The Ant and the Grasshopper (a modern day story)

Alongside a trail in the open Field of Mathe, a Grasshopper was lazing about, discovering and coloring patterns to his heart's content. A lone Ant passed by, bearing along, with great toil and rigor, a bag of corn he was planning to store, yet singing as he labored. Because he knew the facts, he had accurately analyzed his problem, quickly calculated his needs, while carefully considering the 'rithms' of the seasons.

"Why not stop and discover patterns with me," said the Grasshopper, "instead of toiling and moiling (grasshopper speak for "drilling and killing") your life away? You'll be much happier sitting and chitting about your work than making your repetitious rounds." The Grasshopper continued, "That Trail of Boredom you are on becomes the Path of Monotony just over the hill, you know." (Surely, he thought, this was a factor the Ant had not considered.)

But the Ant was undeterred. And he wasted not a step, still singing as he 'plotted' along, for practice had made his search for the missing factors all the easier.

"By the way, my feelers are 'Connected' to my feelings, you see?" said the Grasshopper, everyday still lazing on the trail. "I'm feeling soooo good about my discoveries. I need to be writing about the work I'm going to be doing after I finish discovering what I'm supposed to learn to do. Care to join me?"

"No, thank you. I'm preparing for what's ahead," said the Ant, as he 'Sing(ingly)Poured' his full bag into his storage bin. And off he went to glean more from even higher p(m)aths. "Quit your Trail lazin' days, Mr. Grasshopper. Get up and get to work."

"Oh," said the Grasshopper, "there's still plenty of time. But for now, this writing about my discoveries is just about filling up my hours. But I'll be 'Blazing this Trail' later."

Sadly, the Grasshopper never got past his Trail lazing days. You see, though over 'Andover' he tried, he found that he had to continually 'spiral back' to the place of his discoveries in the '(o)Penfield'. He never made it off the Lazing Trail nor did he ever reach the top of the Field of Mathe.

However, the Ant had all he needed. Not only did he successfully reach the top of the Field, he became the King Ant of the Field of Mathe, all due to his rigorous work and efficient use of time. And he ruled over the Grasshopper all his remaining days.

Hilarious Math

This is just a light-hearted attempt to give us all a few good laughs today.

So, I share with you some quotes from an old book entitled Kids Sure Rite Funny, written by Art Linkletter of Kid's Say the Darndest Things TV fame. Growing up, I always enjoyed watching him interview those little children, knowing that the parents were in the audience and that Art Linkletter was going to do his best to get those children to say something funny or embarrassing.

In his book, Kids Sure Rite Funny, Linkletter shares from the writings of kids across the country. The book was published in 1962. If you should come across one at a garage sale, grab it. You will be entertained for years to come. I don't know how many times I've read these quotes, but they are funny every time.

(All spellings are the children's, as so spelled in the book.)

Hope you have a good laugh.

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"A hypotenoose is a humane device for hanging criminals from a 90 degree angle."

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"If your triangles get four sides, you have wrectangles."

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"In area, a circle is a pie or square."

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"Why I am taking algebra is because I hear that some thoughts cannot be thought without thinking in algebra. Although I have never had such thoughts, I am expecting."

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"The minuend is the number from which the minuet is subtracted."

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"Square objects are rectangular while round ones are tubercular."

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"When rulers are not human, they have twelve inched feet."

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"A tangerine is a line going past a circle."

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"If you are looking for an X, algebra will help you find it."

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"Axioms and postulates are the same. We have both of them in case we forget the word for one of them."

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NCTM Focal Points (Standards)

According to this bulletin from the National Council of Teachers of Mathematics, there are apparently three (3) focal points identified for each grade level, PreK-8.

Here are the focal points, followed by excerpts from NCTM's explanation.

• the use of the mathematics to solve problems;
• an application of logical reasoning to justify procedures and solutions; and
• an involvement in the design and analysis of multiple representations to learn, make connections among, and communicate about the ideas within and outside of mathematics.

"These curriculum focal points should be considered as major instructional goals and desirable learning expectations, not as a list of objectives for students to master."

". . .this set of curriculum focal points has been designed with the intention of providing a three-year middle school program that includes a full year of general mathematics in each of grades 6, 7, and 8."

Go here to read entire bulletin.

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My observations:

No matter how good these focal points (expectations) sound, don't kid yourself that they intend for your child to master them. These focal points are not to be considered "as a list of objectives for students to master." That isn't considered to be important to NCTM. Notice, however, that it is considered important that they learn to communicate about the ideas they are learning. And don't be surprised if that communication is written.

And . . .

If states and districts follow these suggested standards, your students will likely fall short of the goal of most families -- completion of pre-algebra and algebra by the end of 8th grade.

Group Discovery ???

One of the marks of "new math" is discovery learning. But it's more than just discovery learning; it's "group discovery learning". Students are expected to come to knowledge of mathematical facts and procedures through "communal discovery".

You've heard it before. Only what your child discovers himself will he really grasp and remember. He must "own it" to be able to "use it and truly know it".

Now, let me see. If we want everyone to make the discovery himself, why would we put them in a group? That is exactly the opposite way you have every person discovery on his own. The whole group discusses and comes to an action, tried and agreed upon by the group. (Of course, someone in the group had to make the suggestion, to begin with, but never mind that.)

Now, remember through all of this, the purpose is for each student to make the discovery for himself -- so he can "own" that knowledge.

If you're in a group, how can all students come to "discovery" -- of a math procedure -- at the exact same moment? Someone in the group will come to the discovery first. Must he or she remain absolutely quiet? Must he hide his procedure? Remember this is a group discovery.

So, if someone else in the group makes the discovery, I guess no one else ever gets to "own" anything! Be First or Lose Your Shot at It.

By the way, the teacher is very quietly staying to the side, offering no guidance. And once the answer is agreed upon, the teacher is not allowed to check to see that the procedures were used correctly. Later, no drill will be given for students to use for practice. And there is no memorization of facts -- the facts can be 'rediscovered" on the spot, whenever they are needed.

In a traditional classroom, a teacher gives students an opportunity for small discoveries throughout the lesson, but he or she carefully directs the lesson and teaches precise methods, so that all students learn the most efficient procedures. Then the students practice and practice the steps under the supervision of the teacher. Following the guided practice is the independent practice, during which kids do more practice. Teachers are available to guide students and reteach if necessary. It is during this practice that some students "discover" and learn. And as the procedures are practiced over the next days and weeks, still other students may finally make the discovery on their own. But that "discovery" is just as real as the "discovery" made earlier by other students. In the meantime, the student has been able to successfully get the proper answer on the work because the teacher had taught just that.

I disagree that there is no discovery in a traditional classroom. There is. I see it every day.

And I don't think that it is possible for all students in a new math room to "discover" every single fact and procedure without input from the teacher.

Pay Attention to a Child's Individual Bent

This is an especially fun story for me to recall because it involves my precious nephew. He was my first nephew and I was in college. We were all amazed to see his early interest in numbers and marveled at his insatiable appetite to learn about them. No one forced him to sit down and listen. No one made him memorize anything. His parents just sort of scratched their heads and watched it all unfold.

He had an early desire to know more and more. We soon figured out that as he was mulling over the answers to his questions, he was properly putting it all together in a way none of us imagined. On his own he figured out that the symbols for the numbers meant something.

He didn't fit any of the models I was studying in my college childhood education and psychology courses. None of us knew how he did it. He was just too little to explain to us what was happening in his mind.

Here's how it played out. It started with his asking us "What is 3 and 5?" "What is 4 and 7?" We never knew where that idea to ask those questions came from, but as long as we were willing to answer, he kept firing those questions. It was nonstop and it never seemed to be enough. What 3 1/2 - 4 year old normally does this?

I think we all thought it would run its course and play itself out after a while. But it didn't. His mind was an unquenchable little sponge and the questions, always related to numbers and groups of numbers, kept coming.

But those simplle questions soon became "Daddy, what's 38 and 5?" "What's 57 and 8?" And when his daddy was tired, aunts and uncles and grandparents were the next victims. And all that we told him, he took in and somehow analyzed it correctly. And the questions became more precise, showing that he was honing his skill.

We started turning the tables on him. We started asking him questions and were just blown away with what happened. This little guy had somehow figured out sets of "tens" in his mind. He knew how to regroup and go past tens. He could answer anything we asked him. And he was right on the money -- every time.

He could subtract, too. He could add and subtract 8's or 7's just as easily as you or I.

He was not really any smarter than his siblings. He just had a bent toward math concepts. For some reason it was a challenge, it was compelling, it was satisfying to him. To him, the mental computations were enjoyable, and even something difficult didn't set him back. He kept wanting to know more.

His questions turned to money and how much things cost, and even multi-step procedures didn't make him waver.

One evening, my husband and I took him to a small airport in our area to watch the commercial planes come and go. He asked us a question (number related, of course).

"How much would it cost to fly to . . . (a nearby town)?"

My husband said, "Oh, about $5," mostly just to give an answer, rather than to be accurate. Immediately, my nephew shot back, "Well, I'm saving my money till I have $30, and I'm going to take you both to . . . . (the town)."

He had added and multiplied so rapidly, we were stunned. He even planned on our return flight!

If your child has such an appetite, such a natural bent toward math, nourish and encourage it. Provide opportunities for him to grow. Make sure he doesn't languish in a "fuzzy math", "non traditional" classroom. You may not think that the misguided approach of "new math" can have a detrimental affect him, but it can.

Or if your child doesn't show an early interest, provide opportunity for little baby steps of counting and of understanding numbers and make it fun. He/she could be a late bloomer. He can make good progress with just a little encouragement and interest from a parent. And make sure he doesn't languish in a "fuzzy math", "non traditional" classroom. He could likely be at the most risk because he is neither strong or weak and can easily slip through the cracks.

And if your child really does struggle and it never gets easier, just keep being your kid's biggest fan and biggest encourager. Don't ever show your disappointment in him, and don't give up just because he/she doesn't have a natural bent. Support him and praise him for even small gains. And for certain make sure he doesn't languish in a "fuzzy math," "non traditional" classroom. Be prepared for an early intervention with tutoring and evaluations. Make sure he learns to ask questions. And he must know he can come to you. And give him room to excell in something else -- art, or music, or helping people.

Money, the Logical Hands-on Teaching Tool

Having taught at the same private school for many years, it only follows that I have taught several brothers and sisters of many families. I've never expected younger brothers and sisters to mirror their older siblings, but I've also noticed that they often do because of the parents' expectations. Parents who expect diligence and responsibility from their children usually have an entire household of diligent and responsible children.

One such case that puzzled me for several years was the inability of children from one particular family to count money, to make money exchanges accurately, to make change involving bills and coins, and to solve general word problems involving money. This was especially baffling considering I teach 5th grade and it is unusual to have one child, let alone several, who can't make small change accurately and easily.

And then I figured it out. These children apparently were never required to be responsible for the money they were given by their parents or for bringing accurate change back home. If one of them needed money for something, the student would be given a $20 bill with no thought for what he/she was expected to bring back home. In this one case, the lunch cost $4.50, and the student had no idea what change to expect. I discussed it with her and she wouldn't even try to reason through it. All she did was just shrug and say "I have no idea" or "I don't know" to every query I made.

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So begin early and teach your children how to exchange 5 pennies for 1 nickel, etc. Have them count money (and remember to start at zero when counting). Give them small amounts of money to purchase something and discuss ahead of time overpaying and expecting change from the clerk.


Do you save pennies in a jar? This is a good way to involve your child in a family project. Count the money by the week or month, depending on the child's age. Have the child help you swap nickels for the pennies. Then later swap dimes for nickels, etc. This helps teach your child equivalent amounts.

One game that is fun for children can be played in the car as easily as it can at the breakfast table with coins. Say to the children, "How much money do I have? I have 1 quarter and 1 dime." As children learn to solve these problems, ratchet the game up a notch to this next level: "How much money do I have if I have 2 less than 3 quarters, 1 dime and 1 nickel?" Children in second and third grade can get these problems. Then try going past the dollar, but do not mention dollar bills. "How much money do I have if I have 3 more than 6 quarters, 2 dimes and a nickel?"

Give students reasoning problems, but call them "riddles". "Here's a riddle for you. Bill has 5 coins. What five coins did he have if he has 38 cents?" At first, this can be done with coins in hand, but as children get older, they need to be able to solve this type of problem mentally.

Use money to begin teaching tally marks. More on tally marks in another post, but money is an easy way to get your children to think of counting by sets of numbers. And remember counting is the beginning of solving abstract concepts in numbers. Make sure you do a lot of counting.

Have a pretend store at home. Tell your young children that you are practicing what to do at a real store. Teach them to make change. However, the true goal should be more. You want your children to recognize incorrect change. Explain to them that this is "Wrong Change" day at the store. Tell them you are going give incorrect change for a pretend purchase, and have your child figure out what is incorrect.

Expect your child to be responsible and accountable with any money, even his own. Do not pass it out to him/her like it was candy. Children will value its importance if you value it, and will learn to value it enough to want to know how to use it wisely.

Geometry of Today is Not Geometry of Yesterday

I'm pulling up old stuff, I know, but you can't get better than Barry Garelick.

Barry posted this comment on the first Kitchen Table Math back on June 5, 2005.

"From NCTM's PSSM, here's what NCTM has to say about their geometry standard: 'Geometry: Geometry has long been regarded as the place in high school where students learn to prove geometric theorems. The Geometry Standard takes a broader view of the power of geometry by calling on students to analyze characteristics of geometric shapes and make mathematical arguments about the geometric relationship, as well as to use visualization, spatial reasoning, and geometric modeling to solve problems. Geometry is a natural area of mathematics for the development of stusdents' reasoning and justification skills.'"

Translation: High school geometry used to emphasize proofs. Now it just emphasizes shapes and formulae, with an occasional proof and in general is not much more advanced than the geometry presented in 7th grade, except for the fact that not much geometry is presented in 7th grade."

My observations and thoughts:

NCTM is the National Council of Teachers of Mathematics. They are a body of educationists (my word) who are responsible for writing the national math Standards which are supposed to define the expectations for students in each subject area at each grade level. I say "are supposed" because the expectations are so watered down and are so vague that no one can actually identify a specific expectation.

You have to hunt far and wide to find geometry in textbooks today. This "broader view" is part of that "1/8 inch deep and a mile wide" approach to teaching Math. The subject of Geometry is spread all through other textbooks and is no longer taught in a coherent fashion semester by semester.

A high school teacher commented to me 2-3 years ago how much he wished he could teach geometry as an isolated subject so he could concentrate the students' focus on geometry.

Your student is probably being robbed of the opportunity to learn to prove geometric theorems. It's no wonder our high schools students score so much lower than Asian students.

Expectations Need to Be Measureable and Concepts need Time

Standards are "expectations". They are the targets. They are what you are shooting for. They define what is expected of the students at specific grade levels in specific subject areas. (In the United States, curricular expectations are defined as "standards".) In order for standards to be effective, they must be specific.

Specific expectations are easy to measure. If your expectation is that your child know the capitals of all of the states, that is specific and measurable. You can easily discern if a student has met the standards. How? Have him demonstrate that he knows the capitals.

And that is why so many people are concerned about "standards". It is the vagueness of the standards that troubles us.

I came across an article by William H. Schmidt entitled "What's Missing from Math Standards?" which was published at the American Educator website in the Spring of 2008. He discussed the findings of the Third International Mathematics and Science Study (TIMSS).

According to Schmidt, TIMSS found that "student performance is directly related to the nature of the curricular expectations." He explains that he does not mean the instructional practices, but rather "the nature of what it is that children are to learn within schools."

"The TIMSS research has revealed that there are three aspects of math expectations, or standards, that are really important: focus, rigor, and coherence."

Here are Schmidt's comments about all three of these aspects of expectations.

Focus:

"Focus is the most straightforward. Standards need to focus on a small enough number of topics so that teachers can spend months, not days, on them. . . [i]n the early grades, top-achieving countries usually cover about four to six topics related to basic numeracy, measurement, and arithmetic operations . . . In contrast, in the U.S., state and district standards, as well as textbooks, often cram 20 topics into the first and second grades."

It is Schmidt' s opinion that this number of topics is far more than any primary grade student can absorb.

Rigor:

"Rigor is also pretty straightforward -- and we don't have enough of it.... .[I]n the middle grades, the rest of the world is teaching algebra and geometry. The U.S. is still, for most children, teaching arithmetic. . . [O]ther countries outperform us in the middle and upper grades because their curricular expectations are so much more demanding, so much more rigorous."

Coherence:

Coherence may not be as easy to grasp as focus and rigor, but according to Schmidt, "it is the most important element." He explains that there is a formal academic body of knowledge that has been parsed out and sequenced from kindergarten through 12th grade, and he describes how especially important this parsing and sequencing is in the subject of math.

"Topics in math really need to flow in a certain logical sequence in order to have coherent instruction. If you look at the math curriculum of top-achieving countries, you see a very logical sequence. The more advanced topics are not covered in the early grades. Now that seems obvious -- until you look at state and district standards in the U.S. Everything is covered everywhere. Far from coherent, typical math standards in the U.S. often appear arbitrary, like a laundry list of topics."

Some of you may want to consult the entire article to see why our country has such unfocused, undemanding, and incoherent math standards.

Two related articles can be found here and here.

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Some additional thoughts . . .

By spreading topics all through the curriculum, nothing is covered in depth. Teachers are expected to cover so much material during the year that they must fly quickly to the next topic, meaning that there isn't time for a student to grasp the coherence of one concept with another. There is rarely time for the feeling of "Aahhh! I get it!"

Depth is better than shallowness. Kids can get their thoughts around a concept and understand connections when a topic is covered deeply and thoroughly.

For some reason this picture is going through my mind right now: I'm thinking of trying to drink lemonade through a straw -- after we have spread it 1/8 inch thick all over the table or counter top. And we only have one minute to do it!

That might be how many of our children feel when they are trying to "get hold of a math concept". This produces frustration and a feeling of "there's something wrong with me -- I didn't get it."

Laying the Foundation for the Abstract Concept of Numbers

Learning the abstract concept of numbers starts long before the usual introduction of pre-algebra and algebra. It begins when your child is first introduced to the meaning of numbers by counting. Yes, counting is where the development of number sense all begins.

I remember when our son was two and I was participating in an exercise class. We all took our children along and they played next to us while we exercised. Our son, interested in numbers from early on, was fascinated by the class leader's counting out loud, "One, two, three, four; two, two, three, four; three, two, three, four; four, two, three, four; ("again") one, two, three, four; two, two, three, four; three, two, three, four; four, two, three, four . . ." And our son was fascinated by the words. We adults understood that there were two sets of counting patterns being used at once, but to our son, these were just words, recited along in a pattern, with a nice rhythm, and he was intrigued, taking it all in.

A few days later, I heard him repeating the number words, in random order. He had no sense of counting; to him the words were meaningless. Ooops!

Children love counting. Counting games, counting songs, and counting rhymes all serve to draw our children's interest to the fun of counting, and at an early age.

Counting is where the understanding of the concepts and the meaning of a number begins. And it is important to know that there is much more to counting than merely calling out the numbers.

Here are a few ways to properly start your child counting so that early on, he/she will not only begin to grasp the meaning of numbers and will also apply "counting" to objects around him

Assemble a set of objects in a plastic bowl or tub. They may be pennies, blocks, craft sticks, toys, anything. You also need a second empty container. Have your child pick up an object, place it in the empty bin, while saying its number, beginning with "one". (Demonstrate this yourself. Children love to copy.) It is important that your child actually picks up the object and moves it, rather than sliding it across to another pile. Begin with 10 objects or less. Be sure that your child only picks up one object at a time.

After your child has moved the last object to the other bowl, ask, "How may trucks are in your other tub?" Your child should learn to properly say the last number he counted. Vary the number of objects in the tub. It is also important to vary the objects you use so that he/she understands that numbers are used to refer to different types of objects.

This early activity of counting, using concrete objects, though simple, is part of the learning of the abstract number concepts to come later.

And this is important: have one tub on one side of his body and the other on the other side. We want his/her hand to cross the midline of the body with each motion. We know that the brain cells fire away each time an arm or leg crosses that midline.


As your child gets older, and after he/she masters this first counting activity, there is something else very important that you must do. You must develop a sense of "zero". So this is the next step in the counting exercise: Teach your child to start counting with "zero". Before any objects are moved, teach your child that there are "zero" objects in the second tub. Teach him to say "zero" before he begins, followed by "one" as the counting and moving of objects begins.

Next, using blocks that are numbered, or making your own on pieces of paper of cardboard, teach your child the symbols (digits, numerals) that stand for each number from 0 to 20. Line them up in order and point to them as you recite the numbers. Have your child recite them with you as you point to the symbols. Then have your child recite them alone.

Later on, with the blocks still in proper order, point to the symbols in random order and have your child name them.

Much later, mix them up and have your child find them in order and say them as you and he line them up in order.

As your child learns, you can gradually increase the number set up to 20.

Books that have counting pictures can be useful. Point to the symbol and ask your child what it is. Sing counting songs with him/her. Avoid songs that count backwards at first.

Count the silverware as you place it by his plate, or as you remove it from the dishwasher, remembering to begin with "zero". Count his/her socks as you place them into his drawer. For an older child who has mastered the counting of single objects, count the socks in groups of two: "one, two," "three, four", etc., which will lead to counting by multiples of two.

Where to Start

OK, Time to discuss your child's needs.

Does you child struggle in math? Is he in the 4th - 6th grades? OK, then something needs to be done and something can be done. It needs to be started now and it can be started now.

What you know, you can address. What you don't know or aren't sure about, you must find out. You must delve into until you do know. You must identify what it is, or hire someone to tell you what it is that your child needs help on. But I think parents are smart enough to figure some of it out.

With what does your student struggle? Ask yourself and be specific. Or ask him/her. They can usually tell you something at this age.

Is it number facts? You, the parent, can do something about that and you must. It is a lie to say that memorizing facts isn't important and doesn't really matter. It matters, and you can and must do something about that. A child in intermediate grades that doesn't know the facts, all of them, is like a kid trying to ride a bicycle with two very flat tires. He takes so long just stuck at a stand still at the starting gate, never getting very far. Knowing facts gives him a jump start in solving the more difficult, multi-step problems.

There are some very simple methods to use that will help your child learn the facts, and there are some methods that will do very little good. If this is what your child needs, speak up! I'll help you NOW.

Is it 2-digit multiplication? If so, I would bet your student has a gap somewhere in understanding the concept of place value. If you can't help him/her with that, get assistance from someone who can. Two-digit multiplication is usually taught in the 4th grade and reviewed in 5th before 3-digit multiplication is taught. Work on place value with them now. Or hire someone to do it. Students need to be able to correctly solve 2-digit multiplication quickly. And if your student hasn't mastered multiplication facts, back up and work on that with them. And then tackle the multiplication. And if it's one-digit multiplication, start there.

Is it long division? It could be because your student hasn't mastered multiplication facts once again. Division is really just searching for the missing factor. FACTS -- FACTOR -- get the connection? And long division should be mastered in the 5th grade. Two-digit long division should be mastered in the 5th grade. And it can be.

Is it fractions? Is it decimals? Is it percent? If your student is moving into 6th grade in the fall, a good, clear understanding of these 3 is a must. Get help now, over the summer. If your student struggled with any of these, hire a tutor, check out some on-line sites that will help you help them. A good understanding of fractions is the one single determiner of success in pre-algebra and algebra.

And parents, if you don't understand the four operations of fractions, converting fractions to decimals, decimals to fractions, and converting both to percent, DO NOT TELL your child that you won't be able to help them because you were never good at math.

Tell them that even though you were not good at math, you are going to learn it now so you can both work on it together this summer. Show your student that you are willing to do that. You are not too old to learn it. Go to the tutoring class with your student. Your student will take note of your decision. It will speak volumes. It will say, "Math is important in our family and because it's so important, I'm going to make sure we all know it, and that includes me."

So . . . Reality Check right now. Math troubles are real. Admit and identify at least something, something small perhaps, but something that you can work on. And do it. Don't wait until another school year starts. If you wait, then you'll be tempted to put it off another year. It will not be easier come fall. You student will begin next fall at a lower level than he finished at the end of this past spring. It just happens -- a month or two off can cause a set back, especially for a struggling student.

(This is where I wish I was your next door neighbor. Or your child's aunt or uncle. I would only need to watch him/her work for about a half an hour and I would know enough to be ready to start. Yes, it frustrates me to know that your child is there somewhere and perhaps you don't know what to do. But please try.)

Properly Building Self Esteem

Self esteem.

We are told, at every turn, that our children need to have self esteem. You want that for your children. I want that for my children. I want that for your child when he/she is a student in my classroom.

Question: How do we build self esteem? Or better, How do we build proper self esteem? Well, I'll tell you how NOT to build self esteem. Give your child NOTHING to do that requires effort on his/her part. And then, tell him how wonderfully he did it!!!!!

Here are some examples:

Give your 5th grade student 2nd grade spelling words
Have your upper grade student work a 1st grade math story problem.

You get the idea.

Will he succeed? Of course, but you have required no effort on his part. You've guaranteed that he will succeed, yes, but you've really required nothing from him.

Why am I mentioning this? One of the trends of modern education is to "dumb down" everything. It is pervasive in all subject areas, but I'm particularly addressing math in this post. They feel they need to make math less stressful and so the tests are made extremely easy or students are given art projects, on which children are given inflated grades, all in an effort to "even out" the grades.

A primary aim of these educators is to have a completely noncompetitive environment in the classroom. It must be, according to them, a place where all students are judged equal, and judged to be equally successful. And the only way to make all children equally successful is of course, to set the bar so low that no one appears better/smarter than others. All students can then feel good about themselves because no one is better. And this is supposed to give each student "self esteem" -- good "self esteem".

We see this effort all across the country when math teachers do not grade homework. Getting the correct answer on a problem is not required. In fact, if students don't know how to solve the problem, all they need to do is show an effort, just try to solve it. Everyone gets the same credit.

Let's back track to when your child was very young. Remember how he/she loved to help with little chores such as unloading the dishwasher, or putting the silverware on the table, or taking the folded clothes to each bedroom. Why? Because it showed that he/she was able to do a "big" person's job. And then, Moms, what did you do? Perhaps you said to your husband, "Daddy, look at the table. Guess who helped Mommy put the dishes on the table!" And then you and Daddy, made a big "to do" and praised your child for his efforts.

I remember teaching my daughter how to use the dust mop up and down the tile hallway to get all the dust. It took several trips down and back, but she quickly learned to do it without missing a speck of grass or dirt. She was really little and it wasn't easy for her, but she learned it and was so pleased that she had done it properly and well!!

Now back to the classroom. How do we build a student's self esteem? We DON'T give them assignments that are below grade level and tell them how well they've done. We give them something hard to do, something that is a challenge for them. And then when they do it, we praise them for it. This is what builds self esteem. And this is what makes them feel successful.

We need to set high goals and push them to reach those goals. Tell them that you know it won't be easy, but that all other 5th graders have been able to do it and that you know they can do it too, and that you will help them till they get it.

Dr. Wickelgren, in his book Math Coach, A Parent's Guide to helping Children Succeed in Math, confirms this approach. And it is the lack of high goals which causes "new math" methods to be so harmful, according to him. The new way doesn't encourage students properly. It certainly doesn't encourage a student who has put forth an effort to do the work and to solve a difficult problem.

I have seen first hand some of these methods used as my own children came through school. They were discouraged when students who had done no work got equal credit as students who had worked hard. What incentive does that give a diligent student to work hard next time? (We all grew to hate group projects where all students got the same grade, regardless of the effort put forth by each student.) Rewarding a student for no effort does damage, not only to that student, but to the entire class.

Dr. Wickelgren also believes that judging all students the same can have "deleterious psychological consequences for children." Why? Because it isn't the truth and it can cause those children to "have an unrealistic view of their abilities." According to Dr. Wickelgren, this can mislead children "to think they have skills they lack" because they have acquired the reward without doing the work. "Such children may not do the hard work necessary to succeed later in life." They have been trained to believe they will be rewarded irregardless of the work or effort.

So giving students hard tasks is important. It builds self esteem because the children know they have succeeded at something difficult. They know the difference. They know when they have worked hard and when they have not.

It also develops in them the character quality of perseverance, which they will need later in life, not only in high school, but also on the job.

Will all children be able to be successful on all of the work I assign in class? Maybe not. Probably not, because there will always be some things that are hard for some children. There are problems and concepts that I know some children struggle with.

I consider it part of my job to know what each student finds difficult in math. And I must be careful to praise, praise, praise when I know they have worked hard and solved a problem that was difficult for them. I need to give verbal praise immediately. In addition, I also like to write words of praise on a paper as I grade it -- "Good for you!" or "Wow! This is wonderful!" or "Yeah!!! You got it!" This is something that a parent will later see and gives them an opportunity to add their own praise

Even if a student fails to get the correct answer, if they remembered and used the proper procedure, I will write, "Thank you for working so hard on this." or "Keep trying. You'll get it." or "Come see me. I'll help you." This provides encouragement to the struggling child. It tells them they you are noticing their efforts and that their effort has not been wasted.

Every child needs encouragement. They also need to be challenged and rewarded when they have worked hard. Students can be/ should be recognized for excellent and perfect math work. They should also be recognized for improvement.

For a struggling child, recognition for improvement is HUGE. And it can build self esteem because the child knows the task wasn't easy for him. And it gives him hope. Hope that things are getting better and hope that he can do it and hope that he will succeed.

That builds self esteem. Proper self esteem.

A Few More Gleanings: Dispelling Myths

Here are more snippets from the Math Coach, A Parent's Guide to Helping Children Succeed in Math, responding to several "myths" being promoted by the "new math" proponents:

"Concrete and abstract ideas are not separated in the brain, but lie on a continuum. So the implied mental leap necessary to cross from concrete ideas to abstract ones is fictitious." (written in response to "new math" supporters' claim that children under twelve years of age are not capable of learning abstract concepts and operations)

"Object-oriented activities are very useful for young children who are just getting a grasp of the concepts of number, addition, and subtraction. But kids get a grip on such abstractions quickly, and they rapidly outgrow the need to manipulate beads and lay rods end to end every time they are asked to add and subtract. Once children are ready to start doing math on paper, such activities are a tedious waste of time." (responding to the overuse of exercises that involve manipulating objects to teach arithmetic)

"Age is not an important factor; knowledge is." (responding to the myth that the brains of middle school students are too immature to learn algebra, and elementary students have limited ability to understand story problems because they have trouble understanding the meaning of addition or subtraction)

"Over the past two decades, cognitive psychology researchers have repeatedly shown that problem-solving methods and other higher order thinking skills can be used effectively only when a person has a large body of knowledge on which the thinking skills can operate." (responding to the big push to teach students general problem-solving methods)

Dr. Wickelgren points out that these general methods are not really beneficial because they are replacing math facts and specific problem solving skills.

"Creativity is an outgrowth of learning, and a lot of it. . . the more a person knows about a subject, the more creative he or she can be in it. . . A student's ability to be creative in any area of knowledge increases with his or her knowledge of that area. . . Thus, a desire to enhance creativity should not move a curriculum away from the math basics -- but closer to them." (responding to the idea that developing creativity frees a child to let his/her thoughts blossom and thus allows him/her to gain more knowledge)

. . . . . . . .

In response to a comment by Concerned, I need to make clear that Dr. Wickelgren's book is not only about identifying "new math" teaching. The book is divided into two parts.

Part 1 covers such topics as Setting Goals for Your Child, Evaluating Schools, Strategies for Excellence. In Part 2, he covers such topics as Teaching Tips for Parents, Basic Arithmetic, Basic Story Problems, Fractions, and Algebra.

Hangin' It All Together

In a previous post, I referenced Dr. Wickelgren's comparison of the traditional approach and "new math's discovery" approach to teaching math concepts. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he made the following observation about the traditional approach:

"The traditional approach, in which classes are arranged by topics such as arithmetic, fractions, algebra, and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

I've been blessed to teach a very structured, incremental math program. Students who are used to struggling in math tell me over and over that they "get math" for the first time. Now, I see why. Closely related pieces of knowledge "hang together" easily and make sense to them. The slow, incremental steps mean that they practice and master the small pieces, one at a time. Then those pieces are carefully linked to other small pieces and the students see the logical relationship and how they are connected.

Contrast that to a sixth grade book I was using recently to tutor a student. All of the definitions and all of the formulas were thrown together in quick succession, so that my student was thoroughly overwhelmed and confused. (The students are given only that chapter to memorize them all, while trying to learn how to use them, all at the same time. ) It's madness!

And sadly there's not usually much practice of each small piece of knowledge, isolated from all of the other pieces. (Remember lack of practice is a characteristic of "new math.") Here's the line up: Teach one formula, and maybe do a little work. Then the next day, along comes the next formula, and then the next day, another formula. The students are soon doing problems of each formula, before they have mastered and are comfortable with the first one. This is NOT how kids learn.

There was nothing to "hang" the second , or the third, on. It was as if there were pieces floating around in the air and the student saw them all and had no idea which of them was related to what he was being expected to to next. There were no hooks to hang anything on because there had been no time for mastery.

Teachers probably feel that they don't have time for mastery because of the number of chapters they know they must cover. This book was huge!

And if smart kids have trouble with it all hitting them at once, imagine the weaker student's response. They are overwhelmed.

This is not how kids learn!!

Students must see how the pieces fit and hang together. And traditional math helps them do that.

Traditional math is wonderful! It's beautiful!! It's joyful! It's exciting!! It's liberating!! It's confidence-building! It's knowing I can succeed!!

It's like a kid who can finally ride a bicycle by himself!!

Traditional math helps kids do what they all want to do -- learn! I've never seen a kid who didn't want to learn. Learning new things is fun. Learning how to do something hard is even better!

Using "Guess and Check" rather than the reliable "long division" algorithm

I came across an article I had discovered three years ago, written by C. Bradley Thompson on the Teach Math site. Thompson is currently a professor at Clemson University.

In this article entitled "Cognitive Child Abuse in Our Math Classrooms", Thompson discusses the cause of the dropping math scores in the United States. While discussing "whole-math", he makes these two observations about its proponents . . .

Advocates of 'new math' reject the idea that there are right and wrong answers and that . . . there are basic skills that students must be taught.

Advocates also believe that each student should invent his or her own math "strategies" by using the "guess and check" methods tauted by the "fuzzy math" supporters.

Here's the entire article.

In his descriptions of the activities in the "whole-math" (or "new math") classroom, you'll see students making piles of marshmallows to multiply, counting a million birdseeds in order to grasp the concept of "a million". And rather than have six-graders use multiplication or division facts to solve a problem, students are told that their strategies of "guess and check" are just as good as the logically proven principles of long division.

Would you want to be treated by a surgeon who learned his procedures by "guess and check"?

I'm amazed every day at the lack of grade school math being taught, when proficiency in the basic algorithms are desperately needed. It is these algorithms that form the foundation for higher forms of math knowledge.

. . . . . . . . .

Sadly not much has changed since this original article was published. School districts are sill adopting "fuzzy math" curriculum. Students are still struggling and math test scores are still low. Congress is still trying to decide how get our country's math scores up with the rest of the world. And the writers of curriculum are still resisting making needed changes because they refuse to acknowledge that the blame rests with their methods.

And any changes in Standards that thankfully are now being recommended will be slow in coming. It could be years before it trickles down to students. And in the meantime, another generation of students is seen their hopes of getting into engineering school dashed.

Perils of Discovery Learning, Part III: 'Interdisciplinary' Activities

One of the big "buzz words" in mathematic circles today is "interdisciplinary" activities and projects.

If I'm teaching a unit in Science, students might do research in my class or the Library, write papers using skills taught in Writing class, and then generate the paper in the Computer Lab. Or if students are learning about Indian Villages in History, they might work in groups, or individually, to make a village in Art Class.

I often feel that teachers are evaluated (unofficially perhaps) on how much content crosses over into other disciplines, although no requirements have ever been made of me in my private school. It has been "suggested" that I find ways to involve other disciplines, but that's been the extent of it.

I have seen instances when I feel an interdisciplinary project has been very effective and where students are completely immersed. If the other discipline is a favorite of a student, if he loves art, or if she loves to write, he or she will really be engaged. I just don't like the pressure of forcing the project where it doesn't naturally go, where time is lost, all for the sake of "show". We can now brag at how many other disciplines were involved!!!

Dr. Wayne Wickelgren has made studies of interdisciplinary projects and I respect his opinion. In his book, Math Coach, A Parent's Guide to Helping Children Succeed in Math, he contrasts the traditional approach with the interdisciplinary approach:

"The traditional approach, in which classes are arranged by topics, such as arithmetic, fractions, algebra and geometry that build from one level to the next, has been used for decades for good reason. The material within each subject hangs together in logical ways, and is typically broken down into smaller units within which knowledge is even more tightly linked. Teaching students to hang together closely related pieces of knowledge makes sense and produces a deep understanding of a subject."

[That is just beautiful! And it makes such good sense.]

Hangs together in logical ways.

Now for Dr. Wickelgren's assessment of the interdisciplinary approach:

"Teaching across subject boundaries lacks depth. It may be fun for the students, but it doesn't help the mind organize the knowledge in a logical way, making it harder to remember."

There is also the likelihood that a teacher will overlook an important basic fact or principle that would usually be included in an incremental, structured approach.

I just have to say this one more time:
Hangs together in logical ways.

Perils of Discovery Learning, Part II

Dr. Wickelgren, author of Math Coach, A parent's Guide to Helping Children Succeed in Math, explains that it is not possible for children to "discover" much of what is important for children to learn in elementary grades. How, he asks, can a student deduce, through discovery, the meaning of "3 to the 4th power" or that there are 5,280 feet in a mile. There are many things which must be taught by the teacher and then memorized by the student. Now you may avow that these are things that a student could conceivebly discover, so let Dr. Wickelgren describe how a typical "discovery" session works.

"Using the discovery method, students are given very little of the information necessary to solve problems. Solving problems thus requires gigantic leaps of intuition that virtuallly no students possess, so they flounder. A class typically spends half an hour -- and sometimes as long as two and a half hours -- on a single problem."

Because the "discovery" time is largely unsupervised by the teacher, it proves to be a very inefficient use of students' time.

Dr. Wickelgren tells of a student who "complained to me that his class spent an entire week on one problem before the teacher told the students how to solve it."

And remember that in addition to the inefficiency of this procedure, students are also asked to spend time discussing what to do, and then to spend more time writing about how they arrived at their solution. And what if their solution isn't even correct??!! Remember, the teacher is not involved, is not guiding students back on track when they wander off on a rabbit trail.

As a result of all of the wasted time, very few problems actually are solved by the students. Teachers are giving little or no information, and student proficiency takes a nose-dive.

Contrast this scenerio to the traditional classroom, where children are taught by direct instruction. Students are given most, not all, but most of the information needed for solutions. Smaller amount of insight is needed and thus some students, though not all, solve the problem quickly. Several of these problem-solving activities are given throughout the course of the lesson, affording students additional opportunities to use small bits of insight to solve other problems. (And a teacher steps in to help, giving additional information when needed, if the problem takes too long.) Many students gain confidence because of the successes they make each lesson. And they see the point in what they are doing. The goal is short ranged.

Solutions are found quickly, not hours later, and students do not lose interest because the problem's answer is so long in coming. I'd rather have students engaged for several small problems they have hopes of solving than disengaged for an hour because they see no point in what they are doing.

Perils of Discovery Learning, Part I

I love to watch children learn. When people ask me why I like teaching, my answer is always the same: "I love to see that light bulb go off, to see that elation on a child's face when he 'gets it'". So there is a part of every math lesson I teach where I give students a chance to discover on their own.

But these are not huge, time-consuming tasks which involve groups of students. (I never figured out how you know which child is making the discovery when students are working in groups!) Instead, these are short moments built into every lesson where children begin with a known concept and are led to discover and figure out the new concept.

Later into the lesson, for the students who have failed to make that connection to the new, teacher intervention in a one-on-one basis with that student can pinpoint where the connection failed and the student has another opportunity to discover. I have many memories which I prize, memories of struggling children smiling with great joy. They got it. They figured it out without my actually telling them. I love these "light bulb" moments.

Dr. Wickelgren also acknowledges that students need these moments, but he also describes the perils of great units of time spent on "discovery learning projects".

The troubles of "fuzzy math" failures, which have created an uproar and lead to "math wars" around the country throughout the 90's and since the turn of the century are well documented. Parents have risen up to protest and to request changes in curriculum because of the failures of the programs to teach their children rigorous math.

Dr. Wickelgren describes the failures linked to Standards math (fuzzy math, new math) as follows:

"It dumbs down class content and lowers expectations for all kids. It doesn't adequately tests kids' knowledge. It wastes far too much time on activities that have little to do with math. And despite good intentions, it can actually decrease student participation.

"But the most important downfall of the approach is that it often results in only cursory knowledge of the nuts and bolts of math -- including basic aritimetic facts . . . and how to solve a variety of problems. This severely weakens the math curriculum because basic mathematical knowledge and problem-solving skill are the key ingredients of math proficiency. Mastering basic facts early is critical becase they form the basis for a huge amount of mathematics that follows. A child who doesn't know those facts by heart -- and how to use them in problems -- is at a serious disadvantage, even if he or she understands the concepts of addition, subtraction, multiplication, and division."

And this stunning statement from Dr. Wickelgren:

"The primary reason for the downfall: excessive reliance on student discovery of facts and principles instead of explicitly teaching them."

And another:

"Discovery sounds good on paper. In practice, it is time-consuming, inefficient, and results in little learning."

Discovery Learning; More from Math Coach

Dr. Wayne Wickelgren, author of Math Coach; a Parent's Guide to Helping Children Succeed in Math, has much to say about "discovery learning" and the "new math", often called "fuzzy math" (and which he calls "Standards math").

Here are some of his insights:

"The most pervasive theme echoed throughout the Standards is their emphasis on student exploration and discovery. Instead of presenting information to the class, Standards math teachers ask their students to discover mathematical concepts while solving math problems. "Typically, students break up into small groups of four or so to solve a problem. The teacher circulates among the groups to observe the discussions but otherwise does not interfere with their learning by providing too much information."

"This framework is intended to let each child's natural creativity in math blossom, enabling children to discover important concepts and problem solving methods on their own. . . Teachers encourage students in the groups to do a lot of talking and writing about their thinking process that led to a solution . . ."

"To facilitate discussion, the groups are often working on math problems that are somewhat different from traditional computational or story problems. . . grounded in real-world situations. They are open-ended and contain many parts and many possible answers. . ."

"One example: A middle school math teacher demonstrates a pendulum made from a string and a weight and asks students to construct a pendulum, investigate how it functions, and formulate questions that arise."

"Traditional problems by contrast, have a single correct answer and focus on closely related mathematical ideas and facts."

Dr. Wickelgren explains that hands-on, real-world activities dominates "Standards" math classrooms. Students are never encouraged to memorize addition, subtraction, multiplication, and division facts, but rather to manipulate objects to find the answers, rather than using pencil and paper, repeating the activity each time a set of facts are presented.

"Memorization is regarded as dull for kids and also ineffective as a learning method, as it seems at odds with really understanding the material."

All teachers know the value of manipulating objects to teach the four basic operations, but for a child to have to repeatedly group objects to solve 8 X 6 is standard operating procedure in a "fuzzy math" classroom.

Dr. Wickelgren also describes the organization of the textbooks:

Rather than dividing a year by topics, such as two-digit multiplication, fractions, long division, and decimals, the year is organized into group projects which link to other subjects to create interdisciplinary studies. Rather look at how the textbook is organized:

"A year's worth of mathematics might look like this: A Wagon Train's Journey West, A Genetic Study of Fruit Fly Reproduction, Managing a Supermarket, A Month in the Life of a Real Estate Broker, A Voyage to Mars."

The "fuzzy math" crowd design these activities because they believe that prior to adolescence, children's minds can handle such "concrete" mathematical concepts, mentioned above, "but are not mature enough to handle abstract numbers and operations."

And so your child, being too immature to memorize facts, may be asked to design and create a pretty portfolio with attractive bindings to hold their writings about how they feel about math. All of this is during math class!

Dr. Wickelgren acknowledges such positive features of Standards math as "emphasis on understanding, solving more challenging problems, enriching math curriculum with more probability. . . early study of coordinate geometry". But he also warns of the perils of the "discovery learning", which will need to be discussed in a later post.

[Dr. Wickelgren also acknowledges that the Standards have been modified, but warns that the changes are slow to make their way down to the classroom, especially since there are so many who refuse to see the error in the "new math" approach. Curriculum writers take it very personally. Their "works" are "their babies". ]

More from Math Coach . . . Know the Warning Signs of Inferior Education

Back to reading more from Math Coach, A Parent's Guide to Helping Children Succeed in Math, by Wayne Wickelgren.

Wickelgren, bemoaning the new math curriculum which had been introduced just before his daughter entered sixth grade, said he was able to identify the problem quickly because he knew the warning signs of inferior math education "...despite the claims of the teachers that this new program would be superior to the previous one."

At first, according to Wickelgren, he tried "to counter the new math agenda."

When that failed, he "implemented a plan to teach (his daughter) the math she would suddenly not be taught in school."

Wickelgren went on to say that their "stopgap measures prevented any long-term damage."

Here is one last paragraph written by Wickelgren on the subject of inferior math education and preventing its damage:

"Oddly, the methods that so frustrated (his daughter) and left (his son) far behind are part of the latest fashion in math education, one based on the 1989 Standards of the National Council of Teachers of Mathematics (NCTM) -- which I call Standards math. Virtually all parents of children going to school in the present era will encounter teachers and other educators influenced by Standards math. Thus, it is critical to understand these methods and how they might shortchange your child's math education."

Three things that I want parents to notice:

1. Wickelgren was able to quickly identify the problem because he knew what "Standards math" involved. This is where most of us as parents unintentionally drop the ball (I was one many years ago myself.) and this is when most of our children "get off the track." And sadly, we might not even know they are off the track because we are lulled by the teachers' promises of a better way coming.

2. He tried to counter the new math. (more on this later) I think this is what most of us do, if we are knowledgeable enough to be able to identify the problem at the start.

3. He identified what was not being taught and intervened. He implemented a plan which taught his daughter what she would fail to learn.


My first encounter with "new math" (but I didn't know it):

Our own son, coming through high school in the late 80's and early 90's, was very smart in math, and was on a fast track. He started geometry, and quickly told me "Mom, I'm not enjoying math at all."

I looked through his geometry textbook. Geometry had been about my favorite math, and I was incredulous at what I saw, or more specifically, what I didn't see. This math did not even resemble what I expected! Students were given problems to solve but had not been taught theorems nor anything about using theorems in the proofs. Folks, that's what plane geometry is -- using/listing theorems needed to solve the problem and then (proofs) why that theorem helps get the answer. The theorems and proofs take you step by step through the problem until you get the desired solution.

I tried to explain theorems and proofs to our son and how they worked, (I guess I was at step 2 above -- trying to counter what he was learning -- but I made the mistake of using the school's textbook and I just couldn't help him. I knew our son was terribly off track. It was week two and damage had already been done. His self confidence was zero.

So I had identified a problem, although I knew nothing of "new math". And after trying to help, I had been pretty quick to identify what was NOT being taught (step 3), but my mistake was trying to counter what was being done by using his textbook as it was being taught. It just didn't work at that point because there were gaps and I couldn't fill in the gaps using that "new math" system.

And so from my experience, I will agree with Wickelgren that trying to counter what a child is learning probably won't work by following what's in the "new math" book. I needed to start all over again and teach him plane geometry the "traditional" way.

(Let me call your attention to this: It is typical of "new math" curriculum to require students to solve problems before they are exposed to the information and concepts they will need -- to give them the chance to "discover" and to build their own methods and plans. All it did was upset and confuse my smart son.)

I called the teacher, who was very nice and assured me that our son was not alone, that it was typical for students to be confused at this point. She also asked that I give them a few more days to get through the chapter and then it would get better. It did get better, thank goodness, and we had our old son back! Remember, he was a good math student. I wonder now what happened to students for whom math didn't come easily.

I'm convinced in hindsight, that this was one of the very first "fuzzy math" textbooks. However, I had never heard of that, and it never entered my mind that anyone would be so foolish as to skew the math I had loved so much into a twisted bunch of disconnected ideas.

It is my desire to help parents quickly identify bad math teaching and implement a plan to correct it and fill in the gaps. No matter how old your child is, it's not too late. It's doable.

Helping Children Succeed

I'm reading Math Coach, A Parent's Guide to Helping Children Succeed in Math, by Wayne A. Wickelgren, Ph.D.

This is a wonderful book, directed at parents, on how to help children who need more instruction, who need to get caught up, or who need to be challenged.

He says it's not too late, even if they are in 5th or 6th, to catch them up by helping them at home in time for Algebra in 8th grade.

Here are some quotes that are especially appropriate for our intermediate grades:

"Most children can learn math much faster than the school system prescribes. That means that even if your child is currently behind, there is plenty of wiggle room for catching up, especially if you start early. Math education proceeds slowly in school before eighth or ninth grade, so those are the ideal years to improve your child's mastery of math or speed him or her ahead through the curriculum (though starting later is okay too)."

" . . .{by taking prescribed steps}, your efforts will change your child forever. Investing time in your child's education -- whether it's tutoring, consulting with educators, or helping your child set goals -- will show your child that you think education is important. Even when your child complains about doing extra schoolwork, rest assured that he or she is lerning, however, unconsciously, to value learning. It is a value your child will carry throughout life and, in all likelihood, into the next generation."

". . . even if your child is behind in math now, you can get him or her back on track with only moderate effort. If your child is not behind, you can propel him or her substantially ahead in math."

"It is the math prior to algebra that is taught especially slowly in the U.S. Thus, the pre-algebra years are the ones in which you can most easily speed your child ahead, or help him or her catch up in math, giving the child a chance to learn as much math as possible."

Wickelgren has some fascinating studies that show how soon children can grasp concepts of adding and subtracting from a group. And there is great information here on when children can begin sounding out words, although that is a different subject.

He discusses age-related limits to math learning (I was surprised to find that there are very few.) and biological barriers which are connected to processing math, and (one that I've always heard thrown out here) maturational limits.

Here's a great quote from Wickelgren, related to that last statement:

"Of course your child may have temporary trouble understanding certain math concepts, but the reason for that is unlikely to be a maturation issue. It could be lack of verbal skills."

". . . it is the verbal encoding of addition and subtraction that is often lacking in young children and not the nonverbal understanding of these concepts."

Did you get that?

"NOT THE NONVERBAL UNDERSTANDING OF THESE CONCEPTS."

Children can understand math if we find another way of showing students what the digits and symbols actually mean.

I'm loving this book!!