Category: Curriculum Reform

Posted in Algebra, Curriculum Reform

Teaching algebra – it pays to start early

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I believe in early algebraization. I have posted a few articles in this blog on ways it can be taught in the early grades. Check out for example  Teaching  Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it is a number or geometry or pre-algebra lesson always aim at developing students’ algebraic thinking. What do research say about early algebraization? How do can we integrate it in the grades without necessarily adding new mathematics content?

“Traditionally, most school mathematics curricula separate the study of arithmetic and algebra—arithmetic being the primary focus of elementary school mathematics and algebra the primary focus of middle and high school mathematics. There is a growing consensus, however, that this separation makes it more difficult for students to learn algebra in the later grades (Kieran 2007). Moreover, based on recent research on learning, there are many obvious and widely accepted reasons for developing algebraic ideas in the earlier grades (Cai and Knuth 2005). The field has gradually reached consensus that students can learn and should be exposed to algebraic ideas as they develop the computational proficiency emphasized in arithmetic. In addition, it is agreed that the means for developing algebraic ideas in earlier grades is not to simply push the traditional secondary school algebra curriculum down into the elementary school mathematics curriculum. Rather, developing algebraic ideas in the earlier grades requires fundamentally reforming how arithmetic should be viewed and taught as well as a better understanding of the various factors that make the transition from arithmetic to algebra difficult for students.

The transition from arithmetic to algebra is difficult for many students, even for those students who are quite proficient in arithmetic, as it often requires them to think in very different ways (Kieran 2007; Kilpatrick et al. 2001). Kieran, for example, suggested the following shifts from thinking arithmetically to thinking algebraically:

  1. A focus on relations and not merely on the calculation of a numerical answer;
  2. A focus on operations as well as their inverses, and on the related idea of doing/undoing;
  3. A focus on both representing and solving a problem rather than on merely solving it;
  4. A focus on both numbers and letters, rather than on numbers alone; and
  5. A refocusing of the meaning of the equal sign from a signifier to calculate to a symbol that denotes an equivalence relationship between quantities.
These five shifts certainly fall within the domain of arithmetic, yet, they also represent a movement toward developing ideas fundamental to the study of algebra. Thus, in this view, the boundary between arithmetic and algebra is not as distinct as often is believed to be the case.
What is algebraic thinking in earlier grades then? Algebraic thinking in earlier grades should go beyond mastery of arithmetic and computational fluency to attend to the deeper underlying structure of mathematics. The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting. That is, early algebra learning develops not only new tools to understand mathematical relationships, but also new habits of mind.”

The foregoing paragraphs were from the book Early Algebraization edited by Jinfa Cai and Eric Knuth. The book is a must read for all those doing or intending to do research about teaching algebra in the elementary grades. Educators and textbook writers should also find a wealth of ideas on how algebra can be taught and integrated in the early years. Of course it would be a great read for teachers.  The book is rather expensive but if you have the money, why not? Here are some section titles:
  • Functional thinking as a route in algebra in the elementary grades
  • Developing algebraic thinking in the early grades: Lessons from China and Singapore
  • Developing algebraic thinking in the context of arithmetic
  • Algebraic thinking with and without algebraic representation: A pathway to learning
  • Year 2 to 6 students’ ability to generalize: Models, representations, and theory for teaching and learning
  • Middle school students’ understanding of core algebraic concepts: equivalence & variable”

Check out the table of contents for more.

The following books also provide excellent materials for developing algebraic thinking.

 

 

 

 

Please share this post to those you think might find this helpful.

Posted in Curriculum Reform, Elementary School Math

English edition Japanese Grades 1- 6 Math textbooks 50% off until September

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Last year I had the chance to do a few days work for a Japanese Mathematics textbook series translated in English. I just received an e-mail from one of the editors that the publisher is selling the books at 50% off. It will only cost about US$75 for the set of 11 books until September 2011. After September the set will cost about $US150. 00, the latest edition. I don’t have a copy yet of the books so I cannot show here a page  but my colleague and I did some editing for Grades 5 and 6 and we think that the Japanese really have a very different approach in teaching mathematics be it in textbooks and in teaching. There’s a very strong emphasis on developing mathematical thinking (I saw several lesson study demonstration teaching while there. Our Institute also had a 5-year math and science teaching project with Japanese educators). The solutions shown and the structure of the books model how students in that grade level think and would approach the problems. The books are a product of years of doing lesson  study in mathematics. I love the section “Let’s think about how to…”

The schools where you teach may already have a prescribed textbook but you will have a great resource in designing your instruction with this set. Your school might even decide to use the textbooks.

For ordering, please send your order to Mr. Serizawa: Continue reading “English edition Japanese Grades 1- 6 Math textbooks 50% off until September” →

Posted in Algebra, Curriculum Reform

What is algebra? Why study it?

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I’m doing some  literature review for my research and I came across this article by L.A Steen in Middle Matters. He was arguing about the Algebra for All standard in the US and part of the article includes description of what is algebra. I thought I should share them in this blog because it is something very important teachers should be aware of when they teach algebra or what they conceive what algebra is and for. Oftentimes, when students ask what algebra is and what they are going to need it for, teachers lazy answer is “Algebra is just like your math in the grades only that this time you work with letters instead of numbers!”

  1. Algebra is the language of mathematics, which itself is the language of the information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995).
  2. It is the logical structure of algebra, not the solutions of its equations, that made algebra a central component of classical education.
  3. As a language, algebra is better learned earlier and harder, when learned later.
  4. In the Middle Ages, algebra meant calculating by rules (algorithms). During the Renaissance, it came to mean calculation with signs and symbols–using x‘s and y‘s instead of numbers. (Even today, lay persons tend to judge algebra books by the symbols they contain: they believe that more symbols mean more algebra, more words, less.) I think that many algebra classes still promote this view.
  5. In subsequent centuries, algebra came to be primarily about solving equations and determining unknowns. School algebra still focuses on these three aspects: employing letters, following procedures, and solving equations. This is still very true. You can tell by the test items and exercises used in classes.
  6. In the twentieth century algebra moved rapidly and powerfully beyond its historical roots. First it became what we might call the science of arithmetic–the abstract study of the operations of arithmetic (addition, subtraction, multiplication, etc.). As the power of this “abstract algebra” became evident in such diverse fields as economics and quantum mechanics, algebra evolved into the study of all operations, not just the four found in arithmetic.
  7. Algebra is said to be the great gatekeeper because knowledge and understanding of which can let people into rewarding careers.
  8. Algebra is the new civil right (Robert Moses). It means access. It means success. It unlocks doors to productive careers and gives everyone access to big ideas.

And I like the education battle cry Algebra for All. Of course not everyone is very happy about this. Steen for example wrote in 1999:

No doubt about it: algebra for all is a wise educational goal. The challenge for educators is to find means of achieving this goal that are equally wise. Algebra for all in eighth grade is clearly not one of them–at least not at this time, in this nation, under these circumstances. The impediments are virtually insurmountable:

  1. Relatively few students finish seventh grade prepared to study algebra. At this age students’ readiness for algebra–their maturity, motivation, and preparation–is as varied as their height, weight, and sexual maturity. Premature immersion in the abstraction of algebra is a leading source of math anxiety among adults.
  2. Even fewer eighth grade teachers are prepared to teach algebra. Most eighth grade teachers, having migrated upwards from an elementary license, are barely qualified to teach the mix of advanced arithmetic and pre-algebra topics found in traditional eighth grade mathematics. Practically nothing is worse for students’ mathematical growth than instruction by a teacher who is uncomfortable with algebra and insecure about mathematics.
  3. Few algebra courses or textbooks offer sufficient immersion in the kind of concrete, authentic problems that many students require as a bridge from numbers to variables and from arithmetic to algebra. Indeed, despite revolutionary changes in technology and in the practice of mathematics, most algebra courses are still filled with mindless exercises in symbol manipulation that require extraordinary motivation to master.
  4. Most teachers don’t believe that all students can learn algebra in eighth grade. Many studies show that teachers’ beliefs about children and about mathematics significantly influence student learning. Algebra in eighth grade cannot succeed unless teachers believe that all their students can learn it. (all italics, mine)

I shared these here because in my part of the globe  the state of algebra education is very much like what Steen described. You may also want to read about the expressions and equations that makes algebra a little more complicated to students.

L.A Steen is the editor of the book On the Shoulder of Giants, New Approach to Numeracy, a must read for teachers and curriculum developers. The book is published by Mathematical Sciences Education Board and National Research Council.

Posted in Algebra, Curriculum Reform, Elementary School Math, What is mathematics

“New Math” curriculum

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math curriculumIn the video below, Tom Lehrer ‘explains’ and ‘criticizes’ the New Math curriculum in a funny way. I think the New Math curriculum was not that bad really. It helped the teachers to structure their math knowledge in a “mathematical” way. This is a good thing. The trouble was they taught mathematics in the mathematical way. Enjoy the video.

What is the New Math Curriculum?

New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost science education and mathematical skill in the population so that the intellectual threat of Soviet engineers, reputedly highly skilled mathematicians, could be met.

New Math emphasized mathematical structure through abstract concepts like set theory and number bases other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world.

Criticisms of the New Math Curriculum

Parents and teachers who opposed the New Math in the U.S. complained that the new curriculum was too far outside of students’ ordinary experience and was not worth taking time away from more traditional topics, such as arithmetic. The material also put new demands on teachers, many of whom were required to teach material they did not fully understand. Parents were concerned that they did not understand what their children were learning and could not help them with their studies. In the end it was concluded that the experiment was not working, and New Math fell out of favor before the end of the decade, though it continued to be taught for years thereafter in some school districts.

In the Algebra preface of his book “Precalculus Mathematics in a Nutshell,” Professor George F. Simmons wrote that the New Math produced students who had “heard of the commutative law, but did not know the multiplication table.”

— excerpts taken from Wikipedia