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This essay provides a course outline by answering 11 questions regarding teaching math at the seventh-grade level. Teaching Math
This paper doesn’t follow our usual format; instead, it answers specific questions regarding teaching math to seventh-graders. In working on the outline, I’ve also tried to be aware of the commonalities among math, science and technology, and provide meaningful “real life” responses. As a general observation, it seems that many of the resources I’ve found tend to tie these three disciplines together.
1: What should be taught to satisfy the requirements at various levels? What are students interested in? What should they already know? What materials are needed to organize a unit?
Describing state and local standards obviously depends on the location of the school. Searching for “standards seventh-grade math proficiency” has brought 3,800 hits; I’ve compared the first three and found many commonalities. Perhaps the most interesting is that heavy emphasis is put on the idea of solving real world problems.
Math is no longer an abstract concept, but is linked to science and technology, with problems designed to show why math is a strong tool in other disciplines. The sources also mention that students will be expected to use calculators and computers in their courses. Specific competencies that students must demonstrate to satisfy local requirements in these systems include working with integers, fractions, and ratios; solving algebraic equations; analyzing and graphing data; solving geometrical problems; and understanding measurements. Space makes this a very incomplete list; the students are facing a very rigorous curriculum in these three schools.
(The three are Olentangy, Riverdeep and Yokota, and their complete URL’s are listed in the reference section. Yokota is a school for American dependents whose parents are stationed in Japan.)
Since it’s a populous state, I’ve looked at California’s standards. There is a tremendous debate going on because the state standards are less challenging in their use of creativity and initiative than some of the local standards. That is, California seems to have gone “back to basics,” such as learning multiplication tables by rote and doing long division by hand, rather than setting problems that reflect real world situations. At any rate, California expects its seventh-graders to demonstrate number sense; knowledge of algebra and functions; knowledge of measurement and geometry; statistics, data and probability; and mathematical reasoning ability. (“Grade Seven, Mathematics Content Standards,” 2001, PG).
Nationally too the debate over standards goes on; many different groups are writing standards for children throughout the country. The National Council of Teachers of Mathematics has suggested that all children display competence in these areas: number and operations; algebra; geometry; measurement; data analysis and probability, problem solving; reasoning and proof; communications, connections and representation. Obviously this list duplicates almost exactly most of the local and state standards, with the addition of the last three areas, communications, connections and representation. These three subjects deal with fostering the students’ ability to communicate mathematical concepts clearly to others; to understand how mathematical ideas interconnect; and to use representations to communicate mathematical ideas. (“Mathematics,” 19962003, PG).
What do seventh-graders like? I’ll avoid the obvious jokes, what interests most students is the connection between subject matter and actual real world situations. Students today are not like students of the past: many of them have computers, or access to computers. They download music and movie trailers off the Internet; they create CDs, they exchange information via email. They have been raised during a huge boom in technology and they’re comfortable with it. Many of their teachers, coming as they do from an older generation, are not, which can mean that the teachers are unfamiliar and uncomfortable with concepts and learning methods their students already know—they didn’t learn it that way, so they see little value in teaching it that way. What students are interested in, then, is a clear connection between what they’re learning and their future. They are no longer content to sit and endlessly learn multiplication tables when they can use calculators; they find no value in learning how to calculate the volume of a cone when a computer program can do it for them.
Rote learning, drills, and memorization—these are techniques that belong to another age, and they are inappropriate in a modern classroom. Whether the teacher likes it or is comfortable with it or not, students want to learn practical applications, not mere facts. Nor do they need to learn the basics—that’s what they’ve been taught, over and over again. (Bandlow, 2001, PG).
What should they already know? In order to succeed in seventh-grade math, students have to already have successfully understood the foundational material presented up to that level. The material covered in grades K-5, according to the National Council of Teachers of Mathematics, is much the same as it’s always been: learning how to understand, manipulate and use numbers. It appears to be in the seventh grade that the course work expands to (hopefully) be far less repetitive and far more attuned to real-world examples. It is also at that point that more abstract concepts are introduced. Thus, students entering seventh grade need to have basic math clearly understood.
2: Why should this unit be taught at this age? What lesson does it provide? If a student asks, “Why do we have to learn this?” what’s the answer?
Seventh grade is part of the grade range called “middle school.” Middle school is a fairly new concept, and is designed to bridge the years between childhood and adolescence, between elementary school and high school. Most often it encompasses grades 7, 8 and 9. It’s appropriate to teach seventh grade math at this age because the course material is more sophisticated and begins to broaden out, at the same time as the students begin to move away from the core subjects and explore wider options, more sophisticated ways of thinking, and begin evaluating their lives and prospects as a whole. The course material is perfectly timed to reflect a shift from childhood to more mature development.
The lesson provided in seventh grade math is that mathematics encompasses more than just numerical manipulation. At this age, additional concepts should be introduced that illustrate the more abstract reasoning found in higher mathematics. American students in general lag behind the rest of the world in mathematical ability at the end of high school, though they are even with their compatriots at fourth grade. Clearly, something happens during the next seven years that sets them back, and the suggestion is that it is the insistence on teaching the basics, over and over again, rather than going on to algebra, calculus, etc., that is at fault. (Bandlow, 2001, PG).
This observation also applies to the third part of the question, “Why do we have to learn this?” Seventh grade is the place to prepare students for the more rigorous work ahead, something Bandlow says is not being done:
“The rest of the world teaches algebra by the eighth grade and in the middle years begins to introduce challenging concepts in geometry, probability, and statistics. In the United States, most eighth graders are continuing to study arithmetic, even though they are already proficient in arithmetic, because math teachers at that level say that students are “not ready” for algebra.
“And they are right. Eighth-grade students in the United States are ill-equipped for algebra because U.S. sixth-and seventh-grade math curriculum programs generally repeat those of grades 4, 5, and 6, thus precluding students from gaining a familiarity with key algebraic concepts in grades 6 and 7. U.S. students are not ready for algebra despite the fact that children in virtually all other countries are ready, not because they are developmentally incapable, but because the curriculum has not prepared them for challenging mathematics.” (2001, PG).
Seventh grade is the place to start giving children the tools they need to handle higher mathematics, and catch up with the world.
3: According to curriculum guides, content objectives and performance standards, what skills should or should not be taught?
We can answer this by going back to some of the comments in the first question. Over and over again I found suggested content that seems far more “abstract” that such courses used to be. Gone is the rote memorization, drill, and endless regurgitation of facts. In its place is far more abstract thinking, much of which interweaves mathematics, science, technology and communication. Of course basic mathematical skills are essential, and the foundations of the courses are still in learning how to manipulate numbers. But then the content becomes much wider, and brings in concepts that require students to do such things as analyze data, predict probability of events based on mathematical models, and perhaps most importantly, communicate their findings, conclusions and reasoning to others. This requires that they learn to describe mathematical problems and concepts in language.
Things that should not be taught, as mentioned, are those concepts that have been traditionally drummed into students by repetition. It’s really not even necessary that they learn to add and subtract: any calculations required can be done by computers or calculators. What they need to understand is the concept of addition, etc., so that they understand when to add and when to subtract. (It seems to go against the grain to suggest that students not endlessly repeat the multiplication tables, but the bare fact is that technology is not going to go away, and it’s more important now for students to learn to use it well, to manipulate numbers, than it is to learn the numbers themselves.)
4: How do you know when to introduce new vocabulary specific to the subject, and how should you introduce it?
In general, it’s necessary to introduce new vocabulary only when it becomes necessary to do so; when it is introduced, it should be thoroughly explained, used, and then reviewed.
In a 1995 article in Childhood Education, Monroe explains that there are four types of vocabulary: technical, subtechnical, general and symbolic. Mathematical terms are part of the “technical” vocabulary, since mathematics has more concepts and unfamiliar terms in it than any other field of study. Students are constantly facing new and strange terms, and a lack of understand of the words can cripple any effort to work with the numbers. Monroe suggests that teachers should plan to introduce new vocabulary as it comes up in the text; that is, integrate it into the lesson plan when the students begin the material. Then, introduce it by explaining it fully, and if possible, making it more understandable by using real-world examples. (Monroe, 1995, PG).
5: Give four examples of a teacher’s goals for the end of the unit. What should the students have learned?
The answer of course depends upon how the unit was structured and what it was designed to accomplish. Let’s take an example from one of our three middle schools in the first question; the Olentangy school has a clear website with its curriculum for seventh grade math laid out on a grid. In one column it has the unit; the next are the questions that pertain to that unit; and the third lists what the students will be able to do by the time they complete the unit.
The algebra unit, for instance, says that algebra is an “analysis of patterns, relations and functions involving variables,” and that the essential questions regarding algebra are, “How do patterns affect your life?”; “What variables do you encounter in everyday life?”; “What if there were no variables?”; and “How does algebra help us model/explain our world?” Having defined what it is and does, the final column tells us that the students will be able to do the following when they are competent in the subject: “Describe problem situations involving ratios, proportions, and percents with algebraic expressions and equations; solve one-step and two-step equations and inequalities with integers and fractions; create a variety of patterns with tables, graphs, words and symbolic rules; represent functions as ordered pairs and graph; apply simple formulas to problem solving; model use of inverse operations, identify equivalent equations, and examine use of variables.” (“Mathematics Course of Study, Seventh Grade,” 2002, PG).
This is a specific set of competencies; it also incorporates some of the types of abstract thinking we’ve been discussing above.
6: What should the lesson objectives be? Based on the long-term goals for the course, determine how many lessons will cover the unit and give an objective for each lesson.
Lesson objectives as always should be to enable students to learn, grow, develop selfconfidence and self-esteem, as well as mastering the subject matter. If we presuppose that the longterm goal of the course is to give the students a solid grounding in algebra, geometry and the other higher mathematical functions we’ve been discussing, so that they will be able to perform well in high school, then it seems that our three “model” schools, plus others I’ve seen, provide fairly similar curricula.
The standards developed by the National Council of Teachers of Mathematics would appear to help determine the number of units, and include the following: number and operations; algebra; geometry; measurement; data analysis and probability; problem solving; reasoning and proof; communications, connections and representation.
Objectives for each might be enumerated as: for number and operations, the student will understand numbers, the meaning of operations, and be able to compute fluently; for algebra, the student will understand patterns, represent mathematical situations with algebraic symbols, use mathematical models to understand quantitative relationships, and analyze change in various contexts; for geometry, the student will analyze characteristics and properties of two- and three-dimensional shapes, use coordinate geometry to specify locations, use symmetry to analyze mathematical situations, and use visualization to solve problems, and for measurement the student will understand measurable attributes of objects and apply appropriate techniques to determine measurements. In the data analysis and probability unit the student will formulate questions that can be addressed with data, select and use appropriate statistical methods to analyze data, develop and evaluate inferences and predictions based on data, and understand probability; students with a competency in problem solving will build new mathematical knowledge through problem solving, solve problems, apply a variety of strategies, and monitor and reflect upon the use of problem solving techniques in mathematics; students competent in reasoning and proof will be able to evaluate mathematical
argument and proof, and select various types of reasoning and methods of proof; students competent in communications will be able to describe mathematical concepts in precise and appropriate mathematical language and communicate these concepts to others; students adept in connections will be able to recognize connections among mathematical ideas and apply mathematical concepts to fields other than mathematics, and students who achieve competency in representation will be able to use representation to model physical, social and mathematical phenomena. (“Mathematics,” 19962003, PG).
7: What’s a good introduction that will “grab” a student’s attention.
There are as many effective introductions as there are teachers, limited only by imagination (I suppose a brass band is a bit much). One source recalled that his teacher got the attention of the class by “bringing himself down to their level” and using different teaching methods to stimulate learning. He was also available to any student who was having trouble with a concept. (De Smidt, 2000, PG). The “bottom line” is that this teacher made the subject interesting to the students, and also offered help so that none of them was left behind.
The teacher might also consider setting up a mathematical problem that is of relevance to the real world, one that is challenging enough to keep the students interested but not so difficult that it can’t be solved in a reasonable amount of time. It all comes back to keeping the class interested in learning by tying that learning to other concepts of learning such as science and technology.
8: Create learning activities for each lesson objective. Include two introductory, two enabling and two culminating activities and explain each.
This question could be a paper in itself if I were to tackle all nine of the subjects suggested by the National Council of Teachers of Mathematics, and from whom I’ve taken much of the data. Let’s concentrate on one objective of one unit, and discuss it.
Suppose we are moving into algebra, and our objective for the lesson is to enable students to translate mathematical concepts into algebraic symbols. (Although it seems simple now, putting “x” in place of a definite value is an odd construct when one first studies it.)
Introductory activities are designed to introduce the new subject; enabling activities give the students the tools they need to understand and solve the problems; and culminating activities present us with the actual solution to the problem.
Introductory activities might include inviting students to consider the principle behind substitution, and why we use “x” to indicate an unknown quantity. A brief lecture might consider such contemporary icons as “X-men”, superheroes whose identities are not known; another introductory activity might include a tape of some event in which some of the parameters are unknown. (The Columbia disaster might be a potential introduction, as students learn that scientists are trying to figure out what happened without having all the relevant information.) Such examples introduce the concept of substitution algebraic expressions for exact values.
Enabling activities could include a demonstration of the way in which signs work. One source suggests having students walk back and forth along a line as a way to explain negative numbers. (Of course I can’t find that website now!) The idea was that as one student got closer to the other, the distance would decrease; should the first student pass the second, he/she would represent a negative number. Since signs change in algebra, such a physical demonstration might be helpful and imaginative.
Another enabling activity might be to discuss a real-life problem like Columbia, and have the students decide how to begin describing the information they are seeking in mathematical terms. Once the need is clear, the next step will be the culmination of the process.
In this example, the process would culminate when students are able to describe and formulate equations that describe what they want to find out about the failure of the craft. Such equations might not be solvable, but would serve as examples of the type of work done in the real world.
Another culminating activity would be to have a student actually solve a simple equation (x+2=5) at the board.
9: What should you collect from the students to evaluate their standards?
The obvious answer here is homework. Even if students turn the actual computations over to a machine, they still have to be able to write equations that will return meaningful result. The only way to do that is practice.
However, it might also be useful for students to locate and bring in articles from newspapers, magazines, and the Internet that shows the use of mathematics in real-world applications. This tests their understanding of the concepts as well as their ability to manipulate numbers.
10: Name five properties, other than the textbook, used to describe the unit. (Ex: PC, overhead)
I think it’s unfair to assume that every student has a computer at home; it’s a particularly painful reality that most inner city kids don’t have them. But there are now computers available in public libraries, so I think we can assume that, absent the direst circumstances, all students have computer access.
All students also probably have calculators, which have been commonly used in classrooms for decades. Three other devices/properties that might be found in the unit are videotapes, audiotapes, and video links to instructors in remote locations (perhaps professional mathematicians at neighboring universities, or working in industry).
11: Provide a bibliography of any five teacher resources and ten student resources used in this project.
These are separate from the references listed at the end of the paper, and are new sources | found as the result of a search.
Teaching math is like teaching any other subject: despite its fearsome reputation, students can learn it if they’re motivated to do so. It seems that using technology, bringing in real-world examples, and making math relevant to solving actual problems are some of the basic steps that will help teachers present their “dreaded” subject in a way that will make students eager to learn.
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