By Enakshi Choudhuri

(Continuing a discussion on Integrated Math ) 

In the past month I have had the opportunity to delve a little further into the realm of fuzzy math. I must say, I have yet to meet a parent who thinks that the integrated math curriculum is the next best thing since disposable diapers. On talking to parents the common refrain that I heard repeatedly is that supplementing math education is not an option, it is a necessity whether the child attends the local public school or the toniest private school in town. So, in this post, I thought I would touch upon some of the more common options for supplemental math instruction that are available to parents.

However, before I get into the various supplemental programs I would like to define two important terms: spiraling and mastery:

1. A spiraling math program introduces multiple topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics in the next year, the year following that and so on, with the topic becoming more and more complex in each passing year. At every grade level the student works with the same set of 6-8 topics which get more complex as they go up the grade levels. While this approach allows exposure to a topic many times over several grades, the significant gaps in time between exposure and re-exposure to a topic impedes adequate mastery of the core subject matter and concepts.

2. A mastery based math program, on the other hand, simply aims to teach a student to master a single topic or sub-topic, before going on to the next.  So a child will be asked to master addition before going on to subtraction and so on. Topics will change from year to year as the student learns to master different topics. A student may revisit topics from the past but only to learn how to master a topic that builds on the previous topic. For example, a student masters number addition in grade 1. Then in grade 2 or grade 3 he may do fractions and revisit addition in order to learn how to add fractions.

The integrated math program as advocated by the University of Chicago as many of you have already guessed is a spiraling math program. An important factor in this program is that "….students need to construct their own understanding of each mathematical concept, so that the primary role of teaching is not to lecture, explain, or otherwise attempt to 'transfer' mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions” (http://mathforum.org/mathed/constructivism.html). 

Many topics or strands of mathematics are amalgamated with non-math subjects into a real-life context. So, instead of presenting a series of classes in algebra, geometry, trigonometry, and statistics in tracks for advanced, average, and remedial students, they are all combined into one curriculum with a focus on real world, and often politicized, issues such as saving the rainforests. Overall, the integrated math curriculum de-emphasizes memorization of number facts, the learning of proofs, and algebraic skills, but encourages the use of calculators and expects students to discover mathematical formulas and principles on their own or in groups. A hallmark of this kind of program is the lack of textbooks or any other explanatory material.

The integrated math curriculum is heavily funded by the National Science Foundation (NSF). For grades K-5 some common names are Everyday Math, Investigations Math, Math Trailblazers.  Other names in higher grades are Connected Math, Math in Context, Core-Plus Math, SIMMS Integrated Math, MATH Connections and so on.

Increasingly, parents concerned with lack of adequate college level math skills resulting from Integrated Mathematics in schools are turning to supplementary mathematics resources to ensure that their children are better prepared. There are a number of math programs out there for parents to access. Some common ones are Kumon, Saxon Math, and Singapore Math.

Kumon Math: Kumon math was developed by Toru Kumon in Japan almost 50 years ago.  Kumon is not a mainstream math program but more of a supplemental program. Their main goal can be summarized in the following algorithm:  speed + accuracy = mastery. The child is assigned multiple math worksheets and has to complete them in a certain time frame (good practice for timed tests later on in life) in class as well as at home. A child may advance to the next level when he or she has earned a perfect time score + a perfect content score on the worksheets for his or her current level. A child is expected to work about 15-20 minutes each day on Kumon worksheets. In a nutshell, Kumon focuses on repetitive drills and mastery of topics.

Many parents I have talked to have been very happy pairing Kumon with an integrated math program as Kumon addresses some of the shortcomings of the fuzzy math programs, primarily mastery and practice. Kumon also stresses sequential learning. This is where it gets a little tricky. I sat down with a Kumon instructor and she informed me that the child has to master basic math operations in sequence and then go on to more complex topics. So in the Kumon sequence the child may not get to fractions until almost third grade but in Everyday Math with their spiraling structure, children have to learn about fractions in first grade itself. When I brought this up with the instructor, she informed me that I would have to help my child with fractions up to the point when she (the Kumon instructor) would be ready to introduce it to her in class.  In addition, Kumon would not do any problems related to time, money, shapes or measurement, topics that made up the bulk of what my child did in first grade.  The website for Kumon math is www.kumon.com. Some Kumon material is available at Barnes and Noble but for the most part one can only get the material by enrolling in a program (generally $80 – $100 per month).

Saxon Math: Saxon Math was developed by John Saxon in 1981 and is one of the most heavily researched math programs. The Saxon math curriculum is cumulative or incremental in nature and students work towards mastery in math topics by reviewing and building upon previously learned material.  For the most part, students learn a new mathematical concept every day and constantly review old concepts. An interesting study on Saxon Math was done in Anne Arundel County in Maryland, where 14 schools scoring very low in math were switched from Everyday Math and Mathland (two fuzzy math programs) to Saxon Math. In all 14 schools, math performance jumped after a year with Saxon. Saxon has been used successfully in many schools. It has also been used by many parents at home to provide math instruction for children when the math program at school is inadequate. It is very popular with homeschooling families. I know two parents whose children were in schools that used this program, one loved Saxon math and the other did not care for it at all. If you have any comments on this method I would be very interested. You can order Saxon math material at the following website: http://saxonpublishers.harcourtachieve.com/en-US/saxonpublishers.htm.

Singapore Math:As the name suggests, Singapore Math was developed in Singapore, one of the leaders in math ac
hievement. A number of charter, independent or private schools have adopted Singapore Math as their math program. But relatively few mainstream public schools or districts have done so.In Singapore Math: Simple or Complex? by John Hoven and Barry Garelick, the authors explain that the mainstay of the Singapore math program is the Bar Model approach. “…using the bar model approach, Singapore textbooks enable students to solve difficult math problems — and learn how to think symbolically.  Bar modeling is a specific variant of the common Draw a Picture mathematics problem-solving strategy. Because Singapore Math uses this one variant consistently, students know what kind of picture to draw. That's an advantage if the bar model is versatile enough to apply to many complex problems — and it is. It is especially useful for problems that involve comparisons, part- whole calculations, ratios, proportions, and rates of change. It communicates graphically and instantly the information that the learner already knows, and it shows the student how to use that information to solve the problem.”According to the same authors, Singapore's textbooks are used in more than 600 schools in the United States and also by many home-schoolers. The focus is on mastery of topics in a simple, logical manner outlined in a coherent curriculum with a strong focus on skills necessary for success in algebra.  I have not been able to talk to anyone who is familiar with this method and will definitely be interested in hearing from people who have used the Singapore math books. You can order these books at www.singaporemath.com.

Now in additions to these programs, one can also visit your nearest learning store and pick up a bunch of Spectrum Math books that have been developed primarily to help children in integrated math programs practice what they have learned in class. Can I recommend any one program over the other? Well, at the moment, my child is doing the Spectrum workbooks over the summer. I find that her math skills have improved a little bit since she started doing the workbooks. However, I also know that I will look into the other programs to cover the missing gaps in her education. The important thing is to be aware of this issue and make an effort to address it in some way or the other. If your school uses an integrated math curriculum, talk to them about changing it. Be an advocate for your children’s education as no one else will.  In my next post I will look at what some states in the US are doing or have decided to do with their integrated math curriculums. Until then, enjoy the summer.

Enakshi Choudhuri did her Masters in Counseling from Heidelberg College in Ohio and PhD in Counselor Education from the University of Iowa. She has taught and supervised undergraduate and master's level students and has worked with children, adults and college students at various agencies. She is also an author and book publisher. More posts on parenting and education can be found on her blog.