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Sunday, April 1, 2012

A statistician’s view of constructivist math programs

By Nicole O. Stouffer

"It's like these administrators never even met a child."
- Nik Stouffer

I’ve had 4 years of undergraduate math courses, two years of graduate math courses, and I have taught graduate-level math courses. I had never seen the "lattice" method, the Egyptian method, or any of these other alternative algorithms until last year when I looked at Everyday Math homework. Students don’t need them, and those methods will not help a student move onto higher mathematics.

I would have been laughed out of my college classes if I used the "partial sums" method to add. I wouldn’t have been able to take differential equations if I hadn’t mastered long division. There is a reason why traditional algorithms (the math methods you learned in school to add, subtract, multiply and divide) are needed. Traditional algorithms are needed to understand higher mathematics in college. It is extremely important that they are practiced until they are mastered. In fact, the new Common Core State math standards recommend teaching the standard algorithms.

You might think there is no reason not to offer alternative algorithms, as long as they also teach the traditional methods, but I have three reasons why the teaching of alternative programs is a problem.
  • It is a waste of classroom time. There are plenty of other items to be covered and practiced, so students should avoid learning inefficient algorithms.
  • Offering children too many algorithm choices is confusing. I have seen kids straddle two methods, not really understanding either of them and becoming frustrated.
  • Students are not permitted to use these alternative algorithms in college-level math.
Everyday Math does not prepare students for middle school math. There is a huge disconnect between Everyday Math and the rest of your child’s academic career.

My advice to parents… Work with your child to reinforce the traditional math algorithms that you learned. E-mail your principal and board directors to urge them to stop the use of these alternative methods in school and start using a more traditional textbook that uses the most efficient algorithms recommended by the Common Core State Standards.

Spiraling

The core principle behind Medford’s K-5 math program, Everyday Math, is based on a style called “spiraling,” where students continually return to basic ideas as new concepts are added over the entire year. Mostly, spiraling makes sense because kids need to return to topics periodically throughout the school year to remind them of what they have previously learned so they can build on it.

Everyday Math and other reform math programs like it do not use spiraling in this way. They don’t allow time for a child to master a skill before moving onto something else. The topics change so rapidly that kids never get a chance to feel successful in mathematics. Just as they are working toward figuring a concept out, Everyday Math will abruptly move onto another concept completely unrelated to the last. By the time a student returns (spirals back) to a concept, the child has forgotten the skill. Mathematics is a subject better suited toward mastery because the nature of math itself is linear and builds upon skills that are mastered in a sequential and logical fashion.

Everyday Math, Investigations in Number, Data, and Space, and other reform math programs have been highly criticized because of the lack of “focus” that comes with incoherent topic changes from excessive spiraling. In fact, the Common Core State Standards insist that a math curriculum be “more focused.” The final report of the 2008 President’s National Mathematics Advisory Panel (NMAP) recommends, “A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided."

Of course, the authors of Everyday Math sent a letter arguing, “the spiral approach should be more effective than a focused approach.” They are right, it “SHOULD be,” but it isn’t because of the way they do it. This is the reason the NMAP recommended it should be avoided. The authors of Everyday Math decided that they were right and everyone else was wrong. So Everyday Math still continues to randomly bounce from topic to topic each week and never allow enough time for practice or mastery. Even though the authors refused to change Everyday Math to be more focused, that didn’t stop them from slapping a label on their book that says “Common Core State Standards 100% Alignment.”

Shouldn’t our students have a focused textbook that actually aligns to the more-efficient methods recommended by the President’s National Mathematics Advisory Panel and the Common Core State Standards?

Supplementation

Everyday Math and other reform math programs require a lot of patching and fixing. I have talked with several teachers who have to rip pages out of the book and completely re-order the lessons and add additional lessons to make Everyday Math work. They have to practically rewrite an entire curriculum because Everyday Math is so poorly designed. After a few years, teachers can work out the bugs and the kinks and realize what to throw away and what to add, but if they move to a different grade level then they have to do this work all over again. It’s is a real problem when substitutes or new teachers have to experience this inadequate program for the first time because they might not realize how bad this program really is until they actually start using it to teach.

Last year, I had an issue with statistics because Everyday Math has watered down the First Grade lessons in probability to the point where they’re actually wrong. As a statistician, I find it frustrating to see statistical words used incorrectly and statistical concepts presented incorrectly. I wrote to my child’s teacher, who referred me to the principal. He told me that the teachers were required to teach to the curriculum and there was nothing that could be done about the fact that the book was teaching something incorrectly. The principal suggested that I “write the authors of the book.” I’ve been told by the Board of Education not to worry about the bad math program because teachers supplement to make it work. But how much can teachers really supplement and change from the set curriculum? And why would we insist on giving them such a bad tool that requires so much work on their part? Why don’t we give the teachers the best tools to teach our kids?

Minimally Guided Instruction

You could tell your children to “Go clean your room,” or you can tell them, “Put your dirty clothes in the hamper, make your bed, and put your toys in the toy box and books on your book shelf.” Which of the two examples would teach your children how to clean their room? The first is an example of minimally guided instruction and the second is an example of guidance-specific instruction.

There might be examples where minimally guided instruction is an effective teaching tool, but learning how to clean your room and learning grade-school mathematics aren’t two of them. Unfortunately, the math programs Investigations in Number, Data, and Space, Everyday Math and Connected Mathematics, expect children to ‘discover’ mathematics on their own with minimal guidance from the teacher. Many elementary school teachers in NJ don’t buy into that nonsense, so they provide the little kids with more guidance. In middle school, in some districts in New Jersey, such as Medford township, offer a pre-math class where the student is taught directly, with worked examples, before they get to the actual math class where the student is expected to ‘discover’ math for him/herself. Does that make sense to you? Me neither. Why wouldn’t the school just use a math book that directly teaches math instead of offering two math classes to cover up an ineffective Connected Mathematics book that uses the failed method of minimally guided instruction?

Don’t take my word for it, Kirschner, Sweller, & Clark (2006) show that minimally guided instruction goes against more than fifty years of research on human cognitive architecture. There is overwhelming evidence that minimally guided instruction is a less efficient and less effective teaching style.

A bigger problem that affects all students is that it takes time for a child to ‘discover’ mathematics, so Connected Mathematics wastes a lot of classroom time ‘discovering,’ instead of directly teaching efficient methods for solving math problems. Compared to other middle school math books, Connected Mathematics actually covers less material. This means that all students, especially the kids who don’t need the pre-math class, could be learning more than what is presented in class.

Unfortunately, the missing concepts will eventually catch up with a student when they enter high school algebra because you can’t learn what is never taught.

Nik Stouffer is a statistical consultant with more than 15 years of experience. She has a Masters degree in mathematics and has taught graduate school mathematics classes. Website: http://www.medical-research-support.com/



Note from Laurie Rogers: If you would like to submit a guest column on public education, please write to me at wlroge@comcast.net . Please limit columns to about 1,000 words, give or take a few. Columns might be edited for length, content or grammar. You may remain anonymous to the public, however I must know who you are. All decisions on guest columns are the sole right and responsibility of Laurie Rogers.

7 comments:

Anonymous said...

As a math educator who works primarily with secondary and college level students, I agree with the points made here. I also have taken many mathematics courses (as I have a B.S. in mathematics) and have had to ask local students to show me things like the lattice method. I have been approached by teachers who ask how to make a particular method or explanation work when the reality is the method is mathematically flawed. Teaching high school mathematics becomes so much harder when concepts like long division have not been learned so polynomial long division has to start with regular long division lessons. Then that becomes difficult because students never learned basic multiplication facts.

Ken Capron said...

Laurie – Have the math reformists ever explained why THEY feel the "lattice" method, the Egyptian method, etc. should be included? Most people don’t include this superfluous stuff (aka fluff) without some reasons. Has anyone asked them why?

Ken Capron

PS: I learned numerous alternative algorithms in school but they came from teachers and not from books, and they were meant to be curiosities and fun. Even in college, we studied things like Hessian Matrices – which have little or no value except to an elite few programmers at Bell Labs working on queuing theory.

Bruce Price said...

I believe Reform Math (which includes a dozen curricula such as TERC, etc.) is exactly parallel to Whole Word. Both are devious, dishonest pedagogies that do the opposite of what is claimed. Whole Word makes kids illiterate. Reform Math makes kids mathematically illiterate.
I can't believe that the National Council of Teachers of Mathematics (NCTM), which worked to impose Reform Math, is an organization that deserves our respect.

Bruce Deitrick Price

Melanie Kurdys said...

This is an excellent article. I have been a math tutor in four states and have found this horrible math curriculum everywhere. You are especially right that it does not develop a child's confidence in his ability to learn math, which then prevents them from being willing to tackle more advanced concepts. I fear that we now have a generational math phobia where many adults, especially elementary teachers, are personally fearful of math and they simply can not teach it to children. Bold, aggressive intervention is necessary, but I am not sure where we will find the force needed to make it happen.

Nikki said...

Actually Ken, Hessian Matrices are used in statistics. So there is an actual real-life use for them. Sometimes they cause havoc in my modeling programs.
-Nik

Lara said...

There is a wonderful (charity/non-profit) math teaching method developed by John Mighton in Canada called JUMP Math (jumpmath.org). It's the perfect antidote to reform/constructivist math and builds mastery and conceptual understanding as well as CONFIDENCE and passion. Crazily enough he has studies by third parties showing compelling evidence for how effective this method is across all skill levels. His book End of Ignorance is inspiring and very much in line with the comments expressed here.

Anonymous said...

When I entered a public school when I was 10-11 years old my math skills went drastically down even though I was one of the best math students in my previous school.

And today I never learned math since then besides some basic statistic and graph analysis ( I am 22 now, working).

Thank you so much for this article.