Coffee to Theorems

Dan has an quite successful open problem going on over at his blog.

If I give you some cubical candies, what is the least amount of packaging needed for them?

Lots of great problem solving happening in the comments of his thread.  I took a few stabs at it myself. Beginning with making a list of the first few entries and trying to find solutions manually.  1 candy, (1,1,1) cube: surface area 6.  3 candies can be done with (3,1,1).  Surface area 14.  But something like 20 has a few options.  (10,2,1) has a surface area of 64 but (5,2,2) has a surface area of 48.

From that paper-work I was able to generate the beginning of this sequence of minimal Surface Areas: 6, 10, 14, 16, 22, 22, 30, 24… which I then searched on OEIS, resulting with https://oeis.org/A075777 .

I then decided it would be a good exercise in rudimentary python to try to encode that algorithm, so here is my script: 

(caution: this script generates inaccurate results, it is a script of the inaccurate OEIS algorithm.  My improved script is further down in this post)

This algorithm is very similar to some that others were using in Dan’s comment thread.  But here’s where it gets interesting.  Unless I have an error in my code (entirely possible!) then I think we have broken this algorithm.  Dan gives a few frequent algorithm-breaking-numbers here.  And indeed, a few of these break the algorithm on OEIS:

Take n = 1332 using the algorithm described on OEIS:

Cube root is ~11.002

Floor is 11, but neither 11 nor 10 divide n.

9 divides n.  s1 = 9

n / 9 = 148

Square root (148) = ~12.166

Floor is 12, but we need to subtract away 1 at a time until we find a divisor of 148: 4.

s2 = 4

Then s3 = 37

And that gives a surface area of 1034.

However, the minimal surface area is given by a solid of 6*6*37.  Surface Area is 960 in that case.

And here’s one up to 30000 with columns n, s1, s2, s3, minSA: min surfacearea SF upto 30000

As I’m working on my M.A., I plan on sharing my thoughts on some of the articles I read either for a class or for my own research. I hope these serve as both an accessible summary of the article for current classroom teachers and a place for me to share some of my thoughts on the article.  So here is our first one:

Lampert, M. (2001). Teaching while students work independently. In Teaching problems and the problems of teaching (pp. 121–142). New Haven: Yale University Press.

Lampert is a 5th grade teacher who is also a researcher.  During the writing of this book she was teaching part time as a math specialist for an elementary school, but she also has 8 years of full time experience.   Her background includes the philosophy of mathematics and what it truly means to know a mathematical concept. (And she believes that memorization does not constitute knowing.) She describes her teaching style as one that uses problems and therefore problem solving. “Teaching mathematics would have to engage students in doing mathematics as they were learning it.” (p5)

Chapter 6: Teaching While Students Work Independently

Lampert’s students are given an activity that is based around problems of the form (  ) groups of 2 = (  ) groups of 4.  Some of the learning targets in this lesson are for students to use multiplication to create a true statement, and to utilize problem solving strategies.

Notice a few more things here: (1) there are many ways a student can answer this.  (2) its possible but not necessary for students to bring up fractions, (3) the equals sign is not treated as a “do” symbol.  Number 1 means its more open than a normal exercise and provides opportunities for students to analyze each other’s work (great Common Core thinking 12 years early, Lampert!).  Number 2 means we can extend the problem naturally but also we have not put unnatural boundaries on the problem.  2*5 = [  ] is pure arithmetic calculation. But “x groups of 2 = [  ] groups of 4” is all of a sudden touching on algebra.  Finally number 3: the equals sign.  In a problem like “2*5 = [   ]” the equals sign is more of a symbol that the multiplication should be performed and the answer written down.  It is common for elementary students, when faced with “9 + 4 = [  ] + 5” to write, “13” in the blank.  Liping Ma, in her book Knowing and Teaching Elementary Mathematics, even notes that elementary teachers will misunderstand the equals sign: a statement like 3 + 3*4 = 12 = 15 was not flagged as incorrect by teachers.  Back to Lampert’s activity, we have the equals sign properly used as a relation, not an instruction.

Lampert describes some of the actions a teacher performs in order to keep students working on a task, while also getting the most learning out of that task.  In other words, this is not “do the worksheet then check answers at the front while I grade at my computer.”  I broke Lampert’s 11 item list (p140) down into a few categories:

• A – Assessment – finding out what students know
• C – Content – using content knowledge to produce help or challenges from many angles
• S – Structure – keeping the task on track by managing student behavior or task instructions
• P – Problem Solving – providing and modeling tools, suggestions for general problem solving strategies.

Many items had more than one category, such as “providing and maintaining appropriate use of notebooks and seating assignments” which I labelled as P and S.  Or “clarifying, inquiring, probing” as C and A: the teacher must be able to see what the student is thinking and then place themselves in that viewpoint in order to provide the next tool, hint, or question that would best support the student’s thinking.  This mental agility requires deep content knowledge.

Lampert also focused on a few case studies of her interactions with students.  One student, Varouna, had begun a problem like this:  “[ 1 ] groups of 7 = [   ] groups of 21“.  (At this point it is important to note that Varouna had completed parts (a) and (b) which were “[5] groups of 12 = 10 groups of 6” and “30 groups of 2 = [15] groups of 4”.  Note there is only one blank.)  Lampert brings up my favorite takeaway at this moment.  “She had tried an experiment and was now thinking about how to cope with its consequences.” (p123)

Lampert assessed that Varouna was unlikely to use fractions to complete the statement correctly, but to tell Varouna to erase the “1” and choose another number was to devalue Varouna’s efforts and thinking thus far.  It may appear that Varouna hasn’t done much of anything yet, but I believe that Varouna has already accomplished a lot:

• She has applied a strategy: the other problems had only one blank, and I can’t think about this one until I fill something in
• She (perhaps) chose a number with intent: “1” may have been a strategic choice since she assumed the problem would be easier with a smaller number.
• She assessed her own knowledge: she stopped, realizing that she did not know the number for the second part.  But– she knows that she doesn’t know.

Lampert works a little with Varouna around how to make sense of the problem with diagrams (showing that drawing is an acceptable way to do math) and after a short bit, Varouna writes “3”: “[ 1 ] groups of 7 = [3] groups of 21”  Lampert again does not flinch to signal incorrectness:

“My work here was to interpret and respond in a way that taught her something about the mathematics of multiplication and that also respected her efforts to make sense.“

How often are we tempted to make the quick correction that the student is “almost right” or has the answer “backwards” ?  In doing so, we are signaling that the student can stop thinking about the problem.  Furthermore, perhaps the student has constructed the concepts in their heads in a way that is internally consistent, just not in agreement with the outside prompts. By making a quick correction on the result of their thinking, we risk destroying a correct conceptual construction.  We as teachers must take the time to get at the heart of the student’s misconception so that we can bolster their conceptual ideas.  Varouna has knowledge that 3 sevens are 21.  She has written an incorrect statment, but it is the result of that knowledge.  Telling Varouna that she is incorrect may harm her confidence in the foundational fact.

I do recommend reading the rest of the chapter for the other case studies.  Here is a link to the book, or you can find excerpts perhaps on academic journal site.