Friday, May 22, 2009

Math problems of the week: 3rd grade Everyday Math vs. 1920s Math

I. The final word problems in the 3rd grade Everyday Math Student Math Journal 2, p. 317:

1. Rule: × 3

in out
0 ___
1 ___
2 ___
3 ___
4 ___

2. A brachiosaurus is 72 feet long, a diplodocus is 90 ft long, and a stegosaurus is 23 ft long. If they get in line behind one another, how long is the line?

The line is ___ feet long.

Number model:

3. Bary ran 800 meters. Kristen ran 628 meters. How much farther did Gary run?

He ran ___ meters farther.

Number model:

4. I bought a beach ball for $1.49 and a sand toy for $3.96. How much change will I get from a $10 bill?

$ ____

5. Write 3 ways to say 8:45.


6. Complete the Fact Triangle. Write the fact family.

[picture of a triangle with 16 and 2 in two corners, and × and ÷]

___ × ____ = _____
___ × ____ = _____
___ ÷ ____ = _____
___ ÷ ____ = _____

II. The final word problems in Chapter IV, the final chapter for 3rd grade of Hamilton's Essentials of Arithmetic, published in 1919, p. 122:

1. Robert had a plot in a school garden 10 feet long and 8 feet wide. How many square feet were there in his plot?

2. He planted two rows of tomatoes from which he raised 96 pounds. How much did he get for them at 9 cents a pound?

3. He planted two rows of beans which he thinned out to 3 plants to the foot. How many plants did he then have on the 20 feet?

4. How much did he get for 15 pounds of beans at 8 cents a pound?

5. He planted 35 turnips 7 to the foot. How many feet of turnips did he plant?

6. He raised 28 pounds of turnips which he sold at 3 cents a pound. How much did he get for them?

7. He also planted beets, carrots, and Swiss chard. He received 30 cents for his beets, 20 cents for his carrots, and 25 cents for his chard. How much did he get for these vegetables?

8. He raised and sold 10 heads of lettuce at 5 cents apiece and 8 bunches of radishes at 5 cents a bunch. How much did he get for them?

III. Extra Credit:

Discuss the level of higher level thinking in the 1920s.


Anonymous said...

Aside from the differences in complexity, the Hamilton text is using words that are familiar to children of that period (how many children now could identify a turnip?) 1) Its also a primer that would be useful in a farming community. During the 20's most Americans were still living without electricity and automobiles were just beginning to make a difference. People would have no problem approving of this book. 2) The problems are part of a garden theme which implicitly lets the student know how far to work if they are working independently.

The problems also increase in complexity, so I would argue this is a constructivist text because the teacher can easily assess a student's skill level. 3) Working the problems requires paper and the students would have to be primed with some framework for where to begin. The primer would provide examples.

Everyday is more problematic. So I will stick to my objections. 1) Problem 1 is a function (t-table). The rule is incorrectly defined: out = in x 3. The diagram is a scaffold, so students would copy this onto paper. Its more of a test on decoding (was the student able to follow the instructions?) Multiplying by 3 is trivial - most students would learn this in first or second grade. The answers you might see are filling the blanks with "x3 = ANS"

2. The question here is do students know what b, d, and s are. They stand in line and what do you know: 3 dinosaurs add up to 185 feet. Trivial. The explanation or number model? I'm not sure? Matchsticks?

3. So now its subtraction. This is just for fun, but I've seen Ruth Parker use it in her examples as an alternative method for subtracting without borrowing.
800 = 799 + 1
(799 - 628) + 1
171 + 1
You have to decide what method works best for you. Most of what algorithms do is teach students to recognize patterns and regroup numbers. Traditional algorithms streamline the process so

4. This is already a lost cause, but we'll complete the process.
4 is a two-step problem.

This is how you might compute the difficulty of each problem set if you were to weigh each problem equally.

1. Start with something children know already. Five multiplications (5 pts/<1 min)

2. Two sums, no carries, 2+3 = 5 (kindergarden) (5 pts/<1 min)

3. subtraction (two methods: long and short) borrowing requires more teaching, but for third grade this borders on trivial for most students. (10 pts/<1 min)

4. so here we have 6+9 = 15, 40+90 =1.30; 1 + 3 = 4; method of partial sums. And then to subtract these sums from a $10 bill. Again, following Ruth Parker's treatment - counting back.

10 - 4 = 6
6 - 1.30 = 4.70
4.70 - .15 = 4.55

Six adds and subtracts. So this might take 3 minutes? (15 pts/<3 mins)

So altogether you have a class of third graders done with a set of 'word' problems in about 6minutes.

A class of today's 3rd graders raised on Everyday Math and attempting to do a set of problems from Hamilton would have to spend considerably more time doing those problems. In Everyday the number model was asked for, with Hamilton the number model was a real thing (vegetable rows). Most children these days, don't even know what I'm talking about.

Everyday is not constructivist - because each problem has equal weight. Each problem takes roughly the same amount of thinking which in every case is almost trivial. Problem four is a lesson in counting change. Something a cashier would learn. I'm trying to think like an Everyday thinker.

Arne Duncan calling for more time in school is missing the point. Children already spend more time in school than their great, grandparents. They need better textbooks.

bky said...

I go for the extra credit. Basically the problems from the older book are simple arithmetic, and it is easy to glance through them, see the numbers, do the obvious operations, and get the answer.

In this sense the seem like the odious dinosaur problem from the newer book, which goes like this: "Dinosaurs blah blah blah. Add 72 + 90 + 23".

But on problems 3 and 5 you have to understand what the problem-writer means by "3 plants to the foot." At first I thought you have to look out for fence-posting error, which would be the case if this means the plants are spaced every 1/3 foot. But maybe you don't put a plant on the very end of the row. I dunno. This might not have been a big deal to the 1920's students, but it would require a higher level of thinking if some of the problems are not so straight forward, so that you have to stop and determine whether fence-post error is a worry hear or not.

In other words, get the student used to determining what is an easy problem and what is a hard(er) problem.

Anonymous said...

3. Yes, if you were taking Everyday Math, a class might be overly concerned with how each plant was spaced in a foot. But in reality does it matter? Goldie should go plant a garden. Most farmers don't worry about where to begin planting. Its the spacing between the plants that matters? Right?

On average there are 3 plants per foot and that is all we need to know: 3 x 20 = 60 plants. Furthermore this is a rate problem (plants/foot)(foot) = numbers of plants. So it is much more complex than what today's third graders are doing and it can be expanded ad infinitum.

If you got 48 pounds of tomatoes from one row of plants how many tomatoes are produced by each plant? What is the value of one tomato plant? One tomato? How much is 10 feet of tomato plants worth at 9 cents a pound. etc...

bky said...

Anonymous brings up an interesting point, which is that one level of mathematical sophistication you might want to cultivate in students is knowing when an exact answer is required and when an estimate is as good as it gets. For example:

(1) Fence in a rectangular garden of size 9' x 15'. How many fence posts?

(2) Plant rutabagas 3 to a foot down a 15' row; how many plants is that?

In (1) you want an exact count and in (2) given the nature of plants and dirt you would not wonder "is there a plant on the very end?"

In general when you do a problem in a math book the default is to give an exact answer unless asked for an approximation. The issue becomes what assumptions are you making in getting the exact answer. The most common assumption is probably "constant rate" and you would want students to know they are relying on that assumption for those kinds of problems.

Anonymous said...

Yes - this is a point that math reform does emphasize discrete math or counting.

We can disagree with the analysis-however, I would contend that most classrooms are not that sophisticated or interested in having the finer details pointed out to them.

If the goal of the lesson was to teach students how to multiply then why go into the nuances of counting which would only serve to confuse and annoy people?

Why does math reform prefer using a statistical (discrete) model to teach the concept of lines which are continuous? It seems reversed to our way of thinking.

Anonymous said...

While I'll admit the fence post problem is a good one - here's the catch. Traditional textbook authors would use it to supplement or add to instruction. Its not a key problem, students could still be taught mathematics without it.

1. Where does it fit in the math standards? Logic and Discrete geometry?

2. Can one build and extend knowledge from learning how to do the problem? Only if they understood what they had learned.

3. Which brings up my third point. How do you know if students learned what you taught them?

Are there standard algorithms that help us count that can be taught explicitly? Should we begin rewarding students for counting well? Counting is what makes us unique. Its a personal activity. Must we know how to count well in order to learn math? Is this what math reform teaches us?

Tina said...

Thanks for the 3rd grade math worksheets - I added it to my online binder where I've been collecting math word problems for my 3rd grader. You can view the binder here: