My students come to me for math tutoring because they continue to flounder with the “new math” curriculum. For a complete description of what is being taught and how it feels for students, see **Part I** of this series. **Part I – The New Math: Why We Have It**

**If expert mathematicians have redesigned the curriculum, why aren’t the results better?**

I believe it’s because the experts aren’t taking into account the developmental stages of most students, and because they really aren’t aware of the problems most classroom teachers are faced with.

The new math teaching methods are mainly designed to create:

1.) *the ability to work in cross-disciplinary teams;*

2.) * understanding* (now viewed as even more important than being able to compute); and

3.) * innovative and divergent math thinkers*–the three characteristics increasingly required of white-collar jobs in industry today.

Yet the new math curriculum is failing to achieve these goals. Let’s take a look at WHY, by seeing how these things actually play out in most classrooms.

**How These Three Goals Actually Work Out in Classrooms**:

1.** Creating an ability to work in cross-disciplinary teams**. The idea is clearly that “putting students in groups to solve problems” will create this ability. However, there are TWO IMPORTANT REASONS why this is not happening in most classrooms. The first reason is BULLYING, and the second reason is STUDENT ATTITUDE and LACK OF MATURITY.

Middle-school, when most students are first put into math-solution groups, is the age of the MOST EXTREME BULLYING (although bullying starts in Kindergarten). Students are usually left to sort themselves into groups, and usually, in-crowd friends choose each other, while the remaining students are randomly forced into groups with students who regularly bully them. This same situation continues in many high-school classes, and is sometimes worst of all in the smallest schools where there is only one math class per grade.

It takes an extremely effective teacher who can give groups precise tasks, direction, and rewards based on individual effort to get a group to make effective progress. Generally what happens is one of several things. The students don’t understand what they are doing at all and therefore have no idea (or motivation) even to try. They end up wasting time and talking about non-math-related matters. Or, at best, one or two students do understand and do the work, while the others loaf and do nothing, but coast on the group grade (if there is one), having not done the work, and not understanding the work that was done by the others. Or, those who are friends in the group use the hour as a social time, while the unwanted group members spend the time staring at their papers, feeling excluded, and just wasting the whole hour.

Requirements for effective group work are: 1.) being in a group with others you like or respect, and others who like or respect you; 2.) Having enough background in the subject, that when given A SPECIFIC TASK, all the individuals in the group can work on it; 3.) Being able to effectively subdivide tasks; and 4.) Having individual accountability for one’s contributions to the group. Most teachers do not have either sufficient time or experience to be effective in all these ways and rely on immature students who are not willing/able to these things themselves (as an adult work group would be able to do).

2. **Creating understanding of WHY methods work, rather than merely learning computational solutions**. This is an admirable goal, but it is not being correctly implemented at the proper ages, in the proper stages, or in the proper ways.

Mental maturity, and ability to deal with abstract concepts arrives at different times for different students. Abstract thinking arrives for a very few students in the lower elementary grades, for a few more students in the upper elementary grades, for about half of students by middle school, and for at best two-thirds of students by high school and early adulthood. For some people, it never arrives at all. Having taught a great variety of math topics over the years, some students grasp one topic at a young age, but don’t grasp another until many years later, if at all. Since every student has a unique profile of what they grasp or don’t grasp, this is the origin of the “spiral curriculum,” where each year, many topics are introduced, and each year, the math texts cut slightly deeper into each topic (assuming the school is still using math texts).

Let us take telling time as an example. A few students are able to grasp telling time well in kindergarten, while others, no matter HOW much time is spent in the classroom in grades two and three, just cannot grasp it until fifth grade. Then suddenly, something “clicks.” Their brain has arrived at the right level of mental maturity.

Unfortunately, today’s curriculum introduces so many topics that few are actually mastered. Thus, many students move up through the grades NEITHER understanding, NOR being proficient in calculating. Most students need and WANT to become proficient at calculating and getting the right answer in the elementary grades. This builds their confidence. They also want to know in what situations they might use those skills (which gives learners motivation, and is often an area neglected by teachers). Those who do not become proficient at calculating lose confidence in themselves and are certainly even LESS likely to be open to any discussions of “understanding.”

A current controversial topic in the math field is whether students need a certain amount of proficiency before they can understand “why” things work. After two decades of experience teaching math at the elementary and middle-school levels, I come down hard on the side that it IS necessary. Young elementary students can appreciate that a correct answer can be found through several different methods, but it is a waste of precious class time AT THAT AGE to spend a lot of time on WHY (an abstract concept which despite the weeks spent on it does not actually increase their understanding) instead of on developing proficiency and thereby building students’ confidence and excitement about learning more.

It was not the intent of the math experts, I am sure, in revising math curriculum, to have students wind up being neither able to understand, NOR be able to calculate! Their intent was to WIDEN the curriculum to INCLUDE more understanding. But with only four-to-five hours a week (at best) of classroom time to teach math per week, at least half of the available time is being taken up with “understanding” (which is not being understood by the majority of students), and not enough time for most students to become proficient at calculating. Those who do become proficient are generally having additional support from parents and tutors. Furthermore, homework has been greatly reduced from a decade ago (approximately cut in half) which means that more students than ever before are not mastering basic procedures. When students get into middle school and one-third of them still cannot determine the answer to 3 x 8 without consulting their calculators, it is highly unlikely they will gain any “higher understanding.”

3. C**reating innovative and divergent math thinkers**. Criticisms of the past were that students were memorizing times tables and learning to calculate, but not understanding what those calculations meant; students were unable to take even a simple story problem and know which calculations to perform.

After two decades in the classroom, I can easily see this problem did not stem from memorizing or calculating. This problem stemmed from teachers throughout school not teaching children how to TRANSLATE between English words, and math language. In most cases, elementary teachers are not math majors. In fact, most became elementary teachers because they are math-phobic! They teach the calculations, and generally skip all the story problems (as did I when I first began to teach math). Yes, it is partly a time problem, but the REAL problem is that most teachers are afraid they will not be able to explain to students how to do story problems, because they never learned themselves! Speaking as someone who did not learn this skill myself until I was an adult, I see that this is the number one area that students need the MOST help with. I find myself wondering if students in India, China, and Japan are getting this sort of help from a young age, while students in the West are not?

Rather than wasting precious elementary time on esoteric math subjects, and making “arrays” for WEEKS in order to “understand” multiplication, students would be much better served learning to calculate, and having DAILY GUIDED PRACTICE on particular types of story problems, both in order to recognize types of problems, and to be able to readily understand how to translate the English language into MATH language.

What the math “experts” who design curriculum are not realizing is that showing students all the different possible ways to solve every type of math problem does NOT create the “divergent” innovative thinkers they are looking for.

As for math majors, sometimes (not always), those who were brilliant in math are unable to explain it clearly to those who are having trouble, because the teachers never experienced those same troubles themselves. Sometimes (not always) teachers who were not good math students are able to master math, and are far better at figuring out where and why students are “stuck.” Lucky children with difficulties have those teachers! The very first requirement for becoming a divergent thinker is self-confidence in one’s own abilities. This comes from being sure that one knows at least ONE way to get the right answer every time, even if one knows that other ways do exist. The main thing is to MASTER at least one method.

Beyond competence, creating divergent thinkers is more of a personality-trait question. This question has more to do with motivation and stimulating interest, and comes from the sort of child who always asks, “Why?” Most children don’t ask why, and most don’t care about why. To create more innovative, divergent thinkers, every teacher in every classroom, in every subject, needs to challenge ideas and get students excited about learning. And yes, teachers need to be “entertaining,” too! Innovative thinkers aren’t usually innovative in just one area (such as math). Most innovative thinkers draw their ideas from multiple sources and synthesis of ideas from multiple disciplines. Students need help becoming competent, and beyond that, to be inspired enough to pursue their own interests in a self-directed way. Curriculum which forces students to calculate by many different methods fatigues many students and actually de-motivates them from further self-directed learning.

It is difficult for a new or average teacher to overcome these difficulties. Hopefully with time and experience, Western society will adjust to the new math curriculum, but I am afraid it will be later, rather than sooner.

**–Lynne Diligent**

**The NEW Math: Part I – WHY We Have It**

Tags: David Krumholtz, effects of bullying on math achievement, Einstein, home schooling in math, how to improve math teaching, If expert mathematicians have redesigned the curriculum, Keith Devlin, math teaching in international schools, new math, new math teaching methods, problem with translating English-language words into math language, the best math teachers, the three goals of the new math teaching, the wrong kind of math homework, too little math homework, too much math homework, wasting time studying arrays, what makes a good math teacher, why aren't the results better, why students need computational competence before they need understanding, why the new math is unsuccessful in creating divergent innovative thinkers

September 5, 2013 at 8:29 pm |

This is very insightful. I’m sending to a friend who is member of the local school, hoping that she will share it with other school board member, both hers and in the area. Thank you. Priscilla in Indiana

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September 5, 2013 at 8:33 pm |

Thanks for reading, Prisicilla!

September 5, 2013 at 11:22 pm |

I’m currently taking the class How To Teach Math. http://online.stanford.edu/course/how-to-learn-math. I’m a homeschooling parent. I am finding the ideas so far similar to what you are writing about and (I’m only on section 4) fascinating. But I hear your point too. I think there is relevance in both and in most cases balance is needed. The beginning of How To Teach Math is about how math has become so stripped down to operations only, everyone hates it. They are saying math is an art and there are better ways to teach it so that everyone enjoys and benefits and excels. And yes they talk about how the workforce needs problem solvers. I think back to my school days. We did less math and at a slower pace. I didn’t love math but I would say that I love to problem-solve in general. And it was the problem-solving that improved things and resulted in promotions. I do think there are other ways to teach creative problem-solving skills other than through math. Lots to think about here. The final issue, I believe, is that there is too much kids need to learn within a school year to let it sink in and remember and understand it…maybe even enjoy it.

September 6, 2013 at 12:10 am |

I appreciate your reading and thank you for taking the time to leave such a nice comment.

September 5, 2013 at 11:24 pm |

Oh and to your point about teaching students how to solve something multiple ways…we gave up on that in our homeschooling! Too many different ways and my kids just gave up. They just wanted to know one way that worked for them.

September 6, 2013 at 12:11 am |

Thanks for sharing your own kids’ experience with this in homeschooling.

September 6, 2013 at 1:08 am |

We struggled with this concept of spiral teaching at my son’s elementary school. He would bring home 10 math problems a night and the 10 problems would cover 3 or four different concepts. He never “mastered” any concepts and with the constant use of grouping students to solve math problems, he rode along on the results of the group instead of being able to do the work on his own. The second issue for us was in 4th grade, the district implemented “Everyday Math” for all elementary grades. Now he was told that he was doing it “wrong” and here is the “correct” way. He was all mixed up.. Also, the principal told us that “drill” is an unacceptable teaching concept. Yet when the Pennsylvania School Assessment testing comes around, he brings home 30 plus math problems that are samples of what is on the assessment tests. The final straw that broke the camel’s back is his 5th grade teacher couldn’t control the class; less than 50% of the class completed their homework on time. So, we ended up taking our son out of public school and he now in a private Christian school setting.

September 6, 2013 at 2:00 am |

Thank you for sharing your experience. All I can say is WOW! Clearly you did the right thing for your son.

September 6, 2013 at 2:16 am |

Ugh, I hated Everyday Math…and I think a lot of the teachers did too. Now that we’re homeschooling, we’re doing mastery programs. MUCH better!

September 7, 2013 at 3:26 pm |

I am a retired Secondary Math teacher and spent most of my career at the junior high teaching Algebra and Pre-Algebra. I witnessed the decline of math ability in my school with the advent of a new curriculum that was introduced partly and then fully in elementary school. This was done because of the National Science Foundation Grant Money available and this allowed free texts and materials, but no real long term support. Our upper admin insisted and forced all grades to incorporate these same ideas and it was a dismal failure. The state test changed as scores fell . Through the same publisher as the new math material, I took the new revamped test at a development conference and witnessed that the publishers did not really want the teachers’ opinions. I wondered by I had wasted three days with these people just to see them make every cut for all levels of questions for determining student mastery. But, I did see the test was easier and our math scores rebounded. We slipped in the rankings of math performance, but not enough for parents to band together and stop it. I felt powerless and deflated daily trying my best to form the best groups and rectify the student’s confusion. I became labeled as a trouble maker, but one that was not afraid to speak out at any meeting so teachers use me as their sounding person. The points you made here are right on and I do not see a way to help America with local control by school boards and states. Our school board does exactly as the Super wants and parents on the whole are oblivious of what is going on.

September 8, 2013 at 7:17 pm |

Debbie, I’ve been living overseas and taught for many years in an international school. I wasn’t even aware that these “math wars” were going on until about 6-7 years ago when our school changed text books, and I began blogging about math education. I think overseas schools sometimes employ teachers who have been out of the home country for 20 years, and so the changes such as have happened in math education or handwriting (lack of teaching it) didn’t hit our school and other international schools until perhaps a decade later than they hit the U.S. schools. Now all the older teachers are out and replaced by younger teachers in their 20s who all seem to be using the new methods. Looks like the fight between teachers who keep trying to boost accomplishment with older methods is now over; the methods and teachers are replaced, and we just look on sadly at the continued decline…..

September 8, 2013 at 9:29 pm |

(K.L., from LinkedIn Discussion): No. 1 is a skill needed for expert mathematicians because they get frustrated with others than don’t comprehend – but this type of instruction really slows down everyone.

However, when it comes to No. 2 , students advance further in math if they do understand why methods work. When students struggle in algebra, geometry and higher levels, etc., I have them work problems using very simple numbers so that they do not have to worry about incorrect computation. I also do not let them use a calculator. I spent a long time demonstrating how matrices work to one student. After doing so, it was much easier for her to memorized the procedure for adding/multiplying matrices and knowing when to use them.

September 11, 2013 at 8:09 pm |

Lynne,

What’s your recommendation for teaching story problems–translating the English into math? Or more realistically, translating the problem situation into math? Since English is so ambiguous, wouldn’t it be better to state the “quantitative reasoning” problem in mathematical terms from the start, rather than convert it to a “story problem” which must then be retranslated back into math? I suspect most “story problems” are not real “problems” at all but are made-up examples to elicit a particular mathematical solution using the skills covered in the day’s lesson.

I would love to see an anthology of genuine “everyday” problems that can be solved with elementary arithmetic: managing time and money and space, allocating limited resources, constructing objects or meals, planning events, evaluating completion or quality or success, computing sports scores, etc.

Then, if we can teach the “formula-tion” part, why not let the students use calculators for the “arithmetic” part?

September 15, 2013 at 12:09 am |

Joe, I’m not against using calculators once students are proficient at calculating. Many students (and many adults, too) aren’t able realize that they put the wrong numbers into the calculator, when their answers don’t make sense. They don’t even recognize WHEN their answers don’t make sense. As far as teaching the English part, that should be done extensively in grades two and three (speaking as someone who taught third grade for ten years).