I am somewhat confused by this post and looking forward to your next post which describes the integrated math curriculum and differences with other curriculum. I live in India and trying to teach math concepts to my 11 year old.
In India, while teaching math, the focus seems to be on memorizing multiplication tables and rules such as “carry over addition” without a lot of emphasis on connecting to the real world.
Your post suggests that connecting to the real world does not promote learning- if a concept is taught using the trains example, the student may not be able to apply that concept to a water level example.
However, isn’t the purpose of learning math to learn how to solve real world problems ? When faced with a real world problem, like a water level problem, doesn’t the child have to understand and distill it down to the applicable rule which he or she has learned. Without the connection to real world examples how can he or she figure out which rule to apply ?
Am I missing something ?

Hi,
My main point in the post is that children who learn math concepts abstractly are better able to apply that knowledge in a variety of situations than children who learn the concept through a real world example. In the latter case, the details of the real world example detract from learning the actual math concept and the child is less likely to be able to apply the same rule in another situation.

I think a better process is to teach the abstract concept first and then teach the child how to apply the concept to real world examples. The transfer of learning is much easier then. Real world examples are very important and the child should be able to connect to that. But first the child needs to learn the concept without anything else to distract her. Then give her as many real world examples as possible to help her practice using the concept.

I will give an example- this is a problem that my first grade child had to do:

Draw different color crayons in the box. 1/2 the crayons in the box are blue. 1/4 of the crayons are red and 1/4 are yellow. There was no total number of crayons given.

My child has not learned that 1/2 = 1/4 +1/4 (the abstract concept).

She knows separately that 1/2 + 1/2 = 1 and that
1/4 + 1/4 + 1/4 + 1/4= 1. The connections between 1/4 and 1/2 has never been made for her in class.

The child is asked to ‘construct’ this problem and then solve it. The questions following were- how many blue crayons did you draw?, how many red and how many yellow? Then they are asked- Explain how you know that half the crayons you drew are blue.

So the child does not have an abstract concept guiding her. Every child can have a slightly different answer depending upon how many crayons they started out with. I let my daughter solve it in her own way but the concept that 1/2 = 1/4 + 1/4 did not sink in very well. If I had not explained the concept in more detail and had just asked her to do a similar problem using fruits or boats she would have gone through the same process of part guess work and part past experience to “solve” the problem. Her knowledge was dependent upon the ‘context’ in which she had learned it. To differentiate the concept from the context is a big jump for kids at this age.
If she had been taught the concept first she would have had no problem connecting and applying it to various situations.

My daughter is also asked to do problems requiring division or multiplication without having learned the concept of division or multiplication in class. They are expected to construct the answer using manipulatives or tally marks!

So, I hope this makes my point a little clearer.

Thanks for your comment.
Enakshi

As a teacher I never imagined I would be pulling hair to teach Math to my first graders. Some kids just get them as easy as 1,2,3 while some (try all the visual and kinesthetic modes) will be the very same who at the end of the year has shown very little progress. Once again it also depends on the district adopted curriculum, teacher expertise, and comfort level. An integrated curriculum makes sense to me at a later stage when the students can use all the concepts they have learned early on and apply it as needed. As a first grade teacher my focus is on getting their number sense well rooted and I try all modes of teaching I can possibly use to get them to understand the beauty of numbers. Recently I attended a conference on Singapore Math which talked about breaking the tens and ones to help students manipulate big numbers. At the conference I saw most of my fellow teachers struggling to keep up with the speaker. Maybe that is the big problem!

Though I am in favor of an integrated curriculum I also agree with Enakshi that the basic concept has to come first. Looks like the Everyday Math curriculum is not doing that justice

The premise for using real world examples whether in Math or Science is to make the learning relevant and hence engaging to the students. I think a real world example is an appropriate way to introduce the concept because it acts as a hook and gets the students to pay attention. Once students know how a concept applies to their everyday world (real world) they will be more willing to learn the abstract theory that underlies the concept. Helping students make the tranformation from real world to abstract involves using good instructional strategies. It doesn’t happen automatically. Sometimes it helps to start with abstract, move to real world, and then back to abstract. The problem with the Math and Science education in the US lies with the fact that teachers do the fun,realworld, inquiry-based activities and forget to follow it up with the abstract reasoning. I wouldn’t completely discount the use of concrete examples, hands-on activities, or manipulatives. They are cognitively appropriate for younger students and cater to visual-kinesthetic learners.

The study by Kaminski that you referred to talks about applicatioons of underlying concepts to new problems. Regarding story problems like the train problems, an application of an underlying basic concept (distance=time* speed) is applied to a new domain. Essentially, it is a “transfer” . but it is difficult for students to make such transfer. Ideally, students should be able to apply any learned concept to new situations, and solving word problems–or applying them to “real-world” situations– are the preferred outcomes. The article discussed HOW TO TEACH, and not whether such problems should be tackled after initial learning. Clearly, by choosing a transfer situation to asses prior learning, the authors implicitly agree that problems like the train example are proper to solve after initial learning. Just not for initial teaching.

The difficulty of transfer of knowledge to new situations, is called “inflexible knowledge” which is addressed by Willingham in an article he wrote about it:

http://www.aft.org/pubs-reports/american_educator/winter2002/CogSci.html