Tag Archives: Common Core

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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Dreams of an Integrated Curriculum

We are in the midst of a transition year.

We have the luxury of experimenting, while we are in transition to Common Core. We are not sure if the state and the feds will agree to give a state exam – the state has voted not to –  but in the meantime, we are tweaking the curriculum with more freedom than we ever had before. Practically, this means eliminating a lot of stuff that teachers feel the kids do not need. (What it actually means is that we are eliminating things we feel our kids will not get.)

We are also looking for materials for an integrated curriculum that meets Common Core – we have decided as a department that we want integrated. The integrated books labeled Common Core are really not, so we are staying with the old sequence (Algebra 1, Geometry, Algebra 2) until decent textbooks appear. (My fellow teachers also want the ancillary materials – they want to be able to make tests automatically, not to make the problems themselves. Too much work I guess.)

These two circumstances led me to research what a decent Common Core integrated curriculum would look like. I have done some “Googling” and SIMMS, put out by Kendall- Hunt, caught my attention. In my opinion, SIMMS (Systemic Initiative for Montana Mathematics and Science) has two attributes that deserve attention.

First, SIMMS not only integrates the different branches of mathematics, but it also does so within applications and projects. Second, for me the most important attribute is that SIMMS is divided into levels geared to different kinds of students. There are levels “recommended for all students”, those “recommended for students in non-mathematics and non-sciences fields”, and those “recommended for students who plan to major in mathematics or sciences”. There is the possibility for students to change their minds, i.e. go from the more “non-mathy” levels to the more “mathy” ones and vice-versa, so the curriculum does not track students.

This is an immensely appealing idea for a teacher like myself, who works in a school where there is a broad spectrum of students. Currently, in our school, we are forcing students to take Algebra 1 and Geometry even if they have not done well in middle school. We are mandated to do this because the state wants to “expose” ALL students to as high level of mathematics as possible. Did they get a D in Algebra 1? No matter – let them take Geometry. Did they get a D or F in science in middle school? No matter – we are not going to put them in Earth Science in high school, we are going to put them in a laboratory course – Biology.

The results are as one might have expected – lots of D’s and F’s in Geometry and Biology. So the students are exposed to a science and math course they are not ready for, they fail and they incorporate in themselves a sense of failure, a sense of “I am not good at this” and worse, a  feeling of dislike for math and science.

In Geometry, which I am currently teaching, when are students going to use congruence of triangles, if they do not major in math or science? When are they going to use proofs? For these students, calculating areas and volumes are probably the most useful applications of geometry.

Even for a dyed-in-the-wool math person like me, proofs are beginning to look like a form of torture for most kids. I know the argument that proofs are a way of teaching logical thinking. But, are they the only way or the best way of teaching logical thinking? I can envision a case-based course that would teach elements of logic more successfully than two-columns proofs can.

Same thing in Algebra. If I am not going into math or science, do I really need to know the difference between slope-intercept and point-slope? Why are we doing all these line problems? The real reason is that lines offer the simplest predictive models for a variety of situation, but this gets lost in the dicing-and-slicing that we do in Algebra 1. Much better to offer a course where we take data, make a scatter plot and see what a straight-line model does for the data. We can talk about slope and y-intercept and residuals in that context.

In a way, it seems to me that transitioning to Common Core offers us an opportunity to do truly differentiated teaching – not the one about different learners types (visual, tactile, etc.) which has been shown to be cognitively false, but differentiated by aptitude and attitude.

One can only hope. In the meantime, I am going back to grading the latest Geometry exam on proofs  and marvel at the inventiveness of students who make up non-existent properties.