters are written in the order that the sounds appear when the word is spoken.  It's not as straightforward as a reading Spanish, of course, in which each letter has a single sound, but for that you can blame all the early settlers and conquerors of the inhabitants of the little island known as England.  They all left their marks on the language.

The problem, then, is that you can't simply learn 26 letters and sounds and be done with learning to read, much less to spell.  Last time I counted, there were about 70 phonograms in the English language.

A phonogram is the written representation of a single vocal sound-- a letter or combination of letters  (from the Greek:  phono=sound + gram=writing).  The letters "ph" make a phonogram for the sound  /f/.  A phoneme is a distinct vocal sound, like /f/, /ah/ or /sh/, that you combine with others to speak words.  A baby's first word is often a combination of two phonemes, /m/ and /ah/, repeated: "mama."

Although phonograms and phonemes are different things, every spoken word will have as many phonemes as there are phonograms when the word is written.  There is a 1:1 correspondence-- in the word "cat," for example, when spoken, there are three distinct sounds /k/ /a/ /t/ each represented by a single letter in the written word.  Three phonemes, three phonograms.

The confusion for reading and spelling English stems from the following issues:

1. Many phonemes can be spelled in several different ways;  and
2. Many phonograms can represent more than one sound.

For example, the single letter "a" is a phonogram for the vowel sound in the words April, amber, and all.  Meanwhile, the sound /ay/ can be represented by a, ay, ai, ei, and eigh.

So how do you teach a child to read English without confusing him?  Should you teach him all the phonograms and each sound they can represent, or should you teach him all the phonemes and all the ways they can be spelled?

Well, if done correctly, the child eventually ends up with the same information either way.  And in general, you're going to start off simply, introducing a few short words with distinct phonemes that can be represented by single-letter phonograms.  Cat, bat, sat.  So in the beginning, it isn't a huge issue.

Sometimes where you go from there depends on which curriculum you use.  For example, The Writing Road to Reading is organized by phonograms.  The student learns all the sounds represented by "a," all the sounds represented by "b," all the sounds represented by "ough," and everything in between.  On the other hand, Reading Reflex focuses more on the phonemes.  The student learns all the ways to spell the long /a/ sound, the /k/ sound, etc.  You may prefer one or the other, or a combination of both.

Still, 70 phonograms!  That seems like a lot.  But it is not all done in a single week, or even a single year.  How do you eat an elephant?  One bite at a time.  (A very user-friendly way to eat the elephant is the new Spelling You See program.  I have only seen the samples online, but they look excellent!)

Which bite you offer first is up to you.  My favorite introduction to reading is Dr Seuss' ABC.  It introduces the basic alphabet and the concept of letter-sound representation, and throws in those very important rhymes that are essential to developing phonemic awareness.  But wherever you start, you can make it fun!

Here's where the hands-on part comes in.  When a child is beginning to read, she can increase her mastery of the reading and spelling when she:

1. manipulates letter tiles or puzzle pieces to represent the sounds in the order they occur as they are spoken:

The moveable letters are perfect for showing how rhyming patterns work.  Simply by changing out a tile, you can turn "dish" into "fish,"  "dawn" into "pawn."  Internal and end sounds can be changed, too, so that the student learns to focus on each phonogram in the word-- "dish" and "dash" are different words, as are "fish" and "fin."  When the student has mastered three-phoneme words, he can combine them to make compound words: pigpen, dishpan.  Later he can use the tiles to break longer words into syllables:  car pen ter,  so fa, sing er.

2. forms phonograms out of clay, string, or pipe cleaners to spell words;

In this case, you might have the child form the letters to spell "pan."  Then have him add the sound /t/ at the end and read the new word.  Finally, he can add the /s/ and have another word.  This kind of word-building is used in the series Sequential Spelling, which has wonderful lists of words that students learn to spell one sound at a time.

3. writes letters in a tray of sand or baking soda as she says each distinct sound of a word;

Writing in the sand is a also great way to work on the fine-motor skills of forming letters.  Just be sure the student is forming the letters correctly!  If your child struggles with printing or cursive, a great handwriting program that is developed by a physical therapist is Handwriting Without Tears.

4. divides words into syllables and mark each distinct phonogram in a word by circling or underlining with a separate color.  In this example, the student has first divided all the multi-syllable words, then circled all the multi-letter phonograms (missing the "ee" in Queen), and has begun to mark the separate single-letter phonograms:

The pencil-marking activities are probably best done after a student has had practice using the letter tiles.  What you have your student marking depends on what they have learned.   Some reading programs have the student indicating certain spelling rules they recognize in the words. In Writing Road,  for example, the students learn 5 reasons for using a "silent e," and when they find one they underline it twice and identify it with a number, 1-5.  They also number different sounds for multi-sound phonograms:  the ch in "choir" gets a 2, the ch in "Charlotte" gets a 3.

Each of these four techniques can be used during different practice sessions, to reinforce the student's mastery of phonogram-phoneme patterns.  Some activities may be more beneficial than others for particular students at different points in their learning.  For example, the clay models of the letters are suggested in Ron Davis' book The Gift of Dyslexia because, as he theorizes, the dyslexic student sees the letters 3-dimensionally, and forming them in clay helps the brain transition from the 3-D image to the printed 2-D version.  So this might be a critical activity for certain children. On the other hand, you wouldn't expect a beginner to take a pencil and mark all the phonograms in an entire paragraph.  But all of these activities can help the child focus on the details of the words, add variety and interest, and ultimately make the student a better reader and speller.


A complete list of phonograms (accurate to the degree that your dialect of English matches mine) can be found here.

Fractions, part 1: modeling equivalents and beginning addition

Fractions can be very confusing to students who have no concrete experience with them.  One 6th grade remedial math class was learning to reduce fractions, and they learned the hard way that half of 1/4 is not 1/2.  We were halving a brownie recipe (countertop ovens are an essential in my math classroom) and as the brownies were baking, the room smelled deliciously chocolatey... the students could not wait to taste them.  To their dismay, the miscalculation of the salt fraction made the entire batch completely inedible.

So, before any operations can be done with fractions, students need hands-on experience with the concept.  Cooking is a great way to accomplish this, and even toddlers can help measure ingredients for a batch of cookies.  Click here for a lovely sugar cookie recipe full of fractions!  Measuring cups can also be played with in the sandbox, bathtub, or at the kitchen table with some dry rice or beans.

The basic purpose of playing with measuring cups is to note the name of the fraction and its size relative to the other measurements, and to compare the quantities.  Which is bigger, 1/2 or 1/4?  How many 1/4 cups does it take to make 1 cup?  Can 1/3 go into 1/2 evenly? How many cups can you make out of five 1/4 cups?  How many 1/3 cups can you get out of 2 cups?





The more exposure your child has to measuring, the more natural the concepts will be for him.  Once your child is comfortable with fractions in measuring, try pie circles.  These can be purchased or made at home from paper, fabric, craft foam, or anything else you might have.  Templates are available online.

 The first objective here is to get students familiar with what is meant by 1/2,  1/3, 1/4, etc.  Concepts to emphasize include:

1. The fraction's denominator (bottom number) refers to the size of the piece-- how many pieces it takes to complete the whole (in this case, the whole circle).

2. The greater the denominator, the smaller the fraction size:  1/2 is larger than 1/4, even though "4" is greater than "2". (This is very important when you're making brownies!)

3. The numerator (top number) refers to how many pieces you have in the fraction.  2/3  means two pieces that are 1/3 each.

4. When the numerator and denominator in a fraction are the same number, you have all the pieces you need to complete the circle.  So 2/2 = 3/3 = 4/4 = 5/5 = 6/6 = 1

5. Some fractions can be put together to be the same size as a different fraction.  1/2 = 2/4 = 3/6, and  2/6 = 1/3.  These are called equivalent fractions.  But you can't make 1/2 out of thirds or fifths, and you can't make 1/3 out of fourths or fifths.

6. Sometimes different fractions can be added together to make whole circles.  1/2 and 3/6 make a whole, as do 2/4 and 1/2, or 2/3 and 2/6.

Plenty of time should be allowed for these concepts to sink in.  A child who is first learning about fractions may want to play with them for a few minutes every day, for a week or two, before you even start labeling them as fractions or specifying numerators and denominators.  For preschoolers, just playing with the fractions is enough.




At this point, you may help your school-aged child start a list (or booklet) of Fraction Discoveries (Or Fraction Facts, Fraction Truths, etc).  This is where she records what she sees in "math code"-- also known as math sentences or equations.  She can illustrate each equation with pictures of the fraction circles (tracing jar lids or using a compass to make the circles).  Her discoveries may include:

1 = 2/2
3/3 = 1
4/4 = 1
1/2 = 2/4
1/3 = 2/6
1/ 3 + 1/3 + 1/3 = 1
1/2 + 1/2 = 1
2/4 + 1/2 = 1
2/3 + 2/6 = 1

When circle fractions are mastered, you may go over the same concepts using bar models or rods.  These are Cuisenaire rods,

but you can also use homemade fraction strips.  These can be printed out pre-colored or ready to color; having each size a different color makes them easier to tell apart.  Then all you have to do is cut them out.

Warning: with the Cuisenaire rods, the orange "one" that is equivalent to two yellow "halves" is a different size than the blue "one" that is the same as three green "thirds."  The black "one" can only be evenly divided into seven white "sevenths."  That is great for demonstrating prime and composite numbers, but if you think it might confuse your student, you probably would be safer with the fraction strips.

What new Fraction Discoveries might your child come up with now?

Hands-on ways to encourage reading, Part One

There are many ways to discourage reading, but are there any hands-on ways to encourage reading? Happily, yes!  While reading skills themselves can be developed using manipulatives, sometimes a reluctant reader knows how to read but just can't seem to connect with a book.  Or maybe reading is just difficult enough that it doesn't seem worth the trouble, let alone fun.  It is generally better to supply the reluctant reader with books he is comfortable reading-- not too difficult-- although a challenging book that the child is very interested in is better than an easy one that bores him.  The key is to decrease the frustration level by raising familiarity with, and interest in, the reading material.

Here are a few hands-on ways to help your children connect with their books:

1.  Hands-on books:  The youngest pre-readers, toddlers, enjoy books like Pat the Bunny that motivate them to interact with the text.  In their preschool years, my children enjoyed the humor of Pat the Beastie, in which Paul and Judy poke the poor monster's boogers and pull his fur.  Lift-the-flap books, and chapter books with interactive elements, such as the Captain Underpants series, all encourage children to associate books with fun.

2. Tie-ins to videogames.  What???  Yes, my daughter read her first Nancy Drew book after a friend introduced her to the Nancy Drew computer games.  She loved the games, then she read the whole series of books.    My son became interested in reading about history after playing Civilization on the computer.  How did this work? Because of the games, they had some knowledge of the characters and/or events before they even picked up the books, so it made the reading less of a struggle.  The same idea worked for me when I was studying Spanish in college-- the first novel I read completely in Spanish was a translation of The Empire Strikes Back.  Because I knew who I was reading about and the basic plot line, the vocabulary and grammar were less of a struggle.

3. Field Trips and gimmicks.  Find opportunities to tie in real experiences with the books your children are reading.  When we read Red Sails to Capri, we celebrated by drinking Capri Sun (as we looked at the Blue Grotto online).  When we read The Secret of the Andes-- what else?  Andes candies  (while we found the real Andes in the atlas and looked at pictures online, of course).  Capyboppy, Mr Popper's Penguins, Dr. Doolittle-- these animal tales could inspire a trip to the zoo. Whether silly or serious, a tie-in can add fun to reading.

4. Creative projects: Sometimes it's exciting to do the things the characters in your books are doing.  The popular craft and recipe books that go along with the American Girl and Little House books are testimony to this.  When your child is reading a Magic Treehouse book about Ancient Greece, maybe he would like to stage an Olympics.  Perhaps he could build a chariot or a temple out of Legos.   Planting something green indoors or out, or even planning a garden on paper, could help a child connect to The Secret Garden or Princess Chamomile's Garden.  Let your child come up with her own idea of a project based on what she's reading.

5. Detective work: Finding background information on a character can add interest to the story.  If the location and time period are important to the story, the student can find them on a map and timeline. What other things were happening at the same time in history?  What clothes would they have worn, and how would they travel?  What technology did the character have access to?  Could Henry Huggins search for Ribsy on the internet?  How would life be different for the characters if their story was set in today's time?  Would Sarah Plain and Tall have found her husband on match.com?  A child might be challenged to collect things in his own house that would (or would not) be a part of the character's daily life, using clues from the story, or live "a day in the life" by dressing, eating, or in other ways doing what the character would have done on a typical day.

6.  How-to books.  There are tons of non-fiction books out there that can teach your child something he or she would like to learn.  My daughter enjoyed recipe books and craft books.  One pre-teen summer, we went through a book on making homemade beauty products, and we made several of them.  Yes, the internet has videos and websites for this sort of thing, but instead of spending hours browsing Pinterest, sometimes just the right book can be less overwhelming.

It's not easy to motivate reluctant readers, when there is much easier entertainment that competes for their time.  And every child is different, of course; what works wonders for one may fall flat for another.  What didn't work for my children was the very popular Accelerated Reader (AR) program at their school, in which students took tests on the books they read to earn points and win prizes.  They hated the tests, and the whole points-and-prizes system felt manipulative, as if reading was so terrible that the students had to be bribed to do it.  Perhaps none of these ideas I have listed will spark an interest in reading for your child.  However, I hope I may have sparked an idea in your head that you can use to encourage your reluctant reader.