Best Wrong Answer Ever!! How to not graph a function

I never laugh at a student, though I often try to laugh with a student.

Today, we had our first test in our intermediate algebra class.  In this class, I like to extend the very simplistic work the textbook does with graphing functions; we cover this in class, and students have a small set of practice problems.

Well, on one student’s test, I see this:

 Join Dev Math Revival on Facebook:

Problem Solving … and Learning Mathematics

Our Math – Applications for Living course is sometimes used as a last option; students try passing the intermediate algebra class, and (after 2 or 3 tries) an adviser says that they have another option.  This is not true for all students in the course, though it is a common path to my door.  The result is a class with some very anxious students, and many who doubt their ability to solve ‘word problems’.

Math – Applications for living is all about problem solving; all topics are verbally stated.  We had an interesting experience last week when we did an example with a simple statement:

The distance from the Moon to the Earth is 3.8 x 10^5 km.  A light-year is 9.5 x 10^12 km; in one second, light travels 3 x 10^8 meters. How long does it take light to travel from the Moon to the Earth?

The problem presents to issues to resolve: the operation to perform, and making the units consistent (meters and km).  A few students knew to divide distance by speed to get time; if they did not already know this, it did not help much to solve the D=rt formula for t.  We explored the problem by working with rates (as we have been doing for most unit conversions); this helped a little more.

We got frustrated, however, with the km and meter conversion in the same problem.  After about 10 minutes of discussion, some progress was made.

In working through these struggles, more than one student said something like:

Can’t you just show us how to solve these in a way that we already understand?

Of course, it is exactly this gap between current understanding and present need that causes learning to happen.  As a problem solving issue, this is essentially a statement of what problem solving is … as opposed to exercises.  In the most encouraging manner, I told the class that this tension they are frustrated with — is the zone where we will learn something.  I stated, with emphasis, that if I did not create situations where there was a gap like this that they would leave the course with the same abilities as when they started.

I’ve been talking with faculty in some other programs at my college about the mathematical needs of their students.  The first thing they say is always ‘problem solving’, and they don’t mean solving a page of 20 ‘problems’ using the same steps.  The second thing they say depends on their program, and a surprisingly large number of them say ‘algebra’ is the next priority — in spite of the fact that algebra is often de-emphasized outside of the STEM-path.  In the Math – Applications for Living course, we use algebraic methods when useful, as it is when solving problems with percents.

In the larger context, all learning is problem solving.  A learner faces a situation where existing knowledge is not sufficient, and the gap is completed by some additional learning.  I believe that this statement is true regardless of the pedagogy a teacher uses, whether active or passive for the learner.   I do not agree with a constructivist viewpoint, especially the more radical forms; however, there is a basic element in the constructivist view that is true, I believe — knowledge is built as a result of gaps.  I believe that teachers can (and should) model the process of filling the gaps, and explaining the reasoning behind ideas that can help.  Learning math does not need to involve students stumbling through to discover centuries of mathematics; we can both guide and be a sage in the process.

 Join Dev Math Revival on Facebook:

Math – Applications for Living: The Point of It

In my Math – Applications for Living class, a couple of students did something humorous (and sad) with a problem on a quiz yesterday.

Here is the problem:

Computer sales for a certain company were reported to be $40.3 million in 2009.  This was stated to be a 12% increase over the prior year.  Find the computer sales for 2008 (round to the nearest tenth of a million).

We are working on translating a percent change to a multiplying factor (1.12 in this case), and most students are not there yet.  However, here is the thing these particular students did:

40,000,000.3   is ‘40.3 million’

I have not talked to these students about what led them to make this mistake; it looks like they think that ‘.3’ means that part is stuck after the decimal point.  Since we just started working with scientific notation, that type of error will be a large issue.

 Join Dev Math Revival on Facebook:

Mathematical Literacy: Growing Pains?

I am sharing some of the experiences in offering our new Mathematical Literacy course here at Lansing CC (Math105); we have 2 sections of the class, and I’ll be sharing from my section in general.

A normal class day (2 hours, twice per week) involves about 50 minutes of small group work.  The text we are using (a class test of the Almy and Foes Math Lit text) organizes the lessons around this approach; the group work is well designed, and the authors even include time estimates for each activity.  We usually cover 2 lessons per day, and the pace is reasonably comfortable for students.  I experience more stress about the pace than students, because that normal class day involves 4 separate small group activities followed by sharing results and often completed by whole-class discussion or lecture.

Tuesday, we dealt with a problem that involved megabytes and gigabytes … and a conversion between those units.  Each group had people who thought that a gigabyte was exactly 1000 megabytes, and each group had somebody who checked this with their phone using an internet search to provide the correct value (1024).  I was hoping this would happen, though I did not mention the possible problem; the text did not mention a possible need to search for an answer.  We used this problem to introduce a ‘multiply or divide’ approach to converting units; simultaneously, we are building our understanding of rates so we can use the more sophisticated process later.

Yesterday, we had a salary simulation with two different plans for raises; the groups did a lot of numerical work with the two plans and several cases, and discussed how we could tell when one option would be better than the other in a given case.  We then made a transition to writing algebraic expressions as a template for the numeric work, and showed a little bit of combining like terms.  I used these expressions to create a spreadsheet for the example salaries, and also showed the process on a graphing calculator.  Most students did not have a computer to bring to class, and only a few had a graphing calculator yet;  this is one issue that we will have to deal with soon, as a phone or smart phone is not a good device for doing mathematics (especially when we need to proctor tests).

Attendance is a little strange, because it is clear that most students do actually enjoy the work in class; most days, I am only getting about 70% attendance, which is low for my classes.  Since we have a test in two weeks, the absences are a concern.  I don’t think the students in this class have a significantly different lifestyle than my beginning algebra classes, where I normally have 80% to 90% attendance.  This is a puzzle.

The largest problem so far is doing homework.  Assignments include just 6 to 10 problems in the printed textbook, and (usually) another 6 to 10 in the online homework system.  This is pretty light, and we talked in class about the importance of studying for learning … to include these steps.  Only a few students are doing anything outside of class (5 or fewer, out of 18).  This has led me to modify the participation point strategy for each class — starting with the next class, students will lose half of their ‘daily’ points if they do not complete the assignments.  I’ll check the text problems during the first group time, and the online system before class.  I’ll report on how that goes in a couple of weeks.

 Join Dev Math Revival on Facebook: