Connections

Overview

Curriculum Comparison

Textbook Review

Lesson Plan

(note: Italicized text excerpted from NCTM Principals and Standards for School Mathematics.)

NCTM Connections Standard for Grades 6-8

Middle school students should be provided numerous opportunities to develop a “big picture” view of mathematics – to see that it builds on previous experience and connects to itself and every other discipline. Real learning doesn’t happen in isolation and deep understanding develops only with connection to other ideas. Math teachers need to actively work with teachers in other disciplines to find appropriate activities.

Students should:
• recognize and use connections among mathematical ideas;
• develop connections so ideas build on one another rather than being learned in isolation
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• begin to see how challenging problems can be tackled using familiar ideas
• notice how ideas are related
• students should be asked to explain
• students’ answers reveal their connections:
• Why did do you think that?
• Have you seen something like this before?
• Did anyone think of this in a different way?
• recognize and apply mathematics in contexts outside of mathematics.
•  teachers need to choose appropriate problems that extend beyond mathematics into other disciplines and require the use of a range of knowledge to solve
NCTM Connections Standard for Grades 9–12

For all students, mathematical concepts should not be taught in isolation from each other or from other disciplines. Connecting to previous knowledge, to other experiences outside the classroom, and to itself, allows students to build a deeper, more meaningful mathematical understanding.

Students should:
• recognize and use connections among mathematical ideas;
• should understand that more than one approach can be successful and work in an environment that encourages such exploration
• use insights from one context to prove or disprove another
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.
• Teachers must choose problems that provide the opportunity to tie into other areas
NCTM Standards and North Carolina Standard Course of Study

 NCTM Expectations NC SCOS Objectives Match? Connections for Grades 6-12 recognize and use connections among mathematical ideas Connections are never specifically mentioned in a objective, however they are addressed in their main goals for mathematics education: "Connections within mathematics and with other disciplines" In high school, Integrated Mathematics, "provides students the opportunity to study traditional topics from algebra, geometry, probability, and statistics in a problem-centered, connected approach. not directly understand how mathematical ideas interconnect and build on one another to produce a coherent whole recognize and apply mathematics in contexts outside of mathematics

Complete text of North Carolina Standard Course of Study

Summary

It is disappointing how little reference is given to this important process standard in the North Carolina curriculum. Though connections are stated as an over arching goal for math education in the department's philosophy, there are scant references to building them in the stated objectives for courses. Making sure that teachers kept to this idea in practice would be a challenge without direct tie-in to the curriculum requirements.
In both middle and high school classes, courses include objectives from a variety of mathematical strands. But are these courses taught in a way that shows students the value of these connections? Why are there no references in the entire curriculum to "contexts outside of mathematics?" We miss a wonderful opportunity to make math not only more understandable, but more meaningful.
The NCTM Standards do an admirable job of emphasizing the importance of connections. Their goal that students develop a big picture view of mathematics, along with an understanding that it is connected to a huge range of other disciplines may well be key to engaging many students. Taken to heart, the middle school teacher would be building a repertoire of lessons integrated with all of their students' other areas of study, from science to literature. In high school, NCTM fails to note the importance of these cross-discipline connections and focuses wholly on those within the field of mathematics.

Textbook Review
Integrated Mathematics 1

Overview
This would be an ideal text for the average 9th grader. While there's good coverage of basic concepts things like number sense, it's range extends to introductory Trigonometry. Filled with colorful pictures, interesting projects, and encouragement to communicate and think about mathematical ideas, it is, in many ways, an expression of the NCTM Principals and Standards. At first I was put off by how "busy" it was, I think it would have a lot of appeal to an adolescent. Perhaps it panders to the MTV/fast-cut presentation style kids are so used to, but it certainly makes math seem fun.
Each unit begins by introducing a unit group project, things like "Planning a Music Store" for the unit on data representation or "Design a Sports Arena" when exploring direct variation. It even directs students to upcoming exercises that will help with the project and then the unit ends by "completing" the project. Rather than simply offering an idea (actually, they also offer alternative choices too), the text guides students through the process, making it far more likely teachers would include these projects in their course.
At the end of each section there are 30 to 40 exercises including open-ended questions, writing exercises, process practice, and even a preview of what's to come. Units end with a review and assessment and a summary of the main ideas/formulas learned and a list of key terms. In addition to the usual answers to "selected questions" and a glossary, the back of the book also contains a student "toolbox" with good summaries of a wide variety of concepts like the properties of addition and multiplication and percents.

Connections Standard
This book is epitomizes the Connection concept. It connects across mathematical areas, with units on Algebra, Geometry, Statistics and Probability, Logical Reasoning, and Discrete Math. It connects across disciplines to virtually every field of study. Each unit contains one or more "connections to" problems that direct students to explore math in the context of language arts, history, physics, geography, business education, and science. It also connects to the real world and is filled with problems students might have direct experience with -- from looking at geometric patterns in quilt design to scatterplots of rainfall amounts to plotting the path of a baseball to finding the circumference of the family pizza.
The text also presents lessons in ways that emphasizes respect for different learning preferences. Many units include reading, writing, and hands-on activities, along with lots of group activities and even larger-scale group projects. Noticeably absent are the long sections of homework problems and some problems are actually labeled "open-ended." Discovery, exactly as described by NCTM, is used here to develop conceptual understanding and formulas.
The only obvious major fault is that for all its connections, there's little that seems to connect internally. Ideas and units jump from processes and problems with virtually no transitional text. Teaching from with it would require a great deal of careful attention to making things flow and I would hope that the teacher's manual should provide a great deal of guidance in helping to explain the big picture. Though students may seem to just want to be entertained, I think they still like a story line to carry them along.
Philosophically, I think that integrated mathematics is the way we should present the subject. Disconnect between courses no doubt contributes to students' lack of enthusiasm for math. However, this would require a significant realignment of current curriculum along with the added challenge of fitting it into our standardized testing scheme. In light of our current struggles, it's a challenge we ought to take on.

Note: This text is also reviewed in relation to the Measurement Standard.

Rubenstien, R.N., Craine, T.V., and Butts, T.R. (1995). Integrated Mathematics 1. Boston, MA. Houghton Mifflin Company.