A friend of mine commented that he is sure that Americans are  being educated to a lower level in math education than they were in the past, and after considering it carefully, I’ve decided that I agree with him.

I see several causes for the decline in U.S. math education.   Although many major and minor changes have taken place (which probably started about 1962, and have greatly accelerated since that time), here are the four I think are most important.

Change One:

The “new” math seems to have been introduced (when I was a child) between 1962 and 1964.  While there is only a finite amount of time which can be alloted to math each day, some of that curriculum time became devoted to things such as teaching “bases” which I NOW understand clearly, but which I could not make heads or tails of at the age of 9. Even NOW, after teaching for over two decades, I see no reason to waste time on such things at that age (when the time could be much better spent memorizing multiplication tables, or learning long division).   I also think the sections in many elementary math books which require young students to waste 2+ weeks of the curriculum time on geometric figures at the age of 7-8-9 are a complete waste of curriculum time.  In the 1960’s we were not taught how to tell time, count money, or count back change in school.  That was taught at home by our own parents. The primary reason I am opposed to the “new” math” (having gone through it myself, and only overcoming my own “math anxiety” problems after my first two years of teaching elementary-school math)  is that I see it doesn’t leave enough time in the curriculum for most children to master basic arithmetic.

Change Two:

Mainstreaming handicapped children into the classroom has had some extremely positive benefits for both those children, and society.  However,  it has caused a change in THINKING in another important way–it has now become “politically incorrect” to separate classes according to ability (ability grouping for math and reading was highly prevalent before I finished elementary school in 1967; I’m not sure what year it was done away with). Actually, although I’ve lived overseas since 1993, when I went back to observe in Englewood, Colorado in 1999, I found they WERE doing ability grouping for math and reading. my overseas American school, we had only one class for each grade (with no team-teaching or expandable classrooms) and had all abilities in the class grouped together.  The reality of this generally means teaching to the slowest members of the class, or at best, the middle, while the highest members languish in boredom.

Change Three:

Math homework has recently decreased by more than half.  For example, at our school we used the Houghton Mifflin text book series. The accompanying workbook gave 24 calculation problems on the same subject as the lesson, with two story problems at the end, for a total of 26 problems. Over the years, some parents complained it was “too much homework.” From what subsequently happened, I presume the same complaints were made by parents in America.

About two years ago we updated to a newer version of the same Houghton Mifflin series. Some the concepts had been made more difficult by taking each subject to a slightly higher level (such as teaching place value to 100,000 instead of just to 1000 in Grade Three).  In addition, the book made a number of “cognitive leaps” which many student were not able to comprehend.  In some lessons, I had to subdivide each lesson into three separate lessons, and make up worksheets of my own, to take students step-by-step through those cognitive leaps.  The new workbook, however, cut the number of homework problems for each night HALF.

This lesser amount of homework is enough for the brightest students to master the concepts.  However,  it is INSUFFICIENT for many average students (who catch on in class, but forget without enough drill to support recall), and ENTIRELY INSUFFICIENT for the weakest students.

So, this problem can be summed up by calling it INSUFFICIENT DRILL. I used to spend two months of my Grade 3 class just getting kids to recite multiplication tables (yes it took time, but knowing my multiplication tables has turned out to be one of my most useful life skills, and I know they will eventually feel the same way–far more important than studying figures for geometry, which COULD be learned later on….).  The Houghton Mifflin book gives only about two weeks to get through all the tables (one each day), which is no where NEAR what any child less than a genius would need in order to learn the tables by heart (or perhaps many of today’s “experts” who are writing the text books don’t feel that’s important!–Is it any wonder kids get to Grades 6, 7, 8, and beyond, and still don’t know their tables?)

Math “drill” also needs to include ten times as many story problems, but with methods for the problems being TAUGHT (most teachers are skipping the story problems and have no idea how to teach them or explain them…I didn’t either when I first started, but did get very good at it after several years). Knowing multiplication tables is drill.  I spent between two and three months of the Grade Three year (sometimes at the exclusion of time, money, and geometry) drilling daily on multiplication so that the kids would get out of Grade Three REALLY KNOWING their times tables.  The reason I did this was not to embarrass them, but so that they WOULD actually spend time working on learning them at home, and to communicate their IMPORTANCE.  I did my best to make it fun, in game formats, with classroom rewards.

Change Four:

TEACHING methods in math seem to have undergone a MAJOR change in recent years. In the past, math was taught with a DECUCTIVE method–in other words, the material was presented, explained; then followed step-by-step by the teacher while students attempted the work.  When the students mastered the concepts sufficiently in class, they were then sent home with homework to do (which ideally was GONE OVER the next day in class, and NOT JUST GRADED and handed back).

Now, it seems more and more teachers are trying to use an INDUCTIVE method which is both frustrating and ineffective for the great majority of students. I recently heard of a high school teacher who is apparently NOT TEACHING the class at all (in any class, as I hear it from the other students in all high school classes). Instead, he does stuff like this: “Here’s a page of new problems, see if you can figure them out.” He sits at his desk, and when asked questions by students who have no idea what to do, responds, “Just try it, try it..” and doesn’t help them at ALL. So many students are paying private tutors to TEACH them the material they are supposed to be learning at school, while the teacher gives a TEST on the material, after NOT EVEN HAVING TAUGHT IT!  I’ve now heard of teachers like this in both high schools and universities.  My conclusion is that the mathemeticians who are no doubt the text book writers, and promoters of their “new” methods in schools of education,  simply do not understand how the mind of the typical and average child works.  These mathemeticians have no doubt never struggled with math concepts the way many kids do, and do not understand that the majority of kids cannot “figure things out on their own.”  Nor should they have to!  That IS after all what the teacher is being PAID for, to make concepts CLEAR to the students, not to make it more difficult for them!

I also think that a teacher who may have struggled with math as a child, but who has overcome math anxiety, can be a BETTER teacher than someone who has always been brilliant at math.  The reason?  The brilliant teacher often cannot understand how or why his students “cannot understand,” and doesn’t have as much patience with their difficulties.

While there are many other small contributors (such as mothers no longer being at home to make sure kids get their homework done) I would attribute the majority of changes to the four points discussed above.

What do others think? I’d be interested in hearing both from math specialists, as well as general teachers who also teach math.

–Mary Mimouna

One thing that quickly strikes you when you look at Singapore math textbooks and workbooks are the bar models. The bar model is to illustrate the “Part-Whole” concept.  Number bonds are part whole concept too. The idea is that a simple pictorial presentation provides a tool to understand fundamental math concepts better and use it as a foundational tool to attack complex problems without the use of more abstract tools like algebra. Let us take a look at the bar model :

At the www.lionmath.com/forum, further discussion can be found at “The Bar Models.”

Singapore’s education system broken to its components are : framework, texts, tests, teachers prep. The sum of the whole is the key or more appropriately the execution of all the components are important for its success.

Singapore Math is the execution of the above and not just based on its texts. There is a lot more to it as learning go beyond the texts. “Teach less learn more” is the goal and the ideal scenario. Singapore Math texts thus are slim and oriented towards the child learning in the classroom. The teacher ideally should be able to demonstrate and assist the children in learning core and advance concepts.

In practice, tests are way beyond the texts, so it is expected the child has to work beyond the texts. In criticism of the system (Singapore context), the tests are tough and not all teachers are equal.

It is tough going for Singapore children to “Master” Singapore Math which is tested at a level much higher than what is presented in the textbooks. The parents are encouraged to be much involved in educating the child. At the end of grade 6,there is a nation wide test — Primary School Leaving Examination (PSLE.)

Find out more and join us at www.lionmath.com/forum.

Pragmatic Mom finds that no matter how good your elementary school is, there is always a need to supplement your child’s math curriculum at home.  For some of my kids, they needed additional challenge that their school was not providing.  For another child, I found that she needed more individualized attention to really “get it.”  All of my kids benefited from additional math fact drills, and home supplementation helped me to understand how my child learns and exactly what each child needed additional practice on.

I find these math workbooks to be very helpful.  I like to start my kindergartener in a word problem book by Evan-Moore which emulates the Chicago Math program, Everyday Math.  I find that introducing word problems at an early age helps my kids get an intuitive feel for translating words into number sentences.  At this stage, most of the problems are adding or subtracting so word problem translation to number sentences is simpler.  Then, when my kids face word problems in 3rd grade, they are not scared of them.

I like to start my  kindergartener on the First Grade book usually the summer before 1st grade.  This is a really fun book that my kids all actually loved doing.  They generally would do about half the book during the summer and finish up by year-end.  We could do 1-3 weeks in a sitting with my child begging to do more.  Really!  True story!

Daily Word Problems:  Student Practice Books. Grade 1 for incoming Grade 1 summer practice.  Publisher:  Evan-Moor.  Also at Lakeshore Learning. I recommend using the incoming grade book for summer practice  (i.e. going into 4th grade summer, use 4th grade book). Click on image of book to buy through Amazon.com.

I  have my kids doing these books as both summer supplementation and supplementation through the year.  I also recommend the Singapore Math curriculum for additional supplementation.  Singapore Math is the curriculum used in Singapore which ranks usually 1st or 2nd in the world in math test scores.   Once you see their system, it’s easy to understand why.  The concepts are presented in a VISUAL and easy-to-understand way.  There is also adequate drill for mastering concepts which Everyday Math does not have.

Singapore Versus U. S. Math:  We Lose.  Singaporean students rank 1st in the world in mathematics on the TIMS and U.S. students rank 16th. …  http://edbizstrategy.blogspot.com/2010/01/singapore-vs-us-math-we-lose.html

Here’s a link to a study that rates Singapore best in Math, Science for 7th & 8th graders:  http://www.bc.edu/bc_org/rvp/pubaf/chronicle/v5/N27/timss.html

Compare your city’s math scores to the rest of the worlds.  FYI, Singapore scored the highest.  http://www.usatoday.com/news/education/2008-10-22-math-cities-international_N.htm

This is an interesting article from the New York Times by Benedict Carey published on December 20, 2009, about how children are capable of learning math at a young age.  They note that some of these methods have been successful in overcoming dyslexia.   Here’s the link: http://www.nytimes.com/2009/12/21/health/research/21brain.html

Pragmatic Mom understands that it’s a lot to get your child to do two different workbooks at home.  I suggest using just the Singapore Math Text Book which is in color and has some drill as well.  Because the Singapore Math curriculum is also in a different sequence than Everyday Math, you will need to pick and choose chapters that reflect what your child is learning.  Use the Singapore Math Text Book to explain a concept your child is confused about and then do the problems in that section.  Alternately, follow the curriculum to challenge your child who is bored with their math curriculum at school and go one grade ahead.

Pragmatic Mom recommends using the book number as correlated to your child’s grade even though the books state one grade lower than the number listed.  If you click on the book, it will take you to Amazon to purchase.

For preschool, 1 year before kindergarten.  Earlybird Kindergarten 1A and 1B.

For incoming Kindergarten.  Earlybird Kindergarten 2A and 2B.

For incoming 1st grade, Primary Mathematics 1A and 1B.

For incoming 2nd grade, Primary Mathematics Textbook 2A and 2B.

For incoming grade 3.

For incoming grade 4.  (It says on the book that this is for grade 5, but it correlates to grade 4 at my school).

For incoming grade 5.

It is funny how we build things up in our minds to be one thing, and then reality turns out to be quite different. A zen monk named Zuishin that I used to know once told us a story about his going to the dentist, and how the whole trip there he was agonizing about going. He told us how he hated sitting in the chair, rinsing his mouth and watching water dribble across his chin, feeling a needle as they put anesthetic in his mouth, or just feeling them cleaning his teeth. Then he realized that although he could not change his experience while at the dentist, he did not have to live through it twice, which is what he was doing by fretting.

The point? Our mind colors our perspectives with a million thoughts about what we think reality is, rather than seeing what it really is.

I had a vision of Singapore Math as a drill and kill type program which focused on procedural fluency at the expense of all the other things I feel are important in mathematics. Because this is a program some of our parents – and some very vocal critics of the current program we have in our elementary school – have pushed, I decided to purchase the book, The Singapore Model Method for Learning Mathematics, put out by the Singapore Ministry of Education.

Was I in for a surprise! I’m still reading the body of the book, but the initial chapter talks about the Singapore Mathematics Framework. Here is a key thingthey say,

Mathematical problem solving is central to mathematics learning… [it] is dependent on five inter-related components, namely, Concepts, Skills, Processes, Attitudes, and Metacognition.

This describes exactly what I would like the focus of our math program to be. They go on to break down what is meant by each of the components, but really I’m still digesting it all.  Maybe there is something to this after all.

Perhaps the trip to the dentist won’t be as bad as I feared…

[Bill Watterson identifies the trouble with math problems, through the eyes of Calvin and Hobbes.]

It’s time to revive and (hopefully!) finish my long-neglected series on solving word problems in elementary mathematics. I’ve been having fun making up the problems, so now I just have to write the posts. Coming up soon:

• 4th grade math in The Lion, the Witch and the Wardrobe.
• 5th grade math in The Hobbit.
• 6th grade math in… I haven’t decided yet. Any suggestions?

Since it has been more than two years since the last entry, however, I wanted to take a few minutes to recap our progress so far and to refer new readers back to the original posts:

• Pre-Algebra Problem Solving: The Tools
  An introduction to word algebra and bar diagrams.
• Pre-Algebra Problem Solving: 2nd Grade
  Math problems from Mr. Popper’s Penguins.
• Pre-Algebra Problem Solving: 3rd Grade
  Math Problems from Poor Richard.

Bar Diagrams are Visual Algebra

For the rest of the problem-solving series, I will focus on bar diagrams (also called rod models, or simply models) as taught in the Singapore Primary Math series. These diagrams are a pictorial form of algebra, with the length of each bar representing some quantity in the word problem, either known or unknown.

In creating a bar diagram of the word problem, the student must think through the same type of questions that are required in writing an algebra equation:

• What quantities are related to each other, and how are they related?
• How are the known quantities related to the unknowns?
• What is the basic unit (or variable) that I need to find?

The Whole is the Sum of its Parts

All bar diagrams descend from one very basic diagram showing the inverse relationship between addition and subtraction: The whole is the sum of its parts. If you know the value of both parts, you can add them up to get the whole. If you know the whole total and one of the parts, you subtract the part you know in order to find the other part.

When all the parts are the same size, students can use a (number of units) x (size of units) = total diagram to represent both multiplication and division problems, because multiplication and division are also inverse operations. Because the units are all the same, we only need to write a number in the first one.

Practice Makes Perfect

It is important for students to practice applying this method to simple problems, even though they may be able to solve the problems without a diagram. As with any tool, skill with bar diagrams must develop over time, through repeated use.

The 4th and 5th grade problems coming up will require relatively complicated diagrams with multi-step solutions. If you wish to teach this method to your students, be sure to step back to simpler problems (like those in the 2nd grade and 3rd grade posts) at first.

To get more practice creating bar diagrams, your students may enjoy these online tutorials:

• Thinking Blocks Addition and Subtraction Word Problems
• Thinking Blocks Multiplication and Division Word Problems

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Have more fun on Let’s Play Math! blog:

• Word Problems in Russia and America
• Reading to Learn Math
• Solving Complex Story Problems
• Solving Complex Story Problems II
• Story Problem Challenge Revisited

This two-minute video  shows how to multiply 21 x 13, and  123 x 321, with an easy line-drawing and angle-counting method.  No times tables needed!

Here is another very clear example showing 432 x 312, done with the same method.  However it shows a way to “carry” digits.

Seeing these videos makes me wonder anew whether math is really about “shapes,” and about describing all the “shapes” in the universe.

If there is anyone from China reading this blog, I’d really appreciate knowing how extensively this method is actually used in schools.

–Eileen

After struggling to find a way to learn basic math, I’ve finally found a system that may work for me! Singapore Math was developed in Singapore (of all places) and has consistently brought that little country to the top of the math achievement charts ever since its inception. I ordered almost a hundred dollars’ worth of it for myself earlier today. I’m on my way to mastering calculus (after I get through addition, subtraction, multiplication, division . . .)!

This is a general update.  Things have been busy…

Nathan was supposed to take his Hebrew midterm last Tuesday.  After studying like mad, he went to the church where he’d arranged to take the test….got the test…and it was in Greek!  They sent the wrong test!  Talk about “It’s all Greek to me!”  So he’s still studying for his Hebrew midterm which he’ll take next week.  He passed his Church History test and now has Intro NT and Hebrew left.  I’ve been helping him study and I think I know the Hebrew alphabet, and definitely some of the vocab.  Almost a 2-for-1 education!

Nathan started working at UPS part time a few weeks ago.  So far so good, and I really like the schedule – he’s home from 10:30 am onwards…’course, he’s leaving for work at 5 am but that seems to work for him.  I can’t imagine doing it!

We’ve been sick the last couple of weeks with a miserable head cold so not much happening on the homeschooling front.  I’ve been doing reading with the kids (good snuggling at least on sick days!) and reading some books myself about homeschooling philosophies and options.  I was in Montessori schools til Grade 4 so I looked into that first.  I like a lot of the ideas but some of the tasks seem like makework compared to what actually happens at home.  I think I’ll incorporate some ideas into our daily routines.  I already adding doing dishes to Eliana’s chores and she really loves it.  The problem is keeping Sophia and Jonathan out of the sink while Elly and I wash!  I have also been looking into Charlotte Mason methods and unit studies.  I’m hoping to read a little more on KONOS curriculum, I think that sounds like the best fit so far for our multi-level kinesthetic group!  I ordered “Get Ready for the Code” and Singapore Math’s Earlybird 1A and 1B books.  I think I need a little more structure than what I’ve been doing so far, but less makework.  We’ll see how it goes.  I’m feeling a lot more committed to homeschooling as the best choice for our children, so the break has been good.

Next weekend (Canadian thanksgiving) we’ll be heading to Edmonton, AB for Amara’s baptism.  It will be very good to see old friends again!  I’m really looking forward to it, and hopefully 3 whole days away from Hebrew will allow Nathan’s brain to congeal. 

So I’m looking at the web site for Singapore Math textbooks (I only picked up workbooks today, apparently I also need textbooks and a teacher’s guide), but there are two different versions we can choose from: there’s the US Edition and the new Standards Edition. Their FAQ says that you have to choose one (you can’t mix and match). The Standards Edition is not only most recent (published 2008 instead of 2003) but more expensive; instead of $9 for the first textbook, it’s $11/$13 list (and the teacher’s guides are even more, like $16 to $44 instead of $16 to $21).

So I guess my first question is: how strongly do you feel we need the kindergarten book? Because I’m not sure we do; the stuff we’re working on right now, I think is covered in the first grade book, but I’m afraid of leaving a whole in his knowledge.

My second question is: Which series of books do you think we should get? The pricing of the US Edition is nicer, but I’m afraid that they’ll stop carrying that one and I’ll be stuck in the middle of teaching the kids and not be able to get the next textbook and leave something out of their education.

I sure wish we were already in Alaska. At least then we’d be able to get the cost of this reimbursed.

Three indicators that our schools should emphasize depth instead of breadth:

1. Singapore has the best performing math students in the world.  “Singapore math” is starting to be adopted by California and Oregon schools because of the results it achieves.  The difference is that American curriculum “spirals” – e.g., we touch on the same topics over and over, year after year, hoping that by reviewing something repeatedly, students will catch on.

How does Singapore achieve such great results?  In 1999, Singapore reduced curriculum content by up to 30% for most subjects, focusing on conceptual understanding by eliminating content that overlaps with other grades or subjects and skills that rely on plain recall.

2. Dr. James Webb, one of the 25 most influential psychologists in gifted education , when asked about acceleration at the TAGT (Texas Association for the Gifted and Talented) Parent Conference on 2/28/09, recommended that students who need radical acceleration skip middle school entirely.  Why?  First, it’s a terribly difficult time for socio-emotional development.   Secondly, it can easily be skipped because so much of the curriculum is repetitive.

3. Today Education Week reported results of a new study, first reported in Science Education in December 2008, that concludes a focus on breadth is misguided:

A central finding is that “breadth-based learning, as commonly applied in high school classrooms, does not appear to offer students any advantage when they enroll in introductory college science courses,” the authors conclude, “although it may contribute to scores on standardized tests.”

The authors build their research on a national survey of 8,310 undergraduates enrolled in their first college science course. Students were asked how much time they spent in high school biology, chemistry, and physics classes on various subtopics.

In each subject, the researchers said students had been exposed to a topic in depth if they reported spending at least one month on it—for instance, mechanics or electromagnetism in physics or evolution in biology. They controlled for other factors, such as students’ socioeconomic background and math proficiency.

How much does depth matter?

The results show that students who had spent at least one month on one particular topic earned higher grades in college science courses than students who had not. By contrast, those who had been exposed to a relatively long list of topics, but not in depth, did not have any advantage in college chemistry or physics and were at a disadvantage in biology… Students who experience deeper coverage of physics in high school perform in college as if they had received two-thirds of a year more preparation than those who had the opposite mix of depth and breadth.

One of the complaints voiced about Advanced Placement (AP) classes is that they are “a mile wide and an inch deep”, especially as compared to International Baccalaureate, but that is starting to change:

The College Board, which has been criticized for promoting what some say is a diffuse approach on its Advanced Placement science exams, is redesigning them to emphasize depth and scientific reasoning, said Trevor Packer, a vice president of the New York City-based nonprofit organization. The first of those revisions, to the AP Biology test, will be unveiled in September.

quotes from Education Week, “‘Depth’ Matters in High School Science Studies”

So what can a teacher do?  The study authors conclude:

…teachers should use their judgment to reduce coverage in high school science courses and aim for mastery by extending at least 1 topic in depth over an extended period of time.

So far so good with the choice of getting The Big Kahuna the Singapore wb’s. He’s enjoying them! We still use Saxon, since it only takes about 15 minutes. We continue working on the calendar & jumped ahead a bunch of chapters in Saxon. I’ve given him the assessments suggested in the program, to be sure he has the material before skipping the chapters. He knows how to use the manipulatives, so he uses them during his playtimes & when I want to give him a quick assessment for the Singapore wb’s. Now that I have a math & phonics routine going well, I’ve added some “history/social studies” into his curriculum. He now is sitting alongside The Pop Tart during her History readings (books from The Complete Book of… series), then I have them look at photo’s or videos online. I will also print a coloring page that corresponds to the reading for both kids.  He gets most of the discussions, I don’t force him to memorize anything or answer the textbooks questions, like the Pop Tart does.

The Pop Tart is enjoying Saxon, especially now that the second grade review is pretty much over. The set up of the program works very well for her and I will not change a thing. I was not a fan of math throughout my school career, and to see someone looking forward to it is a major blessing for me

My new focus for the upcoming semester, is finding all things FREE!!!!!! Everything has become very expensive and I will have to be VERY thrifty when it comes to enrolling the kids into their extracurricular activities. If it were up to the children they would continue all of their classes, but with the economy being the way it is, I’m left holding the axe For me all their activities offer important skills and purposes. I haven’t a clue as to what will have to be cut out! At the very least, the economy & our cuts will serve as a very REAL LIFE math experience. UGH!!

Well, life is happening in between the homeschooling. Let me rephrase that… homeschooling is happening despite life. Some family emergencies have occured and juggling priorities has gotten a bit difficult for me. Yes! I did have a moment where I wondered “Why the hell am I homeschooling??” My answer was found in a 5 year old man when he simply told me “You are a good teacher, mom.” He said this with such pride, like I was HIS student. I just laughed, and thanked him. We then closed up the books and jumped in bed and read a story. Life is homeschooling & homeschooling is life. The Big Kahuna set me back on track and later on in life I will tell him about the day he saved homeschooling.

The kids are still attending their various enrichment programs. Unfortunately, educational trips have been put on hold due to our family obligations.

Pop Tart & The Big Kahuna are working on a display for a local shop for their NYC Arts & Culture class. All the children are very excited about the project. Of course, the usual fall & halloween theme is alive and well at home with pumpkin carving & decorations.

On an academic note: Saxon Math (for Kindergarten) is going rather slow for the Big Kahuna. I think the program is a bit behind for his level. We’ve had to skip far ahead in the program, to find some challenging work. I’ve ordered Singapore Math, to supplement and maybe eventually take over for math. Saxon Math K, may work well for a child that has never had “school” experience. But, he has been in a schooling environment since birth, so this program is not a good fit for him. He has a strong foundation of math concepts & enjoys being challenged.

Saxon for the Pop Tart is going well, she as well is a bit ahead, but I don’t want to push her too fast. She believes she is a math genius, so shall she remain. LOL.

I will be updating the blog regularly, now that the colder weather is upon us

[Photo by One Laptop Per Child.]

Once again, I am adding to my Free (Mostly) Math Resources page. Here are a handful of helpful websites for teaching math…

Master the basics

Professor Pig’s Magic Math
Cartoons, hands-on games, and mental math practice — these pdf lessons are excellent to help students understand number bonds, rather than just memorizing math facts. (Only the first two lessons are available now; more coming soon.) See also: Math Games to Download.

Word problems step by step

The Singapore Maths Teacher
These slide shows demonstrate Singapore-style math models (also known as bar diagrams) step by step, beginning with relatively easy grade 3 word problems and working through to grade 6 stumpers. Excellent for elementary teachers who need to learn how to teach this method. See also: Problem Solving Strategies.

No, it isn’t infinity

Why We Don’t Divide By Zero
Professor Homunculus of The Math Mojo Chronicles explains the mathematics of dividing by zero.

I love these logic problems

Logic-Grid Brain Teasers
I love logic grid problems, and Braingle offers 34 pages of them. Have fun! See also: Math Brain Teasers.

Challenge yourself with math and logic classics

Problem Solving Island
“The inhabitants of the island are a little different from you or me. Rather than pursuing standard enjoyments, like television, aerobics, and web surfing, they spend most of their time posing and solving puzzles…”

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Since we are discussing the pros and cons of math programs. I thought I would share some evaluations of different ones.

The following is an evaluation of Singapore Mathematics by the University of Michigan Education Department

Singapore Mathematics

Mission Statement:
To produce an understanding of the way mathematical processes work.

Singapore Mathematics came into use in the United States around the year 2000, when Singapore ’s students ranked first in the TIMSS (Trends in International Mathematics and Science Study).  The Primary Mathematics Series was first published in 1982 and was the only series used in Singapore elementary schools up until 2001.  Today, this curriculum is growing in popularity.  Several thousand books are sold annually, and sales have increased by 30% each year.

- Focus of curriculum - the focus of Singapore math is on teaching mathematical thinking, as opposed to rote problem solving (ex- after learning a skill [counting from 1-10 and then backwards], students are asked immediately to solve a problem)
**this is very different from American math programs where students are assigned a single task and asked to repeat it a number of times before asking the child to use that task to solve a problem**

Provided Materials:
- teachers guide – provides daily lessons (not very thorough though – just gives some tips on possible student difficulties and what should be done)

- textbooks – gives lots of pictures of what is trying to be communicated (gives the impression that if the teacher wasn’t there, information can still be learned because the textbook provides a lot of examples of how to solved these problems). Textbook is not very wordy – most information is learned through the illustrations. There are many practice problems in each section.

- Content order - multiplication and division are begun very early (in the beginning of 2^nd grade) so students are aware of the relationship between multiplication/addition and subtraction/division

- Materials – books are colorful (with cartoon-like illustrations and pictures showing each new concept worked out with actual objects)

- Teaching is thorough and complete -  students are walked through all the parts of a problem and then are presented with the entire problem to solve
**this teaches them to think through the different steps of a problem rather than trying to plug it into a formula (therefore introduces problem solving earlier)**

- Teaching of rote repetition strategies – Singapore does not teach this, therefore many children who need these skills before they are able to put them into use will be disadvantaged

- Review of concepts learned in previous grades - there isn’t very much time alloted to review ideas from previous grades. Teacher will need to review these concepts themselves.

- Teacher guidance – there does not seem to be very much guidance in teaching these concepts. Singapore Mathematics curriculum would be more difficult to teach because it is not as thorough.  Teachers need to be very knowledgeable on the ideas being taught beforehand so that they can clearly explain it to the students. The amount of guidance that Singapore Mathematics gives is unsuitable for first year teachers.
-    Use of extra materials – Singapore Mathematics does not provide students with any extra manipulatives

Ok, who has used Singapore? What did you think of it? Would you recommend it? Why not?

Two quick heart-warming homeschooling moments – at least they are heart-warming to the teacher!

1 -  Into our 3rd week of Singapore math and we’re moving into more difficult territory. Like adding real numbers.  For those not familiar with Singapore, the general idea is to progress the student from adding 2 apples and 2 apples, then 2 apples + 2 apples, then 2+2 (concrete to abstract).  Singapore is big on “mental math” – so shifting from concrete to abstract comes fairly quickly.  So I spent lots of time on it, helped, used extra concrete manipulatives; I tried explaining it a couple of different ways, writing it different ways, playing games.  Claire gets the concrete stuff great,  and she gets that the numbers represent the actual objects being added.  But when pressed to just do a problem, she is UNHAPPY! It’s going to tough at Auburn carting around all those teddy bear counters.

Then we had a good math day on Monday – she just seemed to “get it” more. When I tucked her into bed, I said “goodnight math genius.”  She asked why I said that, and I explained that I was so impressed that she’d somehow figured it out. Her answer: “I just realized in my brain that 0 means nothing and then it was easy.”  Now somehow, she knows her 0, 1, 2, double math facts! 

2 – We had our first piano lessons today.  Personally, I loved using new brain cells.  I am concerned, however, about my ability to memorize 8 measures of music.  That’s my homework, and I’m not 10 anymore!  Claire was playing with both hands and finding middle C on her own after 30 min.  She could hardly wait to try it out on our piano (teacher has a keyboard) and then teach Daddy.

Maybe, just maybe, the lack of sleep, the untidy house, the constant togetherness is actually all worth it!

Algebra is an abstract subject.  It’s also a proven predictor of a successful college career.  Further, it grants problem solving skills to those who master its precepts.  Good problem solvers are the people we want to be our doctors, to make our laws, to defend our country and to discover new things to maintain our well being.

On July 9, 2008, California State Board of Education decided that all California’s eighth-graders must take Algebra I by 2011. This is a monumental change in graduation requirements. For it affirmed that some California citizens are not receiving a “good” education and  … click the link to continue reading

INTRODUCTION

Why this proposal?

The United States is facing a critical shortage of graduates who major in mathematics, science, engineering, and technology. As children progress through the educational system, far too many shy away from science and mathematics. Other nations, particularly in Asia, have far more success in producing successful students with strong backgrounds in science and mathematics.

This proposal presents The Mathematics Room Project, a three-week enrichment program that stresses mastering mathematical skills that underpin success in higher mathematics and ultimately, in high tech and mathematics centered jobs. But to no lesser degree, it addresses social responsibilities and the City’s rich cultural heritage.

THE PROJECT DESCRIPTION

What is The Mathematics Room Project?

The Mathematics Room Project (TMRP) is a three-week summer enrichment program that stresses

· fluency with whole numbers,

· mastery of fractions,

· relationships to others and to the XXXXXX neighborhood,

· the City’s rich musical heritage.

Why are these emphases over others?

In a recent Washington Post article, a presidential panel of mathematicians, education experts and psychologists reported that, students need a deeper understanding of fluency with whole numbers and fractions, if they are to be success in completing Algebra II. Further, the panel stressed, that many students are befuddled by fractions. A fundamental focus of TMRP is to develop fluency of whole numbers and mastery of fractions, thereby responding, to correct the problems identified by the presidential panel.

“Market emphasis” is a fact of life. We approach education, the arts and even our spiritual institutions with a business plan. This approach has its advantages but it reduces all things to a bottom line mentality. Often in human endeavors, the process leading to the bottom line is as important as the bottom line, for example, in building relationships, in working in team, in caring for others, in caring for things over which we have charge, in developing plans for the future, etc. In all a spirit is built that is necessary to relate to others as well as our community. TMRP will teach participants to solve mathematical problems within a group structure, thus highlighting the importance of relationships and highlighting our responsibilities in human affairs.

What is the teaching philosophy of TMRP?

The teaching philosophy of TMRP is rooted in the famed Singapore Method of teaching mathematics. Like the Singapore approach, TMRP focuses on teaching mathematical thinking to produce mathematical thinkers. It does this, by walking the learner through all the components of a problem and then presenting her with the whole problem to solve; all done, with the learner’s preparation in mind while poised to elevate her to the next level.

Is this method better than the traditional?

The Mathematics Room Project believes and data shows this method is optimal for learning “how to” frame a problem in a concrete stage, and then, “ how to” usher the learner into the abstract stage, thereby enabling him to learn mathematics in a meaningful way. It encourages an active thinking process, effective problem solving skills and clear communication of mathematical ideas. All of the attributes are necessary to develop a foundation that students need to successfully study more advanced mathematics.

TMRP supports this method with outstanding, experienced, caring teachers and with excellent textbooks and workbooks that stresses the

Concrete-> Pictorial-> Abstract approach.

Is the Singapore Method proven?

Since 1995, Trends in International Mathematics and Science Study tests (TIMSS) have been administered to students of thirty-six developed nations. These nations used the TIMSS test to measure the success of its student preparations in mathematics and the sciences. Singapore, a small nation of 45 million, has never scored less than first position in the TIMSS test, while the United States has never placed higher than 16th position.

Follow-up results of the TIMSS attest that students taught by the Singapore method have a higher success rate in completing algebra and higher mathematics.

Who will lead the Mathematics Room Project?

The Director and Coordinator of The Mathematics Room Project, XXXXX, have acclaimed success in directing mathematics enrichment programs.

The support staff includes XXXXXX – a retired grade teacher of 33 years experience, having taught 25 years at the XXXXXX, one of California’s Distinguished Schools.

Where will the program be held?

The XXXXXX Church has committed to host The Mathematics Room Project. This Christian institution has ministered to the XXXXXXXX Community for XXX years.

How will the participants be recruited?

Thirty students will be recruited by invitations sent to:

· Area churches

· Area recreations facilities

· Advertisements in a weekly newspaper

· The CITY NEWSPAPER

PROJECT NEEDS

The Mathematics Room Project needs:

• meeting space to accommodate 30 children
• eight tables: seating four per table
• Singapore Method mathematics workbooks: grades 4 through 6
• Singapore Method teacher manuals: grades 4 through 6
• cost of admittance for 30 participants to:
• one of the City’s Philharmonic performance and
  
• one of the City’s Choral performance

BENEFICARIES OF THE PROJECT

The Students: The immediate beneficiaries of the project are its participants. Mastery of problem solving skills enhances student self-esteem and cementing student–school bonds. Studies have demonstrated that girls having strong mathematics skills tend to resist predatory male advances and amoral peer pressure. In like manner, boys having strong mathematics skills and strong parental guidance tend to successfully shun gang advances. Thus, TMRP will be another effort to engage students in academic life and to build a student more resistant to social distractions.

Over the years, students trained under the Singapore Method boast high rates of success in higher mathematics. Algebra II, the gateway to higher mathematics, is a proven predictor of success in college. The recent Washington Post article reported that The National Mathematics Advisory Panel – the Presidential Panel composed of mathematicians, educators and psychologists – declared that success in Algebra II is predicated upon mastering fractions and fluency in whole number operations; primary focus topics of The Mathematics Room Project. Thus, TMRP is another local effort to increase the number of students who earn successful completion in Algebra II, and thus earn success in college.

The Society is a long-range beneficiary of TMRP. As problem solving is a necessary prelude to develop a thoughtful citizenry, and as thoughtful citizens build strong and safe communities, The Mathematics Room Project will take part in providing the citizens for better Sacramento communities.

The Country: The science and medical communities of the United States have experienced a severe shortage of prepared graduates in science and mathematics, due in large part to the poor performances in the sciences and mathematics. This is visibly reflected in the medical, computer science and engineering communities. The Mathematics Room Project is a Sacramento effort to increase the number of U.S. students who are successful in Algebra II, and hence increasing the number of candidates in the medical, mathematics and science pipeline.

FITS TO FUNDING AGENT MISSION.

As The Mathematics Room Project is a program with an academic,

cultural and social impact that targets bettering students in a particular Sacramento neighborhood, it is consonant with the vision of FUNDING AGENT. to develop awareness and care for neighborhoods.

It does this:

· By teaching its participants to master mathematics identified as a building block to success in college.

· Through its planned ice cream social in the local park.

· With scheduled attendances to a performance of the Sacramento Philharmonic and the Sacramento Chorale Society.

Familiarity breeds acceptance. So during its instruction time, TMRP will play as background music those selections designated by the City Philharmonic and City Chorale Society to be performed during the scheduled attendance of TMRP. This predisposition to the selected music provides for greater student enjoyment as they witness familiar music dramatized in a cultural setting.

Our activities will be held in the XXXXXXXX Church facilities. This structure is a recognized historic landmark to the neighborhood. Gaining life changing academic experiences, while experiencing positive events and joyful times in a place of historic value in one’s neighborhood promotes positive and manifold connections to the neighborhood, thus this program fits the vision of FUNDING AGENT.

AFTER TMRP, WHAT THEN?

After the program, our participants will have:

· learned “how to” learn mathematics.

· mastered mathematics crucial to learn higher mathematics.

· a better appreciation of mathematics, and stronger self-esteem.

· experienced great music performed by local musicians.

· recognized the historic value and importance of the building that housed project.

In summary, our participants will have exposures that communities desire of its youth.

REQUESTED FUNDS

The Mathematics Room Project requests $ xxxxxx to cover operational expenses, student textbooks, student workbooks, teacher instruction manuals, 12 nutritious snack for 30 students, transportation cost to the music events, t-shirts that stress TMRP common focus and fosters a sense of belonging; and certificates of completion.

FIELD TRIPS

The Project Director will coordinate all field trips during the project year.

PROPERTIES OF THE MATHEMATICS ROOM PROJECT

All properties of The Mathematics Room Project, except student workbooks, revert to FUNDING AGENT. upon cessation of the project.

TOTAL PROJECT BUDGET $XXXXXX

PROJECT START/COMPLETION DATE

ORIENTATION: June 7, 2008

Accept applications, distribute rules of operation, administer pretests to measure student preparedness and measure for t-shirts.

PROJECT TIME FRAME

MESUREMENTS AND EVALUATIONS

Upon entering the program, all students will be administered a pretest to determine whole number fluency, mastery of fractions, and problem solving skills.

TMRP will permit its learners to advance to a new topic only upon 85% or better mastery of his latest workbook assignment.

During the last three days of the enrichment program, the Director will assign to each student a problem he will present at a closing ceremony. Parents and the public will be encouraged to attend the closing ceremony. The presentations will be graded on clarity, correctness and problem insights.

On the final day of TMRP, student will take a post-test. Its results will be compared with the pretest results to measure the impact of the enrichment intervention.

At the completion of the first year of operation, the Director will submit an annual report to the Board of FUNDING AGENT. The report will summarize student performances and the project’s impact on student educational and cultural growth.

CONTACT INFORMATION

other topics: click a “category” or use search box

Intensive Five-day Summer Institutes in teaching “Singapore Math” are scheduled in Orlando (June 23-27), San Francisco (July 14-18), and Boston (July 28-August 1). 

These three will focus on the fundamentals, and are geared to teachers of grades 1-6 who have little or no experience in Singapore Math, “especially those who intend to supplement a non-Singapore Math program or adopt a Singapore Math program.”

A fourth Institute titled “Advanced” will be held in Boston from August 4-8.  The advanced seminar will be directed at grade 1-6 educators with in-depth training experience in teaching Singapore Math who are either supplementing their current program or adopting the Singapore Math curriculum.

Call toll free 1-800-462-1478 or www.SDE.com/revolutionary.

Conference registration is $1395.  It includes a tool kit containing $500 worth of customizable materials: Singapore Math for US Handout Book; an 8-step Model Drawing Book;  the Teaching Elementary Mathematics book; a voucher to redeem one “sample set” from the Primary Mathematics book series (includes one copy of both A & B versions of a Teacher’s guide, student textbook and student workbook — your choice of grade level).

Trainees receive 26.5 total contact hours plus 2 additional hours upon completion of the on-demand electronic follow-up training.  For futher information about graduate credit through Antioch University Seattle and Chapman University College, visit www.SDE.com/revolutionary and follow the “credit” link.

WHAT IS “SINGAPORE MATH?” 

When the US Department of Education commissioned a study in 2005 to find out why Singapore, a country with a population half the size of New York City, ALWAYS scores No. 1 in a widely accepted comparison of global math skills, it concluded, “Singapore’s textbooks build deep understanding of mathematical concepts through multi-step problems and concrete illustrations that demonstrate how abstract mathematical concepts are used to solve problems from different perspectives.”

By contrast, the study said, “traditional US textbooks rarely get beyond definitions and formulas, developing only students’ mechanical ability to apply mathematical concepts.”

Many eminent mathematicians agree.  In fact, according to an LA Times article on March 9 2008, it is hard to find a mathematician who likes the standard American texts, or dislikes Singapore’s.

An article by Barry Garelick from Stanford’s Hoover Institution shows a US textbook side by side with the Singapore Math book.  “Singapore Math’s textbook is thin, and contains only mathematics — no games.  Students are given brief explanations, then confronted with problems which become more complex as the unit progresses.”

By contrast, the US text, “typical of many math textbooks in the US is thick, multicolored, and full of games, puzzles and activities to help teachers pass the time” but which rarely challenge students.

Singapore’s text contains no graphics other than occasional cartoons pertaining to the lesson at hand, no spreadsheet problems, and no problems asking students to use a calculator to find the “mean number of dogs in a US household”.

With SM, students are required to show their mathematical work, not explain in essays how they did the problems or how they felt about them.

According to the Hopkins article, “While a single lesson in a US textbook might span two pages and take one class period to go through, a lesson in a Singapore textbook might use five to ten pages and take several days to complete.  The Singapore texts contain no narrative explanation of how a procedure or concept works; instead there are problems and questions accompanied by pictures that provide hints about what is going on.  According to the AIR (American Institutes for Research, a government group) report, the Singapore program ‘provides rich problem sets that give students many and varied opportunities to apply the concepts they have learned.’ “

Another key difference: the number of topics covered for a single grade: the AIR study frequently criticizes American math texts for being an inch deep and a mile wide, covering a great range of topics with little time spent on developing the material, including mastery of math facts.

The Singapore texts also present material in a logical sequence throughout the grades, and expect mastery of the material before the move to the next level.  “In contrast,” writes Garelick in the Hoover article, “mainstream American math texts and curricula frequently rely on a ’spiral’ approach, in which topics are revisited and reviewed.  The expectation is that not all students achieve mastery the first time around.”

An Ohio schoolteacher summed up the “spiral” approach on an Internet math forum by writing that students “can’t remember how to do it when they do return — or if they do remember it, it’s now being taught in a different way.”

But the most important feature of Singapore’s texts, according to Garelick, is an ingenious problem-solving strategy built into the curriculum.  Word problems are for most students the most difficult part of any mathematics course.

Singapore’s texts help solve the problem.  A key step in problem-solving is model-drawing.

Typically, in US texts, students are taught to use a method called “Guess and Check” — trying combinations of numbers until the right numbers are found that satify the conditions of the problem (a method that many mathematicians consider inefficient). 

The Singapore bar-modelling technique not only provides a powerful method for solving problems, but also serves as a link to algebra.  Symbolic representation of problems, the mainstay of algebra, emerges as a logical extension of the bar-modeling technique.

On Your Mark — Get Set — Think!

The LA Times article describes the classroom of Arpie Liparian, a first grade teacher.  She stands in front of her class with a stopwatch.  The only sound is of pencils scratching paper, as the students race through the daily “sprint,” a 60-second drill that is a key part of the Singpore system.  The idea, once commonplace in math classrooms, is to practice problems until they become second nature.

Critics call this “drill and kill,” but one math coach calls it “drill and thrill.”  Liparian’s students don’t all finish all the problems in 60 seconds; only one girl gets all the answers right.  But they are all bubbling with excitement, and Liparian praises every effort.  “Give yourselve a hand, boys and girls,” she says when all the drills have been corrected.  “You did a wonderful job.”

That math coach, Robin Ramos, says that what isn’t obvious to the casual observer is that this drill is carefully thought out to reinforce patterns of mathematical thinking that carry through the curriculum.

“These are ‘procedures with connections,’ “  Ramos says, arranged to convey sometimes subtle points.  This thoughfulness — some say brilliance — is the true hallmark of the Singapore books, according to advocates.

Yoram Sagher, a math professor at the University of Illinois, says that after 10 years of studying the Singapore curriculum, he still has “very pleasant surprises and realizations” while reading the books.  He is constantly amazed, he says, by the “gentle, clever ways that the mathematics is brought to the intuition of the students.”

It is true that the Singapore texts are not as teacher friendly as most American texts.  “They don’t come with teahers’ editions, or two-page foldouts with comments, or step-by-step instructions about how to give the lessons.  And in our math-phobic society, many teachers lack a strong math foundation to begin with.

Key to implementing the Singapore Math curriculum –  and achieving the spectacular results seen in a few select schools — will be intensive training of teachers.

sources: Barry Garelick’s Hoover Institution journal article  on “Miracle Math”from fall 2006; also Mitchell Landsberg’s article in the LA times on 3/9/08 (www.latimes.com ); and SDE flyers promoting the Singapore Math Intensive Institutes.  

[Photo by armigeress.]

In 4th grade, math problems take a large step up on the difficulty scale. Students are more mature and can read and follow more complex stories. Multi-step word problems become the new norm, and proportional relationships (like “three times as many”) show up frequently. As the year progresses, fractions grow to be a dominant theme.

As a math teacher, one of my top goals is that my students learn to solve word problems. Arithmetic is (relatively) easy, but many children struggle in applying it to “real world” situations.

In previous posts, I introduced the problem-solving tools of word algebra and bar diagrams, either of which can help students organize the information in a word problem and translate it into a mathematical calculation. The earlier posts in this series are:

• Pre-algebra Problem Solving: Preschool and Early Elementary
• Pre-Algebra Problem Solving: The Tools
• Pre-Algebra Problem Solving: 2nd Grade
• Pre-Algebra Problem Solving: 3rd Grade

In this installment, I will continue to demonstrate the problem-solving tool of bar diagrams through a series of ten 4th grade problems based on the Singapore Primary Math series, level 4A. For your reading pleasure, I have translated the problems into the universe of a family-favorite story by C. S. Lewis, The Lion, the Witch and the Wardrobe.

Tea with Mr. Tumnus

Mr. Tumnus told Lucy about a midnight party of fauns and Dryads in the forest of Narnia. 35 fauns came to the party. There were 3 times as many Dryads as fauns. How many creatures were at the party?

If the whole group split equally into 5 large circles for dancing, how many were in each circle?

For students who are not used to bar diagrams, the natural tendency is to multiply creatures at the party. A quickly sketched diagram will show their error:

There are four units in all, and .
So 140 creatures were at the party.

The second part of the problem is straight-forward. A diagram may not be needed, but here it is anyway:

.
There were 28 dancers in each circle.

Back on This Side of the Door

The Professor had 486 books at his house, some in the library room and some in his study. There were 50 books more in the library than in the Professor’s study. How many books were in the study?

In my experience, students who have not learned to think in bar diagrams usually divide the books in half and then subtract 50 from that number to get their answer. The diagram shows the correct path to a solution:

First, we need to subtract the 50 extra books:
.
Then we split the rest of the books evenly between the two rooms:
.
There were 218 books in the study.

In the Witch’s House

The White Witch had 300 servants at her house, which was really a small castle. There were 10 more wolves than Red Dwarfs. The number of Red Dwarfs was twice the number of Black Dwarfs. How many Black Dwarfs worked at the Witch’s house?

For most students, a problem like this is easiest to work backwards, because “twice the number” is a familiar diagram:

Next we add the wolves, which match the Red Dwarfs plus 10 extra:

Now we can see that we have five of our unknown unit (which is the number of Black Dwarfs), plus 10 more, to make a total of 300:

.
.
.
There were 58 Black Dwarfs.

A Day with the Beavers

Mrs. Beaver baked a “great and gloriously sticky marmalade roll” for dessert. She cut 1/6 of the roll for her and Mr. Beaver to share, and then she sliced up 4/6 of the roll for the children. What fraction of the marmalade roll was left?

Fraction problems start very simple, but don’t let the easy numbers make you lazy. Draw those bars! They will teach your students an intuitive understanding of the connections between math concepts: fractions are intimately related to multiplication and division.

The Beavers’ 1/6 of the roll, plus the children’s 4/6, make a total of 5/6 of the roll that gets eaten. So 1/6 of the marmalade roll remains.

What Happened After Dinner

Mrs. Beaver had a pitcher of milk. She poured 1/2 of it into glasses for the children to drink with dinner. Then she poured 1/8 of the pitcher into their cups of after-dinner tea. How much of the pitcher of milk did Mrs. Beaver use?

The fractions in this problem are a little more difficult than the last one, having different (but related) denominators. The first part of the story is easy to draw:

But how can we show the extra 1/8 of the pitcher that gets poured into the tea? We must go back and sub-divide the bar so that each half of it becomes 4/8. As we create the equivalent fraction, can you see why the numerator goes up in direct proportion with the denominator? As the size of the pieces gets smaller, the number of pieces increases by the same factor.

Now we can mark the extra 1/8 used:

Mrs. Beaver used 5/8 of the pitcher of milk.

The Spell Begins to Break

The Witch’s sledge got stuck in the mud and slush 24 times before she gave up and decided to walk. 2/3 of those times, the Witch made Edmund get out and help push. How many times did Edmund have to push the sledge?

Bar diagrams are almost like magic for fraction (and later, percent) problems, because the “whole” bar can represent ANY amount. The process of cutting the whole amount into fractional pieces is clearly related to division, and the picture of the bar communicates what is happening more clearly than an abstract equation like ever could:

The whole 24 is split into 3 units:
.
We need 2 units:
.
Edmund pushed the sledge 16 times.

Peter’s First Battle I

2/5 of the creatures waiting with Aslan at his pavilion beside the Stone Table were Dryads and Naiads. There were 20 Dryads and Naiads in all. How many creatures were waiting with Aslan at his pavilion?

With problems like this, we begin to reap the full benefit of our work on learning bar diagrams. Instead of struggling to understand the algebraic reasoning required to solve something like , our students can draw:

.
.
There were 50 creatures with Aslan at his pavilion.

Peter’s First Battle II

Aslan sent 20 of the swiftest creatures to follow the wolf and rescue Edmund. 2/5 of these creatures were eagles, griffins, and other flying fighters. The rest were centaurs, leopards, and other fast-running beasts. How many of the creatures that Aslan sent could not fly?

This problem uses the same numbers and the same basic diagram as the last problem, but students must read and understand the story to know how the numbers relate to the diagram. Also, students need to be aware of how little, easy-to-miss words like “not” can change their answer.

.
.
So 12 of the creatures could not fly.

The Triumph of the Witch

The White Witch’s evil minions used 4 2/5 m of rope to bind Aslan’s legs together. They used 3/10 m less of rope to tie him tightly to the Stone Table. How many meters of rope did the wicked creatures use in all?

Measurements lead naturally to mixed-number problems. We again have different (but related) denominators — this time, 5ths and 10ths. Start by drawing one bar that is 5 units long. Each unit will represent one meter, and the last unit will be divided into 5ths to show the mixed number:

By making an equivalent fraction, splitting the 5ths into 10ths, we can easily see how long the second rope must be:

.
.
Always remember to put your answer into simplest form! They used 8 1/2 m of rope.

At Cair Paravel by the Sea

4/5 of the sea people who sang and played music for the coronation party were mermen. If there were 8 mermaids, how many sea people came to the party?

The end-of-workbook review takes the students through several multi-step fraction problems. In algebra, this calculation would look like . With a bar diagram, it’s easy:

There were sea people at the coronation party.

Conclusion

In the beginning, bar diagrams often take up more space and require more pencil-to-the-paper work from the student than other approaches to solving simple problems. Many simple word problems can be solved mentally, which makes drawing a bar seem like tedious busy work. But as word problems become more complex, the bar diagrams offer significant help for students who struggle with the question, “What do I do?” Diagrams make visible the abstract relationships between numbers, enabling the student to decide which arithmetical operation makes sense in the context of the problem.

One clear advantage of bar diagrams, in my opinion, is how well they lead students to understand fractions, a topic which will continue to haunt our students in problems of ever-increasing difficulty through 5th and 6th grade. Multiplication and division problems will also grow more challenging as students progress, until they become the dreaded ratios and proportions of algebra. These topics are notoriously difficult for students [JSTOR access required, or try this article instead], but I believe the bar diagrams provide a much better foundation for understanding than any other method I have seen.

To get more practice creating bar diagrams, your students may enjoy these online tutorials:

• Thinking Blocks Addition and Subtraction Word Problems
• Thinking Blocks Multiplication and Division Word Problems

And for some fun practice with equivalent fractions:

• Fraction Eaters game

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More great math games!

Have more fun on Let’s Play Math! blog:

• Math Game: What Number Am I?
• More Backwards Math
• Writing to Learn Math
• Writing to Learn Math II
• Buddy Math