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!!

Real World Math

There is as great thread discussing "Real World Math" on Kitchen Table Math. If you are confused as to how "real world" activities can be detrimental, see the comments posted by Barry Garelick and subsequent discussion.

Intervention, Part 2

Should you decide to "intervene" on behalf of your student, do not be surprised to find that some people in your school may not be happy about it.

Schools often design their own plans for intervention. There may be a distinct intervention "order" which you and your student should/must follow: You start at point A (probably your student's math teacher); if that isn't helpful, proceed to point B (another math teacher), etc.; then on up the ladder to perhaps the head of the math department. This is how the school has decided you must do it.

I know many teachers who gladly help students before school, after school. To them, the goal is student understanding and they are willing to help.

And don't misunderstand; I commend schools for becoming involved in the tutoring process to help students.

But let's consider that your student's math teacher is not competent in teaching new concepts or in identifying gaps in reasoning. What happens if your student goes to this teacher for tutoring (because that is where you must start)? This teacher will use the same approach used in class. What didn't work in class will be tried during the tutoring and your student will probably not get much benefit.

And how does that make your student feel? I'll tell you (but you know). He feels no better, maybe worse! Now he sees that even with extra help, he still doesn't get it!! He may start saying that he is "dumb".

So you desperately attempt to find alternative support and help. Woe be to you, if the school Math Department finds that you have, on your own, asked another teacher to help your child. You could be soundly scolded for not taking advantage of the proper recommended "line" of help.

I know because I was once an "alternative" choice by parents. They had tried the school's plan for after school tutoring by the student's math teacher. The student continued to fail tests, and the parents were desperate for some help. (He had gone from A's to F's in less than a year.) They pleaded with me to help. When I agreed, the parents asked for an extra book for me, but were chastised sharply for not following the "school's plan".

"You should have gone to . . .", the parents were told, and the plan of choice was explained.

This was so baffling to the parents. They just wanted help for their student.

Wouldn't you think that schools would appreciate a parent's desire to get help for a child? Well, schools may tell you that they appreciate it, but you may find out that they are grateful only if the school's "plan" is followed.

(By the way, I did tutor the above-mentioned student, though only for a couple of weeks, and I was able to help him. He passed his first test of the entire year (and this was April) after only two tutoring sessions.) The most important thing, however, according to his parents, was his self image. He knew he "got it", he understood it, and his confidence soared. This student wasn't used to failure. He needed to feel successful again.

All kids want to feel successful. Success give confidence. If your student needs intervention, take the steps. Go to battle, if necessary, for your kids.

Intervention

Perhaps you have identified a few concepts that your student hasn't nailed down (maybe it's learning simple facts, or perhaps it's more complex like long division or multiplying fractions) and you have decided it's time for you to step up and try to do something. Where do you go? What do you do? Is it something you can do yourself? What if the school tells you that in a few months they will cycle back through and that perhaps your child will get it then. They tell you not to worry. Should you even try?

Yes, please try. You must intervene and the longer you wait, the harder it will be. There are so many web sites to help now, and so many families who have gone before you. Help is all around and I urge you to start with sites and blogs, such as Kitchen Table Math so you can have support and help. Locate a few websites, if you need a little review, or even hard copies of helpful books, such as Math on Call, which will give you procedures and confidence.

Be willing to try. Then, tell your child that you know it's been hard but that "we" are going to do something now to help.

Decide on an amount of time. If it's a tutor, the time will be decided by the tutor, or by your budget. (See below on suggestions for tutors.) If it's your time, set aside a certain amount each day. Explain to your child the time requirements so that it will be clear. Stick to it.

Decide on a short range goal, one concept, that you can work on, and go for just that. Do everything to ensure that your child experiences immediate success of some sort, even if it's something small, the very first day. Success builds confidence. I assure you your child wants to feel and be successful. So make sure that your student sees that he/she has made progress that first session.

Be willing to make this a long-term commitment, if it is needed. I mean for a year, or two, maybe more. Catherine Johnson, from Kitchen Table Math worked with her child, using an entirely different textbook in the afternoons, in addition to the school assignment. If you're looking for how much she did a week in the other textbook, I'm sure she'll help and encourage you.

Your child is looking to you for help. You can do it. At least try. Many have and you will be able to. Whether you do it yourself, or whether you support a tutor. Do it.

Many times I intervened, short term, for our own children when they needed a little help. I always looked over all of their work. I did that early on in primary grades and they got used to it and accepted it. If I saw trouble, I made up some problems right then and there on the spot and watched them work. I could tell by watching where they got off the track. Once the problem areas were identified and addressed, we worked a little more, maybe several nights. I usually had one rule: show me that you can do 5 problems correctly without any mistakes, and we will stop. What does that do? It makes them focus on all the little things you've identified and it makes them work carefully. A few days was usually all it took.

Now for tutoring: Any time I've agreed to tutor a student, I explain to the parents that I want more than one session a week. I will agree to tutor only do it if I can have a minimum of 2 times a week, but 3 is even better. Why? Well, usually, I find that some of the trouble is procedural and simple carelessness. And what good can I do if a student has one hour with me to work on correct procedures and then the rest of the week he reverts back to incorrect methods? It is said that a person has to do something about 34 times for it to become a habit. So practice becomes very important, but it takes not just practice, but the correct kind of practice, and consistency. The work must be done properly every time. So keep this in mind, whether you are the tutor, or you are considering hiring someone. I usually find that after about 6 weeks, students may not even need me anymore because they have corrected the bad habits and see the value of working with care.

Pencil and Paper; the Benefits

One of the usual identifiers of a "new math" program/teacher is that there is very little pencil work. Fuzzy math people have been all over the map on this idea, straddling the line, trying to appear neutral, denying that they are against pencil and paper work, denying that they are opposed to learning and drilling on math facts. Now, some are making a turn around and publicly say they do not oppose use of pen and paper practice, but look closely and they provide little to none.

Now, I must explain that "new math" proponents are not opposed to pencil and paper work. They assign students work involving lists after lists after lists and all sorts of writing assignments. It's pervasive in every chapter throughout the books they love. But that use of pencil and paper is OK, because the "new math" teachers are doing it. Not only is it OK, it's desired! It's just not OK if I do it in my traditional classroom, using traditional books and traditional methods.

It's been several years now, but I took a little class called "Teaching in Texas Today" offered by a local school district. It was designed for people who were teachers but who had been out of the classroom 10 years or more or for people who were interested in becoming certified. It sort of brought you up on things that were new in education and each week we had a different instructor. One instructor showed us new ways to be sure that students were learning new concepts. (She used math as her example) Students were all given cards to raise in the air to give their answers. Teachers could easily identify the students who had the correct answer by looking at the card the child held up. No pen or pencil work was required. Students were encouraged to do the work mentally. And I'll never forget her saying, "Once you see that the student has done several correctly, you don't need to burden them by requiring that they do any more." (I'm not opposed to mental math -- I love it, my children love it and some even thrive on it. I use it every day, but not at the expense of proper pencil and paper practice.)

Soon afterward, I returned to teaching, and I've thought of the instructor's words many times, primarily, when I see how quickly some students forget newly taught material. Even with practice and practice, there are those students who will still struggle. All it takes is a 2-3 day sick spell, and they're lost. (I'm not talking about that student who can think it through, figure it out mentally in 10 seconds. If that is your student, you are not reading this blog. I'm talking about the students who can't figure it out instantly, but who need to be given a method to use to get the answer until they can figure it out by themselves.)

There is something about the "doing of the problem", about the "using of the pen or pencil" that helps children learn. I don't meant that "helps them 'remember'". We've all known that for a long time. But now we know that "the doing" actually helps children "learn". We now know that there are some children who learn by doing, by moving, by writing, by some sort of physical motion. And for them, it is in the actual doing of the work that they "get it". Using and moving concrete items (blocks, counting sticks, pennies) to solve problem is a must for these kids. Walking, physically moving, through the steps in the problem is also important for these children. But along with these actions, we know that there are those children, for whom it it a terrible mistake to restrict the pencil and paper. We rob them of their way of learning. For these students, there is a need for clear step by step procedures to follow and practice, practice, until they figure it out because this is the way they figure it out.

Oh, I forgot! I'm not supposed to be teaching them procedures or methods. We should be allowing them a chance to figure it out and discover their own methods of getting the right answer. You, the parent, may have been given such an answer when you've explained that your student doesn't understand what's being taught. You may have even been told NOT to teach them your old methods because children need to figure it out. Never mind that it might take them months, and that the rest of the class has long gone on to something else. So you're told not to help yor student and your student may even be publicly embarrassed in class if you do. So you are told NOT to intervene. Teachers are told not to intervene.

But I can't do that to your student. And as a parent, in your heart of hearts, you don't want to do that either because you see what it does to their confidence. You see the frustration and the discouragement and you want to help, NOW, not next semester, or next year.

And I do not agree that only if a person is mentally thinking on his own, deciding on his own methods to try to use solve a problem, that only then will he be engaged. Students who are going carefully through a step-by-step method can be very "engaged" because that is the method that gets their brain going. You will see them whispering, talking through the steps, and that is good. Yes, they can be very engaged. And they are learning.

(I'm going to cut my original post here and post the remainder on a new page, "Intervention".)

Is Your Student Being Taught "Fuzzy" Math?

Parents may wonder if their student's math curriculum is "fuzzy" math. Here are some general identifying marks of "fuzzy" math:

1. Very little practice (You might have someone tell you that new studies show that too much drill kills interest in math. That is not true -- it makes students efficient and confident.) Lack of pencil and paper work -- in the "fuzzies' minds it's bad to do a lot of pencil and paper work.
2. Individual problems on the homework are not actually graded. Students are given credit if they show that they "tried" to do it. It doesn't even matter if they successfully solved the problem.
3. Calculators are used to do what children should be learning to do mentally. Some people may even be so bold as to claim that there are students who will never learn all of these math facts anyway. Calculators are said to be sufficient.
4. (Almost exclusively) student-centered activities. Students work in groups to figure out something, and the teacher may not even be involved at all.
5. There will be little or no instructions, certainly no explicit instructions. Activities are student directed and teacher input is lacking. Students are sent home to figure it out by themselves and the next day the class is asked by the teacher if there were any questions.
6. "Real world math" is what it's all about. It's one of the buzz words. After all, who doesn't want students prepared for the "real world".
7. Students are told to write paragraphs explaining all of the steps that are used to solve the problem. Or students are expected to write about how they feel about math. Or as Walter Willaims referenced, students write on such topics as "If Math were a color, what what that be?" In other words, students are writing their math. So much for the benefit of symbols and digits to help solve a problem efficiently, in the least amount of time.
7. Math books are huge, thick books, and pages are filled with visual "clutter" which distract students' thoughts. (For some "adhd" or "add" students, these pages can be a disaster. It is no wonder it takes them 50 minutes to do a 15 minute assignment.) There is a lot of "non-math" content such as photographs in color and motivational stories which are meant to "inspire" kids to greatness, I suppose, but which have no place in the middle of a math assignment.
8. Patterns, patterns, patterns. Looking for patterns, drawing patterns. Probability will be prevalent by 4th or 5th grade. And much time spent on data analysis projects (finding the mode, the mean, the median) starting in elementary school. What is appropriate for a high school course is inappropriate for elementary chldren. The time spent on one "projects" may be days and will indeed use lots of pencil and paper. Notice here: It's OK for them to do pencil and paper work. It's just not OK for me or you to do pencil and paper work on traditional math practice.
9. And here's one dead give-away that the curriculum takes a "fuzzy" math approach:
Students are given problems in one lesson, for which they are not at all prepared. Students spend much, much time on these problems, only to discover that the concept is taught a lesson or two later So if your student has no idea what is being asked, look ahead and you'll probably find that the concept is coming up. This is called "discovery learning" because students have a wonderful opportunity to "discover" the concept on their own. Imagine your 4th grader being forced to "discover" how to do long division with no input from anyone else!!

I'm sure there are more I've overlooked. One more thing that is obvious: Your student is (perhaps suddenly) discouraged, thinks he/she is not smart, gives up trying. I know one student who had been in my 5th grade class and was an excellent and a very diligent student. What didn't come naturally, she learned by shear determination and perseverance. How sad I was to hear that she was threatening to kill herself because her high school math was so hard for her and she didn't think she was ever going to get it. Her teacher's approach was to assign the algebra/geometry lesson, forcing the students to teach themselves, and then ask if any of the stusdents had any questions or problems. If there were no questions, they proceeded on to the next lesson.)

So, do any of these ring a bell to you? Well, if so, you must rescue your student as fast as possible. Go to Kitchen Table Math for insights how a family rescued their own student. It took lots of hours after school, but the results made it worth it.

Some Math Stories, Same Troubles

A year or so after I had first discovered that not all math was the traditional math I had been taught, it began to hit close to home. I was celebrating Thanksgiving with my husband's family and sadly not all family members were there. In talking to my brother-in-law about my new found information on "fuzzy" math and "math wars", I was told, "Well, it's interesting that you should mention that." He then told me that was why his daughter and her husband were not with our family. They had used that extended weekend to visit her husband's family 200+ miles away because his young sister was suddenly having a lot of trouble with math and they were going to see if they could help her. And, he continued, this young sister was really smart and had always made really good grades in math and scored well on tests and the parents couldn't figure out what was wrong. How I wished I had been able to help this young girl or at least talk to her family. I didn't really know her, but I felt empathy for her and her family. And this is what often happens -- the whole family becomes involved trying to help, but because terminology is different, because parents are told NOT to try to help students by using the parents' old, out-of-date methods (because you're going to confuse them), parent involvement is often delayed until much damage has been done. And another thing, the student get's "down" on themselves and start telling themselves "I'm dumb".

In a second "family" story, I was able to talk to parents, but not until the students were in college. The mother was bemoaning the fact that they didn't know what they were going to do with their boys, my nephews, because they both were having so much trouble with college algebra. I knew they had had trouble in high school. It had been a source of bits of discussion off and on as their troubles were mentioned, as tutors had been hired, etc. We all caught on that, but for the tutors, those boys might not have made it through high school algebra. But now here they were in college, still struggling with algebra. I told the mom, "Well, it may not be your boys fault. It may be that the boys were taught "fuzzy" math." She went on to describe how they themselves had tried to help them during middle and high school, but the math didn't make sense and so they had to use tutors. She didn't know she was describing a situation that was repeating itself all across the country.

Has anyone noticed the number of Sylvan, Kumon, Huntington, etc., Math Centers popping up all around the countries. Parents used to be able to help children, prior to the first "new math" intrusions of the 60's and 70's. My own father helped me occasionally, but it didn't take much because the base/rate/percentage problems, or the algebra were the same as he had learned. My 7th and 8th grade "math" teacher was really an "art" teacher who had been asked to teach math for a time and unfortunately for me, it was during my time. No one in the class was getting it, including me, and I had to go home every evening and have my dad help me until I got it. Then, before school the next morning, we students would all meet together to go over math and I'd reteach the lesson, as my dad had helped me. We all made it through middle school math because of my dad. That was before the days of "tutors". Family could help family then. But not any more. At least, it's not easy to do now. Thankfully there are parents who will not give up on their kids, who try to find help when they don't know what to do. But my dad got me ready for algebra and I never had a bit of trouble with algebra (I'm sure it was "traditional algebra").

There are several variables: is it the math curriculum? is it the math teacher? is it the student? It could be more than one. If your student has a young math teacher, chances are he/she came through college having been taught to use the "new math" approach (Don't teach the students methods or algorithms! Make them figure it out by themselves.). But it could also be the curriculum and I'm going to add a post shortly with some identifying marks of "fuzzy" math so you can determine if that is the type of curriculum your student is using.

How Can It Happen?

A few years back I had the blessed privilege of teaching one boy in particular. Now, I have to say that I could say that about every student I teach, but this is a particularly sweet story for me to remember. If you knew his family history, you'd understand why I say this. I was told ahead of time what his limitations would be and what troubles to expect. (And indeed those were descriptions of what his 4th grade year had been like after he arrived.) I will tell you now, before I relate some of his story, that he was one of the smartest students I've ever taught.

As the first few days went by, I noticed that none of the anticipated problems were presenting themselves. None, at all! His grandmother/guardian asked me how he was doing, and I remember my answer to her. "He is knocking himself out to please me. He is working so hard, not complaining one bit. Nothing is difficult for him, he's really smart!" And it was true. I'm still convinced he was brilliant. His young life may have had a hard beginning, but as his confidence grew, his self-image grew and he just blossomed that year.

About 3 weeks into the year, we had our first Math Test. The next day, as I returned the tests for the students to look at, his response was one I'll never forget. As I gave him his paper, upon seeing the grade (95), he said out loud, "I've never made an A on a Math Test before!" That was the first of many "A's". In fact he made an A on his report card every reporting period throughout the year. Math wasn't hard for him, he didn't struggle to "get" a single concept, and he wavered only once, when he had been sick (and even then a little intensive input from me was all it took). He was so confident as he worked, and his family and I watched him grow and bloom even more. He left my classroom a different child, with his head held high. He knew he was smart. (I told him I thought he was brilliant!)

So, fastforward one year. Imagine my shock and surprise when I discovered that he was failing Math the following year. He was the same student, with the same brain, same brilliant reasoning abilities. So what happened?? I only have some ideas because I wasn't there with him each day, but I knew enough to believe that something was wrong. Terribly wrong. And I wished I had known earlier. I wished I could have done something to help.

As I enter this post, I am thinking that perhaps this might be happening in your family, to one or more of your children. I read the stories on other Math blogs of students who suddenly seem to fall apart in Math. I read the words of college math professors who bemoan the fact that students are ill prepared for college. (I do want to write more about this in later blogs.) I believe that if there is a predictor of which students will struggle with higher maths, it would be how well they transition into the abstracts and unknowns of pre- pre-algebra in 5th and 6th grade. (Yes, I mean pre- pre-algebra.)

On another matter, and in closing this post, I want to state that all students need to see their tests and have an opportunity to discuss the tests, within a day or two. And I believe that parents should be able to see the tests. I'm always baffled when I hear that a teacher won't let a test out of the room. How can a school say, "we want you, the parent, to help us, to support us with your child" but then not allow the parent to see a test!!?? This makes no sense to me. I'm a teacher, but I'm a parent also. I'm the one who might/will/can be helping him/her prepare for the next Math test. Do I want him to still be "unclear" on why he missed a problem? (I tutored a student this past spring semester, one who was failing math, one who had not passed a single Math test the entire year. I asked the parent if they had any tests I could look at, and they had never seen a test because students were not allowed to bring them home. The dad was trying to help him and reviewed with him each evening. But they were at a loss because they didn't know where they or he had failed.)

So what do you do when tests aren't coming home and you, the parent, are not seeing them. Here are a couple of suggestions. First go to the teacher, respectfully, and ask if the tests are being returned. (Students are notorious for "losing" papers.) If you discover that the teacher doesn't allow them to be taken home, ask about how much time a student has to study over the mistakes on that test. And then ask if the teacher would consider changing the policy of keeping tests. (Anticipate that you will probably be told something about needing to use the test the next year and that siblings coming up could have an advantage. If you get this excuse, respectfully tell them that you understand and that you don't want to compromise the test at all.) Sometimes a teacher will reluctantly, quietly, give you or your student the test. If they do, know you are blessed. But what if you try and still get no response? . . .

There is one last step that I used once when my son was in middle school. After having a very pleasant phone conversation with the teacher during which he explained why he didn't return tests, I felt that I still needed to see the test, if only for a brief while. So I calmly called the principal, and asked about the school policy. He explained that some teachers had that policy and others didn't, but it wasn't a school policy. I explained to him that I wanted to respect the teacher's desire to keep the test confidential, and so I requested that he, the principal, get the test and allow me to come up into his (the principal's) office to look over it in his presence. Guess what! It worked. I got to look over the test, I saw how my son had studied improperly, and got a glimpse at what the test looked like. And better than that -- I didn't make the teacher or the principal angry. I hope this works for other families.

Look, it's all about trying to help a student.

There are many reasons why a student's grades can drop, but if the parent can eliminate one of them, a parent should be and usually is willing to do it. The test will be used by the teacher to determine a grade. Hopefully, it is also used to help determine teaching strategies which can improve students' understanding of concepts. In other words, the test can be a tool to help the teacher learn better methods for teaching concepts. And if teachers can learn anything from the test, why not allow the student to learn from the test? Or the parent? Wow, that would be a new approach -- teachers and parents and students all benefiting from the test!

As I said, it's all about trying to help a kid!