After this introduction, Manzi delves into perhaps the strongest section of the book where he outlines how he used randomized experiments in business to determine strategy.   When doing this, he looked into the literature of random trials in medicine, as well as random trials in social sciences (for example, leading up to welfare reform in the 90s, the federal government required a number of random trials for states which wanted to try different strategies.).  

There is another section of political trials, which reminded me of Rick Perry and His Eggheads and the better Get Out the Vote, which deal with experimental tests in politics of what strategies work and which don’t in get out the vote efforts.   

Perhaps the weakest section of the book is where the author attempts to make policy suggestions.   But overall, I would recommend reading this book.  If nothing else, it is helpful in increasing skepticism of predictions which lack experimental support.

The Problem

The introduction starts out with a fun math class that the author has attended.    The author  enters and the students are solving a problem.  Imagine that you have a merry go round and a skate boarder attached.  At some point, she lets go–calculate how long it will take for her to hit a padded wall.    She later gives her answer in in the solutions.   It’s a fun problem.   It’s broad enough to pull in some geometry, trig., and algebra.  The class she discusses is very involved, with students arguing about different approaches to the problem.  They go the board and show their ideas.  You could imagine an experimental session being included…She notes that it is the students rather than the teacher who ultimately solve the problem (it takes them 10 minutes).

This example allows her to raise the debate between the more traditional approach to math education, where the teacher lectures, then the students silently work problems, and the more “modern” approach where students are more involved.   This is a debate I often had with my dad about the advantages of drilling/memorization compared to understanding…The school system I grew up in leaned more towards understanding and experimented with many different systems (ahh….Cemral).   I have to say that I went through a period where math just somehow didn’t seem rather relevant–until Calculus, where I could see how it applied to physics…But, back to the book, the author’s worry is that far too many students in the US aren’t just apathetic about math, but actually hate and fear it.   Far too few of us continue to advanced levels and given the trajectory of jobs, this is a disaster in the making….She then goes on to give a number of rather depressing statistics about how few students are going into math, our relative ranking in the world, etc.     Beyond jobs, you just need math to make sense of how the world works today–be it looking at the results of a medical study, to understand basic economics, etc.   She also discusses the disconnect between the math that people are often taught in school and how they apply it in everyday life (for example, even when students have math to do in their part time jobs, few use school based methods for it…).   Sadly, the teacher whose class was observed was told that she was no longer teach in the way she was doing–with a problem solving based approach.   Instead, classes had to be given with teachers lecturing and students quietly solving problems individually.   Tragic…

Some of the reviewers in Amazon made a rather unfair criticism of this book, claiming that much of the discussion is anecdotal.   In reality, the author has conducted studies of thousands of American and British students through middle and high school.   She then draws on these observations to make generalizations.   She uses a few particular representative cases to highlight general trends–because most people relate well to stories.

In the first chapter, she sets up the problem.   Namely, there is a huge disconnect between the way that mathematics is taught in many schools (in the US) and how it is practiced by mathematicians (and anyone who uses math professionally).   In real life, mathematicians pull across a number of sub-disciplines to solve a problem (she gives the example of Fermat’s last theorem).  The other thing that she notes is that there is a certain persistence that professionals use that is not taught.   Finally, she notes that in the real world, people collaborate!    This is definitely true in my field of physics.  I’ve been extraordinarily fortunate to have had very good collaborators.   While each individual needs to do their best, the modern world involves a great deal of collaboration.   I think that the main thing to note is that mathematics, engineering, physics, etc. are “living” arts.   We learn a number of basics, but when we are confronted by new challenges, we have to use whatever’s in our arsenal to solve a problem.   Often, the problems we’re solving are hard and require the use of different people with different areas of expertise.  We need to bounce ideas of of each other.    It’s not just a lifeless set of rules to be memorized for a quiz and forgotten…While the basics are definitely important (like kata are important for martial arts)–if that’s all you do, then you miss the art, get bored, and move on….

In chapter 2, she mentions more about the math wars.  Here, she introduces us to her first encounter with a strange group of traditionalists  (see www.mathematicallycorrect.com) who hate modern approaches to teaching math.   Again, while I doubt that anyone would question the need to practice–different students need different approaches.  When I was in grad. school, I was an NSF GK-12 fellow for awhile, where we would go into middle schools and try to help math/science teachers improve their classes.   I remember demoing one lesson in a 6th grade classroom about probability.   I’d notice that one student didn’t understand something, so I’d present it another way, and if she still didn’t understand, I’d try yet again (we also did activities to drive concepts across).

What she worries about is passive teaching approaches, where a teacher lectures and then students just work problems, without real understanding.   After years of this, students lose their ability to do creative problem solving.  Here is a simple problem that she gives:

A woman is on a diet and goes into a shop to buy some turkey slices.  She is given 3 slices which together weigh 1/3 of a pound, but her diet says that she is only allowed to eat 1/4 of a pound.  How much of the 3 slices she bought can she eat while staying true to her diet?

Unfortunately, many adults have problems with this.  You could set up an equation to solve it–but pictorial solutions are also a good approach.  The most important thing is to set up the problem…

She also brings  up the interesting idea of contexts.   Apparently, in the 70s and 80s, math educators in the US thought that they could make math more relevant by making it less abstract and more concrete and applying it to everyday situations–but they abstracted the everyday situations too much so that people familiar with the real situation got into trouble with the abstract situation.   Here, the better approach might be to start with the fully complex problem and show how we abstract it to something “tractable” to get an idea for what may be going on.   This is not easy to do.   To maintain enough “richness” of the original problem, but to make it something tractable to someone less experienced…I may discuss this in another post.   Of course, the author mentions that there are some problems that are without context, but still beautiful…

Solutions?

In Chapter 3, the author begins to move towards possible solutions and better approaches to teaching math.  One approach she looks at is one where the students try to communicate their solutions using multiple representations (graphs, words, tables, etc.).   It’s an interesting approach and also encompasses group work.   In the author’s research, they compared performance of students taught in this manner, with those taught in a more traditional manner and found  superior results from this approach–and a higher enjoyment of math.

Another approach is the project based approach.   Here, projects lasted about 3 weeks and focused on a given theme.   For example, students might be told that the volume of an object was 216 and then they would think about what kind of object it could be.   Concepts were taught on an as needed basis…The difficulties I see with this approach are many.   It requires the teachers to be careful to get a good coverage of different areas of mathematics and the end of the day.   It also requires a lot of effort to choose the right problems–in short, it requires a high calibre of teacher.    However, on national exams, the project based approach school outperformed a standard school with similar demographics.

The key here, I think is that if the projects are well chosen, it would be more engaging than a drill/kill approach, so retention is probably higher–but it would take more involvement on the part of the teacher…This is also the view of the author–though she does recommend some books in her Appendix.

Testing

Multiple guess–Ack!   Her first comment is that national assessments in Europe are rarely multiple guess.  There’s only so much you can learn from multiple guess—you can try to choose common mistakes–but I think you’d learn more from free response–but that would be much more costly and harder to standardize….

The other issue the author finds with the standardized tests in the US is that they emphasize procedure under timed conditions rather than actual problem solving ability…Also, the feedback is limited–they are just given a score–not a listing of areas where they are strong and where they need to improve.

The author suggests an alternative that she calls, “Assessment for learning”:

1)  Communicate what is being learned and where they are going

(For example, understand the difference between mean and median.._

2)  Let them know where they are (how they’re doing)

3)  Let them know what they need to do to improve…

Generally, comments are more effective than grades in getting students to actually improve…(Which may suggest that leader boards and such comparing students in math games and such may be counterproductive…)

Tracking

Ability based grouping is an interesting phenomena.   It’s something that I used to argue a lot with my dad and I have to say that I’m coming more around to his way of thinking.   In the US, tracking starts from a fairly early age and by high school, you have remedial, average, college, accelerated, and gifted tracks in subjects such as math.   It’s very hard to move up in tracks…The idea is that it would make it easier for a teacher to focus on kids if they are roughly at the same level.  It gives more time to focus on the “slow” kids and doesn’t hold the fast kids back.   I think it can be nice for the fast track–but the harm for slow track can be great.    In point of practice–a number of countries that are much more successful in math education on average, such as Finland and Japan do not employ it.  I think you run into all sorts of problems, ranging from different development rates (“late bloomers”) to the fact that people in the lower tracks may simply give up…

Girls

Here is a chapter that I won’t go into too much–but one observation of the author is that a number of girls are discouraged in math due to a lack of explanations.   Her claim is that they especially suffer when just given instructions, without clear reasoning, whereas boys are more likely to soldier on.

Abstraction

One important point that the author looks at is the difference between successful and less successful students–flexibility.   Low performing students are rather good at counting, but as problems get more complex, they don’t abstract.   As time goes on, this becomes more and more of a challenge for them.  She then highlights some individual cases where she led a class to try to help low-performing students over a summer learn to treat numbers more fleibility.

A good start

Problems, puzzles, and toys oh my!   The author observes that a number of people who use mathematics professionally, had a lot of out of school experience with math.   They were encouraged to ask questions…

So, overall, I thought it was a good book.   There are some specifics that I’ve glossed over, but the most important thing is the approach.   Instruction which engages students and leads them to communicate with each other is more effective.   There aren’t quick fixes…