Developing Grit and Recognizing Grit

One of the recent emphases in education, especially college mathematics, is ‘student grit’.  Grit is what allows students to succeed when there are barriers, and we can recognize success (usually).  However, the concept of grit is not productive unless we can recognize and develop grit prior to that point.  #gritmath

The context for this post is a recent test in our quantitative reasoning course (Math119).  Our first set of topics dealt with dimensional analysis; every conversion in class was completed by that method.  Overall, students did about as well as I’ve seen.

However, some students did their work in an indirect fashion.  Take a look at this first example:

And, this example:

In both cases, many of our math classes would say “Just move the decimal point”.  I did have a few students complete the problem that way.

More importantly, many of us would tell these students that their method is wrong.  However, the first example is conceptually perfect; the error in the answer is strictly due to the rounding of the conversion facts.  The second example is also pretty good … except for the inversion of a basic conversion.

I think both students showed significant ‘grit’ in working these problems.  Although I don’t generally want students to do a problem in a complicated way when a simpler way exists, it is impressive that both students were able to salvage a problem begun in a non-standard way.

I’m not suggesting that any grit shown in these two cases is equivalent to the level needed to complete a math course.  However, I do think that developing grit is the same as developing other traits:  We start small, make it explicit, and practice.

One of the wonderful things about a good quantitative reasoning course is that there is a focus on non-standard problems.  Methods are emphasized, but we don’t focus on procedure as much as we do reasoning.  This environment lets students explore and develop in ways that traditional math courses don’t.

I suspect that our traditional math courses either discourage grit or prevent much development.  With such a strong focus on procedures and correct answers, students are often doing the ‘instructor dance’ — following steps because it will please the instructor.  Student traits can not develop in a overly structured environment.

It is important that we recognize the difference between “incorrect thinking” and “different thinking”.  Different thinking is part of trait development, like grit.  Students can not show, nor develop, grit unless I provide them opportunities to work differently.

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National Math Summit … final schedule & session information

Next month, many of us will be in Anaheim (March 15-16) for the second National Math Summit; the first summit (2013) had the word “developmental” in the title, and that is still the primary intent for this year.

The organizers have released what will likely be the finalized schedule and session information.

Sessions:  NMS Strand Presentations R for Web Display Final version 1-27-16

Schedule:  National Mathematic Summit Tentative SchR2

I’m involved with two sessions.  On the first day, there is a panel of dev math reform, where I will be joined by 3 respected colleagues (Brian Mercer from Illinois, Kim Granger from Missouri, and Laura Bracken from Idaho).  The four of us have experience with reform, and also have state policy changes to discuss.

On the second day, I am doing a brand new presentation on the New Life model for developmental mathematics.  In this presentation, I will connect this work with broader changes in college mathematics.  A highlight will be some models for implementing the courses (Mathematical Literacy, Algebraic Literacy) to fit the local needs.

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Co-Requisite Remediation and CCA (Saving Mathematics, Part V)

Complete College America (CCA) released a new report on co-requisite remediation this week.  Actually, that statement is not true … the CCA released a web site which shows some data on co-requisite remediation, with some user interaction.  What’s missing?  Anything that would help a practitioner judge whether they should consider co-requisite remediation!  #CCA #Corequisite #SaveMath

Many of us are dealing with policy makers in our states or institutions who see co-requisite remediation as the solution to the “developmental math problem”.  There are, in fact, serious problems in developmental mathematics; there are also serious problems with how ‘college math’ has been defined, and how policy makers are defining a problem away instead of solving it.

Within developmental mathematics, we have been working hard teaching the wrong stuff to our students, frequently using less-than-ideal methods to help them learn.  Our curriculum has too many courses, and the combination is lethal … not many students reach their dream.  When students proceed from developmental math to college algebra or pre-calculus, they often find that the gap in expectations between the two levels is very difficult to deal with.

Co-requisite remediation steps in to this complex problem domain, and declares that all will be fine if we just put students into college math with some support.  The most common (and sometimes the ONLY) co-requisite remediation done is in Intro Statistics and Quantitative Reasoning [QR] (or Liberal Arts Math).  The history, frequently, is that students had to pass intermediate algebra prior to these courses … even though that background has nothing to do with the learning; the requirement was to establish “college level”.

So, the CCA and allies declare that students can take Stat or QR instead of developmental math.  Of course this is ‘successful’; the old prerequisite was unreasonable, and the co-requisite method puts students directly in to courses they are relatively ready for, and also provides extra support (in some cases).  Many colleges, including mine, had already lowered the prerequisite for Stat and QR years ago; our results from both Stat and QR are better than what the CCA states for their co-requisite model.

The co-requisite ‘movement’ is an illusion.  The work succeeds (almost totally) because students are placed in to math courses that have minimal needs for algebra.  I get better results by just changing the prerequisite to Stat and QR.

We also face a risk to mathematics in this illusion:  students with dreams that involve STEM are frequently told that this dream is being shelved in favor of co-requisite remediation, that they will take either Stat or QR.  The path to calculus is either not available or involves work that is not articulated well to students.  Policy makers are treating math as a barrier to cope with, a problem to solve with the least remediation.  The need for mid- and high-skill STEM workers is well documented, but the co-requisite ‘solution’ often blocks the largest pool of students from those fields … the minorities, the poor, the students served by under-performing schools.

Society needs our work to succeed for all students.  We can not accept a solution which reduces upward mobility; a solution which does not provide ‘2nd chances’ is a risk to both mathematics and to a democratic society.

Don’t get me wrong — Stat and QR have a major role to play in our curriculum, and these courses might be the most common math courses students should take in college.  My main message is that we need to question the illusion called ‘co-requisite remediation’, AND we need to articulate a vision of our curriculum which enables ALL students to consider STEM and STEM-like careers.   [The New Life Project provides a vision of such a curriculum.]

If you really want to read the CCA “Report”, go to http://completecollege.org/spanningthedivide/#the-bridge-builders

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Building Understanding in Algebra

Like most of us, I have a tendency to assess student learning with an emphasis on “doing” problems … simplify this, solve that, etc.  We risk missing critical information by this practice — information that would help us build a stronger understanding of algebra.

On a small-group activity this week in our beginning algebra class, I used this question:

Paraphrase the expression  5x² – 4x + 5

This was the first problem of a set of 3, with a heading that included “properties and order of terms & factors”.  Because students have a hard time accepting a math expression as an object (and not always a directive to ‘do something’) many students struggled with the problem.

However, there was one particular error that was quite common, leading to this answer;

21x + 5

Since no work was shown, I was puzzled; I asked each student how they got this result.  Their answer?  The square meant 5 squared, so 25x … then 25x – 4x = 21x.

This is exactly the same issue we deal with when we present “-8²” and “(-8)²”; many of us see those problems as unnecessary.  I don’t agree, as many of my students have struggled mightily with “what does that exponent apply to”. These students can get a majority of correct answers when we say “simplify” because they have memorized the rule about like terms; it’s not that they believe it is wrong to get 21x for the problem — they just know that they are not supposed to do that when the directions are ‘simplify’.

If our students are not clear on “what the exponent applies to”, their understanding is limited to  first degree objects.  Now, we waste a lot of time on polynomial arithmetic that would be better spent on exponential models & numeric methods (to complement symbolic methods).  I have to say, though, that a beginning understanding of our symbolic language is based on the answer to that question “what does the exponent apply to”.

If you teach any algebra (beginning, intermediate, college, or pre-calculus), consider giving your students some open-ended questions about the meaning of our expressions.  Don’t assume that  correct answers is an indication of correct knowledge; the human mind is capable of much memorization and disconnected information.

Helping students build a strong understanding is a labor-intensive process.  Individual and small group dialogues are the most powerful tools to correct bad ideas; just getting feedback like “not correct, it means this” will not be effective.   [This is the reason why about 33% of my class time is spent using those tools.]

Remember that assessments don’t have to involve points or grades.  The best learning in my classes occurs when individuals and small groups struggle through stuff they did not understand correctly.  Every human comes with a drive to understand, and that can be harnessed in our math classes — if we use assessments that create those opportunities for deeper learning.

 
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