## Thursday, July 21, 2011

### Math problems of the week: 4th grade Investigations vs. Singapore Math

I. A 4th grade Investigations (TERC) assignment, assigned at the end of September [click to enlarge]:

II. From a similar point in the 4th grade Singapore Math Primary Mathematics 4A Workbook (p. 30) [click to enlarge]:

III. Extra Credit

Which is more mathematically engaging: "naming" numbers, or simplifying the "number names" provided to you by your educators?

Aly V said...

If you have answered this question previously, please forgive me. Will you please define "mathematically engaging?"

It would also help this reader if you would clarify your rhetorical question by stating for whom you are addressing your query. In other words, which is more mathematically engaging to the average fourth grader? to a talented math student? to an adult who has already grasped the concept of expressions having the same value?

Katharine Beals said...

Aly V,

Thanks for your questions, which I assume aren't merely rhetorical.

By "mathematically engaging" I mean engaging students in mathematics (as opposed to boring them, confusing them, or engaging them in something other than mathematics).

Since this is a 4th grade problem,
I have 4th graders in mind (4th graders in general, not just American 4th graders, who may only know Reform Math and perhaps have been held behind by it).

One way to assess which problem set is more mathematically engaging would be to observe a variety of 4th graders doing each problem set. How interested vs. unmotivated do they appear to be? How well do they do and how much effort did they put in? Which problem set did most students prefer doing?

It's extremely important that we get feedback from students on these issues. (And if it's a homework assignment, parents might have some feedback to offer as well).

Barry Garelick said...

I would add that "mathematical engagement" allows students to engage their mathematical knowledge (both procedural and conceptual). Some point to the traditional algebra problems of trains leaving statiosn at different times, or coin problems, number problems, work problems or mixture problems as not being mathematically engaging because they are not "relevant" to students--i.e., that students are not motivated to want to answer the questions asked. Or another criticism is that the problem already asks the question--the student doesn't have to formulate it--and the data are contained in the problem itself. The antidote proposed by various reformers is to make the problems "interesting" and "relevant", ignoring the fact that when students are given proper instruction so that they have the appropriate "schemas" to solve problems, relevance becomes irrelevant--they are too busy solving the problem. This doesn't mean that they are simply applying a previously learned procedure in an automatic fashion. They still need to reason, to identify what the variable is, and how that variable is expressed in the context of the problem and so forth. Reformers, however, try to engage by giving students a problem for which they have little or no previous knowledge and then "facilitate" the student to learn what is needed in order to solve the problem in a "just in time" manner. Problem solving skills are built by developing schema and doing many problems. Proper scaffolding can keep the problems engaging regardless of the subject matter.

The so-called "engaging" problems that require students to collect their own data frequently amount to tedious exercises in data mongering. The mathematical content is generally very small, and in many cases relies upon statistical estimation.

LTW said...

My nephew is using Investigations at school. My sister hates it and is basically teaching a whole math curriculum herself at home. After seeing this example, I can see why. I have no idea what students are actually supposed to be learning from these problems.

Aly V said...

I do not think either is mathematically engaging. If students (and their teachers) understood arithmetic concepts when they were first introduced, there would be no need to teach "order of operations" later.

Katharine Beals said...

While the fact that order matters is a mathematical concept, what we call "order of operations" is a matter of convention, not of conceptual understanding. Understanding an essential concept or concepts (as in arithmetic) doesn't, by itself, lead to understanding arbitrary conventions (as in order of operations).

Therefore, no matter what, order of operations must be explicitly taught. And, no, the details of this convention are particularly mathematically engaging.

But a careful look at the Singapore Math problem set reveals that it's not simply having students order of operations. It's also engaging students in math. In particular, it allows them to see, in action, actual examples of why order matters. What better way to teach that!

20 + (8+4) / 3 is not 24! It is 10.6666666666667. My little girls were trying to figure it out and they are the ones that came up with that after doing one of your posted math problems.

Katharine Beals said...

Are your daughters familiar with the Order of Operations (PEMDAS)?

Hainish said...

Hm, I would have simplified the term (8+4)/3 first, before performing addition, since MD come before AS.

Is the expression supposed to be read as (terms before divider)/(terms after divider)?