Learning Hierarchy (Gagne)
Teach Kids Algebra
Starting in the 1st grade, students learned about expressions, equality, equation structure (expression = expression), and letters like x and y that can represent unknown numbers. The students solved x using guess and check. They started with true/false: 3 + 4 = 10 - 2. The statement is false because the left side is 7, and the right side is 8. In short, 7 ≠ 8, therefore it is false. An equation is like a balance. The left side must equal the right side in value. For the equation x - 3 = 8, x must equal 11 to make a true statement: 8 = 8. Unlike science, math is absolute. It is not an opinion. It consists of true statements made from other true statements. For example, if 3 + 4 = 7, then 7 - 3 = 4. (Note that the equation 7 - 3 = 4 is true because 4 = 4.)
Teach Kids Algebra
Starting in the 1st grade, students learned about expressions, equality, equation structure (expression = expression), and letters like x and y that can represent unknown numbers. The students solved x using guess and check. They started with true/false: 3 + 4 = 10 - 2. The statement is false because the left side is 7, and the right side is 8. In short, 7 ≠ 8, therefore it is false. An equation is like a balance. The left side must equal the right side in value. For the equation x - 3 = 8, x must equal 11 to make a true statement: 8 = 8. Unlike science, math is absolute. It is not an opinion. It consists of true statements made from other true statements. For example, if 3 + 4 = 7, then 7 - 3 = 4. (Note that the equation 7 - 3 = 4 is true because 4 = 4.)
Many teachers don't know how to explain math to kids. They don’t know how to write behavioral learning objectives (Mager) or construct a hierarchical-based curriculum (Gagne). Often, elementary school teachers, even middle school teachers, are weak in some necessary math skills such as fractions, long division, and algebra. It's not about understanding, which is difficult to quantify; it's about knowledge in long-term memory and applying it.
When I first started giving algebra lessons to 1st through 3rd grades, I made a topic list. Then, I developed specific behavioral objectives (i.e., performance-based) (Mager) and sample problems (i.e., worked examples) to explain the performance. I often asked the students questions and gave them a couple of problems to work on their own (guided practice). Finally, I handed out the lesson's practice sheet (independent practice), which included both current and problems from previous lessons (review). For the remaining 30 minutes, I walk around the room to give encouragement and corrective feedback to individual students. I called my algebra program Teach Kids Algebra or TKA. Sessions were weekly for an hour. In short, I constructed my own curriculum.
In the 2018-2019 school year, I gave TKA lessons to two 4th grade classes. No group work. No Common Core. No manipulatives. No calculators. The only crutch was an integer number line. In the Spring, I gave algebra lessons to a class of 2nd-grade students. They received 6 hours of instruction. The 7th lesson was a culminating activity.
Gagne’s idea of curriculum development was hierarchical and indicated specific prerequisites and background knowledge. See the example below.
Break a problem into smaller problems.
It is a fundamental idea taught in mathematics, and it can carry over to everyday life. The idea that new knowledge builds on old knowledge is central to learning math. Because math builds in long-term memory, the proper sequencing that creates coherence (a learning hierarchy) in a math curriculum is paramount. Below is a sequencing example from Science--A Process Approach (SAPA), which had used Gagne’s hierarchical approach.
Part C is 2nd grade in the K-6 SAPA science sequence, but I used it in 1st-grade TKA. Incidentally, 4 of the 6 processes taught in the 1st-grade SAPA lessons (Part B) were arithmetic or math-related: using numbers (arithmetic), communicating (graphing), measuring (metric units: g, cm, m, mL), and using space/time relationships (geometry).
There are multiple problems in math education today. One is the lack of a coherent learning hierarchy (Gagne). The same is true for science.
Not only did SAPA upgrade science education, but it also shifted higher level math content down to lower grades. SAPA taught the math kids needed to know to do the science. In other words, the math in SAPA was much more advanced than K-6 students traditionally learned. In first-grade SAPA materials, 4 of the 6 processes were math-related. There has been nothing like it since the 60s.
In math, experienced teachers know what’s essential and what’s not. They can figure out prerequisites and develop curriculum. Most teachers can’t do this. They can't teach what they don't know well.
When I left K-8 classroom teaching (2000), I started tutoring high school mathematics, especially Algebra-2 and precalculus. Tutoring led me to this idea: If you can’t calculate it, then you don’t know it.
Teachers must use effective instructional methods, such as explicit teaching, memorization, practice (drill), and continual review so that fundamentals stick in long-term memory, not minimal guidance methods that are ineffective and lack scientific support.
Minimal Guidance = Minimal Learning
(Kirschner, Sweller, and Clark, 2006)
(Kirschner, Sweller, and Clark, 2006)
Lastly, if learning is remembering from long-term memory, then we have not been teaching children to learn and master essential content. Much is taught, but little is learned.
Also, we should not limit students to so-called grade-level content.
Reading Real Books
I think that much learning can be attained by reading books--history books, science books, math books, geography books, literature books, art books, and so on.
Some of the responsibility for learning should be placed on the shoulders of students, too, not just the classroom educator. Reading books outside the classroom was significant in my learning. It still is.
One GATE student expressed to me, "I hate reading screens or one to two-page handouts; I want to read real books, with physical pages I can feel and turn." Where are the books? Students don't have subject matter books in digital classrooms--not real books. Also, it's hard to focus (pay attention) because the GATE classroom is so noisy.
Reading Real Books
I think that much learning can be attained by reading books--history books, science books, math books, geography books, literature books, art books, and so on.
Some of the responsibility for learning should be placed on the shoulders of students, too, not just the classroom educator. Reading books outside the classroom was significant in my learning. It still is.
One GATE student expressed to me, "I hate reading screens or one to two-page handouts; I want to read real books, with physical pages I can feel and turn." Where are the books? Students don't have subject matter books in digital classrooms--not real books. Also, it's hard to focus (pay attention) because the GATE classroom is so noisy.
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