Examine the curriculum guide carefully. Does it give a day-by-day plan or just a collection of units or topics? Remember, this is a curriculum guide, and you can modify it to fit with your expertise. Some guides give suggestions in detail. For example, the guide might suggest that you teach algebraic factoring in a specific order in an Algebra 1 course:
- Find the common factor in the expression 3xy - 5x. The factor common to both terms is x. The factored expression is then x(3y - 5). Then they might have you consider factoring the expression:
5(x + y) - x(x + y),
where the common factor is (x + y), so the factoring gives
(x + y) (5 - x).
Remember, this is really the distributive property of multiplication over addition/subtraction in reverse.
- Factoring the difference of two perfect squares, using, for example, x2 - 16. Because both terms are perfect squares, we get (x + 4)(x - 4).
- To combine the two previous factoring techniques, the students examine the expression for factoring procedures in the order they have been taught, that is, first for the common factor, then for the difference of two squares. For example, to factor the expression 3x2 - 27, first check for a common factor (here it is 3) to get 3(x2 - 9). Then, because both terms in the parentheses are perfect squares, we get
3(x + 3)(x - 3)
Trinomial factoring can also be simplified by first finding the common factor and then doing the usual trinomial factoring. Consider the trinomial 2x2 + 24x + 54. Factor the expression for the common factor 2: 2(x2 + 12x + 27). Then, factor the trinomial to get 2(x + 9)(x + 3).
In some cases, you may discover that there is no curriculum guide; consider asking a more experienced teacher if you can borrow a lesson plan book from a previous year. This can provide a tremendous amount of material to direct your teaching. Although the content of this teacher's plan book may not fit your teaching style, it at least provides a guide regarding the amount of material that can be presented during a given time span. It also provides a lesson-by-lesson sequence you can examine as you plan your lesson. A word of caution is necessary: Don't directly adopt someone else's lesson plans (tempting as this may be). That will not work. Regard this borrowed plan book simply as a guide to help plan your lessons. You'll see more about lesson planning in Chapter 5.
Above all, don't hesitate to ask for help. Try to find a teacher who can act as a mentor for you for each course you are teaching for the first time. Most experienced teachers will be glad to share their knowledge and experience with you—it is flattering to be asked—as an acknowledgment of mastery!
Is Your Textbook the Curriculum?
Not all curriculum guides go into detail. In fact, it is possible that your school may not even have a curriculum guide. Don't worry. You always have your old friend the textbook to rely on. The textbook is an excellent guide to tell you which topics to teach and in what order. If all else fails, examine the Teacher's Edition of your textbook. It should give you ideas for teaching and many other features to help you. Most textbook publishers provide potentially useful supplementary materials to accompany their textbooks. A word of caution: Do not use these materials just because they are available. Always use your personal judgment so that the instruction is yours and not one provided by prescription. (For more on this, see Chapter 4.)
Your textbook also gives many exercises you can use with your students. Although not a curriculum guide per se, your textbook is definitely a curriculum guide of last resort. Also look at the standards of your local district. The state often generates these because they are ultimately responsible for enforcing the standards in all fields and at all grade levels. Under the No Child Left Behind law, states now require teachers to adhere to the state standards, yet local districts might have some modifications for you to follow. Do not confuse the standards with a curriculum guide. The latter is designed to help you teach the material by providing a suggested order of topics, indicating the depth of your responsibility for covering the material, and providing suggestions for teaching: possible motivational activities, developmental suggestions, and assessment options.
Differentiation Within the Curriculum
Your curriculum may have special sections for teaching gifted or special education students. Yet, you are responsible for providing instruction for all student types: English language learners, “average” students, struggling students, gifted students, as well as special education students. Rest assured your classes will include students from many of these groups. (Note: The Appendix includes more technical information about special education law and the inclusive mathematics classroom.)
The English Language Learner
Teaching mathematics effectively is a daunting task for even the most experienced teacher. However, teaching English language learners is particularly challenging because the teacher must teach both mathematics and English at the same time. The task is more difficult if teachers and students cannot communicate in a common language.
There are several strategies you might employ to create a classroom that is warm, nonthreatening, and rewarding for English language learners. Consider using small groups to allow students who share the same first language to communicate with each other in a relaxed environment. This gives them a chance to ask questions of each other and clarify concepts in both languages while you “manage” the groups. To address the varying levels of understanding a group of English language learners may bring to the classroom, employ a variety of instructional strategies in your classroom not only to keep the classes lively but also to reach more students. Manipulatives enable the English language learner to discover relationships and learn concepts while circumventing the language barrier. You can then ask the student to express the relationship using informal language that does not stress grammatical structure but rather focuses on the mathematical concepts.
Teachers should be sensitive to the frustrations of English language learners and present activities that are both interesting and relevant to the students' lives. English language learners can relate to situations that they are experiencing and are more likely to respond when relevant material is presented. Activities involving sports, music, movies, and games are likely to capture their interest. English language learners can benefit greatly from visual aids, so try to reinforce concepts and skills using charts, graphs, diagrams, and pictures.
Another important factor in the effective instruction of English language learners is the simultaneous acquisition of a mathematics vocabulary and the English language. You should spend some time each day building the English language learners' mathematics vocabulary and making certain that they are well versed in the vocabulary words essential to the day's lesson. You may have students keep a separate vocabulary journal so they can review vocabulary. English language learners may feel more comfortable writing in a journal than speaking up in class. This is natural, and you may wish to first check the journal entries and then call on students to share entries aloud with the class. Knowing that their responses are correct will instill confidence in them and allow them to contribute in a nonthreatening environment.
Teachers must constantly monitor their teaching habits when working with English language learners. Remember to speak slowly and pause often to allow students to thoroughly comprehend what they are saying. Paraphrase your thoughts using different vocabulary and always write key words on the blackboard. Keep spoken sentences short and build in wait time to allow students to process the information before proceeding with the lesson. By following these guidelines, you are more likely to provide your English language learners rich mathematical experiences.
The Average Student
Working with the average student presents teachers a great opportunity to gauge their own teaching effectiveness. Many teachers of average learners are complacent and forget to challenge these students. Outstanding teaching, however, can transform the average student into an above-average student by engaging the student in interesting and relevant activities that will better reinforce conceptual understanding. In addition, the new mathematics standards require that students have a deeper understanding of mathematics and the ability to apply it to various problem-solving activities. Mathematics instruction is no longer restricted to presenting simple procedural tasks; rather, it has broadened into formulating a process to solve provocative problems using careful analysis and the synthesis of many skills.
Although average students may not be expert mathematicians, each average student is an expert at something. The effective teacher finds a way to include opportunities for the average student to show off personal strengths. By incorporating writing, art, and even sports statistics into your activities, you give each student a chance to shine. The average student's motivation in the mathematics classroom increases with this opportunity to feel confident.
Consider asking average students to pair up with struggling students to provide support in some activities. Being asked to explain a concept to a struggling student can be an effective means of motivating an average student to acquire a fuller grasp of the associated concepts; this often results in a valuable learning exercise for both students. In short, the average student, like all students, should be exposed to a mathematics classroom that is lively, engaging, and rich in content.
The Struggling Student
Teaching is not an exact science. Although teachers plan to reach all students with a single clear explanation, it would be naïve to think that just because you've explained something all students necessarily get it. The realization that some of your students are struggling does not imply that you are failing as a teacher. The real failure comes from refusing to accept the challenge of reaching those struggling students.
Struggling students may need to be retaught, and there are two ways of doing that: Either use the same approach or use a new approach. Simply reteaching using the same approach may not be too inventive, but it can be effective for some students. Mathematics can be considered like a language, and in language learning, repetition can bring benefits. Just rehearing an explanation may help concepts “sink in.” However, if you can make the same points in a modified fashion, then there is a good chance that you will avoid student boredom and possibly strike a chord that resonates with the student. Such action can even be the break in the learning frustration that may have begun to set in.
If your re-explanation doesn't work, then it is time to search for a new angle. Try to incorporate visuals, manipulatives, or real-world examples to which struggling students can relate. Your goal might be to enable the struggling student to help himself. If the textbook is beyond that student's reading level, then provide a book that is more appropriate.
If the student is still struggling, consider teaming the student with an average or accelerated student for peer tutoring. The tutoring process benefits the struggling student as well as the student tutor by reinforcing both students' understanding. Working with the parents to devise strategies is another way to effectively address the needs of the struggling student. You might ask parents to monitor their child's study and homework time. This may reveal that simple measures such as increasing study time might improve performance in mathematics. Such parental involvement also sends an important message to the student: There is genuine interest in improved performance. This prominent parent involvement has proved successful in increasing student effort. Without increased effort, improved student achievement might prove evasive.
As you identify struggling students, you can try to prevent future difficulties by anticipating which prerequisite skills each student may lack. An astute teacher will also consider many factors that could impinge on student achievement. Some of the considerations are as follows:
- Are there any undetected learning disabilities that may need to be addressed?
- Is there a language problem (e.g., for an English language learner)?
- Is there adequate support in the home, where many feel that true learning really takes place while doing homework?
- Are there any psychological issues that need to be addressed?
- Does the student have the proper prerequisites for the course?
- Are optimum methods of instruction in use for this student?
The Gifted Student
Teaching the gifted student can be as difficult as teaching the weaker student. The challenge is different, but it is a challenge nevertheless.
You will be able to identify gifted students by their creative talent, curiosity, and ability to achieve at a high level. The gifted student will often come up with an innovative or unusual method for solving problems, reflecting a rare insight into mathematics. You can use these unexpected responses to exhibit to the rest of the class alternative ways to look at the mathematics under discussion. Often, the gifted youngsters take pride in sharing their ideas with the rest of the class.
In some schools, gifted students are moved ahead or accelerated. They may begin their algebra work in 7th grade or earlier and complete geometry by the end of 8th grade. This enables them to continue taking mathematics courses and complete a year of college-level mathematics, such as calculus, while still in high school. In some cases, students may accelerate beyond the capabilities of the high school and be forced to take courses at a nearby college or to take a “vacation” from math. The latter would have deleterious effects on the development of a talented student. A high school may offer advanced (i.e., college-level) courses that do not involve calculus (e.g., probability, number theory, advanced Euclidean geometry, etc.). These are all options to consider.
Within your class, however, you will want to challenge gifted students. You do not want them to be bored by moving along with the class at the regular pace. One way to challenge gifted students is to have them delve more deeply into certain topics. For example, the class may be studying the Pythagorean theorem. You might challenge your gifted students to extend the theorem to non-right triangles. (For an obtuse-angled triangle, a2 + b2 <
c2; for an acute-angled triangle, a2 +
b2 > c2.) Or, you might ask them to generate Pythagorean triples using the following parametric equations:
a = u2 - v2
b = 2uv
c = u2 + v2
Then, have them examine some of the triples that result from these equations:
3, 4, 5
7, 24, 25
5, 12, 13
8, 15, 17
9, 40, 41
Will all primitive Pythagorean triples1
have exactly one leg of even measure? Will the product of the three members of a primitive Pythagorean triple always be a multiple of 60? And so on.
In a geometry class, for example, introduce gifted students to the elements of simple topology. The four-color map problem, the Möbius strip, and the bridges of Königsberg are all topics that will interest gifted students. (See the References and Resources for sources of material on topology.)
There are many questions that can be assigned to the gifted students to interest and intrigue them. Here are some you might consider:
When is 1/x > x?
If a2 = b2, then will a = b?
For what value(s) of x will x2 + 6x + 6 be a negative number?
If x lies between 0 and 1, then can x be less than
x2? Explain your answer.
When does x = 1/x + 1?
The Special Education Student
The special education student comes into your class under guidelines of an entirely different set of rules. In addition to the rules set by your school or school district, these students are governed by the Individuals with Disabilities Education Act (IDEA). (See the Appendix for more information regarding IDEA and your classroom.) If you have a classified special education student in your class, then that student will probably come with an Individual Education Program, or IEP. This has been prepared by the student's previous teacher together with a child study team and the child's parents.
You must follow what appears in this plan; it is a legal document. If a problem arises, you may need to consult with the child study team and discuss modifying the IEP. However, you should always plan to modify your instruction to accommodate the needs of the learning disabled (LD) child. A typical procedure is to have a teacher aid or special education teacher work with the individual student while you are working with the rest of the class. Be certain, however, that the teacher aid has a good understanding of the mathematics taught.
There are other techniques you can use to help these children achieve in your class. For example, you should obtain permission to give the LD child a grade of Pass or Fail rather than a letter or numerical grade. This is often specified in the IEP. You may have to modify what you teach. For instance, when you teach factoring in an algebra class, use simple numbers. Instead of asking the LD child to factor an expression such as 2x2 - x - 6, which factors into (2x + 3)(x - 2), you could ask the child to factor x2 + 5x + 6, which factors into (x + 3)(x + 2).
Remember, students with disabilities are not necessarily slow. They can learn mathematics. Consider reducing the number of problems expected of the child for a class or home assignment. Ask a bright child to work with the LD child and assist with the work. (This might be an excellent way to challenge the gifted child, who might otherwise be bored.) You might consider the following problem:
Ian has $1.35 in nickels and dimes. He has 15 coins altogether. How many nickels and how many dimes does Ian have?
The majority of students in your algebra class would immediately resort to a system of two equations with two variables (where x
= the number of dimes, y
= the number of nickels):
x + y = 15
10x + 5y = 135
The student working with the special education child might instead encourage a guess at the answer. After all, intelligent guessing and testing is a valid problem-solving strategy and should be encouraged for everyone. Then, the two students could consider how to move on to the pair of equations solved simultaneously as a more efficient method of solution.
Many special education students may not have a good command of the basic arithmetic facts. For example, they may not always recall the multiplication facts. It's usually wise to make a calculator available for the special student. (All students might well have a calculator available all the time.) At the same time, break the various tasks into smaller pieces. Instead of an entire proof of a theorem in geometry, you might have the special education child just do the first part, such as simply proving triangles ABC and DCB congruent (see Figure 2.1). Then, in a second assignment, have the child prove that the line segments AC and BD are parallel by establishing that ∠ACB ≌ ∠DBC.
It is good to have models of geometric figures available. Many LD students need to physically touch geometric shapes to understand them. For example, you might help them grasp the concept of congruence by asking them to place one triangle on the other and make the parts coincide. The student may even have problems associating the symbol for congruence (≌) with the word congruence or the concept of congruence. The word and symbol may have to be placed side by side for several days. Some special education students may forget material within a day or two. For these children, make reteaching and extended drill and practice a regular part of their lessons. Some special education students have trouble organizing numbers on their papers. For example, they might easily misalign numbers in a simple subtraction problem or when adding a column of figures. To help them, always provide graph paper.
How Do You Get Ideas to Teach Something Beyond the Textbook?
There are numerous books that can provide you with ideas appropriate for your class and yet not included in your textbook. These enrichment units, along with student materials, can be found in several books about teaching algebra and geometry (see Posamentier, 2000a, 2000b, 2000c).
Teaching Secondary Mathematics: Techniques and Enrichment Units
(Posamentier, Smith, & Stepelman, 2006) provides enrichment units for all secondary grades, as well as methods of teaching mathematics in the secondary school.
As a new teacher, begin to collect books for a resource library, which will serve you well as you select topics that can be used to enhance your lessons and will enrich your students. A little secret is to find topics and small units you find exciting and you will continue to rejuvenate your professional outlook and become a more enthusiastic teacher. Students appreciate when you take time to show them math “things” not necessarily part of the standard curriculum. You shouldn't think of these short digressions as “wasting time” that could otherwise be used to move ahead in your syllabus; rather, the time spent on these activities will serve you well because they will motivate your students, making them more receptive learners. In that spirit, a list of some of the books that you might obtain as you build your professional library is provided in the References and Resources.
A primitive Pythagorean triple has no common factor throughout.