## Why So Many Elementary Students Aren’t Mastering Basic Math Facts — Part I (of 2)

Remedial math in college

A friend of mine teaches remedial math at the Community College level.  We were discussing the problem of a number of  students who never seem to have their addition facts mastered (much less their multiplication facts).  He wrote:

“I remember as a young math teacher wondering how many hours of flash card drill it takes in the elementary grades to become fluent in the addition and multiplication facts. I could imagine ten minutes a day of actual flash card drill, five days a week, for 45 weeks in one grade, a total of 50 hours if I multiplied correctly, might be a reasonable guess.  Surely it has been studied.  Well, if it has been studied I have never seen any evidence of it in the last fifty years.  I thought of that as probably pretty basic knowledge about the teaching of arithmetic.”

I’d estimate that at least 30 hours of drill, spread over a period of time would be required for an average child to learn addition facts, and an equal amount of time later on to learn multiplication facts.  But there is NO classroom time provided for this in the math curriculum.

The problem here, speaking as a third-grade teacher of 8-and-9-year-old students for a decade, is that the elementary math curriculum (in America) is not structured to provide ANY time for drill such as he describes.  Even when I was a child in the early 1960’s, we did not have drill of that type in elementary school.  My mother worked on flash cards with me 10-15 minutes every day before I was allowed to play.  I HATED every moment of it, but saw the value of it when I got into the working world in my 20’s and was using multiplication every day, knowing my multiplication tables by heart thanks to her efforts.

I’ve been discussing math teaching with other math teachers for several years now, and I find there are several trends I highly disagree with.

Houghton Mifflin Grade 3 Math Homework prior to 2007

Trend 1:  The amount of math homework has been cut in half from 24-30 problems nightly, to 12-15.  I can only guess that this has come about from parents complaining about too much homework over the years.  While I am in favor of not giving more homework than necessary, unfortunately, the current lighter homework often does not give sufficient practice in a certain type of problem for the students to be able to understand or master that type of problem.  One or two examples of a certain type of problem are just not sufficient.

Houghton Mifflin NEW Amount of Grade 3 math homework, starting in 2007 (taken from end of a Grade 2 workbook)

Trend 2:  Drill practice is considered “old-fashioned.”  Never mind that the teacher can make drill practice into a fun lesson, just like any other type of lesson can be made fun by a a dedicated teacher.  Without any drill, and without parents practicing or drilling children at home (such as the type of flash card practice my mother did with me as a child), many children are just NEVER mastering even the basic addition facts, let alone multiplication facts.

I no longer teach Grade 3; I am now a private tutor.  Unfortunately, I am now running across a number of 14-year-olds who are using calculators to add 5 + 3, or 7 + 6, or 9 + 2.  What’s even worse, THEIR TEACHERS LET THEM!!!!  I personally think calculators should just be thrown out until about Grade 11, or whenever math involving higher functions on calculators is started.  Prior to that time, they shouldn’t be allowed in school at all.

When I taught Grade 3, I made students show all of their work on their homework, including every carry number, and every cross-out for borrowing; I didn’t allow them to say, “I did it in my head.”  (See photo above of example homework prior to 2007.)   One reason for making students show all of their work (I had several reasons)  is that I knew perfectly well many of them had calculators at home.  However, even if they did their homework with calculators, they would have to redo it to mark all the carry numbers and borrowing cross-outs.  This makes it better to just do it by hand in the first place.  I then spent 30 minutes of my teaching time DAILY, going over these homework problems.  It’s so satisfying to a teacher to hear, “Oh!  Now I see my mistake!”  It’s a big mistake for a teacher just to mark answers right or wrong, as students learn nothing from that.

Trend 3  (mostly at the high school level, I haven’t yet seen it appearing in middle schools, although I could be mistaken):  Don’t instruct and explain, and then follow up with practice to master the skills.  Instead, put students into groups, and let them see if they can “figure out themselves” how to do problems.  Don’t give much feedback, but of course, students will have the same test as if you taught them the traditional way.  (So the parents who can afford it get math tutors to do at home the job that the teacher should be doing;  the parents who cannot afford tutors or understand the math themselves have children who completely fail math).

High school math class, with students working in groups.

Trend 4 (has been around for at least 25 years):  It doesn’t matter if children don’t master a unit.  Just move through all the units, and the same units will be covered next year in a little more detail.  If they still don’t get it, the same thing will happen the following year, and hopefully they will get it then.  This idea has a name, which is called something like “spiraling.”

Even though I’ve never seen it, in the past couple of years I’ve become aware that “Singapore math” requires mastery of each math subject to a certain degree before moving on to the next math subject.

I think students would be far better served by having HALF the number of math topics (eliminating topics in Grade 2 such as Data, Graphing and Probablility; Congruent Shapes and Symmetry; etc.) and making sure they have mastered basic addition facts (by heart), addition  and subtraction of  two-digit numbers, and multiplication tables up to 5 (by heart) before moving into Grade 3.  If parents don’t have time to drill children at home on these facts, then some time for it should be allowed in the school curriculum.

–Lynne Diligent

Other Math Posts by Lynne Diligent:

Why So Many Elementary Students Aren’t Mastering Basic Math Facts — Part 2 (of 2)

### 29 Responses to “Why So Many Elementary Students Aren’t Mastering Basic Math Facts — Part I (of 2)”

1. JB Says:

I (taught the old, rote, boring, blah blah blah way) can cope with all the math I encounter in my daily life. Algebra 1 and 2 were easy. My daughter, taught with modern methods, is completely flummoxed.

2. Bill Says:

I agree with JB, I learned math that way also, rote memorization, endless drills, and numerous homework problems (of course, this was back in the 70’s when scientific calculators weren’t affordable).

I’ve seen modern methods of math today, and it would confuse the beejesus out of anyone who learned math the old fashioned way.

I’ve seen persons who cannot make change from a \$4.80 order given \$5.15 (it’s 25 cents)…I’ve seen deli clerks who cannot convert 2/3rds of a pound to a decimal value (which is .66) on a digital scale (which has no fractions).

It’s sad what has happened to the state of math education in the US.

• nancypeskenkp Says:

I, too, learned math in the 70s–which is what’s being described here. I totally agree with the article!
One thing I’d point out is that if you want kids to SEE their mistakes, you must provide enough writing space to do so. When I was a kid, we had to copy the problems from the book onto looseleaf paper–no visual clutter to distract us, and some penmanship practice. Then we were to cross out the carryover numbers if there was going to be another round of carryovers. This taught attention to detail, an important skill. Nowadays, they cram so many problems onto a worksheet that any child would be more likely to make a calculation error; a child with visual processing issues (a common hidden disability these days) will be completely distracted by the clutter. That’s why I have in my son’s IEP that he is to have scratch paper for math–and I send it to school with him. What I’d really like is an appropriate curriculum like Singapore Math for him given his language disability but there are 1001 excuses for why that’s impossible. Thus, I’m stuck homeschooling him in math where I can–and I use all the methods described above!

• Sara Webb Says:

Those same people who were unable to make change, or do basic operations with fractions are the ones who were taught using procedural methods. There is a reason that they can’t do basic skills. Give it time, and students will have a much better number sense. Already, I can see the positive change in my third grade students.
We are preparing students to participate in a global market where many of the careers not only do not exist, but have not been thought of. Teaching kids the way we used to was great for factory workers, and yes, many adults today can say that they learned fine that way. However, I wish I had kept a tally of all the parents who have sat in a conference with me and said, of course my child struggles with math, I always did. or I never understood math. I would estimate it to be between 60-75% of parents I have worked with who have said this. Clearly “the old way” or the “old standards” were not working.

3. Lynne Diligent Says:

Is it any wonder that the white-collar jobs are being shipped off to India and China?

4. Brian Rude Says:

The idea that there are too many topics in the arithmetic curriculum today certainly makes sense to me, but I have no experience of my own to know about that. “Data, graphing, and probability” in grade 2? ? ?Why? ? ? What possible sense could it make to second graders? But I suppose it’s modern, it’s trendy. I am somewhat aware of what the NCTM put into their 2000 Standards, so I can see how that could do a lot of damage to the arithmetic curriculum. The idea that every part of the math curriculum should be present in some form in every grade is probably a big mistake.

I have given some thought to the math curriculum, particularly the arithmetic curriculum, and have written up my results in an article, The Math Curriculum” on my website. It’s at http://www.brianrude.com/mthcur.htm.

I remember when the “spiral method” was going to be a great thing in education. Of course it turned out to be just another fad. However giving it some thought I concluded that it would be wrong to dismiss the concept of spiraling in learning. I concluded that spiraling is inevitable and useful in various ways in many subjects. The question is not whether to spiral or not, but whether the spiraling we do, or don’t do, is optimal for a given situation. I developed these ideas a bit in another article, “Patterns Of Coherence” at http://www.brianrude.com/Tchap07.htm. The discussion of spiraling is at the end of this article.

5. Lynne Diligent Says:

Brian,

Instead of replying about the Grade 2 curriculum here, I will do it in a follow-up post which will be up today or tomorrow.

Regarding your article dealing with spiral curriculum, you’ve made a good case for why topics spiral to some degree–providing for review (helps the problem of kids forgetting material over the summer), and how it could be viewed as a natural method of learning–learning a bit, moving on to a new topic, etc.

However, I would propose that a better spiral curriculum would NOT attempt to cover every possible math topic, and should instead focus on only SOME math topics, taking mastery of each to a higher level before moving on. Here’s an example. Many students are getting in to algebra and having trouble, because they have never mastered fractions!

In Grade 3, only two weeks are provided in the spiral curriculum to learning and mastering times tables (one day for each table). I’m sorry, but following this is only sufficient for understanding how times tables work and possibly being able to count up the answers either with fingers or by “counting on.” In my classes, I devoted 8-10 weeks to mastering times tables (but at the expense of leaving out certain units such as geometry or time).

Time is certainly an important unit, but I found that for the kids who don’t get it in Grade 3, you can work on it for weeks, and they STILL don’t get it. Suddenly, those kids get into Grade 5 and it “clicks.” So I finally decided it’s a brain development thing. So I decided it’s better to spend the time helping as many of them learn their times tables as possible, which means they are able to do division, fractions, etc. Those who do not master times tables are at a far greater disadvantage than those who do not master time in Grade 3.

6. Teacher Says:

Students must be allowed to ‘direct’ their own learning. If that means doing no math at all, they are allowed.

Get the UN out of our schools and all the social engineering hogwash and get back to teaching skills!

– 35 year teacher

• Sara Webb Says:

I just want to make sure that I am reading this correctly. Did you mean that you think students can choose whatever they want to do? So, if a child doesn’t like reading, it is ok if they never learn to be literate? If they don’t want to do math, they could just read all day?

7. Lynne Diligent Says:

My article was picked up and written about by Joanne Jacobs, and many comments were left on her blog:
http://www.joannejacobs.com/2011/07/why-math-tutors-prosper/

Here is a sampling:

Kate says:
July 15, 2011 at 12:05 pm

I would say trend 3 mentioned in the article is going on in elementary schools to some extent now if they have a constructivist curriculum like Investigations/TERC.

Crimson Wife says:
July 15, 2011 at 1:00 pm

“One or two examples of a certain type of problem are just not sufficient.”

This is so variable from child to child. 30 problems is not always better than 3. In fact, 3 really challenging word problems would much better for a bright student than 30 easy-peasy equations.

The problem is that we’re wedded to a “one size fits all” instructional approach that bores bright students with “busywork” while at the same time may not provide enough practice for slower students.

We’ve got the technology available to individualize math practice. Why don’t more schools adopt it?

Crimson Wife says:
July 15, 2011 at 1:05 pm

“I then spent 30 minutes of my teaching time DAILY, going over these homework problems.

Causing bright students who got it the first time to become resentful of the slower kids in the class.

As an adult, I realize these feelings weren’t fair because it wasn’t my classmates’ fault that I was placed into a class that had a too-slow pace and not challenging enough material. But that’s how I felt at the time.

BB says:
July 15, 2011 at 1:47 pm

On the other hand, CW, if the students are all matched for ability and readiness, you do still need to go over homework in order to get the full benefit of it.

SuperSub says:
July 15, 2011 at 2:08 pm

I guess you could say I was a bright student, and looking back on my education, while I disliked the 30 minutes of HW review and was bored out of my mind, it did serve two purposes.
One, although I grasped the material after only a few problems, doing (and going over) 10-20 of them taught me one big lesson – Math is 80% process, and 20% focus. Without it I would likely be a lot more careless with my calculations nowadays.
Two, everyone needs to learn to be bored. Really. Removing boredom simply furthers the aggravation that individuals feel when they have to be bored later in life. Heck, there were points during my wedding ceremony (a long Catholic mass) that I was bored and my mind wandered.

SuperSub says:
July 15, 2011 at 2:09 pm

Agreed with the spiraling criticism, though, I’ve tutored enough victims of that scam philosophy.

momof4 says:
July 15, 2011 at 2:13 pm

I’m with CW on the issue of the one-size-fits-all model in the public schools; it’s inherently unfair and discriminatory, not to mention ineffective and inefficient. (do public schools have any awareness of efficiency?) It’s one thing to spend 30 minutes going over homework that was appropriate for the child’s level and entirely another to spend 30 minutes going over homework that was inappropriately easy for some and inappropriately difficult for others, while assuming (hoping?) that it was appropriate for some. That’s the fundamental problem with heterogeneous grouping; it ignores the educational needs of real, individual kids.

Susan says:
July 15, 2011 at 2:18 pm

My son is a math tutor at his high school and he has had more than a few kids (not special ed) come in not knowing how to add single digits without a calculator.

Roger Sweeny says:
July 15, 2011 at 2:36 pm

I’ve had lots of 9th graders who need a calculator to multiply a number by ten.

Crimson Wife says:
July 15, 2011 at 2:44 pm

On the other hand, CW, if the students are all matched for ability and readiness, you do still need to go over homework in order to get the full benefit of it.

It is worth going over problems with a student that he/she has missed. Again, this is something better done by a computer since each child is likely to have missed different problems. And if the student got 100% correct, then there is little gained by reviewing the answers. If I have calculated the proper answer, what more is there to say about it?

Robert Wright says:
July 15, 2011 at 3:11 pm

Lynne Diligent is right that many elementary students do not learn basic facts and I believes she’s right that learning basic arithmetic is greatly improved with flash card support at home.

My guess is that assigning 25 problem to do for homework is better than assigning only 12.

As for her criticism of spiraling, I’m not so sure. I’ve seen alternatives that backfired.

Postman derided “the measles theory” of education. If you had it once, you can’t have it again.

I only taught math for a few years and but I’d have to say that the most common, serious obstacle to learning was student laziness. Those who fell behind weren’t suffering from math anxiety or had a learning style I wasn’t accommodating. The problem was they didn’t do their homework. Because it was too hard? No. Because they didn’t feel like it.

As for saying the use of calculators impedes the memorization of math facts, it’s just no so. That might be somewhat counter-intuitive because you can imagine that a student never memorized the fact that 3 add 5 equal 8 because a calculator was always there to remember for him. But in the real word, it’s just not so. That’s what research will tell you and over 90% of any full time middle school math teacher.

There are a number of obstacles to memorizing math facts, but the use of calculators isn’t one of them.

That has been so widely proven to be the case that when I encounter an anti-calculator point of view, it’s like hearing somebody alarmed about public water fluoridation.

Allow me to summarize a bit.

Knowing basic math facts is a good thing.
There are a number of reasons why some students are slow to mastering basic math facts.
Parents can help their children memorize math fact and need to help their children learn them if/when they’ve fallen behind.
I don not claim to be a math instruction expert.
Calculators do not harm learning not do they delay the mastery of basic facts.
Having an anti-calculator point of view isn’t supported by research or most experienced math teachers.
I’m not sure what is the root of the anti-calculator point of view. Maybe it’s the fluoridation in our water?

Robert Wright says:
July 15, 2011 at 3:26 pm

“My son knows children who can’t add single digits without a calculator.”

I’m sure that’s true.

But that doesn’t mean the use of a calculator caused the problem.

It’s like saying crutches cause people to have broken legs.

Crimson Wife says:
July 15, 2011 at 3:54 pm

“Having an anti-calculator point of view isn’t supported by research.

While correlation doesn’t equal causation, frequent calculator use in math class is negatively associated with test scores.

“Nine-year-olds who reported that they used calculators in class every day had the lowest NAEP scores of any response category, while students using calculators only once or twice per month had the highest scores.

A similar pattern is evident on the TIMSS. Frequent calculator use is negatively correlated with math achievement in several countries. A vast majority of students in the highest-scoring nations (Japan, Singapore, Korea) report that they never use calculators in math class.”

Source is here.

Cal says:
July 15, 2011 at 3:56 pm

While correlation doesn’t equal causation,

Then why mention it?

Achievement in math is not linked to hard work or lack of calculators. It’s linked to cognitive ability.

Susan says:
July 15, 2011 at 4:09 pm

Arithmetic can be handled by special ed kids with low cognitive ability. My other son who is borderline IQ doesn’t need a calculator to add. He can also subract, multiply and do long division thanks to his mother.

Crimson Wife says:
July 15, 2011 at 5:13 pm

IQ sets a maximum limit on achievement, but effort matters too. Given the same level of effort, the smarter kid will outscore the dumber one on a math test every time. But a smart-but-lazy kid could wind up with a lower score than an average IQ kid who works his/her tail off studying for it.

It’s like athletics- talent only takes you so far.

Cal says:
July 15, 2011 at 6:27 pm

But a smart-but-lazy kid could wind up with a lower score than an average IQ kid who works his/her tail off studying for it.

First, what’s more likely true is that the genuinely average IQ kid could, with lots and lots of work, equal–not exceed–what the genuinely smart kid does with no work at all. But so what? Why should some kids have to work and work and work, while others learn it effortlessly? That alone indicates that kids aren’t being given the same challenges at school.

But more importantly, who cares? You really think that average IQ kids not learning math is the problem being discussed here? Seriously? Why the absurd hypothetical?

I’m talking about kids with IQs of 85 being given algebra in 8th grade. You’re talking about a kid with a 125 IQ beating out a kid of 135–such is the woeful ignorance that the upper/middle class have about what actual average IQs are.

Bill says:
July 15, 2011 at 6:35 pm

I graduated in 1981, the first affordable scientific calculators were coming on the market in approximately 1979. I got my TI-55 and while it helped in subjects like Algebra II/Trig, it did not replace the concept of basic skills (Add, Subtract, Multiply, Divide, etc).

My old Algebra I teacher told us once ‘you guys and gals have no problems doing algebra, you just can’t add, subtract, multiply, and divide.

Robert Wright says:
July 15, 2011 at 6:44 pm

Crimson Wife, the source you cite is from an organization of parents who strongly favor “back to basics.”

That’s a source, but it’s a source of opinion, not research.

The research on the subject is overwhelming.

Bill says:
July 15, 2011 at 6:47 pm

Well, we keep trying to reform what works in education from 30-40-50 years ago into stuff which doesn’t work today in our classrooms…Look at the lattice method to try to multiply numbers (I saw a youtube on it)…kid spend 9 mins to try to multiply two numbers and doesn’t succeed.

Using the old fashioned method, the kid solved it inside of a minute…go figure…

Bill says:
July 15, 2011 at 7:13 pm

Another thing which works really well…School House Rock…

Naughty Number 9 was my favorite…

Also liked the 4 legged zoo…

David Wees says:
July 15, 2011 at 8:02 pm

I recommend reading Keith Devlin’s “The Math Instinct.” He makes an interesting argument, which he supports with research, that the math people learn in schools is virtually never used outside of schools, except by a very small percentage of people.

First, he shares research which shows that people have a high rate of accuracy when solving mathematical problems using self-made strategies in such contexts as the supermarket, etc… He then points out that the longer people have been out of school, the MORE successful their strategies are. In other words, remembering the school math strategies for solving problems is a hindrance when trying to solve real life math problems. Further, he points out that only a tiny percentage of people are able to use the school math strategies (which are highly efficient in many ways) to solve problems.

The problem, according to him, is one of an inability to transfer knowledge gained in one domain, and in one style of learning, to another domain in our lives. In other words, knowledge gained about algorithms in schools, regardless of the curriculum used, is too far removed from the actual applications of the math.

So it seems to me that this means that it doesn’t matter if the kids memorize some algorithms as a kid, that this should not be the primary purpose of mathematics education.

Another interesting point to make is that mathematicians, engineers, and other professionals who use mathematics are often not the best at arithmetic, but excel in problem solving and applying math they’ve learned to different contexts. They become a profession that uses mathematics because they are able to use it creatively.

I’m really sick to death of everyone worrying about whether or not kids know how to multiply 6 by 4 and worried that no one seems to be concerned about whether or not kids know WHY we would want to multiply 6 by 4, and HOW this is useful in their actual lives.

If you don’t think that we have a serious problem with numeracy (the ability to think mathematically and apply mathematics to different contexts), see this map of numeracy levels (including ALL adults aged 16 and older, so like all of you who learned mathematics the old way): http://www.ccl-cca.ca/cclflash/numeracy/map_canada_e.html

It paints a pretty scary picture of the problems in numeracy across Canada, in one of the best education systems in the world.

Maybe if we spent a lot more time doing engaging mathematics and applying what the kids learn in context, we might actually have a generation of people who USE mathematics, rather than a generation that complains about how awful math was when they grew up, and how horrible they were at it, but then asks their kids to do the same thing they did.

Some people actually LIKE mathematics, believe it or not.

Malcolm Kirkpatrick says:
July 15, 2011 at 8:03 pm

Robert): “Having an anti-calculator point of view isn’t supported by research or most experienced math teachers.”
“The research on the subject is overwhelming.”
Cite? What research?
I’m pretty experienced and I oppose calculators before Calc II.

Sean Mays says:
July 15, 2011 at 8:17 pm

FWIW: I wrote a position paper against calculators and computers for my math methods class for the ed masters. Much of the research supporting calculators was done by think tanks supported TI, with former TI execs in senior positions. Hardly what I’d call an arms length relationship.

It’s illustrative, though hardly conclusive, that many nations beating the US on TIMMS and PISA use calculators far less.

Bill says:
July 15, 2011 at 8:18 pm

Well, I’d say prior to 1980 or so, it was pretty rare to find calculators that commonplace, let alone ones which could assist in doing higher math as well. I guess those of us who went to school before then just got forced to do it the hard way…

Miriam Kurtzig Freedman says:
July 15, 2011 at 9:50 pm

Thank you for this article! I remember being so happy that my children’s school did not allow calculators…. now I see why.

Kate says:
July 15, 2011 at 10:49 pm

My kids have been exposed to the Investigations/Terc curriculum and Everyday Math.

My own opinion regarding the use of calculators, is that if you are using some of the suggested alternative strategies, at a certain point with larger numbers these strategies become to cumbersome and at that point they switch to calculators.

Lynne says:
July 16, 2011 at 4:48 am

I’d like to thank everyone for so many thoughtful comments on Part 1 of this subject.

Here is Lynne Diligent’s Part 2 discussing the current problems in math education:

https://expattutor.wordpress.com/2011/07/16/why-so-many-elementary-students-arent-mastering-basic-math-facts-part-2-of-2/

–Lynne Diligent
Lynne says:
July 16, 2011 at 5:10 am

I’d like to respond to the comments above that I taught in a private American school overseas where we only had one class in each grade, combining all levels of ability.

When going over homework, I gave the answers to each problem (which students corrected themselves). Then for the six problems in each row, I asked by a show of hands how many students missed each particular problem and chose the two to work on the board which were missed by the greatest number of students. We did this for each row of problems. I felt this was a good middle strategy to prevent too much boredom, yet all students seemed to think was most fair for most of the class.

In my class, I did succeed in getting all my children to do their homework. I used a system I observed in a Grade 3 class in Colorado. First thing was to check who did their homework. Those who did their homework (right or wrong) got a BIG “A+” on their paper (because having done it, they were ready to learn from their mistakes in the daily the class review) and those who did not do their homework, or who only did part of it got a BIG “F” on their paper, as well as a “Homework Alert” sent home to parents which had to be signed every night. I did not actually count the A’s and F’s in the grades for the report cards, but it sure did get everyone doing their homework, and everyone learning. (And yes, I did explain the system to parents at the beginning of the year–and yes, most were supportive.)

I’d also like to clarify that I am not anti-technology, nor anti-calculator. Yes, there could be some reasons why a few students have trouble learning times tables, such as learning disabilities. However, by having us work on it daily IN THE CLASS for 2+ months, for about 20 minutes a day, and by asking for student accountablility (in front of peers) it gave students the message that IT IS IMPORTANT TO TRY.

In my school, we were all required to take touch typing. I hated it at the time, learning on blank-key typewriters, yet that has been the number one skill I have used in every facet of my professional life since (and I also use times tables and estimating, measuring and conversions every day of my life as a mother and housewife, as a shopper, as a human being). My point is that schools need to put TIME in the math curriculum to PRACTICE these skills. Not all parents are able to pick up the burden of daily flash-card practice at home, even though it works. However, PRACTICE DOES NOT HAVE TO BE BORING! It can be fun! Any boring subject can be made into a fun game that students love by any dedicated parent or teacher, and this was the way I did it.

I’d like to thank all the commenters above for having taken the time to read and and consider these issues, and to leave such thoughtful comments.

–Lynne Diligent
Dilemmas of an Expat Tutor
expattutor.wordpress.com

j.d. salinger says:
July 16, 2011 at 6:48 am

Crimson Wife, the source you cite is from an organization of parents who strongly favor “back to basics.”

Robert, the source she cites is from a report by the Brown Center for American Education and it was based on data not opinion. What source were you looking at?

Bill says:
July 16, 2011 at 6:54 am

Lynne, wanna entertain third graders, get Multiplication Rock (in the Schoolhouse Rock series), they taught an entire generation of kids math, history, government, english, etc…

8. Danaher M. Dempsey, Jr. Says:

The bizarre “Modern Math Curricula” are not just puzzling to the children. A couple I know were completely stumped by some of the homework for their grade 6 child, who was using “Connected Math Project” materials. This is likely the same thing happening to far too many parents …. except in this case Mom has a BA in Mathematics and Dad has a BE in Mechanical Engineering.

The “Connected Math Project” was rated as “exemplary” in October 1999 by the US Dept of Education based on the opinion of math education experts……. Perhaps these bizarre programs should be evaluated based on actual use and performance in classrooms rather than how experts think they might work in the classroom. Evidence of success should be a requirement for a math program to be rated as either exemplary or promising.

Here is the list of Math duds…
http://www2.ed.gov/PressReleases/10-1999/mathpanel.html

9. Lynne Diligent Says:

More comments on this post from Joanne Jacobs’ blog:

ex0du5 says:
July 16, 2011 at 10:08 am

CW nailed the issue on the head. The problem isn’t that we need more X or less Y, it’s that education needs to be tailored to the individual. All of these discussions about method always seem to assume that the 1 teacher to N students model is necessary. Online education breaks that requirement and allows individualised lessons at the pace of each student, using whatever methods work best for them.

And the discussion of the calculator is silly. Some students would be much more benefitted from using the time spent usually studying three or four digit addition instead investigating other, non numerically oriented math like logic, proof, etc. The algorithm for digit addition makes more sense with algebra anyways.

momof4 says:
July 16, 2011 at 11:05 am

Too many don’t even understand basic math well enough to use a calculator. In the past few years, I have encountered retail clerks/cashiers unable to: (1) calculate 5% sales tax on a \$10 purchase, (2) calculate a 15% tip for a \$40 service and (3) calculate correct change for \$3.02 given for a \$2.82 purchase – WITH CALCULATORS! They had no clue how to format/input the information they had in order to get the answer they needed. Even when I walked them through it, they obviously didn’t understand – and they were all in their 30s or older. I’m not any kind of math whiz, but I do those kinds of calculations in my head; obviously a foreign concept to the above people. I was also assured by deli clerks at two different stores that they were unable to weigh 1/3 or 2/3 pound on their digital scales. I was assured that they could only weigh in increments of 1/4, 1/2, 3/4 and whole pounds

Ze’ev Wurman says:
July 16, 2011 at 12:14 pm

Regarding Cal’s argument that math achievement is not linked to hard work but to cognitive ability, this is no different for math than to any other subject. In other words, *everything* we do, excluding functions of the autonomous nervous system like breathing, is “linked to cognitive ability.” Such bland statements contribute nothing to the discussion.

The more meaningful question is whether typical K-12 mathematics, as taught in typical classrooms across the land, requires particularly advanced cognitive ability. Is it beyond the grasp — with only a reasonable effort — of a large group of school kids? Fortunately, there is an answer to this question, and it clearly indicates that the overwhelming majority of children can reasonably easily learn what we teach in our K-12 schools, given competent teachers and effective teaching methods.

But first, let’s consider the question itself. Does anyone believe that a broad and old system like education would willfully impose on itself goals than are inherently beyond a large fraction of its clients? Is the system suicidal? Doesn’t it know, if not through research then from experience, what can be reasonably expected to be learned by children? Do we teach in this country content that is clearly above and beyond what other countries teach? Anyone that believes that is simply out of touch with reality.

That leads to the obvious answer. Yet there is also data supporting it. In the 1990s UCSMP (yes, that purveyor of the deeply flawed Everyday Mathematics) translated the excellent Japanese math curriculum edited by Kunihiko Kodaira. In is preface it tells the story:

The Japanese school system consists of six-year primary school, a three-year lower secondary school, and a three year upper secondary school. The first nine grades are compulsory, and enrollment is now 99.99%. According to 1990 statistics, 95.1% of age-group children are enrolled in upper secondary school, and the dropout rate is 2.2%. […]

Japanese Grade 7 Mathematics (New Mathematics 1) explores integers, positive and negative numbers, letters and expressions, equations, functions and proportions, plane figures, and figures in space. Chapter headings in Japanese Grade 8 Mathematics include calculating expressions, inequalities, systems of equations, linear functions, parallel lines and congruent figures, parallelograms, similar figures, and organizing data. Japanese Grade 9 Mathematics covers square roots, polynomials, quadratic equations, functions, circles, figures and measurement, and probability and statistics. The material in these three grades (lower secondary school) is compulsory for all students.

The material described is essentially all of U.S. algebra 1 and geometry curriculum. In other words, here is a country where 99.99% take algebra 1 and geometry by the end of 9th grade. And they are generally successful, as over 95% continue to (then) non-compulsory upper secondary.

TIMSS also provides some insights. Students in leading countries like Singapore or Korea have their 25th percentile achievement at a level close to our 75th percentile. Unless someone believes that Chinese, Koreans, or Malays, are genetically superior to us, the cause must be in how we teach our students, both in school and outside it.

The answer to the original question then is that anyone with an IQ within 1-1.5 standard deviation of the average — probably about 80 and above — can master our general K-12 curriculum. Some with more effort, some with less, but none needs unreasonable efforts. Except when they are incompetently taught. Very much like anyone, except a handful of disabled children, can run a 100 yards race. No special “sporting ability” required.

Robert Wright says:
July 16, 2011 at 12:14 pm

Mr. Salinger, the “source” is from the Society for Quality Education which says up front that it opposes “progressive education.” I think the National Council of Teachers of Math is a far better source.

Malcom, if you are an experienced math teacher, I think you are the rare exception.

Vince says, “I see many students who can’t multiply without a calculator because they never learned the multiplication tables.”

Once again, there’s a failure of logic. Though what Vince says is true, it doesn’t mean the use of calculators prevented the learning of multiplication tables. Again, though one often sees a guy with a broken leg using crutches, that doesn’t mean the crutches broke the leg.

Lynne, it was very good to read your post. I’ve tried a strikingly similar technique with homework but I’m afraid I’m had less success. I think I need to alert parents as frequently as you do.

The drudgery of memorizing arithmetic facts is a necessary evil which will prove to have life-long, practical benefits, but it shouldn’t serve as a gate stopper for advancement into higher level math. If you count on your fingers, you can still learn the distributive property.

Again, I think every child should know arithmetic and know it well and when children don’t learn it, there needs to be aggressive, effective remediation.

But keeping calculators out of classes that are a step beyond basic arithmetic doesn’t force remediation. Instead, it thwarts forward development.

I once wrote a math grant. All sixth graders were to be tested in addition, subtraction, multiplication and division. Those who hadn’t mastered all their math facts had a Saturday class taught by college students, 8 AM to noon. They were to go every Saturday until they could prove mastery.

This way, students could stay in their regular math classes yet would learn to master the arithmetic they had failed to memorize in elementary school.

Unfortunately, the grant wasn’t funded.

Bill says:
July 16, 2011 at 12:57 pm

Mom Of 4, I’ve also seen the deli clerk who couldn’t figure out how to enter 2/3rds of a pound on a deli scale (.66) and not being able to figure out what 15% of \$40 is, that’s grade school math (at least it was in my day) without using a calculator (which wasn’t allowed in grade school in the early 70′s).

Robert, the calculator is a tool, if you don’t understand the basic principles behind math, all the tools in the world won’t help you solve a math problem (that’s a fact).

Want to know what employers are looking for in prospective employees?

Critical thinking and problem solving skills (math/philosophy)
Excellent communication skills (English)
Self Starter/motivated
Able to work with minimal or no supervision
Time management skills
Interpersonal skills.

A former supervisor once told me, he never liked to micromanage people, but he had to do it on occasion due to the fact that some people simply cannot function without supervision at all times.

Math (and some other subjects) takes discipline and practice. Giving kids access to a calculator in the 3rd grade is not going to help their math skills what so ever. Which Lynne is making a pretty hefty chunk of change trying to correct.

Crimson Wife says:
July 16, 2011 at 12:57 pm

Mr. Salinger, the “source” is from the Society for Quality Education which says up front that it opposes “progressive education.”

The article was from there, but the source of the research cited was from the Brookings Institution- a think tank with a reputation for being on the liberal end of the political spectrum.

Robert Wright says:
July 16, 2011 at 2:45 pm

Bill, you write:

“the calculator is a tool, if you don’t understand the basic principles behind math, all the tools in the world won’t help you solve a math problem.”

I agree.

And the same can be said for memorization.

If you don’t understand the basic principles behind math, lightening speed recall of the multiplication tables won’t help you solve a math problem.

Crimson Wife, the research that was referred to was embedded in a page with a lot of spin, some subtle, some not so subtle, composed by opinionated parents, not objective educators. The writing was intentionally misleading. The National Council of Teachers of Mathematics, practically all departments of education in respected universities and over 90% of middle school teachers who teach math day in and day out have an opinion about the use of calculators that are far different from the organization you linked to.

It’s difficult to discuss the use of calculators because for some it’s an emotional issue (for reasons that escape me) and no amount of authentic evidence will have any impact.

Really, whenever I bring up the topic to my colleagues who teach math, they look at me, both liberal and conservative teachers, like I’ve just asked them if they think the world is round.

Cal says:
July 16, 2011 at 2:55 pm

this is no different for math than to any other subject. In other words, *everything* we do, excluding functions of the autonomous nervous system like breathing, is “linked to cognitive ability.”

To put it mildly, Duh.

In context:

Assertion: math achievement is aided by practice. Assertion: math achievement is hurt by calculators. Me: Math achievement is determined by cognitive ability.

Nothing in there to make math different from other academic subjects. It’s just that we’re talking about math.

The more meaningful question is whether typical K-12 mathematics, as taught in typical classrooms across the land, requires particularly advanced cognitive ability. Is it beyond the grasp — with only a reasonable effort — of a large group of school kids?…The answer to the original question then is that anyone with an IQ within 1-1.5 standard deviation of the average — probably about 80 and above — can master our general K-12 curriculum.

You are, to put it politely, delusional.

K-12 is an awfully big category for math. So let’s define it as 9-12 math, which is currently defined as geometry, second year algebra, trigonometry, a few odds and ends, and calculus.

And yes, a complete grasp of this math is beyond the effort of a large group of school kids.

I would hypothesize that an even basic understanding of the math subjects listed above is, without question, beyond the ability of anyone with an IQ below 100. That defines well over half of African Americans and Hispanics, and roughly half of all whites and Asians.

Moving down to K-8, there’s no question that low ability students do better with repetition and drills in arithmetic, whereas high ability kids don’t require it. Algebra, ostensibly an 8th grade subject, is bound to the same IQ requirements I mentioned above.

A person with an IQ of 80 is fully able to participate in society, but largely incapable of abstract concepts. High school academics is loaded to the teeth with abstractions.

Does anyone believe that a broad and old system like education would willfully impose on itself goals than are inherently beyond a large fraction of its clients? Is the system suicidal? Doesn’t it know, if not through research then from experience, what can be reasonably expected to be learned by children? Do we teach in this country content that is clearly above and beyond what other countries teach?

To answer the questions in order: yes, yes, yes, and no, the other countries aren’t suicidal and so they don’t impose these goals on all children. They sort and track. And, in some cases, the countries have populations that are entirely white and Asian, and thus aren’t bound by the same lower level constraints that we are–and yet, they still track.

Are you really that naive? Our country refuses to even explore the possibility that the achievement gap is cognitively based, for obvious and understandable reasons. And in fact, the achievement gap is almost certainly based in large part on cognitive differences (and I have no idea what the cause is, so don’t pretend I do).

Yes, we are suicidal when we declare these goals achievable by all. We never used to. We used to accept that only smart kids, regardless of color, went on to study advanced topics. Then we realized, to our horror, that “smart” wasn’t equally distributed.

This is obvious. Are you really that clueless that you don’t understand the distortions the achievement gap forces upon us? Or is it that you just don’t understand what it means to have an IQ of 80?

Daniel Stamm says:
July 16, 2011 at 3:49 pm

{Cal wrote]… the other countries aren’t suicidal and so they don’t impose these goals on all children. They sort and track. And, in some cases, the countries have populations that are entirely white and Asian, and thus aren’t bound by the same lower level constraints that we are–and yet, they still track.
This is absolutely false. Neither Japan nor Korea track students in grades 1-9. In Japan, promotion to the next grade is not dependent on achievement; they are socially promoted. In Japan, there is no gifted education.

Daniel Stamm says:
July 16, 2011 at 4:02 pm

Here’s the reference for no tracking in Japan.

http://www.ed.gov/pubs/JapanCaseStudy/chapter3.html

Crimson Wife says:
July 16, 2011 at 4:03 pm

“practically all departments of education in respected universities.”

Why don’t you try asking the math, engineering, and hard sciences departments whether or not they support the extensive use of calculators in elementary & middle school math classes? I’d be willing to bet that STEM professors are not nearly so enthusiastic about calculators as the ed school profs are.

I don’t have a problem allowing occasional calculator use AFTER students have mastered the pencil & paper algorithms. A calculator is a useful time-saver once students have had enough practice to realize when an input error gives them a garbage answer.

Daniel Stamm says:
July 16, 2011 at 4:25 pm

In regard to the cognitive ability of the Japanese, they Japanese have every bit as much variability in ability as we do in the U.S. As kids progress though the compulsory grades (1-9), the slower kids get further and further behind, but they still learn vastly more than if they had been tracked or ability grouped. (This isn’t done within classes, either. As groups, neither Chinese nor Japanese have been shown to be inherently more intelligent than Americans. This was shown by Stevenson et.al in
Stevenson. H. W, & Lee, S. Y. (1990). Contexts of achievement: A study of American, Chinese, and Japanese children. Monographs of the Society for Research in Child Development, 55(1-2), p. 4

Ze’ev Wurman says:
July 16, 2011 at 4:39 pm

Quoting Cal: “To put it mildly, Duh.”

Exactly my point when referring to your post.

More to the point, when I wrote about typical K-12 content, in math I referred essentially to about Algebra 2. Neither calculus, nor pre-calc, are typical or expected for the majority of high school kids. Check the data — less than a third of graduates take those. Similarly, any AP or true honor course is by definition not typical.

Even more to the point, I appreciate your answers but nothing stands behind them except your own interpretation of reality. Further, you never bothered to explain the Japanese data or the TIMSS results I quoted, in view of your beliefs.

Finally, your sweeping over-generalizations border on the irresponsible. While those countries may “sort and track,” TIMSS samples across all types of schools and hence comparing our 75th percentile to their 25th is entirely appropriate. Similarly, throwing generalizations like “white” or “Asian” indicates gross ignorance. Neither is a homogenous category and the fact that in Singapore, which is at least as diverse as we are, the Malay minority achieves much higher than we do is highly significant. The Malays were once considered in similar terms you refer to our Black and Hispanic minorities.

Again, nobody argues IQ plays no role. I am arguing that what we require in our typical K-12 education does not require IQ above at least one standard deviation below the average.

Robert Wright says:
July 16, 2011 at 5:07 pm

Crimson Wife writes:

“Why don’t you try asking the math, engineering, and hard sciences departments whether or not they support the extensive use of calculators in elementary & middle school math classes? I’d be willing to bet that STEM professors are not nearly so enthusiastic about calculators as the ed school profs are.”

I would imagine that professors of math and engineering would preface any opinion being saying it’s out of their area of expertise.

Well, we might be in agreement on many issues regarding calculators. It’s a broad topic.

When I was a teacher and I assigned for homework a problem of multiplying a two digit number with a three digit number, I wouldn’t accept any answers where the student didn’t show his work. Getting the answer with a calculator in this case doesn’t improve learning. “I didn’t show my work because I did it my head.” I got that excuse and rejected it. Nope. In a case like this, a calculator does the process that the student is supposed to learn. A calculator would still be good to use, but only to check the answer, not to do the problem. If and when calculators are used to multiply when the objective is to learn multiplication then yes, they impede learning.

But let’s say there is a different problem:

Fred, John and Alex are roommates. They all agree to split food expenses three ways even though Alex has an eating disorder and over-consumes products with fat, sugar, and red dye #16. In the kitchen is a hollowed out Buddha that Fred made for his mother in ceramics class for Mother’s Day. But since his mother isn’t what you’d call a universalist, it’s remained in Fred’s possession ever since he saw it on a card table in front of their house during a Saturday yard sale. The asking price was \$5.00. Every time somebody buys food for the house with his own money, he puts the receipt with his name on it in the Buddha. Every time somebody has some extra change to contribute to the food fund, they put it in the Buddha along with a note that includes their name, the date and the amount. Every three months they like to even out the account so nobody has contributed more than anybody else. This means that sometimes John will give Alex a few bucks or Fred will give Alex one amount and John another. It all depends on the receipts and the contributions. In March, April and May of this year, John bought food totaling \$200. Alex’s total was \$175. Fred’s total was \$185. John put in \$15.50. Alex put in nothing and Fred put in \$75.00. At the end of May, the guys empty the Buddha of the receipts, the notes and the currency. The question, is, how do they divide up the currency and then who needs to give whom what in order that everybody has contributed equally for those three months?

In solving this problem, the use of a calculator would do no harm.

Cal says:
July 16, 2011 at 5:32 pm

This is absolutely false. Neither Japan nor Korea track students in grades 1-9. In Japan, promotion to the next grade is not dependent on achievement; they are socially promoted. In Japan, there is no gifted education.

It’s absolutely false that Japan and Korea are Asian? I was assuming that tracking begins in high school.

I am arguing that what we require in our typical K-12 education does not require IQ above at least one standard deviation below the average.

Fine. Show me studies demonstrating that students with an 80 IQ are capable of mastering Algebra, let alone Algebra II. Of course, we don’t have any at all. Which means that your argument, like mine, is based purely on “your own interpretation of reality”. Except mine is much closer to reality, since the test scores show something much closer to my assertions than yours, including higher test scores for whites and Asians, even accounting for poverty.

As for the Japanese studies, not everyone passes, right? And they have far fewer people with IQs below 90 than the US does. So I’m not sure what you think Japan’s experience refutes.

Crimson Wife says:
July 16, 2011 at 5:54 pm

“I would imagine that professors of math and engineering would preface any opinion being saying it’s out of their area of expertise.”

I’ve heard several college professors who teach math, science, or engineering complain about the inadequate preparation of many of their students in recent years compared to in past decades. So yeah, I think their views are relevant.

Ze’ev Wurman says:
July 16, 2011 at 6:10 pm

Cal,

Kodaira writes: The first nine grades are compulsory, and enrollment is now 99.99%. According to 1990 statistics, 95.1% of age-group children are enrolled in upper secondary school, and the dropout rate is 2.2%.

95.1% corresponds to 1.66 standard deviation below average, or IQ of about 75 and up. All those were deemed capable to continue to the non-mandatory upper secondary, after taking Algebra 1 and Geometry by grade 9. Factoring in the 2.2% dropouts brings us to everyone above 1.47 standard deviation below average, or IQ of about 78 and above, successfully completing Japanese upper secondary schools.

Estimates of “national” IQ are rather questionable and highly imprecise, and vary between plus(!) 7 to minus 7 points between United states and Japan. Even accepting the extreme disadvantage for the US, this would adjust the numbers above to 82 and 85 respectively.

10. Lynne Diligent Says:

Vince Lynch says:
July 16, 2011 at 6:39 am

Re: Calculators

I teach in a one year, post-high school prep school. The following are some things I’ve noticed that I think are the result of allowing students to use calculators too early and two often.

I see many students who can’t multiply without a calculator because they never learned the multiplication tables. If you can’t multiply, you can’t divide. If you can’t divide, you can’t factor. If you can’t factor, … I suspect if they didn’t have a calculator available, they would have been much more likely to have learned the multiplication tables and the curriculum would have supported it.

Even more students can’t handle fractions. Because most calculators will perform operations with fractions, the students apparently never spent the time (or possibly had the curriculum time) required to learn how to find common denominators, etc. While calculators work fine when the denominators are numbers, they don’t work when the denominators contain variables. The result is that students can’t perform operations with rational expressions because they never really learned how to perform the operations with rational numbers..

Many students seem to have little feel for numbers – 50 is the same as 5000. I know that somewhere along the way, they are exposed to estimating. However, if you can’t multiply, estimating doesn’t seem to take.

Besides memorizing math facts, understanding all aspects could facilitate the memorization. I am retired now but taught for many years. I have some addition, subtraction, and multiplication models I used in my combined first and second grades, but they could be for any age. Each gives a way to fully understand the processes. After understanding the models of addition, subtraction, and multiplication, memorization should follow. Children loved taking the timed tests by keeping a personal graph of one’s progress.

12. Kerrie Says:

Calculators ARE used in Singapore. Singapore believes that if they teach their children things that calculators can do, then they are putting their children in direct competition with machines and it’s cheaper to buy a machine than hire a person. Therefore, they teach them to “think”, not just memorize facts and formulas. They also feel that if they spend time on “drills”, they aren’t teaching children to think. If they spend time on thinking, children will eventually “remember” their facts, just by using them, not by memorizing them.

Singapore ranks in the top 3 of all nations in Math, while the US ranks the lowest of all industrialized nations.

Of course they way we were taught seems like the right way, because we are familiar with it…but Singapore must know what they are talking about when they produce those types of results.

13. Derek Says:

I can think of nothing more torturous than 50 hours of math drills. The issue with rote memory tests is they inhibit student’s ability to think citically and/or creatively about the math problem they face.

I want students who can find multiple paths to one solution rather than one memorized path and a no understanding of the question.

14. Anamika Sinha Says:

Wow, really well researched and nicely written article. I too believe that practice is the key when it comes to Math. Recently a nice website for Math practice http://www.edugain.com/ph is launched for students in Philippines.

15. Jody Weissler Says:

This topic sure sparked a lot of discussion. I would like to throw the idea of Japanese Math techniques into the equation. Why not have the students use an abacus (Sorban in Japan) instead of a calculator. I have taught this method of learning facts to my own children and have documented it at a website. I even show how you could make your own abacus with basic supplies.

• Sara Webb Says:

I have seen videos of students who had learned with an abacus doing amazing math operations in their heads! It is crazy how awesome it is! I wish I knew how to use an abacus!

16. Tina Says:

An offshoot of the “spiraling” debate (and one reason why some deem spiraling necessary)… One of the unfortunate trends in US math is the “inch deep and mile wide” approach that has taken hold. The teachers are simply required to cover too much material each year. It doesn’t matter if the children haven’t mastered basic arithmetic. They still need to cover a short unit on area & perimeter, metric & English units of measurement, angles and polygons, data & statistics, fractions, and decimals, even by the third grade. I tutor using the Singapore books. By contrast, they don’t even introduce decimals in any context other than money until book 4B (the end of fourth grade). American books try to teach tenths, hundredths, and sometimes thousandths starting in third grade, despite the fact the child hasn’t mastered long multiplication or long division yet.

If a teacher wants to skip a unit on geometry, for instance, to spend more time on long multiplication, she is hard-pressed to do so. The schools are worried about how their students will score on state assessments. What if the state assessment has questions about area & perimeter on it, and that’s the unit Mrs. Smith skipped in order to review long multiplication? Then Mrs. Smith’s class will miss all the area & perimeter problems on the test and won’t score high enough to maintain the school’s “good” reputation!

The tail is wagging the dog, unfortunately. Even the best textbooks that include plenty of arithmetic practice still have this problem – and I have sat on textbook review panels. In the case of the “good” textbook publishers, it’s not always the fault of the authors; it’s the fault of the marketplace demanding they cram too much material in a single school year.

• Lynne Diligent Says:

Tina, I really appreciate your comment. I was a third-grade teacher in an overseas American school and I sometimes skipped areas such as you mention above in order to help children really master their multiplication before moving up from grade three. The text book only allows two weeks for all of the times tables, if you can believe that!! Thankfully, overseas we weren’t hampered by this continual testing problem and were able to get into the important subjects in sufficient depth.

17. Michale Tredinnick Says:

Times have changed….kids only want to play video games and watch youtube….obviously, if the parents let them, they will do it

18. Jewish Watch Australia » Common core curriculum includes having 5th graders learn how to fill out tax forms Says:

[…] an accounting or business law class.  But when many high school graduates today are unable to even count change correctly from behind a retail counter, learning math by simply filling out a form using a set of […]

19. Sara Webb Says: