What is Homework?

Do your students do homework?  Do they know how to get help on homework?  Most likely … no to both questions.  After trying other steps over the years, I am trying an invasive tool this semester:  Every student must take homework to a source of help on campus during the first week of the semester; they get a faculty member signature on a quick little form, and get 10 points for completing this.

Some interesting problems arose very quickly.  Several students asked “What is the homework?”  This might seem like an odd question, until you think about it from their point of view.  Most math books have reading, examples, section problems, chapter problems, review problems and more … some texts have ‘margin’ problems (or equivalent).  In my beginning algebra class, students were having a difficult time discriminating among these components — and a difficult time generalizing these in to a category called ‘homework’.  My students were thinking that ‘homework’ meant the written steps they did for a set of problems on one page (similar to the section problems); I was thinking of homework as being any and all of these components.

“Getting help” holds some mysteries for our students.  Without direction from faculty, students may not not see the sources of help as being a resource for them to use; one of my students reported that she is repeating the course because she waited until the last two weeks of the semester to get help.  We might think that “I am not passing” would be a strong enough message to go get some help, but this does not work out for many students.  For whatever reasons, there seems to be a large social gap between students and the sources of help that will get them through their course.

If you are curious, my experiment with requiring students to get help in the first week is working well.  Some of the students are getting comfortable with the process of getting help as a result.  For others, this process also helps uncover misunderstandings about what ‘homework’ means.    The number of points (10 for the assignment, out of 1000 for the course) is high compared to the effort, which makes the assignment more tolerable for students.   The sources of help available include my office hours (not during class time), our campus tutoring office, and the math department help room.

I encourage you to not assume that students will get help when needed.  Look for ways to require students to get help, especially in developmental and gateway courses.

 
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Math – Applications for Living XII

In our ‘math – applications for living’ class, we are reviewing what we have learned this semester.  Some parts (like probability) are still tough for students, partially because there is some memorizing to do with new material.  Truth is … I like to cover probability mostly because the process encourages reasoning about quantities.  [For example, we had a problem to solve about the probability of having 5 children — 2 girls followed by 3 boys; some of us wanted to look at this as dependent probability: 2/5 for the first girl, 1/4 for the second, and then confusion about what to do with the boys.  Clearly, knowing that events are independent is critical.]

The best problem we worked on today was one with almost no practical value: 

We had to really work on this problem.  The intent is to have students focus on the units (we need ‘square feet’ for area; we have cubic feet and feet … how can we do this?).  When students asked how to do this problem, I would ask them “How do you measure area?” (to get them thinking about units).  Every student (individually) said “length times width”; clearly, we are still too focused on one formula, and not thinking about what we are measuring. 

Of course, we could follow up on the “length times width” idea with something more reasonable. 

S: Area is length times width.
I: Okay, for a rectangle we calculate area that way.  How do we calculate the volume of a box?
S: Multiply (writes V = LWH)
I: So, the volume is L*W times H; right?
S: Yes
I: We know that L*W is the area of a rectangle.  Think of that volume formula as “V = area * Height”.  How would we solve this for the height, which is like the depth of the lake?
S: Hmmm (thinking) … we would divide
I: Yep — divide both sides by area.  Does that give you an idea how to solve the lake problem?

Most students originally decided that they had better multiply the numbers in the problem; of course, they only dealt with the value not the units.  They did not think about getting “feet to the fourth power”, and what this might mean.  A couple of students thought that the ‘cubic’ in ‘cubic feet’ meant that that value needed to be cubed.  [More evidence of a ‘messy landscape’ of math knowledge.]

The good news from today’s class was that students actually did a reasonably good job figuring out a complicated ‘unit conversion problem’ (given dimensions of a box, the flow in gallons per minute, and rate of gallons per cubic feet … how long would it take to fill the box).  Prolonged effort on related problems with diverse settings has paid off.  We are having more difficulties with geometry (concepts) than we are with proportional reasoning.

 
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Slope … Fast?

Well, I’ve been somewhat discouraged by the latest test in our ‘math – applications for living’ course.  This test is all about understanding linear (additive) and exponential (multiplicative) change.  In spite of approaching these models in different ways, followed up by repeated work, students had considerable difficulty translating verbal descriptions to anything quantitative or symbolic.

One basic problem seems to be that students did not start with much understanding of slope for linear functions.  Every student in the class had completed beginning algebra (usually recently), where we do the usual work with slope — calculating it, using slope to graph, and finding the equation of a line given information about slope.  When faced with non-standard problems with information on slope, the transition to quantitative or symbolic statements was not easy.

One part of this difficulty is the connection between input  & output units and units in slope.  Like most traditional algebra books, ours does not make an issue out of units for slope in ‘applications’; as you know, slope must involve two units.  Because of this difficulty, students would see a percent change as a linear change.

Mostly, this post is a “note to self”:  Learning slope is not really a fast thing.  Repeated use, alternate wordings, and non-standard problems need to be used to build a more complete understanding.  We too often spend a week or two on all linear function topics in a beginning algebra course; to get that ‘sufficiently good’ understanding, we need to allow more time so we can really dig in to what slope and the linear form are doing.

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Rotman’s Race to the Top

A few year’s ago, Paul Lockhart wrote an essay entitled “A Mathematician’s Lament”, sometimes known as “Lockhart’s Lament”.  I find this essay to be incredibly inspiring.

If you have not read it, here is a copy: lockhartslament

As far as I know, Lockhart never published this essay. I actually know very little about Paul Lockhart, besides the information that he went to teaching in a K-12 school after being a university professor for a time.  Some brief bio stuff is at http://www.maa.org/devlin/devlin_03_08.html

Okay, back to the essay.  Paul Lockhart is presenting a vision of mathematics as an art, not as a science, where learning is very personal and where inspiration is a basic component.  A fundamental process in Lockhart’s vision is the personal need to understand ideas, often ideas separate from reality.  The ‘lament’ aspect of the title for the article refers to the view that school mathematics has made mathematics into ‘paint by numbers’ instead of an exciting personal adventure.  Where we might focus on insight and creativity, we instead deliver an assembly line where students receive upgrade modules to their skill sets; Lockhart presents the view that these skill sets have nothing to do with mathematics.

I have two reasons for highly appreciating this essay.  First of all, people often assume that I believe that mathematics should be presented in contexts that students can relate to (probably because I advocate for “appropriate mathematics”).  Contexts are fun, and we can develop mathematics from context.  However, contexts tend to hide mathematics because they are so complicated; mathematics is an attempt to simplify measurement of reality, so complications are ‘the wrong direction’.  When I speak about appropriate mathematics, I am describing basic mathematical concepts and relationships which reflect an uncluttered view of mathematics; applicability might be now, might be later … or may never come for a given student.  Education is not about situations that students can understand at the time; education is about building the capacities to understand more situations in the future.

The second reason for appreciating Lockhart’s Lament is the immediate usefulness in teaching.  If we believe that students want to understand, then we can engage them with us in exploring ideas and reaching a deeper understanding.  Here is an example … in my intermediate algebra class this week, we introduced complex numbers starting with the ‘imaginary unit’.  Often, we would start this by pointing out that ‘the square root of a negative number is not a real number’ and then saying “the imaginary unit i resolves this problem.”

Think about this way of developing the idea instead (and it worked beautifully in class). 

1. Squaring any real number results in a positive … double check student’s understanding (because quite a few believe that this is not true).
2. The real numbers constitute the real number line, where squaring a number results in a positive.
3. Think of (‘imagine”) a different number line where squaring a number results in a negative.  Perhaps this line is perpendicular to the real number line.
4. We need to tell which number line a given number is on; how about ‘i‘ to label the number?
5. Let’s square an imaginary number and it’s opposite (like 2i and -2i).
6. Squaring an imaginary number produces i²; since the imaginary numbers are those we square to get a negative, i² must be -1

Any context involving imaginary or complex numbers is well beyond anything my students would understand.  I would always want to explore imaginary numbers; first, it provides a situation to further understand real numbers … second, the reasoning behind imaginary numbers is accessible to pretty much all students.

You might be wondering why I gave this post the title “Rotman’s Race to the Top”.  One reason is just not very important … I wanted a alliteration comparable to “Lockhart’s Lament”; well, I tried :).  The important reason is this:  The “Race to the Top” process is based on  some questionable components that do not connect directly with the learning process (common standards [a redundant phrase], standardized testing, and outcomes.  The world we live in, as teachers of mathematics, is a personal one: We interact with our students, we create situations that encourage learning, and we want our students to value learning for its own sake.  We also want our students to understand some mathematics.  Inspiration and reasoning have a lot more to do with this environment than any law or governmental policy.

Rotman’s Race to the Top is the process of teachers showing our enthusiasm, modeling the excitement of understanding ideas, and helping each student reach a higher level of functioning.  Building capacity and a maintaining a love for learning are our goals.  Rotman’s Race to the Top is about our students and their future.

Will you join this Race to the Top?

 
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