I have read a lot of research papers, articles, books, magazines, and blogs about math education. I find it is impossible to keep them all straight! Writing a review helps me process and remember the main points. It's like when they made you write a paper about the reading you did for class in school. Consider this is my free gift of cliff notes to you. Check out all my "reviews".
I picked up an amazing book at NCTM's Boston Conference: Classics in Mathematics Education Research (Amazon). It is a greatest hits album of the best and most influential papers on mathematics education. The editors -- Carpenter, Dossey, and Koehler -- provide a context for each article, and then you get to enjoy...well...a classic!
The first article in the book is "The Place of Meaning in the Teaching of Arithmetic" by William A. Brownell from 1947 (JSTOR). Brownell is recognized as a significant contributor to the argument that memorizing algorithms and quick computation skills does not necessarily reflect understanding and is not the best way to educate our students. In this article he lays out reasons why we should make arithmetic make sense to students and addresses common objections to teaching arithmetic with meaning.
"'Meaningful' arithmetic, in contrast to 'meaningless' arithmetic, refers to instruction which is deliberately planned to teach arithmetical meaning and to make arithmetic sensible to children through its mathematical relationships."
The block quote summarizes quite nicely (and in 1947 no less) what so many of us strive for in our lessons in math class. We want our students to understand the mathematical principles behind the procedures we are teaching. We want our students to understand why, for example, you can "carry a 1" to the next place value and what that represents mathematically.Many teachers now use the term regroup instead of carry to better represent that concept.
In recognizing the need for developing student understanding, Brownell is careful to point out that we cannot prepackage understanding in to a quick digestible bite: "For relatively few aspects of life, for relatively few aspects of the school's curriculum (including arithmetic), do we seek to carry meanings to anything like their fullest development. Moreover, whatever the degree of meaning we want children to have, we cannot engender it all at once. Instead, we stop at different levels with different concepts; we aim now at this level of meaning, later at a higher level, and so on." He rightly points out that students need time to develop meaning and that will mean rethinking the rote practice methodology.
One interesting exercise the author embarks on is to group the "meanings of arithmetic" into four categories. Each was presented as a paragraph but I did my best to summarize them here:
- Basic Concepts (definitions of fraction and percents as well as vocabulary such as divisor, common denominator, etc.)
- Fundamental Operations (When to add, when to multiply, what happens when you apply operations)
- Relationships and Generalizations (additive identity, commutative property, equivalent fractions)
- Role of Decimal Number System (how place value connects to other groups)
I am not sure I would group "meanings of arithmetic" in this way, but it is neat to see how he was thinking about this. In particular, I find it interesting that he gave the role of the number system its own category. His categories do seem to include everything I can think of, but I am not sure that categorizing the concepts brings anything to understanding to the discussion. Perhaps it was just an opportunity to show the reader how much meaning truly exists in this subject.
Unfortunately, Brownell did not have much evidence to support the idea that teaching arithmetic with meaning is better for students (remember, we are still in 1947). He cites anecdotal evidence and common sense as well as the fact that "[meaningless] programs have not produced the kind of arithmetical competence required for intelligent adjustment to our culture."
He shares optimism that "school personnel and, to some extent, the public at large are beginning to awaken to the fallacy of treating arithmetic as a tool subject. To classify arithmetic as a tool subject, or as a skill subject, or as a drill subject is to court disaster." Sadly, I still see and hear this argument coming from parents and teachers -- particularly those focused on Algebra. They see learning arithmetic as building the set of skills needed to succeed in Algebra. To counteract this argument, Brownell outlines 10 benefits to teaching arithmetic meaningfully which could just have easily appeared in a modern paper:
- Improves retention
- Can refigure out forgotten skills from known understanding
- Can apply knowledge more widely -- so skills are more likely to be used
- Makes learning new ideas easier
- Less repetition needed to learn
- Prevents illogical answers
- Emphasizes problem-solving over memorization
- Provides student with diversity of procedures to choose from
- Fosters independence and confidence
- "Presents [arithmetic] in a way which makes it worthy of respect
Brownell, who it should be clear by now likes lists, also outlines four common objections to teaching arithmetic with meaning and attempts to address them.
- Is understanding the arithmetic really essential to learning arithmetic?
- Isn't understanding arithmetic just too hard for young kids?
- Do we really have the time to teach arithmetic so it can be understood?
- If students do understand arithmetic, does it get in the way of their understanding later? As in, thinking about the meaning of an algorithm will make it take longer to complete calculations in the future.
Response to (1): Argues, essentially, that when we memorize through rote practice, we forget quickly. Moreover, it is impossible to practice all skills enough to be proficient at them when we don't understand them.
Response to (2): The goals of understanding should be appropriate for the age. But, even younger students can understand simple examples and begin to see how they can apply/relate to more complex ones. He brings up the point that, if taught for meaning, students do not necessarily have to understand EVERYTHING perfectly. But, whatever is understood will be beneficial.
Response to (3): Takes more time, yes, but the gains are worth it -- notes that there is little evidence of that yet. Again, were still in 1947.
Response to (4): They need to understand it at first, but the goal is still to get them to automaticity.
I definitely want to write more about the goal of automaticity in a later post, but remember that pocket calculators weren't around until the '60s or '70s.Intel had to invent the microprocessor first. Of course, they did have slide rules before then. Being able to calculate quickly and accurately was more essential when everyone didn't have a calculator in their pocket.
In some ways this article was heart-warming, and in others it was disheartening. It is amazing to see how clearly Brownell was able to articulate some aspects of the modern math education movements well before it became mainstream. It is, however, frustrating to still need to rehash so many of these arguments so frequently. If these arguments were being put forth 68 years ago, you would think it would be more settled by now, especially given the overwhelming evidence in support of "meaningful arithmetic" nowadays.
I'll leave you with one final nugget. Brownell really was on the forefront of many of these ideas and one line from his paper struck me as especially predictive: "The culture is highly quantitative and is steadily becoming more so." No kidding! Brownell sure saw the tide turning, I just hope he wouldn't be disappointed in the progress we have made.