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    Mathematics Mathematics Curricula FAQs"In the Middle -- linking Elementary and Secondary" 1. What rationale/philosophy was employed as the foundation of the mathematics curriculum change? Based on Directions, the Ministry of Education is implementing Core Curriculum (with its accompanying CELs and other initiatives) as the basis of the educational program for the students of Saskatchewan. This necessitated a review of all subjects in all grades. In mathematics, the two major influences for the curriculum changes are the National Council of Teachers of Mathematics (NCTM) 1989 publication Curriculum and Evaluation Standards for School Mathematics (1997), and Charting the Course: A Guide for Revising the Mathematics Program in the Province of Saskatchewan, written by Dr. J. Hope of the University of Saskatchewan, and published by SIDRU, 1990. 2. Why are we changing the mathematics curriculum? Do we have to throw everything out that we have been doing? The mathematics curriculum has been changed to incorporate the latest research regarding how students learn and understand mathematics. The curriculum guide is an instructional guide to help teachers guide students from the concrete to the abstract. The content is very similar to what it is has been over the years but the emphasis is on helping students to understand concepts rather than manipulate symbols. Problem solving is the central focus of the courses with the intent of having meaningful, relevant, real-world problems that students are able to solve. Our new curriculum is consistent with that of Western Canada in both content and philosophy. Don't throw out everything that you use, adapt it. Try teaching concepts using a variety of instructional strategies and methods, a variety of resources, and a variety of assessment techniques. 3. How will The Common Curriculum Framework (CCF) for Mathematics K-12 (Western Canadian Protocol for Collaboration in Basic Education) affect our Saskatchewan mathematics curriculums? The Saskatchewan mathematics curricula K-12 were in the development phase when the Western Canadian Protocol was signed. Rather than disrupt the process in this province, it was decided to continue with the implementation of the provincial mathematics curriculum. As maintenance of provincial mathematics courses occurs and the CCF is revised, the two will become aligned further. It is estimated (1996) that the congruency between the CCF and Saskatchewan's new Core Curriculum Mathematics is as follows: Elementary (95%), Middle (90%), Secondary (85%). Saskatchewan was well represented in the development of the CCF and in the evaluation of suitable resources to support it. Provincial courses and the CCF have the same philosophy. Other than the placement of a few concepts indifferent grades, the content is also the same. In the CCF, concepts related to integers are in Grade 7; in Saskatchewan, they are in Grades 7-9. Fraction concepts are also spread over Grades 7-9 in Saskatchewan and are studied in Grade 8 in the CCF. The major differences occur in Grade 9 where trigonometry and factoring are in the CCF whereas in Saskatchewan these topics are delayed until secondary mathematics courses. Many of the examples in the Middle Level mathematics curriculum are the same as the illustrative examples in the CCF. Print resources that were specifically designed for the CCF are also appropriate as support material for the Saskatchewan curriculums. 4. Why was the secondary program developed before the Middle Level program? The Ministry of Education's decision to develop programs for all subjects and all grades precluded having all subjects done in sequence. This would have meant that elementary teachers would have had to implement many new courses in the same year, or alternatively, if only one or two were implemented each year, then the entire process would take too long to work its way through the K-12 system. As a result, some subject areas were designed to be developed "out of sequence" with mathematics being one of these. This does not mean that the Middle Level mathematics was developed in a vacuum. 5. Is the content of the courses being "watered down"? Because the emphasis of the new courses is to develop student understanding rather than rote manipulation of mathematical symbols, some of the content is introduced later than it was traditionally. Abstract concepts are delayed until students are more developmentally ready for them. Also because the traditional pencil-and-paper algorithms for operations with numbers have been replaced to a large extent by calculator algorithms, less time is required to develop student proficiency with concepts such as long division, and more time is spent in solving realistic problems. Operations with fractions have always caused difficulties for many students. In the new courses, more time is spent helping students to understand fraction concepts by using manipulatives and hands-on activities before doing operations and using symbolic representations. 6. What is the position of the Ministry of Education on "class sets" of textbooks? The Ministry suggests that each jurisdiction decide how to best utilize its resources for purchasing texts and support materials. Since most texts in mathematics are similar in content, and since no text will adequately cover all topics in the proposed curriculum, it may be useful to have some copies of each text available. Other materials, such as videotapes, manipulatives, and computer programs should also be considered. A list of recommended resources is available. In addition, more comprehensive curriculum guides will provide good support to teachers.
7. Can I still use a commercial program? Many commerical programs provide useful support for teaching students mathematics. The following cautions, however, need to be observed:
There are several commercial programs that are appropriate for students in the Middle Level grades. Use the curriculum guide to determine which concepts need to be developed and refer to a variety of print materials for the best approach to the teaching and learning of those concepts. 8. Can I use resources that are not in the bibliography? Yes, resources are not limited to those in the bibliography. The bibliography contains the key resource and other resources that have been evaluated and determined to be valuable to supplement the material in the curriculum guide. Human and community resources are also very valuable. We assume that any "key" resource would be in the bibliography, however "Math" activities are not limited to the bibliography. 9. Why are we expecting students to multiply and divide with only one or two-digit multipliers and divisors? The wide access to calculators that are reasonably priced has meant that calculations in everyday life are frequently done with calculators rather than with pencil and paper. Students still need to know basic facts and should be encouraged to estimate answers before using the calculator. However, the efficient use of the calculator is a skill all students will require in their lives. Students need to develop mental mathematics skills to enable them to choose wisely whether to use mental math, calculators, or pencil and paper when confronted with a problem. Rather than spending countless hours helping students to become proficient with long division, we should help them to become proficient problem solvers. 10. Why is the new curriculum recommending the use of manipulatives? Research, learning theories, and classroom practice all support the use of manipulatives when teaching mathematics. Many students at the Middle Level are still at the concrete stage of understanding concepts. The use of manipulatives introduces concepts in a concrete manner. Once students become faimilar with a concept, they are more easily able to visualize it. Although different people are able to move from the concrete to the pictorial and then abstract at different rates, most people benefit from the introduction of a new concept with the use of concrete materials. Teachers should assist students in making the transfer from the concrete to the abstract. 11. Will my really bright students be challenged with manipulatives? Some students will move from the concrete to the abstract more quickly than others. However many complex ideas can be introduced and developed using manipulatives and students, regardless of ability, benefit from their use. Using a variety of manipulative materials to develop a particular concept makes it easier for students to generalize the concept. You can extend the concept for those students who understand it using manipulatives. 12. If students spend all this time "playing", when are they going to learn real mathematics? Manipulatives are used to teach a concept and help move from the concrete to the abstract. We can help bridge the gap between the use of manipulatives and the use of symbols by having students, once they begin to internalize the concept or skill, record their manipulations in symbolic form. What is real math?... is it problem solving, process, and understanding? Parents who are uninformed as to the purpose of manipulatives are more apt to be negative. Show how manipulatives address the curriculum objectives and can be used to help students of all ages and abilities to understand concepts. 13. Will manipulatives work well when teaching in multi-graded classrooms? You can use the manipulatives to teach the same concept at each grade but have the students work at different levels. The model unit for Grades 6/7 in the curriculum guide gives some suggestions for using manipulatives to teach fraction concepts in a mixed Grades 6 and 7 class. The more capable students can help the less capable ones. Often teaching others can help an individual better understand the material him/herself. Manipulatives can be used in activity stations or independently for problem solving. 14. Are manipulatives over used? Will they provide a better understanding of concepts or just confuse students? They are not over used at present. But we must be sure to show transfer from concrete to pictorial to symbolic (pencil and paper) to abstract. Manipulatives allow the students to understand "why". They could be over used if one does manipulative work with no correlation to paper and pencil and or to real world. Manipulatives are used as a tool not as an object of study. 15. How will I manage the storage, use, and distribution of manipulatives? Some suggestions used by other teachers include:
16. Is there going to be money for the required materials for the new courses? It is the responsibility of school boards to allocate necessary resources for the implementation of new courses. Some of the resources are currently in place. We suggest teachers be prepared to share and exchange manipulatives to make the most efficient use of these materials. In an era of multi-resource materials, boards should not purchase a single text per student as the sole resource. Schools should either buy one type of calculator or recommend a particular model for students to purchase. Factors to be considered when purchasing suitable calculators are provided on page 11 of Mathematics for the Middle Level: An Information Bulletin for Administrators (1996). 17. Is it still important to teach basic facts to students? Will using a calculator delay the acquisition of basic facts? We definitely need to teach basic facts. The calculator with you always is the one between your ears. Mistakes on a calculator are easy to pick up if you have the knowledge of basic mathematical facts. A calculator is not a substitute for lack of mastery of basic facts; it is a supplement. A calculator only does what it is instructed to do. Students require knowledge of basic facts to estimate and to determine the reasonableness of the results. The knowledge of the facts leads to more confidence in mathematics and even in the use of the calculator. If students are encouraged to estimate results before doing a question or exercise, they will require knowledge of basic facts early. This need should assist in an earlier rather than later acquisition of basic facts. 18. How do we teach retention of basic facts? Retention of basic facts is assisted with the constant use of mental math and with drill and practice. If the basic facts are introduced through patterns andmeaningful activities, they are more likely to be remembered. Students learn doubles early on and can then use these to develop other basic facts. For example, 6 + 7 is double 6 and then plus 1. Other patterns and relationships to previously learned facts will help students learn and retain additional facts. 19. Is there still a need to teach fractions? Yes, students need the concept "parts of a whole" where the whole can be a single item or a collection. An understanding of fractions helps with the understanding of ratios, proportions, and percents. Also, it provides a picture in their mind of what decimal means. Fractions are not to be taught in isolation but rather be related to real-life situations and other subject areas; e.g., arts (music), science, social studies, etc. Using manipulatives to teach fraction concepts assists students to understand them. The use of the metric system has meant that the fractions we study can be less complex (i.e. halves or tenths rather than seventeenths) but there is still a need to study fractions. 20. Are mathematics skills being taught adequately? Are there gaps? Mathematics skills are being taught adequately but there is always room for improvement. Some of the gaps include:
21. Is there time to teach the required skills and use all the instructional strategies? The required skills and the instructional strategies are not two separate entities. It is recommended that you use a variety of approaches, strategies, or methods to teach the necessary concepts and skills. You will not use every instructional method every day or every week but over the course of the years you will probably use all of the ways. Make a goal of learning one new method every year. 22. In implementing the new courses, how do I plan for ...
Your planning is determined by what resources are available. Don't rush into everything. Start by using a different method of teaching a topic with which you are comfortable. Year
Strand
Tomorrow
23. How are we going to evaluate students? We are going to evaluate the process as well as the product. Keep the old means (e.g. tests) and slowly try other methods. Other methods include:
(Refer to pages 11-34 in the Middle Level mathematics curriculum guide.) Templates are provided in curriculum guide and in Student Evaluation: A Teacher Handbook from the Ministry of Education. By the end of the year be comfortable with two or three new methods. Set a goal. Yes, it is time consuming, but don't try something new once and discard it. Try again (modify if needed). 24. Should we assess things other than content in mathematics? We always are. In the new mathematics curriculum, we are integrating many subject areas and mathematics areas in daily work; i.e., academic areas such as science, social studies, health, language, etc. We are also studying concurrently many strands in mathematics. Due to this, we are assessing more than content. The nature of a nontraditional style of delivery using these mathematics curricula lends itself to peer and self-assessment of processes used: cooperative groups, use of manipulatives, presentations, discussions, journals, etc. As well as all of these things occurring, you have to deal with the mathematics content itself. How does one assess or evaluate a nontraditional approach with a traditional evaluation method? We need to get away from meaningless numbers as representative of accomplishment or mathematics competency. Mathematical processes and skills need to be assessed also. The Western Protocol Curriculum Framework addresses generic mathematical processes similar to CELs. 25. How do I best evaluate my student's performance and report it to parents in a meaningful way? Student performance is best evaluated by using a variety of evaluationstrategies and techniques. Keep up to date records and keep parents/students informed as to what's happening in the classroom. Refer to pages 11-33 of the Middle Level curriculum guide for additional suggestions. 26. How do we get students to focus on process or steps, not just an answer? Some suggestions might be:
27. How can I get students to attempt several strategies before giving up on a problem?
28. How do we get parents more involved with their child's education? Parents are interested in knowing what goes on at school and need to be kept informed regarding changes in the curriculum. It is suggested that an information session be held with parents to inform them about the new mathematics curricula. They need to understand the philosophy of the program, the areas of emphasis and the focus on understanding mathematics. This is a new curriculum for the students and the teachers. Both will require time and effort to feel comfortable with using it. In presenting parents with an overview of the curriculum
The best way to build parental support of a program is to anticipate the concerns, attempt to understand them, and address them directly and quickly. The new courses are designed to help students develop flexible ways of thinking about numbers and number relationships. The students are encouraged to use a variety of strategies and to explain to others (verbally or in writing) their method(s) of solution. Parents need to be informed and provided with activities that explain the philosophy, rationale, and content of any new course. Once parents understand that the intent of the changes is to help their children to better understand mathematical concepts, they are more apt to be supportive of the changes. For further information and suggestions, refer to page 4.27 and the article on page 4.29 in the Leadership Manual for Middle Level Mathematics that teacher leaders and Regional Coordinators - Curriculum and Instruction have. Also available at the Stewart Resources Centre (STF) is a paper entitled Parent and Parent-Child Mathematics Evenings, developed by Lillian Forsythe. 29. Are mathematics classes designed for all students? Many can't cope. Classes need to be adapted to meet all students' needs. Using a variety of instructional strategies and methods will accommodate a larger number of student learning styles. Manipulatives allow for more cooperative group work which helps more students learn especially concrete learners. Socializing and group work go hand in hand with Middle Level students. The use of calculators allows all students to solve more realistic problems. The development of estimation skills gives students more confidence in their ability to do mathematics. 30. How can we make mathematics a "likeable" subject?
31. Where and how do I get good computer software? There is an increasing amount of good software available. Suggestions can beobtained from catalogues, colleagues, Mathematics 6-9: A Bibliography, and publishers' representatives. Prentice-Hall Ginn has Toolkit, Addison-Wesley has Minds on Math Template and Data Disk Package, and Nelson is developing a computer-guided learning (CGL) system. Software is available from local retailers and companies such as Sunburst, Broderbund, MECC, and Software Plus. Many schools have spreadsheet and data management software that can be used productively in mathematics classes. Some suggestions are provided in the Middle Level mathematics curriculum guide. 32. How can I do all this with my high enrollment? Once students have learned to work in groups, they can do many of the activities in these groups. Problem solving is often more productive when done in groups as is learning concepts using manipulatives. When students are working in groups, you can concentrate on students having difficulty, either in small groups or individually. Peer evaluation can be used rather than the teacher marking all the time. Journals will help speed up evaluation and help you to identify student problems. As with any classes having a high enrollment, there is concern over enough resources. Certain concepts can be developed using different manipulatives and different print resources. Not all students need to be using the same resources at the same time. Activity centres can reduce the amount of a particular material needed. 33. How will I be able to plan enrichment for students who need it? Should students be accelerated or provided with enrichment? For enrichment suggestions, use the curricululm guide, seek advice from peers, and refer to the teacher resources of various textbook series. Activity centres with different expectations for different students and with concepts extended to a higher level of understanding can be used to individualize student learning. Enrichment is a very important part of this program. |
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