## Thursday, November 19, 2015

### Math problems of the week: Finnish vs. Common-Core inspire exams

1. A problem from the Common Core-inspired PARCC exam, which a commenter on Dan Meyer's blog thought I would particularly like, and felt was comparable to the sample problems on the high school exam in Finland that I blogged about earlier:

2. The Finnish problems I blogged about earlier:

1 c. Simplify the expression (a2-b2)/(a - b) + (a2-b2)/(a + b) with a not equal to b or –b.

5. A circle is tangent to the line 3x-4y = 0 at the point (8, 6), and it touches the positive x-axis. Define the circle center and radius.

6. Let a1…an be real numbers. What value of the variable x make the sum (x + a1)2 + (x + a2)2 + ….+ (x + an)n as small as possible?

9. The plane 9 + x + 2y + 3z = 6 intersects the positive coordinate axis at the points A, B and C.
a) Determine the volume of a tetrahedron whose vertices are at the origin O and the points A, B and C.
b) Determine the area of the triangle ABC.

13. Let us consider positive integers n and k for which n + (n +1) + (n + 2) + + (n + k) = 1007
a) Prove that these numbers n and k satisfy the equation (k +1)(2n + k) = 2014.

III. Extra Credit

1. Consider what is involved in solving the PARCC problem (recognizing that it's a quadratic; seeing what the a, b, and c coefficients of the standard form of the quadratic equation correspond to here, and seeing that 4ac must be positive). Compare these recognitions, apparent with minimal symbol manipulations, with what's involved in solving the Finnish problems.

2. Some problems can appear mathematically abstract without involving much math. Discuss.

3. About Finnish problem 13, the commenter on Dan Meyer's blog writes:
Problem 13 is a variation on the proof for the sum of an arithmetic sequence. Algebra 1 in the CCSS, btw. We don’t require a proof, but if we were going to require one, that would be one of the easiest to pick. And reform mathematics programs will typically explore that proof (geometrically, and then with algebraic symbols to formalise). Traditional American textbooks just give the formula as something to memorize because, get this, accurate procedures without understanding are considered sufficient.
Discuss.

Brett Gilland said...

Your (mental) symbolic manipulation for the PARCC problem is more than is necessary. It also only establishes a possibility of two solutions (a bit short of full justification, but allowing for verbal explanation with no diagram...). It also almost certainly requires actual symbolic manipulation for most students. The move of m and b and related sign switch isn't a small task, even for the brilliant students of Finland.

The problem can be done with even fewer (mental) symbol manipulations than you used, and likely would be by those with an appreciation for the graphical representations of quadratic and linear functions. The quadratic given is a parabola with vertex at (0,0) that opens up. The second equation is a line with positive slope and a negative y-intercept. Distinct real solutions will be represented as intersections. It is possible to have 2 intersections (secant), 1 intersection (tangent) or 0 intersections (line has comparatively low slope), and thus for f(x)=g(x) to have 2,1, or 0 real solutions. Still a bit short of full justification, but this thing begs for the ability to draw a sketch as part of the justification.

Item 2 of your Extra Credit is where the vast majority of our disagreement comes from, I believe. The phrase "without involving much math," along with your comments about symbolic manipulation in item 1 suggest a willingness to conflate symbol manipulation with mathematics. But they really aren't the same thing. Symbol manipulation occurs in many other fields than mathematics. Mathematics has aspects that go beyond symbol manipulations. As is typical in many classes, my calculus students are often very good at the symbol manipulations involved in calculating a derivative and such, but they often struggle with the relationship between f, f', and f''. Both of those abilities are necessary to be 'good at calculus', even though the latter can be (and often is) assessed independently of symbol manipulation.

Katharine Beals said...

Speaking as a linguistic with a background in foundational logic/model theory, I'd say that symbolic expressions and symbolic manipulations are the syntax of math, and numbers, sets, and graphs the semantics. Both are key. Symbolic expressions and manipulations vs. semantics do, indeed, apply to other fields, including natural languages, where syntax is, again, pretty darn important. One could answer this particular problem symbolically or graphically; graphics aren't necessary. In the math I see mathematicians doing (I'm not one myself, but am closely connected with many), the work is mostly symbolic rather than graphical.

I question whether "the move of m and b and related sign switch" would stump a brilliant math student from any country. It certainly doesn't involve anything like what's involved in constructing the kind of multi-step proof that, here in America, use to be much more commonplace.

Katharine Beals said...

Sorry: syntax error! Speaking as a linguist...

Barry Garelick said...

set K is said to be compact if an only if eery open covering of K contains a finite open subcoering of K."

Using the definition of compact set as "closed and bounded" will suffice for the less rigorous course of real analysis. Similar things are done in introductory calculus courses; for example, students learn an intuitive definition of limit and continuity, and then later learn the more formal delta-epsilon definition and do proofs of limits and continuity using the more formal definition.

Thus, providing algebra students with the formula for sum of consecutive numbers is not egregious; particularly if students are given challenging problems to solve that depend on knowing that definition.

I have seen the famous so-called "young Gauss" proof of sum of consecutive integers. It is sometimes pointed to as "evidence" that we are not teaching young students to think mathematically if they cannot come up with it on their own.

Barry Garelick said...

Oops, the first part of my comment got truncated. I said that the commenter on dy/Dan's blog referred to in the post, laments that students in algebra classes are given the formula for sum of consecutive numbers without proof, and sniffs at that. But whether one knows the proof or not, solving problem 13 of the FInnish test still would not be any easier knowing the proof. It is a challenging problem that makes use of a student's knowledge of the formula.

A similar thing happens in other courses. In a textbook of real analysis for engineers, the definition of "compact set" is given as "a set that is closed and bounded". THis is actually a theorem that derives from the formal definition of compact set which is a set K ... (the rest follows from the above comment).