Tag Archives: mathematical rigor

What is Rigor?

As mentioned in my previous post, I am looking at various publishers’ textbooks for high school integrated math – Common Core compatible. Common Core prides itself on being a rigorous set of standards and the textbooks all mention that they are rigorous. (Pearson’s books are stamped “Rigorous, Engaging and Data – driven” directly under the title). But what is rigor, more specifically what is rigor in the context of a curriculum and a math textbook?

The dictionary defines rigor(s) as “the difficult and unpleasant conditions or experiences that are associated with something; the quality or state of being very exact, careful, or strict”. Of course, we are not going to introduce a course to our students by saying it is difficult and unpleasant. In the context we are interested in, rigor has more to do with exactness, and with a logical and data-driven presentation of the concepts. In other words rigor refers to how the material is presented to the students and it has the connotations of validity and cogency. This is the interpretation of rigor that the publishers and the Common Core authors refer to.

But other definitions of rigor have synonyms such as “hardness”, “difficulty” and “stringency” .  In our context we can perhaps call this “depth” and add this dimension to those of exactness and logic with which we present the material.

None of the curriculums I have seen have much depth to them. None of them offer high-level, challenging problems to the student. Granted, not every student is going to go into a math or science career, but why not include challenging, critical thinking problems – if only to have them available for those students who want a challenge?

As an example, consider radical expressions as introduced by Pearson’s Integrated Mathematics, SIMMS Integrated Mathematics and an European collection of problems. All three texts address 9th  to 10th graders.

Pearson’s text starts with simple binomial radical problems such as

Rigor 1b

and then progresses to solving equations such as

Rigor 2b

The SIMMS text is less traditional in that it develops mathematical concepts entirely within a set of applications. For example, some simple radicals and exponent rules are introduced in a module where students learn about carbon dating, while much more sophisticated radical equations show up in the context of conics with students using a flashlight to create different conic sections.

The SIMMS text does not appear to have “drill and kill” problems similar to Pearson’s and the more sophisticated equations appear rather abruptly.

The European text starts with problems such as:

Show that

Rigor 5b

and continues by asking for solutions of equations such as:

Rigor 6b

Both the Pearson and especially the SIMMS text have multiple activities, investigations and projects, the European text has none.

In terms of rigor defined as “difficult” or by “challenging” problems, the European text is far ahead. The American texts however engage students in activities where they either apply previous knowledge or develop that knowledge in the context of working some “real-life” situations. Under the influence of Common Core, math textbooks will increasingly have activities and/or problems in which students will have to justify and explain their solutions. The hope is that  students will achieve deeper understanding, become used to group work and to presenting  their results.

It appears that in the process of doing projects and modeling, American students lose strictly mathematical rigor – i.e. depth – and do not encounter very challenging problems. (This is perhaps unkind. American texts have lost that kind of rigor since the 1960s.)  Perhaps the argument is that with all the projects and the modeling there is simply not enough space or time for truly challenging problems .

What is better for the students? Why can’t we have both?

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