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This is a soft question perhaps not well suited for the format of the site but I'm interested to hear opinions from this community on this topic.

K-12 mathematics textbooks (understandably) divide their content into chapters. My concern is that the level of compartmentalization from chapter to chapter is so extreme that it hinders the development of real mathematical thinking, and instead fosters the development of what I call flow-chart mathematicians.

The most discouraging example I know comes from the texts used in grades 7 through 9 in my own Atlantic Canadian province. Each book contains a chapter dedicated to "Real Numbers" where students learn to do arithmetic with fractions. In each book this chapter is followed by another called "Measurement", where children work on problems to do with perimeter, (surface) area, and volume. The thing that's shocking to me is that no problems in the Measurement chapter involve objects with fractionally labelled lengths!

I believe this is an error, and I sometimes try to address it when working with kids this age. After establishing the prerequisite skills, I'll draw a rectangle that's $4$m by $\frac{2}{3}$m and ask them to calculate the area. A surprising number of students who are capable of moving from $4 * \frac{2}{3}$ to $\frac{8}{3}$, and who are also able to explain to me how and why the area of a rectangle is equal to its length times its width, still have substantial difficulty putting these skills together in order to find the area of this given rectangle.

I think that this reveals that they don't understand that the multiplication of fractions that they've learned in the previous chapter is the same operation as the familiar multiplication of integers. Rather, they've learned a flow-chart to follow when they come across $\frac{a}{b}*\frac{c}{d}$. This, I think, is a huge problem!

I would like to hear arguments for and against this level of compartmentalization, and if possible, examples of situations where it's particularly beneficial or harmful.

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    $\begingroup$ There is a (troubling?) tendency to cook up problems with "nice" input values and "nice" results. This might be just an example of this, with the author not realizing the non-use of fractions. $\endgroup$ – vonbrand Jul 8 '14 at 18:12
  • $\begingroup$ To me this sounds like a bit narrower problem; namely, not practicing fractions and decimals enough throughout the curriculum. This tends to be a bit of a scam (correlated with locally-drafted "only what they need" materials and assessments being made trivial to boost passing rates). My solution would be to use a good book, like Martin-Gay's Prealgebra & Introductory Algebra, which does consistently touch on those types of numbers in exercises throughout the textbook. $\endgroup$ – Daniel R. Collins Nov 26 '15 at 5:54
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The compartmentalization drives me crazy. I'll have students who can use the distributive law perfectly when we're on the "distributive law" chapter in the book. But if I later give them a problem to solve 37 x 40 + 37 x 48 + 37 x 12 in another context, they will do the 3 multiplications out by hand and never notice that that could have been simplified to 37 x 100 using the distributive law. Likewise, if I give them that problem they will ask me "Do we have to use order of operations?" like that's some special thing you only do when you're explicitly asked to.

At the middle school level, I try to bring in problems from sources like MATHCOUNTS to expose the kids both to problem solving strategies, but also to using the math they know even when no one told them which topic was going to be exercised.

And I try to emphasize (with limited success) that the things we learn are tools that they can use for all sorts of problems going forward, and not just tricks they have to learn to pass the chapter test. And I do try to back that up by bringing them problems that use prior chapters' (and years') skill sets.

So, I can't cite you any research or anything, but my experience tells me that this compartmentalization is bad for students, and contributes to misconceptions as to what mathematics is really all about!

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The opposite of compartmentalization as the question describes it may be connections. Connections is a process standard of the U.S. National Council of Teachers of Mathematics (NCTM), and they have written widely about it, including this article which includes a story about a breakthrough moment in class when a student connected past work with factors and multiples to current work with area.

Elsewhere from the NCTM website:

Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study. When students connect mathematical ideas, their understanding is deeper and more lasting, and they come to view mathematics as a coherent whole. They see mathematical connections in the rich interplay among mathematical topics, in contexts that relate mathematics to other subjects, and in their own interests and experience. Through instruction that emphasizes the interrelatedness of mathematical ideas, students learn not only mathematics but also about the utility of mathematics.

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I agree with you: this is a huge problem and to avoid it, mathematics should be taught in a way completely different that the compartmentalized approach given by textbooks. But, unfortunately, this is the way that our current society wants mathematics to be taught. At least in my country is that way: not only textbooks have this completely compartmentalized structure, but official curriculum tends to be much more compartmentalized every day. Mathematics is a subject with an increasing social pressure in such a way that, if you try to implement some kind of innovative way to teach math to get rid of such compartmentalization (for instance, a problem solving approach to introduce new mathematical contents), you will have lots of parents complaining that you are not doing real mathematics at your classroom.

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Decompartmentalize. Start the new chapter as written, then connect it back to things you did before. Also introduct things that you can't do yet with what they know so that there is mystery and anticipation.

Math is about patterns.

A lot of math is taught just as symbol shuffling. Stress visualization. E.g.

Draw 3 rectangles that are labeled 47 x 37 etc. This works great if you have a presentation screen and can move figures around and rotate them. Show them that they line up end to end with a common 37' length that they can factor the 37 out. Do this with different combinations.

Then show it with one of the 37's divided into a 20 and a 17. So now you apply the factoring twice. in different ways.

Leave them with a packing problem where you have a whole list of rectangles to pack into another rectangle.

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  • $\begingroup$ +1 for making an effort to resolve the problem. I place a huge amount of emphasis on "connecting the dots" if you will. I do this by presenting new knowledge as a continuation of previous knowledge. How this would work in an ordinary classroom, I'm not sure. I mostly teach using the flipped classroom model. $\endgroup$ – Alec Oct 12 '14 at 18:58
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I don't agree with it, but "flow chart mathematics" and "flow chart learning" generally, seems to be the order of the day, with the rampant computerization of society. It started about 40 years ago, and is coming to a culmination as we speak.

The advantage of it is that it is practical for solving linear, or straight line, problems using logical reasoning. Its disadvantage is that it inhibits creative thinking using analogies, or involving hidden relationships, or even connections.

In their book, "Generations," regarding the Anglo-American world, probably including Canada (the late) William Strauss and Neil Howe postulated that linear (flow chart) and critical thinking alternated in roughly 40-year cycles (that is, over the course of a total 80 year cycle). The development of critical thinking in schools has been at a low ebb, and may be about to make a comeback.

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