## Saturday 5 November 2022

### Do We Teach Students to Think About the Base They Are Working In?

Almost three decades ago, I helped my school system create and adopt a class in discrete math, (it was becoming a trend then) and then took the two sections a year at my school giving up two advanced algebra classes.  One of the blocks was about arithmetic in different number bases.  My focus was on making as much as possible about applications, so I wrote a unit on "five-money", pennies, nickels, quarters, and Super Dollars (equal to the euro at 1.25 american).  Since our school was in England, they were  well familiar with the exchange.

I started putting some of their problems on the board to see how "bright" math students in Math Analysis or Stats or Calc would handle them.  These are simple versions of some of them:

If a 5 ft 9 in board is cut from a 9 ft 6 in board, what length is the remaining board?

How long is it between 7:40 this morning and 3:25 this afternoon?

A recipe for bread calls for one pound and three ounces of flour for the extra large loaves.  The cafeteria needs 15 of these loaves for lunch.  How much flour will they need?

What is the area of a rectangular room that is 9 ft 6 in wide and  8 ft 9 in across?

A worker worked 5 hrs 43 min on Monday, 7 hrs 35 min on Tuesday, and 3 hr 20 min on Wednesday.  How many hours have they worked so far?

A 16 ft board is to be cut into boards of 2 ft 9 in length.  How many boards can be made, and what length piece remains.

Notice that none of these involved working in the "New Math" idea of different bases away from units of measure.

The first problem produced lots of 96 -59 = 37 answers, some then converted to 3 ft 1 or 3 ft 7 in.  Many students converted everything to inches and subtracted, then  converted back to ft and in.  None Wrote the problem in the three line subtraction model they used for base ten and simply did it in base 12.  Zero out of 34 students.These were our very best, the bright kids too clever to need the new discrete math. The ones who managed to get correct answers went back to tally marks and scratching around to adapt what they knew about units to get answers that made sense to them.

What surprised me is one of the most common solutions for the time problem was to draw a clock and mark around the perimeter, 20 min + 4 hr + 3 hr + 25 min.  Others did the same with what I call number roads,  7:40 >>>  (20 min) 8:00  >>>>(4 hrs) noon >>>> (3 hr) 3pm  >>> (25 min) 3:25 and get the answer. Unknown to these students, I had introduced the number roads and counting on method to the students in the Discrete math course.I had used  Nobody on the board cutting problem saw the same solution.  My conclusion is that they were algorithm bound, for this kind of problem, you used this tool.

The bread problem had lots of 15 # 45 oz answers, and many adjusted by converting the 45 oz to 2# 13 oz and finishing, while a few got to 15# 45oz and then finished with 19# 5 oz.

The area problem through algebra students into a tizzy because they were sure it would be a problem involving the distributive law, (shall I say algorithm bound again) but didn't know what to do with 6x8 and 9x9. As often happens with young students there was little use of units.  It was a rare result that included square ft or square inches.  Statistics students were more likely than others, perhaps because of my tantrums over unlabeled axes in graphs.  I wanted to hug the ones that got something like 72 sq ft,  129 ft inches, 54 sq inches.  Many resorted to converting to inches, multiplying and dividing by something (144 and 100 were both common,24 appeared a few times as well).  Several just left it as a decimal number of their division, and of those who tried to label it they were split between inches and square inches.

For the last problem I imagined some bright child writing the problem like

2'   9' )  16 ' 0"

or perhaps 16 / (2 3/4)  or even 16 / (2 9/12)

alas, it did not happen.  Most correct solutions went back to a sketch of some kind, cutting the length and adding board lengths one at a time.  Others converted to inches, but some did the calculator and had no idea of what the .8181... decimal fraction indicated (other than they hadn't thought through how they divided any other problem.

My disappointment made me squeeze time in to talk about counting on to subtract, number roads for all kinds of problems, arithmetic in any base including division of ft/in into square ft/sq inches (and the magical inch-foot that makes them like quadratics.  We talked about lot of possible ways to try problems with units, and the beauty and benefit of labels.

I had taught the Discrete Math students to use counting on in subtraction  with problems in compound addition and subtraction and weird subtraction problems with lots of borrowing (ooops, regrouping)  like  20003 - 18354.

A few weeks later a very bright calculus walked into a seminar session where two of the Discrete Math students were working on base 8, a new topic, (but as Tom Lehrer said, it's just like base ten if you cut off two fingers)