Sunday, June 19, 2011

Some thoughts on teaching little kids algebra

Teach Kids Algebra Project (TKA: 1st, 2nd, 3rd Grades)
I shall never forget these kids! 
The idea that little kids cannot learn algebra ideas is nonsense.
TKA is a response to reform math and Common Core.

Special Insert 10-30-17
"Equations are the lifeblood of mathematics, science, and technology," points out mathematician Ian Stewart (In Pursuit of the Unknown, 2012). It is the reason that students need to learn to write equations, rearrange them, deal with them and become skilled in solving them, all paper-pencil, not only for math class but also for chemistry and physics classes, etc. Also, knowing trig is important in physics courses. Right triangle trig used to be taught in good 7th-grade pre-algebra courses, but it is no longer the case. (Also, the 1970 Dolciani Algebra-1 textbook included a chapter on Geometry & Trigonometry. The trig problems were physics problems that are found in algebra-based physics courses. Other problems involved finding the vertical and horizontal components of a vector and the resultant of vectors. Students learned to solve trig equations that solved physics problems.) 

The use of graphing calculators and the shift away from solving equations using traditional algebra have produced weak algebra students over the years. It is one reason that early algebra is important in our schools.
End Insert 

TKA 3rd-Grade Students (2011)

Algebra is arithmetic plus variables. A variable is a symbol, such as x, y, or ☐, that represents an unknown number. For little kids, it is important to stress that a variable is a number
Students start with numerical equations (true/false) and the idea of equality, go to equations in one variable, proceed to equations in two variables (functions: input-output model), etc. Little kids solve equations using guess and check, memorized math facts, number and equality properties, procedures, and logic. This prepares them for algebraic methods for solving equations (e.g., the addition property of equations, etc.) by 3rd or 4th grade. Algebra ideas are not difficult if students know math facts (automaticity) and pay attention in class.  

● Variables, Equations, tables, graphs

Little kids, including typical 1st-grade students, can 
1. solve a range of linear equations using guess and check, number facts, and properties; 
2. write equations in one unknown to model a word problem (The idea is to translate words into symbols);  
3. build tables that show numerical relationships; and 
4. plot points and graphs of linear equations.
The idea of substitution is an important preparation for Algebra I. 
● Student Comments
1. "When you give us work it's fun because we don't know the answers." 
MV, 3rd Grade TKA Student
2. "I look to you to keep my brain working. I was able to understand what you were teaching. You have helped me with math I used to struggle with. The math that you taught me is amazing. You have taught me never give up, keep on trying. Thank you for all you have done for me."  MR, 3 Grade TKA Student

● Effort, persistence, and practice (EPP)
The idea that kids need innate ability to learn arithmetic and algebra is bunk. Kids need the skilled teaching of content and lots of practice to master arithmetic and algebra. Furthermore, they must work hard, i.e.,  effort, persistence, and practice count. Indeed, according to Daniel Willingham, a cognitive scientist, the “vast majority of K-12 students” can learn arithmetic and algebra; however, this should not imply that learning math is easy because learning math does not come naturally. Willingham points out, “It takes time, effort, and mastering increasingly complex skills and content.”  Math is hierarchical and should be taught so that one idea builds on another. It is logic. The logic begins with the transitive property of equality (Think Like A Balance) and true and false statements. Reasoning in math (new true statements are linked to other true statements) is different from reasoning in science (inferences are based on observations). In short, problem-solving is always domain-specific and requires domain knowledge.
● Equivalency (=)
Equivalency, which is a fundamental concept in mathematics, is seldom taught well. For example, 3 + 5, which is 8 {true}, is equivalent to 10 - 2, which is 8 {true}.
Therefore, 3 + 5 = 10 - 2 is a true statement. 
The logic behind the true statement is the transitive property of equality: two things equal to the same thing {8} are equal to each other. In Teach Kids Algebra, 1st, 2nd, and 3rd-grade students apply the "equivalency idea" (Think Like A Balance) and guess and check to find an unknown: 3 + 5 = x - 2. It is not possible to determine whether this statement is true or false without substituting a number. The box is a variable like x, and it can represent any number. But, to make a true statement, x is 10, not 8.
● Algebra, the Higher Arithmetic
The algebra lessons I give to 125 1st, 2nd, and 3rd-grade students require them to use number facts, number procedures, and number properties or laws (arithmetic). In short, algebra is built on arithmetic. As Morris Kline states, “Algebra is the higher arithmetic.” If we want kids to learn algebra, then they must be good at arithmetic. In primary school, kids who have auto recall of number facts and experience with standard algorithms do better and understand more. 
● Mini-Lessons: Lecture And Feedback
Manipulatives and calculators are not used. Students do not color things, paste things, cut things out, or work in groups. I lecture (explain how things work with examples), write stuff on the board as I explain things (visual-auditory), and ask students questions as I go (interaction). Then, I hand out problems for students to try on their own (guided practice) and roam around the room talking to students, giving feedback, and providing individual help. The classroom teacher helps a lot too. All this is accomplished in a 30 minute period, which is often too short. Explicit instruction works best
FYI: I met one 3rd grade class twice a week for at least one hour each time. Typically, the session would last 15 to 20 minutes more than an hour We always ran out of time. (Ms. S., the classroom teacher, was generous with time.) The students work hard for an hour straight, usually longer, but they do not seem to mind. Time passes quickly. I was not only teaching fundamental algebra ideas, but I was also teaching persistence and effort. One 3rd grade student wrote to me, "I had a hard time with algebra, and because of you I got better with algebra. I enjoyed learning because you made my brain work." 
At first, the math is challenging, but I keep reassuring and encouraging students, often one-to-one, giving them support: "Try a different number. Do not give up! You can do this. Let me show you how. Try this. You can learn this--it's great stuff." And, they did. The success of the program hinges on talking to individual students and giving them important feedback and encouragement. Kids need adult support. 

● The Equation As Model: Translating words into symbols
When I present a word problem, let's say in 1st or 2nd grade, I model it with an equation. For example, Jill has some pencils. Bill gives her 5 more pencils. Now Jill has 12 pencils. How many pencils did Jill have before Bill gave her pencils? Jill has "some pencils" is represented by a variable I call x. Step-by-step, I piece together an equation: x + 5 = 12 on the board by asking questions: Do we add or subtract 5? What do we do with 12? The equation is the model. Kids do not need bar models to understand simple problems. They learn to write equations. They learn to translate words into symbols. It is important that students identify the unknown in a word problem. This starts in 1st grade. Identifying the unknown and solving problems from an algebraic perspective makes sense.  

For older kids (3rd grade), the equations become more complicated. Write an equation, then solve it. I am thinking of a number. Six less than triple a number is 15. What is the number? 
Equation: x + x + x - 6 = 15
Solution: x = 7  by inspection (7 + 7 + 7 - 6 = 15; 15 = 15). 

Note. Being able to translate words into symbols (equations) and being able to solve the equations is what algebra is all about.

Note. It is okay for students to struggle because math is not always about getting the right answer. It is about developing young minds and improving their effort, persistence, and reasoning in solving problems. Furthermore, math builds the brain. It makes kids smarter. In short, a cognitive struggle is a good thing.
● Myth
Willingham also observes that “our society has accepted the fact that math is not for most us;” however, he says that this “notion is a myth.” The idea “I am not good at math” is rooted in our culture. We need a radical change in our attitudes toward learning and schooling. 

● Cognitive Horsepower 
Attention is essential for learning. Attention is controlled by something called executive function. Gary Stix (How to Build a Better LearnerScientific American, August 2011) says that executive function encompasses important cognitive attributes such as the ability to "be attentive, hold what you have just seen or heard in the mental scratch pad of working memory, and delay gratification. These skills (being attentive, holding stuff in working memory, and delaying gratification) often predict a child’s success in school. 
Children who have difficulty concentrating also have difficulty learning mathematics. Teaching little kids algebra is a blast, but learning algebra ideas requires mental fluency with arithmetic facts and procedures and sufficient concentration to hold stuff in working memory.  
In Teach Kids Algebra lessons, for example, students deal with a lot of new stuff all at once in a short time. This s t r e t c h e s working memory and pushes students to concentrate. Knowledge of key math facts and procedures in long-term memory (automaticity) helps a lot because of working memory space, although somewhat "plastic," has limitations. In problem-solving, too often students figure out simple facts that should have been memorized (e.g., 3 + 8). This "figuring" wastes time and working memory space and distracts from solving the problem. Facts should be memorized and stored in long-term memory for a child to excel in mathematics. The focus in working memory should be on solving the word problem, not on figuring out simple facts.   
A child's working memory structure, including the mental scratchpad, is in place by age 6. It has less capacity than an adult's working memory, but it improves somewhat as children grow. Working memory, however, has limitations. It can hold only so much stuff at a given time before becoming overloaded. In contrast, long-term memory does not have this limitation. In mathematical problem solving, it is important that key math facts and procedures are in long-term memory (background knowledge) so that working memory can hold all the essential information from the problem to devise a plan for solving. Also, negative thoughts about math (e.g., math anxiety) can often crowd working memory, which is another concern. (Information from From Stix, Willingham, Beilock)
According to Sian Beilock (Choke), "Working memory is your cognitive horsepower. It involves the ability to hold information in mind (and protect that information from disappearing) while doing something else at the same time." Beilock says that "working memory is one of the major building blocks of IQ." Working memory can be developed, so it is important to practice, stretch, and exercise working memory with challenges to improve cognitive muscle. Beilock points out that "practice shapes your brain." To learn math well, for example, requires both practice and challenges. Also, a student will not learn much math if he is easily distracted or has difficulty with attention in class.
Daniel T. Willingham, a cognitive scientist, states that students learn what they are thinking about, which takes sharp attention. The ability to control attention and hold information are skills that can be trained. Bronson & Merryman (Nurture Shock), suggest that "being able to concentrate [cognitive control] is a skill that might be just as valuable as math ability, or reading ability or even raw intelligence." 
Of concern is that a child's attention span or ability to concentrate has been declining over the past 20 years. The Net-Generation tends to bounce from task to task. Click, Click, Click! They often have difficulty focusing on one task and doing it well. They do not reflect or think through things. Nicholas Carr (The Shallows) says the Net causes brain changes. The biggest change is that students have difficulty concentrating. He observes, "Tests of memorization, vocabulary, general knowledge and even basic arithmetic have shown little or no improvement."  The Net does not make you smarter; school makes you smarter. The Net, says Carr, should not be a "replacement for memory." Remembering is a fundamental cognitive skill that is needed for problem-solving. Math facts and procedures must be retrievable from long-term memory so they can be used in working memory to solve problems.
Aimee Cunningham (Kids' Self-Control Is Crucial for Their Future Success, Scientific American, July 25, 2011) points out that a child’s “self-control is crucial for their success.” She writes [Long Quote], “Self-control—the ability to regulate our attention, emotions, and behaviors—emerges in childhood and grows throughout life, but the skill varies widely among individuals. Past studies have reported that self-control is partially inherited and partially learned and that those with less self-control are more likely to be unemployed, engage in unhealthy behaviors such as overeating, and live a shorter life.” 
● Common Brain Myths
Teachers and parents should be aware of common brain myths. Gary Stix (Scientific American, August 2011) lists five myths from Mind, Brain, and Education Science (Takuhama-Espinosa, 2010). Here are two of the myths. 
1. “Left-brain” and “right-brain” people differ. No. “Brain-imaging studies show no evidence of the right hemisphere as the locus of creativity. And the brain recruits both left and right sides for both reading and math.” 
2. Each child has a particular learning style. No. There is little evidence to support this claim. “For this and other myths, public perceptions appear to have outstripped the science.” Many parents tell me their child is a visual learner or a kinesthetic learner (not an auditory learner). These popular perceptions held by many parents and educators are not backed by evidence. 
In education, "there is an enormous supply of totally untested, untried, and not very scientific methods.”   
(To Be Revised)

LT, Guest Teacher
LT, Founder of  ThinkAlgebra
Updates: 6-19-11, 6-20-11, 6-21-11, 6-30-11, 7-1-11, 7-2-11, 7-9-11, 7-20-11, 7-23-11, 7-24-11, 7-25-11, 7-26-11, 7-31-11, 8-1-11, 9-21-11, Minor grammar corrections made on 4-29-17
Photos by 3rd grade classroom teacher: CSmith