Doing Math vs. Understanding Math

By David Ginsburg on March 30, 2015 10:39 PM 

(Excerpt including sample problem from:     
http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2015/03/doing_math_vs_understanding_math.html)
                                                   

(some stuff deleted...)

Consider, for example, this 6th grade problem from the Port Angeles School District website:

In a bowling tournament, Elton scored 188, 212, 214, 196, and 200 in the first five games. In order to qualify for the semifinal round, he must average at least 205 for the six games. What is the least he can bowl in his final game to qualify?

Most students will solve this problem as follows:

Step 1: Find the minimum total score for six games (205 x 6 = 1230)

Step 2: Find Elton's total score for the first five games (188 + 212 + 214 + 196 + 200 = 1010)

Step 3: Find the minimum score required for the sixth game (1230 - 1010 = 220)

I, on the other hand, solved this problem by comparing Elton's scores for each of the first five games with the minimum average score of 205. I then computed the net deficit (amount below 205) or surplus (amount above 205). Here's a visual representation of my thinking:

And since the average after five games is 15 less than 205, I knew the score for the sixth game would need to be at least 15 more than 205, or 220.

Would I say that my approach is better than the first approach? No, but I would say it reflects conceptual understanding of averaging. I would also say it reflects procedural fluency, which NCTM defines as the ability to apply procedures accurately, efficiently, and flexibly. In contrast, it's possible for students to get the right answer using the first method without even knowing what "mean" means. It's also possible for students to follow a procedure without having procedural fluency.

The lesson here for us as math educators is that we need to shift the emphasis from answer-getting to the problem-solving process. We also need to model for students--and encourage them to pursue--multiple solution strategies rather than prescribe a standard procedure.

In essence, we need to make the most of opportunities to deepen students' conceptual grasp of math and build their procedural fluency. We need to help students understand math rather than just do math.

 

I have some serious doubts about this Ginsburg character after reading his EdWeek opinion piece on "Doing Mathematics vs. Understanding Mathematics. "

http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2015/03/doing_math_vs_understanding_math.html

Here's the guy's webpage touting his various accomplishments, including one about being a "hero" in education according to Microsoft (Jeez...)

http://www.ginsburgcoaching.com/Home_Page.html

This guy is apparently some kind of superman when it comes to teaching math in an "urban" setting, too.

Gawd help us all if he ever teaches this crap outside the ghetto.

In this latest entry he tries to show a method of calculating a sixth bowling score so that the average of those scores would be 205 given that the first five scores were already known.

So, basically, the problem is given five numbers, what is the sixth number such that the average of those six numbers equals 205. 

Most people would know that this means the total of the six scores would be 6x205=1230, so
the sixth score would simply be 1230 minus the sum of the first five known scores.

Pretty simple. 

And a good example of clear mathematical reasoning using the fairly simple concept of averages.

His "answer" had me quite stunned for its stupidity when compared to the much simpler method he said (somewhat dismissively) that most of his students (and I, if not totally wasted on some mind altering substance) would use.

He basically turned a fairly simple three step process into about a 10 step process which used addition and subtraction instead of multiplication and subtraction to get an answer. 

Sure he got the same answer, but went all around the problem to do so.  Bad form. Bad math.

Even dumber is the fact that he knew the process the students allegedly would use only had three simple and logical steps while his was much more convoluted and involved computing a "deficit" or "surplus" of each score from the mean as well as keeping a running tally of the summation of those individual "deficits" or "surpluses". 

All kept in a table, no less.  Jeez, talk about analysis paralysis.

The students methods were much more elegant and showed a superior understanding of the material than his did.  I seriously hope that idiots like this are not having a big influence on teaching math in our schools. 

His big "complaint" about the simpler method is that kids might do it without knowing what "mean" means.

Oh, for crying out loud, teacher, TELL THEM WHAT "MEAN" MEANS!  Start by averaging two numbers, if they're that slow.

It certainly makes more sense than your BS method of dancing all around the meaning without making it any clearer.  If anything, YOUR method makes it easier to get the answer without knowing what "mean" or "average" means because most people will tell you that "average" has something to do with SUMMING A BUNCH OF NUMBERS AND DIVIDING THEM BY THE NUMBER OF NUMBERS.

NOT ADDING AND SUBTRACTING "SURPLUSES" AND "DEFICITS" WHILE KEEPING A RUNNING TOTAL IN A TABLE.

That sounds more like something an accountant would do to reconcile a bunch of credits and debits on a spreadsheet.

If I were grading, I'd give his "students" an A and him a C, depending on whether or not I saw him adding and subtracting using his fingers and toes.

Because he is seriously retarded if he thinks his method shows a better understanding of mathematical concepts such as averages.

I guess most teachers nowadays are just so mathematically illiterate that they would actually believe some doofus like this and put him on a pedestal.

This kind of stuff just makes me angry at what passes for "education" today.

And the fact that he labels his stuff  "understanding math" is probably a bit intimidating to those teachers who feel that they probably do not "understand math" well enough to question this BS but who may still feel that his method is cumbersome at best.  And NOT a good example to follow.

Well, I DO "understand math" and he is a doofus for putting kids who may not know any better through this convoluted method.

"Learning Is Fun" with the Dictaphone Electronic Classroom - A Discussion. Beulah E. Brown, Journal of Negro Education

Even though I know you can lead a horse to water...

I'm just going to post this data again anyway.

From the NAEP Publication:

ACHIEVEMENT GAPS

How Black and White Students in Public Schools Perform in Mathematics and Reading On The National Assessment of Educational Progress.

This data is from pg. 8, Figures 7 and 8.

I've summarized the data for 2007 below (similar data is out there for other years as well).

4th Grade scores:

FL=Free Lunch
RL=Reduced Lunch
NE=Not Eligible for Free Lunch

White FL=235 RL=241 NE=252
Black FL=217 RL=228 NE=232

Note that in the 4th grade, the poorest whites (in the FL category) outperform blacks who do not even qualify (NE) for the FRL program.

Now onto the 8th grade:

White Fl=274 RL=280 NE=295
Black FL=253 RL=265 NE=274

Wow, we've made some "progress" here.

Note that the Non-Eligible blacks score exactly the same as the whites who receive Free Lunches.

Finally, we have "poor white redneck"=blacks...

Only it's the middle-class blacks the "poor white rednecks" are apparently equal to.

In these numbers you can see that while poverty does play a role in the scores, it is not enough to explain the gaps between the races.

Simply because the Free Lunch whites either score on par with (8th Grade) or above (4th Grade) the Non-Eligible blacks.

I hope those "courageous conversations" get to the bottom of this little puzzle.

January 3, 2013 at 8:17 AM

Trying to preserve some prior posts from CO "Your School" blog here...


Why have more special programs (or even Project LIFT) when the solution to the academic problem among blacks was found nearly 50 years ago?

Technology was the answer then (especially corporate sponsored, "real-world" technology) just as it often is today.

This could have all been nipped in the bud if only they had used the

Dictaphone Electronic Classroom

(using Dictaphones with Dictabelt technology!)

as recommended by Beulah Brown in the Journal of Negro Education back in the summer of 1966.

Among its reputed benefits were (p. 248):



(3) Teachers can work more closely with the children and help build a more solid foundation for today's complex living.

(4) Teachers and pupils will find the "teaching-learning" experience loads of fun.

(5) Classroom discipline (surface behavior problems) will be almost nonexistent.

(6) Slower and less alert pupils will receive more from this personal belted teaching-learning experience.

(7) Potential dropouts will have a greater incentive to stay in school a while longer.

At least according to this "study":

"Learning is Fun" with the Dictaphone Electronic Classroom -- A Discussion.

By Beulah E. Brown, The Journal of Negro Education, Vol. 35, No. 3, Summer 1966.

August 22, 2013 at 7:44 AM