Kids Can Learn Harder Stuff--if they work hard & if their teachers work hard!
Build Mastery Through Practice!
Some educators say that little kids cannot learn algebra or memorize multiplication facts because it is too hard. I often hear, “Kids are not developmentally ready for that.” Or, “That’s too advanced.” Or, “Our kids cannot compete with Singapore kids.” Or, "Rote learning is bad learning." It’s bunk! It is low expectations! And, it is contrary to experience and cognitive science. Repetition, multiple exposures, effort, and extensive practice are important. Students need strong teacher guidance, explicit instruction, a world-class curriculum, and lots of practice to learn arithmetic and algebra well. In short, our students need to master a world-class curriculum starting in grade 1 to stay competitive. It takes effort, work, and the development of smart thinking. Students cannot apply arithmetic or algebra that they do not know well. Thus, acquiring background knowledge quickly is critically important. Student achievement, which is graded C- by Quality Counts 2012, should be the highest priority in our classrooms. What has gone wrong? It boils down to the curriculum, which is the content in the textbook or the math program. Beverlee Jobrack (Tyranny of the Textbook), observes that "standards, achievement testing, technology, and professional development [and other reforms] have had little effect on improving student achievement" (p. xx). It also boils down to instructional methods.
We need a world-class curriculum that is logically sequenced, organized by familiar course names, and focused on achievement.
Such a math curriculum is not found in many textbooks or popular math programs. The reason students do not do well in arithmetic or algebra is a lack of pivotal background knowledge (Knowing). (The same is true for reading comprehension.) Students cannot think about math ideas, do the math, or solve problems in math if they do not know the key fundamentals of arithmetic and algebra. Unfortunately, most students underperform in math? If teachers do the same things they have done before, then how can they expect major academic improvement? Teachers must analyze what they teach (content), how they teach it (instruction), and how they measure what they teach (assessment). See Common Core on ThinkAlgebra. Updated 1-21-12, 9-2-12
[FYI: Random Thoughts: I do not think Singapore textbooks are sufficient either. Typical students in 1st-grade can learn basic algebra ideas that use, connect to, and reinforce basic arithmetic concepts and skills (see Teach Kids Algebra project in March-May 2010). If I had the technology, every first-grade student would learn to write procedures that make polygons (squares, rectangles, etc.) using the LOGO programming language, which is what my 1st-grade class did in the early 80s. Also, I suggested years ago that teachers break from the textbook and the status quo and explicitly teach content and skills that are above those expected at a given grade level. This takes good instruction and practice. Teach grade level and above because teaching at grade level sets the bar too low. Also, teachers must accept that learning math well requires hard work.] Updated 1-21-12, 9-2-12
[FYI: Random Thoughts: I do not think Singapore textbooks are sufficient either. Typical students in 1st-grade can learn basic algebra ideas that use, connect to, and reinforce basic arithmetic concepts and skills (see Teach Kids Algebra project in March-May 2010). If I had the technology, every first-grade student would learn to write procedures that make polygons (squares, rectangles, etc.) using the LOGO programming language, which is what my 1st-grade class did in the early 80s. Also, I suggested years ago that teachers break from the textbook and the status quo and explicitly teach content and skills that are above those expected at a given grade level. This takes good instruction and practice. Teach grade level and above because teaching at grade level sets the bar too low. Also, teachers must accept that learning math well requires hard work.] Updated 1-21-12, 9-2-12
Current Textbooks & Popular Math Programs "Perpetuate Mediocrity.”
Textbooks have been the de facto curriculum (what is taught in the classroom), and, according to Beverlee Jobrack (Tyranny of the Textbook), textbooks have changed little over the years and "perpetuate mediocrity," regardless of the many math reforms. Textbooks sell, not because they are effective in student learning, but because of their "design, labeling, and teacher appeal" (p. xviii). Jobrack writes, "Textbooks sell based on design and superficial features, not because they are based on the latest research on how children learn [cognitive science] and how well they promote student achievement." In my experience, K-8 math textbooks are cluttered with colorful, eye-catching graphics with extra topics included. But they often lack a logical, well-organized sequence of concepts, skills, and applications (scope and sequence) that lead to algebra in middle school. Moreover, textbooks often lack enough challenging content and sufficient practice exercises for skill mastery. After over 40 years in education, I have observed that math textbooks and popular math programs are not focused on achievement; they also marginalize or ignore the basics of cognitive science. Research-based does not mean that the textbook was tested for its effectiveness because effectiveness is not a priority in publishing. Most teachers tenaciously follow the textbook, no matter what, and, therefore, students get only bits and pieces and practice very little. Jobrack writes that the most effective curriculum materials do not appeal to most teachers and are rarely adopted (p. 1). Updated 1-21-12, 1-22-11
Of course, each state has a rough "curriculum framework" or set of standards [Common Core], but, in reality, teachers use state-approved textbooks for curriculum. Moreover, the difference between what teachers teach (the textbook curriculum) and what students actually learn, know, or master is large, while the difference in Singapore, for example, is much smaller. In short, the minimal guidance instructional methodologies that most teachers were taught in ed school and that permeate most math programs and textbooks have not been effective. Many instructional strategies (e.g., group work, differentiated instruction, learning styles, etc.) are especially ineffective. Furthermore, praising students for no good reason and telling students they are smart have backfired [self-esteem movement]. Teachers should stop inflating grades and focus on the child's competency. Also, teachers should make nested connections (i.e., connections within content) but most do not know how because elementary teachers are trained as generalists, not math people, according to Dr. H. Wu. Students need to practice a lot to master content. For example, math facts must be memorized early and used repeatedly.
Of course, each state has a rough "curriculum framework" or set of standards [Common Core], but, in reality, teachers use state-approved textbooks for curriculum. Moreover, the difference between what teachers teach (the textbook curriculum) and what students actually learn, know, or master is large, while the difference in Singapore, for example, is much smaller. In short, the minimal guidance instructional methodologies that most teachers were taught in ed school and that permeate most math programs and textbooks have not been effective. Many instructional strategies (e.g., group work, differentiated instruction, learning styles, etc.) are especially ineffective. Furthermore, praising students for no good reason and telling students they are smart have backfired [self-esteem movement]. Teachers should stop inflating grades and focus on the child's competency. Also, teachers should make nested connections (i.e., connections within content) but most do not know how because elementary teachers are trained as generalists, not math people, according to Dr. H. Wu. Students need to practice a lot to master content. For example, math facts must be memorized early and used repeatedly.
Curriculum & Instruction
In my view, curriculum and instruction have been the biggest problems we have had in math education. Jobrack says that "educational publishers must play a much greater role in improving education" (p. xii). The textbooks add significantly to the problem of inadequate math learning because they lack content and coherency, are not based on cognitive science and do not "promote student achievement." Jobrack adds, "The [Common Core] standards are not the curriculum. The curriculum is what teachers do every day, introduce a concept, check for understanding, have students practice, and assess it." In short, the math curriculum has been the textbook, and, over the years, the curriculum in classrooms has not changed much. FYI: The Quality Counts 2012 annual report, which grades each state, gives K-12 student achievement in the United States a C-.
Cognitive Science: Practice, The Key to Competency
Clifford Stoll (Berkeley astronomer and physicist) says that explicit practice has fallen out of favor, "yet it ['repeated drill'] is the core of mathematical competency." For example, in 3rd grade, the multiplication algorithm, finding areas (squares, rectangles, triangles), word problems, and counting problems are excellent ways to practice or review multiplication facts, but Common Core delays the mastery of the standard algorithm until 5th grade for no good reason. [Note. Common Core 5th Grade (p. 35): "Fluently multiply multi-digit whole numbers using the standard algorithm."In Common Core, "fluency" does not necessarily imply "automaticity." In my view, the explicit practice has been replaced by manipulatives, calculators, group work, and strategies (e.g., Everyday Mathematics).] Added 1-16-12, Revised 1-17-12
Cognitive Science: Practice, The Key to Competency
Clifford Stoll (Berkeley astronomer and physicist) says that explicit practice has fallen out of favor, "yet it ['repeated drill'] is the core of mathematical competency." For example, in 3rd grade, the multiplication algorithm, finding areas (squares, rectangles, triangles), word problems, and counting problems are excellent ways to practice or review multiplication facts, but Common Core delays the mastery of the standard algorithm until 5th grade for no good reason. [Note. Common Core 5th Grade (p. 35): "Fluently multiply multi-digit whole numbers using the standard algorithm."In Common Core, "fluency" does not necessarily imply "automaticity." In my view, the explicit practice has been replaced by manipulatives, calculators, group work, and strategies (e.g., Everyday Mathematics).] Added 1-16-12, Revised 1-17-12
Cognitive Science: Thinking [Learning] Is Hard Work!
Dr. Art Markman (Smart Thinking) writes that students who do not know content and who are not persistent will have difficulty solving math problems (“Smart Thinking"), such as word problems. Markman, a cognitive scientist, says that, like chess, smart thinking is a skill that can be learned. In short, for students to solve math problems, they need to learn lots of basic content to build a tool kit for problem solving. It should start in 1st grade with the mastery of math facts (auto recall), procedures (algorithms), place value, equations, and properties of numbers. Moreover, Markman says that students learn new stuff better when it is linked to stuff they have already learned and practiced. Thus, coherent sequencing (Bruner), attention to prerequisites (Gagne), and nested connections (Markman) in arithmetic and algebra are important because math is highly structured and hierarchical; i.e., one idea builds on or links to another. In short, you cannot teach math like you teach social studies. (Note. The TIMSS definition of "Applying" in its advanced math framework means routine problems found in textbooks. Students must be good at routine problems before they proceed to non-routine problems.)
Cognitive Science: Math Ability Grows!
Dr. Art Markman (Smart Thinking) writes that students who do not know content and who are not persistent will have difficulty solving math problems (“Smart Thinking"), such as word problems. Markman, a cognitive scientist, says that, like chess, smart thinking is a skill that can be learned. In short, for students to solve math problems, they need to learn lots of basic content to build a tool kit for problem solving. It should start in 1st grade with the mastery of math facts (auto recall), procedures (algorithms), place value, equations, and properties of numbers. Moreover, Markman says that students learn new stuff better when it is linked to stuff they have already learned and practiced. Thus, coherent sequencing (Bruner), attention to prerequisites (Gagne), and nested connections (Markman) in arithmetic and algebra are important because math is highly structured and hierarchical; i.e., one idea builds on or links to another. In short, you cannot teach math like you teach social studies. (Note. The TIMSS definition of "Applying" in its advanced math framework means routine problems found in textbooks. Students must be good at routine problems before they proceed to non-routine problems.)
Cognitive Science: Math Ability Grows!
Understanding Grows Over Time.
If we want our children to excel in mathematics like kids in other nations, then our kids must know arithmetic and algebra flat out. Most don't. There are no short cuts. It takes consistent practice, hard work, and smart thinking (Markman) to learn anything well, especially mathematics. To establish good math habits, students need to build mastery through practice. Thinking, says Markman, is hard work because "your mind is designed to think as little as possible." Learning something new (i.e., content) in arithmetic or algebra takes effort and smart thinking. Repeatedly, cognitive scientists have told educators that math ability is not fixed; it grows through a child's persistence, effort, and hard work to learn new stuff, which grows brain cells. Students don't learn anything unless it is practiced well. Again, this takes effort, persistence, and hard work. Learning something new in arithmetic or algebra can be frustrating at times, but students must learn to persevere and get through it to foster cognitive growth. Thus, students need to develop persistence and an ethic of hard work to learn challenging content. In short, brain cells must grow to achieve in mathematics. It is about time we educators listen.
If we want our children to excel in mathematics like kids in other nations, then our kids must know arithmetic and algebra flat out. Most don't. There are no short cuts. It takes consistent practice, hard work, and smart thinking (Markman) to learn anything well, especially mathematics. To establish good math habits, students need to build mastery through practice. Thinking, says Markman, is hard work because "your mind is designed to think as little as possible." Learning something new (i.e., content) in arithmetic or algebra takes effort and smart thinking. Repeatedly, cognitive scientists have told educators that math ability is not fixed; it grows through a child's persistence, effort, and hard work to learn new stuff, which grows brain cells. Students don't learn anything unless it is practiced well. Again, this takes effort, persistence, and hard work. Learning something new in arithmetic or algebra can be frustrating at times, but students must learn to persevere and get through it to foster cognitive growth. Thus, students need to develop persistence and an ethic of hard work to learn challenging content. In short, brain cells must grow to achieve in mathematics. It is about time we educators listen.
We need to train children for the future, but they won’t have much of a future [very limited postsecondary choices] unless they master mathematics (arithmetic, algebra, and precalculus), which takes discipline, concentration, effort, and hard work. Most children grossly underachieve in mathematics and science. For nearly half a century, we have known that very young children can learn math faster and at a much higher level, but this is not the way math is taught in most schools. I ask educators to shake off the role of "guide on the side" and put aside group work and manipulatives. Teachers should explicitly teach challenging content and help students attain mastery through practice.
©2012 LT/ThinkAlgebra
