Problem Solving

Reform math programs launch into "problem-solving" backward by de-emphasizing the grade-by-grade mastery of basic arithmetic knowledge, both facts, and procedures, which are the very essentials needed for problem-solving. Polya's problem-solving strategies work only if the student has sufficient prerequisite knowledge needed to solve a specific problem. Thus, problem-solving is always deeply rooted in background knowledge in long-term memory. The primacy of background knowledge cannot be over-emphasized. Elementary school teachers should focus on making sure students master the fundamentals of arithmetic, both factual and procedural knowledge, starting in grade 1. A curriculum that is focused on problem-solving strategies and light on content knowledge does not cut it.  ThinkAlgebra [Draft I]

Let’s start here . . .
“Mathematics has the dubious honor of being the least popular subject in the curriculum . . . Future teachers pass through the elementary schools learning to detest mathematics . . . They return to the elementary school to teach a new generation to detest it.” You would think that this was written in 2011, but it wasn't. G. Polya was so concerned about math education that he wrote it in the 2nd edition preface (1956) of his famous book, How to Solve It. Actually, Polya quoted it from a study reported in Time magazine. Not much has changed in the past half-century. I think some teachers are doing a great job teaching math. We just do not have enough of them.    

This cycle has been entrenched in education for at least five decades because schools of education are not selective or academically demanding. I do not blame teachers; I blame those in charge of selecting, training, and certifying teachers. If we want better teachers in elementary and middle school math, starting in 1^st grade, then we need to educate and train them better and weed out teachers who dislike mathematics, who are mathphobic, or who demote the importance of mathematical [content] knowledge. Poyla believes a teacher's knowledge of mathematics and attitude toward mathematics rub off on students. He writes, "Yet it should not be forgotten that a teacher of mathematics should know some mathematics and that a teacher wishing to impart the right attitude of mind toward [math] problems to his students should have acquired that attitude himself." The book was first published in the U.S. in 1945.

Polya poses a problem.
The length of the perimeter of a right triangle is 60 inches and the length of the altitude perpendicular to the hypotenuse is 12 inches. Find the sides?

A student cannot solve this problem without substantial knowledge of high school mathematics (algebra and geometry). But, isn't prerequisite knowledge necessary for any math problem, at any level, even for routine problems? Knowledge first! In mathematics, students should start with basic arithmetic and routine problems to build a storehouse of knowledge and experience in long-term memory before moving to more complex problems that take more insight. The idea that students can do problem-solving without fundamentals in place is illogical and backwards, yet this is what many teachers think. Elementary students should focus on mastering basic arithmetic and routine word problems, grade by grade. This requires solid practice.

I pose a chemistry problem.
Calculate the grams of hydrogen required to produce 82.000 grams of ammonia from nitrogen and hydrogen gasses.

Would you attempt to solve this routine chemistry problem without knowing the fundamentals of high school chemistry? Of course not! Solving problems requires domain-specific knowledge. Moreover, learning to solve routine problems, whether they be in chemistry or elementary school arithmetic, presupposes both knowledge and practice.   


I pose a Latin problem.
Ego vos hortor ut amicitiam ombibus rebus humanis anteponatis. Sentio equidem, excepta sapientia, nihil melius homini a deis immortablibus datum esse. 


Would you attempt to translate Latin without knowing the fundamentals of Latin? Of course not. Translating Latin requires domain-specific knowledge. 

Knowing builds the foundation for higher-level thinking.
You cannot apply something you do not know well.

Knowing builds the foundation for higher-level thinking . . .
The range of cognitive skills (right), starting with a strong base of Knowing, is similar to Bloom's taxonomy. Applying requires Knowing, and Reasoning implies both Knowing and Applying. Teachers should start at the bottom and focus on Knowing (both factual and procedural knowledge in arithmetic and algebra). This builds the foundation for higher thinking, such as Applying and problem-solving. In TIMSS, Applying is solving routine problems. (This is problem-solving.) Furthermore, knowing something takes substantial practice. You cannot apply something you do not know well [in long-term memory]. 

Research

We tend to believe what we think, but our assumptions are often wrong. 
Sweller, Clark, and Kirschner [2] write that the results of research in problem-solving in mathematics are “both counterintuitive and contrary to many widely held views. For example, many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge.” The assumption is wrong and unproven. It is a gross misinterpretation of Polya by many math educators, special interest groups (e.g., P21), and others. 

Practice Fundamentals
Knowledge Is Key

According to Polya, when a student attempts to solve an unfamiliar problem, the student should think of a related problem and then, by analogy, try to solve the original problem. He writes, “We may consider ourselves lucky when, trying to solve a problem, we succeed in discovering a simpler analogous problem.” But, thinking of a simpler analogous mathematical problem, solving it, and applying it (“extrapolating” it to solve the original problem) require specific content knowledge. Moreover, a simpler analogous problem can have both similarities and dissimilarities. As one reads Polya’s book, it becomes clear that students need extensive mathematical knowledge and “determination” to do problem-solving. In short, problem-solving in mathematics requires sufficient mathematical knowledge in long-term memory, practice, and “determination.” There are no shortcuts. 


Sweller, Clark, and Kirschner point out, “There is no body of research based on randomized, controlled experiment indicating that such teaching [generalized problem-solving strategies] leads to better problem-solving.”  They observe, “Recent reform curricula both ignore the absence of supporting data and completely misunderstand the role of problem-solving in cognition.” 

Note. Reform math champions and elevates the idea of “general problem-solving skills,” group work, spiraling of content, and calculator use. At the same time, these “problem-based” programs minimize or demote essential content knowledge. This is no accident; it is by design. In my view, this upside-down relationship is a crucial flaw in reform math programs. Students must practice content to learn it and to apply it. 


Ze'ev Wurman, in a recent blog (7-16-11), writes, "The overwhelming majority of children can reasonably easily learn what we teach in our K-12 schools, given competent teachers and effective teaching methods." But this is not what happens in K-12. Most of our students remain mediocre at best (TIMSS) and only about 30% are proficient in math (NAEP). Furthermore, tens of thousands of incoming students flood remedial math courses at community colleges. Wurman says, "[The] cause must be in how we teach our students in school and outside it." We do not come close to teaching content that nearly all Singapore students learn in grades 1-9. By 9th grade, virtually all Singapore students (99.9%) have covered all of Algebra 1 and Geometry, according to Wurman. (Note. Ze'ev Wurman was one of the writers of the California math standards adopted in 1997. While most states continue to use the NCTM reform math framework, California dropped it in 1997 because its test scores plummeted. The 1997 California math standards were among the best in the United States. Furthermore, they were benchmarked to top-performing nations. Kids in top-performing nations do algebra in middle school. The California 1997 standards put Algebra 1 in 8th grade and Geometry in 9th grade. However, recently, I am sad to say, California replaced its excellent standards with mediocre Common Core math standards. Wurman has been an outspoken critic of Common Core math standards.)

Without explicit guidance

The belief in reform math is that, if students can learn problem-solving strategies and “discover” solutions to problems “without explicit guidance,” knowledge, or instruction, then learning math content (e.g., basic arithmetic) is not that urgent or important. This methodology is not “the most effective or efficient way to learn mathematics” and partly explains why most students are not proficient in mathematics (NAEP). What happens in reform curricula is that students do not learn enough content to support cognitive problem-solving. There are many math programs that emphasize a problem-based approach. This sells textbooks but does not produce “excelling” math students. In fact, reform math programs have produced a flood of remedial math students. For example, in 2009, nearly 90% of incoming students at Pima Community College (Tucson) were required to take remedial mathematics. 

Inverse [Upside Down] Relationship

An emphasis on problem-based curricula often displaces mastery of key math content. This inverse or upside-down relationship is found in most reform math programs. Not only is this inverse idea a basic philosophy in NCTM math standards, but it also carries over to Common Core by delaying fluency. [3] Math educators call it spiraling. 

In reform math, grade-level mastery of arithmetic is not the goal. If a student does not learn addition in 1^st grade, it is repeated in 2^nd, 3^rd, and 4^th grade (it spirals). Indeed, in the new Common Core math standards (left), students are not expected to be fluent in addition and subtraction until 4^th grade. This is a nonsense approach. 

To be effective, a curriculum that is strong is problem-solving must also be strong in computational skills and fundamentals. Knowing is the foundation for higher-level thinking. So-called “generalized problem-solving skills” are not a substitute for mastery of fundamental content.  

In arithmetic and algebra, problem-solving is deeply rooted in the background or content knowledge (both factual and procedural) and not in general [problem-solving] strategies as embraced and advocated by many math educators, textbook writers, ed school professors, and special interest groups.

Problem solving in math is domain-specific, but math educators act as if it is not. 

Skill in problem-solving in mathematics requires substantial domain-specific schema [background knowledge], not “domain-general.” Students cannot apply the mathematics they do not know well. Students should be taught to solve routine problems to build an arsenal of background knowledge for more complex problems.

Worked Examples

Sweller, Clark, and Kirschner write, “But domain-specific mathematical problem-solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies.” In other words, students can learn problem-solving skills by studying worked examples that exemplify them. This requires diligent practice. In mathematics, problem-solving skills are deeply rooted in content [knowledge]. Moreover, these skills take time to develop.  Sweller, Clark, and Kirschner write, “There are no separate, general problem-solving strategies that can be learned.” 

21st-century skills (P21), the latest foolish fad

The idea in 21-century skills is that these skills can be learned outside of domain-specific content knowledge. I think not. 

Endnotes

[1] G. Polya's How To Solve It was first published in the United States in 1945. Polya was concerned about math education, the training of teachers, and how teachers influence the attitudes of students.  

[2] Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics by Sweller, Clark, and Kirschner, in Doceamus, November 2010. 

[3] The Common Core Math Standards (2010) continue the NCTM spiraling approach. Delaying fluency makes no sense. 


Return to ThinkAlgebra

Updates: 7/15/11, 7/16/11, 7-17-11, 8-4-11

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 Learner, Scientific 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

Email

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 

Most kids do not understand science

NAEP recently released science results. Most kids do not understand science. Only 34% of 4th graders, 30% or 8th graders, and 21% of 12th graders are proficient or above in science. Science education is just as bad as math education. Dr. Mark A. McDaniel says that before engaging students in inquiry-based problem-solving in science or mathematics, they should have a sound knowledge base (background knowledge). In short, background knowledge of fundamentals in long-term memory is key to problem-solving in math, science, and other academic disciplines. Furthermore, background knowledge should be domain-specific. (Note. The idea that math or science always has to be fun or a game to play is nonsense. There is nothing wrong with rigor or struggle. To learn math or science well takes plenty of effort and hard work.) 

Unfortunately, elementary teachers are told to focus on math and reading (NCLB) at the expense of science and other subjects. Even so, achievement in math and reading falls short of NAEP proficiency levels for most students. Furthermore, very little science is taught in elementary school; K-8 science textbooks tend to avoid math. In contrast, Science--A Process Approach (1967), from the Sputnik era, focused four of the six processes taught in first grade on mathematics: Using Numbers, Using Time/Space Relationships, Using Communicating (graphing relationships), and Measuring. The math was ahead of the normal 1st-grade math curriculum. 

To improve science achievement at the high school level, students should take more than biology. They should take both chemistry and physics and avoid integrated courses. The coursework taken in high school makes a big difference in science learning. Chemistry and physics also require a good background in algebra and solving word problems. (FYI: No calculators are permitted on AP Chem or AP Physics exams.) 

Systemic Approach, Not Bits & Pieces
The lack of adequate math and science achievement in the United States is partly a result of system failure. One cannot fix one part, such as standards, in isolation from other parts, yet this fragmented approach seems routine in education. Better standards will not fix a system failure without concurrently addressing teacher education and schools of education, instruction and curriculum, textbooks and assessments, policies and teacher support, learning attitudes, and so on, as all are interconnected. In short, a complete system overhaul is needed, not bits and pieces now and then. 

Return to ThinkAlgebra

---