By Enakshi Choudhuri
An oft used idiom, ‘there are two sides to every coin’ seems an appropriate way to start my current and last post on the integrated math curriculum in K-12 schools. It is obvious that a large number of teachers and school officials are in favor of integrated math as it has been approved in over 200,000 classrooms across the country despite protests by parent groups.
In my search for the positive aspects of integrated math, I approached a number of current and former school officials or teachers. Surprisingly, many of my requests for comments were turned down. Those who agreed to talk to me requested absolute anonymity. Consequently in writing this column, I have utilized material from my own independent research and drawn on some very general points that came up in conversations with various people. I have not quoted anyone directly.
The most striking aspect of integrated math appears to be its focus on helping students learn multiple methods and concepts simultaneously or in a spiraling fashion as well as being able to integrate or pull these all together to solve problems. As many proponents have observed, math in the future is not going to be about computation or memorization of basic facts to be used to solve standard problems. In the future, math will be used primarily as a conceptual tool to aid in analyzing, estimating and formulating solutions to practical problems.
To that end, students must be aware of, and be familiar with, a vast array of methods and concepts and be able to weave together disparate topics to come up with a plausible solution, the computational aspect of which will be performed by computers or other machines. This is the main argument that drives the use of calculators even in first grade and the focus on students being able to construct their own solutions to problems. The integrated approach to mathematics enables students to develop highly creative approaches to problem solving.
Another point that was stressed over and over again is that the integrated math curriculum actually teaches math the way kids think. For many children standard algorithms such as carry over addition or long division are abstract meaningless concepts. The alternative methods endorsed by the integrated curricula, however cumbersome, provide a clearer picture of the math concepts underlying the process.
For example, some children find it difficult to comprehend that 2 in the tens place is actually 20 or that 5 in the hundreds place is actually 500. So when they are adding 526 + 210 using the standard ‘carry over’ algorithm, they add 6+0 = 6 in the units place; 2+1 = 3 in tens place; and 5+2 = 7 in the hundreds place. Using the standard algorithm, they are basically adding in terms of ‘units’. The Everyday Math partial sums method on the other hand clearly shows them that they are adding 500 + 200 in hundreds place, 20 + 10 in tens place and 6+0 in the units place to arrive at 700 +30 + 6 = 736. This is what a partial sums solution looks like:
Add hundreds – 500 + 200 700
Add tens – 20 + 10 30
Add units – 6 + 0 6
Final sum 736
The math process is simplified in a way that students find it easy to comprehend and learn, even though it may take a little longer to arrive at the solution. So students really ‘get the math’ and do not have to rely on rote memorization of abstract concepts.
Integrated math also emphasizes the application of math to the real world. Many of the problems are context-rich and students are constantly being asked to connect the math concept to a situation that they may have encountered in their day to day lives. For example, a student may be given an open-ended objective to construct a subtraction problem using bananas as a unit. The student constructs a number story about buying a crate full of 50 bananas in the grocery store, eating 6 bananas for lunch and being left with 44 bananas to share with his friends in school. In addition to math skills, the student also needs to use language skills to build this story problem and the real world context is evident in the story line. Each student constructs his or her own story problem with the only commonality being ‘subtraction’ and ‘bananas’. So in a class of 20 students, the teacher gets 20 different story problems of varying levels of complexity. This method engenders creative problem solving balanced by a real world context that does not occur in traditional math curricula.
These are some of the positive aspects of integrated math curricula that I have not dwelt upon in my previous posts. I would like to point out that while many proponents of integrated math emphasized the advantages of this system, they also acknowledge that such curricula are weaker in promoting mastery of basic facts and skills. The spiraling method also seems to interfere with retention of concepts over time. However, if you were to ask them, the favorable aspects of integrated math far outweigh the slight imperfections that exist. So, while I can perceive the beneficial aspects of integrated math, and even as curriculum developers are ironing out the creases in this system, I still continue to believe that we as parents have the responsibility to help our children by supplementing their math curriculum at home to compensate for the existing deficiencies in the system.
Before I conclude, I would like to discuss the state of Minnesota’s efforts to address math and science proficiency in the K-12 educational system. In the 2007 Trends in International Mathematics and Science Study (TIMSS) the state of Minnesota opted to be treated like a ‘mini nation’ so it could be ranked among all the other 60 participating nations from around the world. Minnesota in the last twelve years has established rigorous state wide standards for math and science that school districts have to abide by regardless of the curriculum they may have chosen to be implemented in their schools. These standards have resulted in increased classroom time for math and science in all schools, especially at the elementary level, and stringent graduation requirements in both math and science.
As a case in point, three years ago, Carver County elementary schools in Minnesota had the worst performance among schools with similar demographics on math achievement tests (http://www.startribune.com/local/east/37480764.html?page=1&c=y). With small changes such as increased math instruction (up to 75 minutes a day), specialized teacher training in math and the addition of a math specialist, they now rank 15th among the 52 schools in the same category. Their decision to increase daily math instruction time using an integrated math curriculum and more content based training for teachers seems to have resulted in big gains.
To get back to the TIMS Study, in 1995 Minnesota’s performance in grades 4 and 8 was not significantly different from the overall US performance. However, in the 2007 TIMSS, Minnesota’s scores at both 4th and 8th grade level were significantly higher that that of the US (i.e. the rest of the 48 states as Massachusetts also opted to be treated like a ‘mini nation’). At 4th grade level, Minnesota’s rank in mathematics (2007) was 5th after Hong Kong, Singapore, Taipei and Japan whereas the overall US rank was 12th. At the 8th grade level, Minnesota was ranked 6th in comparison to the US at 10th place.
The overall gains by Minnesota students point to the fact that despite the variations in individual school district math curricula, students succeeded at an international level primarily because Minnesota chose to implement strict state-wide yardsticks that are comparable to some of the nation-wide math standards set by top performing countries such as Singapore, Hong Kong, China and Japan. In addition, Minnesota developed accountability assessments that would reduce the variation from one school to another in terms of what teachers teach and emphasize and instead focus on what students actually learn. Thus, the winning mantra for Minnesota schools seems to be an increased focus on math, more time spent daily on math instruction, additional teacher training in math and science and the use of rigorous state-wide standards to establish tangible objectives regarding a child’s progress in learning various math concepts and skills.
Minnesota’s gains in the TIMS Study makes a significant case for state-wide or nation-wide benchmarks in key subject areas such as math and science. Such standards will ensure that every child is expected to reach certain benchmarks at different age levels and at that point curriculum and methodology will become secondary to the actual content and focus of math instruction. For example, if the standards require every child to have mastered multiplication by third grade, then it is immaterial which curriculum the school follows or what methods are used to teach multiplication. The more important issue becomes that of mastery of that specific concept by the requisite grade level. Maybe this is what different states and the nation as a whole should focus on in developing K-12 proficiency in math and science so that eventually our children will be able compete effortlessly on an international platform.