In 2015, Siegfried Engelmann wrote that K-8 math "students should be grouped homogeneously, placed in the instructional programs according to their skill level, and taught at a rate that assures they will perform at about 100% by the end of the daily lesson." The methods used to teach content must be efficient and must work. We don't do anything like this in our schools; consequently, our students grossly underperform. I think Engelmann is one of the few people in education who made sense.
Applying Engelmann's example: If the learning objective is for students to "memorize the sums for all pairs of numbers in the shortest period (efficiency), students should memorize the answers of a performance test in which items appear in an unpredictable order. "They are not permitted to count on their fingers or use any other types of counting or calculating. No calculators. If students score nearly perfectly on an exhaustive test of addition problems presented in random order, then the instruction was highly effective."
Arithmetic is not always exciting until it is related to algebra, geometry, number theory, finance, engineering, technology, science, especially physics, etc. Also, it is the foundation for higher mathematics. Learning arithmetic is hard work and not always fun.
Dr. Robert B Davis, Director of The Madison Project, wrote in 1965, "We need to produce well-educated people is becoming more and more the central problem of our society. Within education, mathematics and science are assuming ever greater importance."
Arithmetic is too narrow of a math curriculum for K-8. Students need to learn algebra and geometric fundamentals. Davis proposed that in addition to arithmetic, some of the fundamental concepts of algebra should be taught, including "variable, function, the arithmetic of signed numbers, open sentences...." Also, some fundamental concepts of coordinate geometry should be taught, such as a "graph of a function."
My Teach Kids Algebra (TKA) enrichment algebra program attempted to broaden the math curriculum starting in grades 1 to 4. It included both algebra and geometry concepts. Note: Total instruction time was 7 hours in the 1st and 2nd grades. For grades 3 and 4, the instruction was weekly for an hour.
4. Function Rules & Building x-y Tables
5. Plotting Points in Q-I & Finding Perimeters
6. Graphing Linear Equations in Quadrant I
7. Given y, Find x (Reverse, Undo) & Steepness of a Line (Slope)
Robert Davis writes, "Arithmetic cannot be clearly understood all by itself. It becomes clearer as one sees it in relationship to algebra and coordinate geometry." Davis says that arithmetic becomes significant only when it is "combined with algebra, geometry, and science." According to Davis, "Students need to develop the ability to discover patterns in abstract situations, acquire a reasonable mastery of important techniques (standard algorithms), know basic mathematical facts such as -1 x -1 = 1, and learn basic ideas of mathematics such as variable, table, function, and graph." These are cognitive ideas, but they won't matter much unless attitudes toward math change. Math should be valued, not trivialized. Teaching algebra to very young children has helped me understand how children learn. Lastly, Dr. Davis wrote, "The effectiveness of a program must be judged not by what was taught but rather by what is learned." (Note. Davis' program started in the 3rd grade.)
My idea to broaden math beyond arithmetic for elementary students, especially in grades 1 to 4, was to fuse algebra to conventional arithmetic. Part of my inspiration came from The Madison Project of 1957 by Dr. Robert Davis. However, I discounted Davis' ideas that children should invent algorithms and discover insights or patterns that took geniuses like Gauss and Newton to figure out. Indeed, "minimal guidance" methods such as discovery learning in groups do not assure high performance or efficient learning. Moreover, basic school arithmetic and algebra do not change. Regardless of the rhetoric from educationist leaders, there is a strong attitude against math in our society; consequently, the negative attitude often permeates our classrooms.
Russian students are taught more content earlier.
Applying Engelmann's example: If the learning objective is for students to "memorize the sums for all pairs of numbers in the shortest period (efficiency), students should memorize the answers of a performance test in which items appear in an unpredictable order. "They are not permitted to count on their fingers or use any other types of counting or calculating. No calculators. If students score nearly perfectly on an exhaustive test of addition problems presented in random order, then the instruction was highly effective."
Arithmetic is not always exciting until it is related to algebra, geometry, number theory, finance, engineering, technology, science, especially physics, etc. Also, it is the foundation for higher mathematics. Learning arithmetic is hard work and not always fun.
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| First-Grade Student in My TKA Algebra Class 2011 |
Dr. Robert B Davis, Director of The Madison Project, wrote in 1965, "We need to produce well-educated people is becoming more and more the central problem of our society. Within education, mathematics and science are assuming ever greater importance."
Arithmetic is too narrow of a math curriculum for K-8. Students need to learn algebra and geometric fundamentals. Davis proposed that in addition to arithmetic, some of the fundamental concepts of algebra should be taught, including "variable, function, the arithmetic of signed numbers, open sentences...." Also, some fundamental concepts of coordinate geometry should be taught, such as a "graph of a function."
My Teach Kids Algebra (TKA) enrichment algebra program attempted to broaden the math curriculum starting in grades 1 to 4. It included both algebra and geometry concepts. Note: Total instruction time was 7 hours in the 1st and 2nd grades. For grades 3 and 4, the instruction was weekly for an hour.
➡ 1st Grade Teach Kids Algebra (TKA), Spring 2011
Fusing Algebra to Standard Arithmetic
1. True False & Equality (=) "Think,Like A Balance."
2. Equations in One Variable, Guess & Check
3. Equations in Two Variables (Input-Output Function Model)4. Function Rules & Building x-y Tables
5. Plotting Points in Q-I & Finding Perimeters
6. Graphing Linear Equations in Quadrant I
7. Given y, Find x (Reverse, Undo) & Steepness of a Line (Slope)
Robert Davis writes, "Arithmetic cannot be clearly understood all by itself. It becomes clearer as one sees it in relationship to algebra and coordinate geometry." Davis says that arithmetic becomes significant only when it is "combined with algebra, geometry, and science." According to Davis, "Students need to develop the ability to discover patterns in abstract situations, acquire a reasonable mastery of important techniques (standard algorithms), know basic mathematical facts such as -1 x -1 = 1, and learn basic ideas of mathematics such as variable, table, function, and graph." These are cognitive ideas, but they won't matter much unless attitudes toward math change. Math should be valued, not trivialized. Teaching algebra to very young children has helped me understand how children learn. Lastly, Dr. Davis wrote, "The effectiveness of a program must be judged not by what was taught but rather by what is learned." (Note. Davis' program started in the 3rd grade.)
My idea to broaden math beyond arithmetic for elementary students, especially in grades 1 to 4, was to fuse algebra to conventional arithmetic. Part of my inspiration came from The Madison Project of 1957 by Dr. Robert Davis. However, I discounted Davis' ideas that children should invent algorithms and discover insights or patterns that took geniuses like Gauss and Newton to figure out. Indeed, "minimal guidance" methods such as discovery learning in groups do not assure high performance or efficient learning. Moreover, basic school arithmetic and algebra do not change. Regardless of the rhetoric from educationist leaders, there is a strong attitude against math in our society; consequently, the negative attitude often permeates our classrooms.
Russian students are taught more content earlier.
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| 3rd Grade Algebra Lesson |
Students do not need discussions as a means of solving a word problem. They need a knowledgeable teacher to show them the best way to solve a typical problem type and recognize the patterns of common problem types.
Davis and The Madison Project (1957), Science--A Process Approach (1967), and mavericks like Zig Engelmann* inspired me to teach basic algebra ideas to grades 1 to 3 in 2011. Teach Kids Algebra (TKA), my algebra enrichment program, was an attempt to broaden the math curriculum starting in the 1st grade. Algebra obeys the rules of arithmetic (i.e., properties of numbers and equality). Still, my 3rd-grade algebra program in 2011 did not measure up to the best Russian program. The main reason was that the Russian students were much better in basic arithmetic.
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* Reference is made to Zig Engelmann's 1966 film with pre-first-grade students (Kindergarten children) doing math by memorizing multiples and solving equations. My first-grade students used guess-and-check, addition facts, and rules to find the solution to the equation 3 + x = 12.
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Note: Teachers should focus on the mastery of essential content, not state test-based proficiency. I do not think this will change much because state test proficiency is the reason for Common Core state standards and the Every Student Succeeds Act (ESSA), which replaced No Child Left Behind (NCLB). Also, we have been unwilling to sort students according to their achievement for math class. Instead, we toss kids together, mixing high achieving kids with low achieving kids in elementary school for equity, which is a "fallacy of fairness," says Thomas Sowell.
Davis and The Madison Project (1957), Science--A Process Approach (1967), and mavericks like Zig Engelmann* inspired me to teach basic algebra ideas to grades 1 to 3 in 2011. Teach Kids Algebra (TKA), my algebra enrichment program, was an attempt to broaden the math curriculum starting in the 1st grade. Algebra obeys the rules of arithmetic (i.e., properties of numbers and equality). Still, my 3rd-grade algebra program in 2011 did not measure up to the best Russian program. The main reason was that the Russian students were much better in basic arithmetic.
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* Reference is made to Zig Engelmann's 1966 film with pre-first-grade students (Kindergarten children) doing math by memorizing multiples and solving equations. My first-grade students used guess-and-check, addition facts, and rules to find the solution to the equation 3 + x = 12.
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The major idea for solving an equation is to do the algebra operations in a proper sequence so that the equation is always balanced (Left Side = Right Side). An efficient sequence of steps is important. Also, the idea of equality (=) can be demonstrated on an equal-arm balance. The meaning of the equal sign is not taught well in elementary school arithmetic. 3 + 5 = 8 because the left side is 8, and the right side is 8. Thus, 8 = 8 makes a true statement (Think, Balance). Math deals with true statements, not false statements.
However, by the 3rd grade, the numbers are larger, and "guess and check" is no longer effective. First-grade students solved 12 + 18 - 18 = x by pattern recognition (undo or inverse). The undo idea can be demonstrated on an equal arm balance with smaller numbers. Also, it makes sense that if you start with a number (5) and add 2 to it and then subtract 2, you end with where you started: 5 + 2 - 2 = 5. An inferential leap is needed when using larger numbers or different types of numbers such as negative numbers or fractions: x + 87 - 87 = x, which leads to equation solving: x + 87 = 173 by subtracting 87 on both sides to undo add 87: x + 87 - 87 = 173 - 87 (Therefore, x = 86.)
Note: Each of the two 1st-grade classes had 14 half-hour lessons. Each of the two 2nd-grade classes had seven hour-long lessons. I appreciated the cooperation of the four classroom teachers at the urban Title-1 school in Tucson. Almost everything I introduced to students involves basic arithmetic, such as the auto-recall of single-digit addition facts, number and equality properties rules, and so on.
Performance
Performance can be strengthened by upgrading the curriculum (it's not world-class) and downplaying low-value instructional methods (minimal guidance = minimal learning). Students don't need a steady diet of group work and discovery learning (i.e., reform math). Teaching the "test items" is a fragmented curriculum. Teachers should explain math via explicit instruction with worked examples on the board, and students need to practice and review the basics for mastery. A review should be built into each lesson. But first, teachers need a top-notch curriculum that advances most students to Algebra no later than middle school. A K-8 world-class curriculum with high-impact instructional methods does not exist in many American classrooms.
© 2018 LT/ThinkAlgebra
