The new math curriculum believes, among many things, that:

· Students learn by attaching meaning to what they do, and they need to construct their own meaning of mathematics.

· Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must realize that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable.

· Curiosity about mathematics is fostered when children are engaged in, and talking about, such activities as comparing quantities, searching for patterns, sorting objects, ordering objects, creating designs and building with blocks

Before I critique Anna’s arguments, we must realize that when someone talks about “meeting” or “raising standards”, they are usually referring to applying conventional, or traditional, instructional techniques in the classroom. Usually this really means, “Everything we are doing is OK, we just need to be harder on students”.

Anna Stokke, will have us believe that we have lowered standards in our math classes and we need to raise the standards back up. Her first argument about the new math curriculum is

“… practise and memorization of number facts are no longer a priority in our schools. Children are instructed to use overly complicated "horizontal" methods to work out simple arithmetic problems. Now, simply knowing how to produce an answer to a basic multiplication question, no matter how long it takes, is considered to be a sufficient indicator of fluency.”

The reality is that being told number facts, and forced to regurgitate these facts, will not create an environment which is conducive to deeper learning. The best way a student can understand, not just recite, mathematics is through discovering the facts on their own. For some, this discovery process can take seconds, while for others it may take an entire class. The role the teacher plays in this is by acting as a tour guide. By keeping the students on the “path” towards the discovery, it will ensure that all students create their own individual, and innovative, techniques in learning mathematics.

Next, she says that

“Kids are set up for failure if they are not required to memorize basic number facts. Without the memorized facts, they will become hung up on these simple numbers when they are trying to solve more difficult problems.”

I often ask myself, which skills do I want for my own children? Do I want them to be taught the skills to memorize, and ultimately be able to complete algorithmic tasks, or the ability to create and design which will lead them down a road towards a career with heuristic tasks?

The reality is that the age old saying that “Practice makes perfect”, does not apply to deep learning in mathematics. Traditionally, students have been given worksheets which have multiplication facts on them and are given a strict time limit to complete these (named Mad Minutes), but in actuality the mark given on these “Mad” sheets really do indicate to anyone as to knowledge of the student writing them.

The weak students truly suffer the most from this model of proficiency-driven, because they find these tasks dull, repetitive, and entirely unusable in the world outside the walls of the classroom. I would agree that sometimes, knowing facts as the days of the week are important, but these facts should be a by-product of use and application not out of necessity.

Another fact is, by spending time forcing these facts to be memorized is truly interfering with a child’s innovative and creative ability. Most of these worksheets require low level thinking and usually can lock the thought process of a student into understanding simple algorithms. Eleanor Duckworth summed it up the best,

“Knowing the right answer requires no decisions, carries no risk, and makes no demands. It is automatic. It is thoughtless.”

Is this the environment in which math students should be learning?

Last she makes the statement,

“…children need to practise arithmetic skills, without calculators, do an adequate amount of pencil-and-paper math, and memorize times tables in order to become proficient in math. Children need to be given time to do a reasonable amount of math daily at school and this needs to be a priority.”

This closely sounds like a case for homework at younger levels and there is not one shred of evidence that supports the idea of homework in elementary grades. Assuming this is not her argument, I will critique her argument for repetitive daily work in school.

When we teach math as “routine skills” students may get the correct answer, at an efficient rate, but they will most likely be clueless about the significance of their of their answer. The National Research Council calls it “Mindless mimicry mathematics” and can no longer be the norm in our classrooms. Another math educator, William Brownell, over 70 years ago explained “one needs a fund of meanings, not a myriad of ‘automatic responses..’ Drill does not develop meanings. Repetition does not lead to understanding.

There are many studies that support this view of thinking. If we look at an environment where students are not learning effectively, it is usually due to an overwhelming desire to maintain traditional beliefs and practices. Once again classrooms need to be aware that

It is important to realize that it is acceptable to solve problems in different ways and that solutions may vary depending upon how the problem is understood.

Dear Dave,

ReplyDeleteThank you for posting about my op/ed piece. I would appreciate it if you would post a link to the entire piece rather than simply taking out selected portions and critiquing them. I would like to point out that both conceptual understanding AND procedural skill are essential for success in mathematics. This is not an either/or situation and I have not, at any point, argued for ONLY routine learning. However, I do argue for the inclusion of what I know is missing in the curriculum. Here is a link to the article:

http://www.winnipegfreepress.com/opinion/westview/why-our-kids-fall-behind-in-math-129938903.html

Best regards,

Anna Stokke,

Associate Professor of Mathematics

I do apolgoize, I thought I had included the link before, and it will be added now.

ReplyDeletePerhaps I am misunderstanding your point, but after reading the article it seems as you believe memorization should be the focus of math classrooms not deeper learning. Is this true?

Have you read any of Alfie Kohn's material on teaching mathematics?

Dear Dave,

ReplyDeleteThank you.

Of course I do not believe that memorization should be the entire focus! You would be hard pressed to find a mathematician who thinks that kids should simply memorize math facts and procedures! Obviously I have a deep understanding of math myself and have published several original papers in mathematics (I've "discovered" new math myself) - understanding and discovering mathematics is the lifeblood of my profession.

However, I also know my times tables automatically and can calculate quickly and efficiently when I need to. (I cannot say the same of most fifth graders - and even a large number of university students - in my province.) This frees up working memory so that I am able to concentrate on more difficult problem-solving.

When a person advocates for the inclusion of memorization, automaticity of basic math facts and the use of standard algorithms this does not mean that they are advocating AGAINST understanding. Here are two other quotes from my piece:

"It is important to understand the concepts behind the math that is taught, but this should not be at the expense of basic arithmetic competency. A false dichotomy in math education is that memorization and practice of basic skills must come at the expense of understanding. This is simply not true -- one can, and should, have both."

"Imagine the middle years student who gets stumped with an algebra problem because he or she needs to work out 9 x 4 by repeated addition: 9 + 9 + 9 + 9 = 36!"

Yes, I am familiar with Alfie Kohn.

I think we definitely need to get back to the inclusion of memorization of basic facts before getting into that deeper understanding. I've seen my last couple of elementary school-aged children struggle with math because they've been expected to know their times tables without really being given a chance to learn them properly. It is extremely frustrating for them and for me. Frustrating because they waste so much mental energy on those basics to solve the more complicated problems, and more frustrating because they're stuck with extra homework from taking so long to do their work at school. Why should we all have to put in so much extra time learning these things at home when the kids are in school for that many hours in the day? It's ridiculous.

ReplyDeleteAs a junior high school math teacher I can attest to the FACT that children not memorizing basic math facts is putting them way behind due to the simple fact that it takes them MUCH longer to solve problems of simple to medium difficulty and therefor they get much less practice with the problem solving. They are not comfortable with whole numbers so when you throw fractions or decimals into the mix the wheels literally come off. I find myself spending more time teaching the rudimentary aspects of mathematics than I do with pre-algebraic concepts and skills. It is disheartening to see our children lagging further and further behind because of broken systems and pop curriculum.

ReplyDelete