Learning and Teaching Math

Math Standards in the US. I was “happy” to see that what I’ve been saying is supported there. I urge you to read the report if you want to understand some of things you should do or don’t do with your kids. You can find the complete report, as well as the executive summary here http://www.edexcellence.net/foundation/publication/publication.cfm?id=338&pubsubid=1117#1117
Here is an excerpt from the executive summary, with my comments in italics:

Common Problems

Why do so many state mathematics standards come up short? Nine major problems are widespread.

1. Calculators

One of the most debilitating trends in current state math standards is their excessive emphasis on calculators. Most standards documents call upon students to use them starting in the elementary grades, often beginning with Kindergarten. Calculators enable students to do arithmetic quickly, without thinking about the numbers involved in a calculation. For this reason, using them in a high school science class, for example, is perfectly sensible. But for elementary students, the main goal of math education is to get them to think about numbers and to learn arithmetic. Calculators defeat that purpose. With proper restriction and guidance, calculators can play a positive role in school mathematics, but such direction is almost always missing in state standards documents.

Further more, it prevents practice of #2 below, thus degrading knowledge acquired earlier. Often it also prevents, in later grades, manual operations (division, multiplication, fractions operations, etc.), thus reducing practice of acquired knowledge, and thereby pushing those skills to a lower level.

2. Memorization of Basic Number Facts

Memorizing the “basic number facts,” i.e., the sums and products of single-digit numbers and the equivalent subtraction and division facts, frees up working memory to master the arithmetic algorithms and tackle math applications. Students who do not memorize the basic number facts will founder as more complex operations are required, and their progress will likely grind to a halt by the end of elementary school. There is no real mathematical fluency without memorization of the most basic facts. The many states that do not require such memorization of their students do them a disservice.

It is not just the “working memory” that is reduced. The ability to “see” the path to solving a problem is the part that suffers the most since the student cannot see beyond “what is 31×15″ [or anything of sort] and therefore cannot abstract to the overall path from the collection of calculations and operations required to walk that path.

However, be careful not to stress it too much. Each child would have his/her level of ability to do those things quickly. If your child can answer basic facts (such as multiplication of numbers up to 10, and addition of two digits numbers) reasonably quickly, i.e. without actual calculation, then s/he is good enough. Knowing those type of facts does not bring understanding of Math, but prevents learning difficulties.

3. The Standard Algorithms

Only a minority of states explicitly require knowledge of the standard algorithms of arithmetic for addition, subtraction, multiplication, and division. Many states identify no methods for arithmetic, or, worse, ask students to invent their own algorithms or rely on ad hoc methods. The standard algorithms are powerful theorems and they are standard for a good reason: They are guaranteed to work for all problems of the type for which they were designed. Knowing the standard algorithms, in the sense of being able to use them and understanding how and why they work, is the most sophisticated mathematics that an elementary school student is likely to grasp, and it is a foundational skill.

It is good (and important) to let students come up with ideas BUT, the standard algorithm should be then taught. Additionally, a good discussion requires a good teacher, one who KNOWS the underlying principles and can lead a Socratic method based discussion (i.e. guiding questions). It is also important for the teacher to recognize that most students will not have much to say and will not be able to come up with ideas and allow for that. There is a reason why it took thousands of years to develop our basic mathematical knowledge.

4. Fraction Development

In general, too little attention is paid to the coherent development of fractions in the late elementary and early middle grades, and there is not enough emphasis on paper-and-pencil calculations. A related topic at the high school level that deserves much more attention is the arithmetic of rational functions. This is crucial for students planning university studies in math, science, or engineering-related majors. Many state standards would also benefit from greater emphasis on completing the square of quadratic polynomials, including a derivation of the quadratic formula, and applications to graphs of conic sections.

This is why in my topics map Fractions is an important stepping stone. As I show in the topics map, ratio, proportions, percents, and probability [as well as Decimal fractions], are but extensions of Fractions. I guess there is a reason why my book on it is 110 pages long…;)

5. Patterns

The attention given to patterns in state standards verges on the obsessive. In a typical document, students are asked, across many grade levels, to create, identify, examine, describe, extend, and find “the rule” for repeating, growing, and shrinking patterns, where the patterns may be found in numbers, shapes, tables, and graphs. We are not arguing for elimination of all standards calling upon students to recognize patterns. But the attention given to patterns is far out of balance with the actual importance of patterns in K-12 mathematics.

Pattern recognition is a good ability to have. Practice should be made about it early on. However, it should be home based mostly. It does not contribute much beyond the early grades. It is also an innate ability (like music) that some would be naturally better at and some not as good. However, one can be absolutely daft at pattern recognition and still be an excellent mathematician [not just good at school Math]. There is a relationship between the two but it is not a necessary condition.

6. Manipulatives

Manipulatives are physical objects intended to serve as teaching aids. They can be helpful in introducing new concepts for elementary pupils, but too much use of them runs the risk that students will focus on the manipulatives more than the math, and even come to depend on them. In the higher grades, manipulatives can undermine important educational goals. Yet many state standards recommend and even require the use of a dizzying array of manipulatives in counterproductive ways.

I have been saying exactly that quite strongly to parents, and I am glad there is support! Manipulative, while helpful in understanding, can als hinder abstract thinking development if used too much, not just make the student depend on them for calculation.

7. Estimation

Fostering estimation skills in students is a commendable goal shared by all state standards documents. However, there is a tendency to overemphasize estimation at the expense of exact arithmetic calculations. For simple subtraction, the correct answer is the only reasonable answer. The notion of “reasonableness” might be addressed in the first and second grades in connection with measurement, but not in connection with arithmetic of small whole numbers. Care should be taken not to substitute estimation for exact calculations.

As I said on some boards, estimation is a good skill to have in life BUT it is completely unnecessary for understanding of Mathematics at any level. It is also one of those “innate skills” that can be taught to a certain level. The best way to get your kids to do estimation and see its value is exactly not in Math related work, where it has not value, but in life related issues, such as change, time, how much candy they get, how much work they have to do, etc.

8. Probability and Statistics

With few exceptions, state standards at all grade levels include strands devoted to probability and statistics. Such standards almost invariably begin by Kindergarten. Yet sound math standards delay the introduction of probability until middle school, then proceed quickly by building on students’ knowledge of fractions and ratios. Many states also include data collection standards that are excessive. Statistics and probability requirements often crowd out important topics in algebra and geometry. Students would be better off learning, for example, rational function arithmetic, or how to complete the square for a quadratic polynomial—topics frequently missing or abridged.

Probability, and specifically discrete probability that is taught in school, is actually simple and very difficult at the same time. The basic concept is simple (it’s a fraction, nothing more). Application to problems and understanding how it works can be difficult since it is not so intuitive to us. I have found it to be the most confounding aspect of Math with even college students, and even with my peers at the school of mathematics! You should only introduce it after Fractions, ratios, percents, and decimal fractions, have been mastered. You should do a lot of basic exercises as well. It can get tricky beyond the basic since one has to be a good “accountant” to do it right. BUT, there are some simple rules and ideas that make it much easier. Only they are not taught in most if not all books I’ve seen ☹ [and mine is not yet written…]

9. Mathematical Reasoning and Problem-Solving

Problem-solving is an indispensable part of learning mathematics and, ideally, straightforward practice problems should gradually give way to more difficult problems as students master more skills. Children should solve single-step word problems in the earliest grades and deal with increasingly more challenging, multi-step problems as they progress. Unfortunately, few states offer standards that guide the development of problem-solving in a useful way. Likewise, mathematical reasoning should be an integral part of the content at all grade levels. Too many states fail to develop important prerequisites before introducing advanced topics such as calculus. This degrades mathematics standards into what might be termed “math appreciation.”

The least studied topic, yet the most important. It is a subject/topic that is threaded through ALL of math since it is not subject specific [i.e. it’s not about Fractions, or about equations]. It should be studied as a subject all to itself with applications in specific subjects. See the discussion Denise and I have been having on the “How to solve word problems” blogs I wrote. If your child is good at fundamental, basic problem solving [i.e. regardless of subject matter], s/he is good at all problems. If not, s/he is good only at those subjects s/he mastered.

Having taught at two countries and at multiple levels (elementary to university) I have seen that Math is a universal problem. Many countries struggle with how to teach math, and even those that do well do not argue that it is easy to achieve. Therefore, the conclusion must be that it is not a cultural issue, as many in the US like to believe, although culture and values can lessen or strengthen the problem.

The Math Wars got it right, though neither camp is right. To distill the approaches, we can say that one camp, supported by education researchers, favors discovery mode, where kids discover solution approaches by themselves and learn to explore math, and at its strongest approach relegating algorithms (solution methods) non-existent. The other camp, supported by mathematicians, and sometimes confused with “traditional math”, favors a strong technical foundation. While the first camp has been winning on the curriculum front and most school district in the US are teaching variants of the “New Math” (Discovery Math, etc.), the US test scores on the international test (TIMSS – see www.timss.org) have fell, and as students grow they do worse on the test, lending support to the “algorithmic” camp. The arguments on each side are complex at times and have devolved into what the title implies: a war. It is no longer a discussion but more of a shouting match.

As a mathematician I KNOW that the discovery type of curriculum is bad for kids but at the same time, as a long time teacher, I KNOW that rote learning of algorithms is not enough either. The problem is that each side is answering a different question, and both are the wrong questions!

The “discover/new math” camp asks, “How do kids learn and how can we facilitate such learning?” It is a question that is natural for education researcher but it is the wrong question since ALL of us have flaws in the way we learn. While some of us are better at it than others, without guidance most if not all would end up with some faulty problem solving approaches. As millennia of experience show, learning how to think SYSTMATICALLY requires long and rigorous training. As I have constantly seen with my college students, in some of the top universities in the US, the education system fails in teaching them how to systematically analyze, since it is not taught AT ALL. We can think of the question posed using my favorite metaphor of sports. A coach would not ask, “How do kids shoot and how can we facilitate such?” because the coach knows most kids do not naturally end up with the right method of shooting. While it is a critical and important question to ask, it is not the RIGHT and main question, but rather a supporting question.

The “algorithmic” camp asks, “How do we make sure kids know how to solve the math problems they study in school and college?” While in terms of ensuring performance in Math it is a better question than the “discovery” one, it is also the wrong question. As studies across nations have shown, technical skills are important for understanding math concepts. But, technical skills, while they might ensure good scores on standardized tests, do not ensure deep understanding of concepts. However, it is the wrong question because it focuses on Math skills, and not thinking and learning. Having seen thousands of college students who mostly got high grades in their studies, Math included, I have learned that their general analysis capabilities and skills, as well as understanding of basic mathematical concepts, are severally lacking at best.

To reach the right question we have to examine what the objectives of education (the “thinking” part of it), and not just Math education are, or should be. There are numerous important objectives that have to do with how students approach learning and specific topics (sense of wonder and joy, curiosity, etc.) and even broader ones that has to do with their social and personal life. But to distill the issue at the center of the Math Wars I focus on the intellectual/mental elements only.

There are three elements that education should strive to achieve in students:

1. Ability to learn.
2. Systematic analysis, thinking, and problem solving abilities.
3. Domain specific knowledge and skills.
4. The ability to apply the second to the third.

The first objective is general to all topics and subjects and provides students with the fundamental capabilities they need to approach any domain of knowledge, be it Math or Literature. The second is obvious and arrives from the need to master certain knowledge and skills, although which ones can be argued on. The third is but a merger of the first two, but it is not necessarily trivial.

It is clear that the “discovery” camp tries to answer a question related to the first objective, believing that answering the “how kids learn?” question leads to achieving that first objective, which in turn results in achieving objectives 2-4, when used in a specific domain, such as Math. The problem is that facilitating students learning does not necessarily (and that word is critical) results in students achieving even #1. They may, or they might not learn how to learn. Further more, even if #1 is achieved, #2, #3, and #4 might not be, since the link is not automatic.

It is also clear that the “algorithmic” camp focuses on the third objective. As a result, while this approach would result in higher Math capabilities, it would not guide students to develop to their full potential, and might kill their internal curiosity and sense of joy in learning, due to over emphasis on rote and drills. It would also not teach them how to learn, or systematically analyze and solve problem, not just in Math.

It is important to note that when students know how to learn, and know how to approach problems, they enjoy the specific domain in which they have those capabilities. This is why those who are good at Math tend to enjoy it, and why those who are good at sports tend to enjoy sports. Naturally, enjoying an activity creates a strong incentive to learn and practice it, thereby creating a positive cycle. But it is the ability that drives joy, or can kill it if absent.

The right question is therefore simple:

I will discuss and answer it in my next Blog ☺