kitchen table math, the sequel: Steve H on Common Core in his town

## Friday, October 25, 2013

### Steve H on Common Core in his town

from Steve:
In our town, life is going on just about the same. We still have Everyday Math and only those kids with help at home will get to algebra I in 8th grade and have a chance of getting to a STEM career. High school kids still pack their schedules with honors and AP classes while ignoring the meaningless state tests. The CC PARCC test will be no different. Students worry about the PSAT and SAT, but ignore the state tests. They are meaningless for them.

However, there are still many kids who will now have to meet these state test standards to get their high school degree. These are tests that try to judge one's thinking and understanding abilities, not just mastery of basic skills. They are the ones most hurt by these fuzzy tests. Classroom teachers should be the ones best able to judge these qualities, but it's now turned over to test makers who try to boil that ability into a few understanding-type questions on the state test.

Why should students fail to graduate high school when they pass high school courses but flunk a fuzzy state test? This is a failure of the school or the test. Why should minimal passing grades be based on fuzzy understanding rather than mastery of basic skills? How do these tests give teachers any feedback on how to improve? When my son was in first grade, I was a member of a parent-teacher team that evaluated our state test results. The test gave thinking-type questions where they (magically) split the results into areas like number sense and problem solving. Rather than directly test to see if students can add, subtract, and handle percentages and fractions, they test to see if they "understand".

So here we were sitting around a table discussing what could be done to fix a lower school score in "problem solving". The answer was to spend more time on, you guessed it, problem solving. If they tested something directly, like fractions, that would give them much better feedback. But then again, they should be doing that already. A state test should only be used as a last-resort check for systemic school failures, not as a means to check for "understanding".

The downside to CC is that many more will point to it as rigorous path that is meaningful to the development of their kids. I'm also waiting to see if our 7th and 8th grade math texts are watered down. A few years ago, we managed to get rid of CMP and replace them with the same strong algebra textbooks used by our high school. Common Core might now force us into a less rigorous path.
I attended the SRO Common Core shindig in my district last night.

Irvington kids are going to be inferencing for 13 years.

Inferencing and problem-solving because, in the real world, math has more than one right answer.

in the real world, math has more than one right answer.

Ha ha, try that one on Infernal Revenue.

Barry Garelick said...

in the real world, math has more than one right answer

SteveH said...

Problems can have more than one approach, but the math used to back up that result will have just one correct answer. Even for an interpretation of a best answer, one need to mathematically define a merit function so that others can see where it comes from and see how the variables are weighted.

Unfortunately, many educators mix up the two things. They are bound and determined to unlink mastery of basic one-answer skills with understanding and problem solving.

...unless we're discussing an optimization problem for which there are multiple local optima, and no global optimum... but something tells me that that's not what we're talking about here.

SteveH said...

A typical no-one-right-answer problem in the early grades has students determine how to spend a certain amount of money on a birthday party. They are given a list of things they could buy and their costs. They get to choose what to buy, but adding up the costs and making sure that the total doesn't go over the limit has only one right answer. You could add up the numbers in different ways, but there is only one result. Many educators don't appreciate how math can model the subjective choices using a merit function - which has only one correct solution.

Educators could teach this by having students determine weights for subjective variables when they evaluate a product. There are lots of examples of this sort of thing. They could discuss the choice of weights and their values. They could see how weights could cause completely different results. High school students could evaluate USNWR's college ranking formula. They could guess at the huge fudge factors used for holistic college admissions. (Sorry. My son just had a college interview yesterday.)

In later grades, they could start looking at problems that are non-linear and have one optimum solution, or many local and one optimum solution. They could study feasible domains for the solution.

Math has formal ways to deal with all of these things. They can study what can be done when 'm' equals 'n', 'm' is less than 'n', and 'm' is greater than 'n'. One has to first master the one-right-answer solution skills before applying them to real problems which may have different solutions for different merit functions.

This is all obvious to those who know math. It is also very frustrating when educators blather on about no-one-right-answer. They don't know enough to be embarrassed.

Actually, a problem such as you have described, where there are some number of different items that can be purchased, each at a different price, and a certain amount of money that constrains how much can be spent, can be modeled in an integer programming problem that might have zero solutions, exactly one solution, or multiple solutions. If we weren't picky about the integral nature of items to be purchased (e.g. if one of the items were something like "ice cream" and you were not constrained to purchasing it in integral quantities), then this would just be a linear programming problem, and might have zero solutions, exactly one solution, or infinitely many solutions.

Example:
Helium balloons cost \$2 each. Green plastic army men cost 15 cents each. Rubber baby snakes cost 75 cents each. Felipe wants to take at least one of each item to Miguel's birthday party. Felipe has \$5 to spend. How much of each item can he buy?

If we represent the number of helium balloons, the number of green plastic army men, and the number of rubber baby snakes by the sequence (h,m,s), then the solution set comprises the following sequences:
(1,15,1), (1,10,2), (1,5,3), (2,1,1).

If we change the problem so that Felipe has only \$2.50 to spend, then the solution set is empty: the problem has no solution.

If we change the original problem to add the constraint that there must be enough rubber baby snakes so that each of the three boys at the party gets a snake, then the problem has exactly one solution: (1,5,3).

I suppose you could say, technically, in the first case above, there is only one answer, and that answer is that the solution set contains four possible solutions, but elementary students and their teacher probably wouldn't see it that way.

If this had been an actual problem given to an actual class of elementary students, the problem would have been given enough constraints so that the solution set would contain exactly one solution. However, the real world isn't always this neat.

SteveH said...

My point is that educators don't know that there are formal mathematical techniques available to deal with all types of problems. They view math as some sort of general thinking process. It's not even used as part of a discovery process where they finally introduce more formal techniques and understandings. The general thinking process is the end in itself. Students don't even end up with a simple understanding of the different classes of problems they could see. To them, they are all just "thinking" problems. Once you start getting formal, then knowledge and skills become important and we know what they think about those things.