I was at a session at a small math conference last fall, and the presenter was going through their list of things students must do to learn mathematics. When the presenter said that students had to do things according to the instructor’s method (“They have to do it MY way!”), I am sure my jaw fell to the floor. In my experience you have to let students think and experiment in order for them to learn and understand mathematics.
As the years have passed I have tried to allow students time to think in class, especially how they think they should solve the problem.
When I tell them how to do it I rob them of a chance to think mathematically.
When I tell them that this is the only way to do it I am telling them that they no longer have to think.
For students who have struggled with mathematics this approach takes away motivation and inspiration.
Last semester, in an intermediate algebra class, we were reviewing for a final exam and were discussing solving absolute value inequalities. This is a topic where I only teach one method: convert the inequality to a compound linear inequality and solve. As I reminded students of the strategy we used in the past, one student asked if we could solve this type of inequality in the same way that we solve quadratic inequalities and rational inequalities. (Find critical values and test intervals.) I was so proud of that out-of-the-box thinking and praised the student for being able to tie the concepts together. Even better – several students used this approach on their exam.
I observed a colleague last semester and he was having students come to the board to factor polynomials. One of the polynomials to factor was x^2 – 4x. Many students quickly confuse this with the difference of squares x^2 – 4, missing the fact that there is a factor of x that is common to both terms. The student who went to the board surprised me by adding on “+ 0”, rewriting the expression as a trinomial. He then proceeded to factor it as (x + 0)(x – 4), and ended up writing the factored form as x(x – 4). I loved that the student thought of such a unique way to factor the expression instead of just working their way down a checklist.
When you tell students to find the maximum/minimum value of a quadratic function that they must complete the square, you rob them of trying to find the most efficient way to find the vertex for that function.
When you tell students to graph lines by first rewriting the equation in slope-intercept form, you rob them of perhaps finding a more efficient way to graph the line, such as finding intercepts.
When you tell students to solve a system using the elimination method, you rob them of the chance to think whether the elimination or substitution method would be more efficient.
Let your students try to solve new problems before you tell them how to do it. Thinking leads to understanding and growth. Don’t restrict them to only one approach to a particular type of problem. Don’t tell them the technique to use on an exam. Try changing MY way to YOUR way, and watch your students flourish!
Last month my wife and I visited the Royal Ontario Museum in Toronto. We saved their Pop Art collection for the end because it’s a light way to finish a day at the museum. We were admiring an Andy Warhol piece on Elvis when a guide walked up to us and asked us if we’d like to learn about a piece in the next room. We love to learn, so we took her up on the offer.
It did not go as I had thought. The guide started by saying “Rather than me telling you about the piece, I’m going to give you five minutes to look at the piece and then have you tell me about it.” At first I felt sheer panic, and I immediately thought about how I am sure that I have many students that feel this way in my math classes. Then I did what I suggest to my students – I thought about the piece. I looked for patterns. I looked for symmetry. I looked for parts that stood out. I tried to figure out what the artist was trying to say.
The guide asked for our thoughts and my wife went first. As she spoke I was impressed that she mentioned different characteristics than I had thought of. I shared my observations. The guide acknowledged our contributions and mentioned that she hadn’t thought about the things we mentioned and that they were good impressions. She then told the story of how the piece was created and told us of several other themes that could be found in the work.
It was a great learning experience because she let us do our own discovery first, then she helped us build upon our own observations. I try to do the same in my math classes. Math, like art, needs to be experienced. It should not be something that happens to you, you have to play an active role and become engaged in the process to learn and truly understand. I think I will share this story with my students on the first day of class next semester.
One of my goals for next semester is to encourage my students to think more and for me to “instruct” less. We will be discovering together. It took an unexpected learning experience in an unexpected place to make this all crystal clear to me.
At the beginning of the new year I started a blog project called uCLID. Each weekday, and most weekends as well, I intend to work my way through one of the propositions is Euclid’s Elements. It’s been a lot of fun. You can keep up on my progress through the “Blog” tab on my website, or by going directly to this address: http://www.georgewoodbury.com/euclid/
I just posted Proposition 24 in Book 1 today, so that puts me halfway through Book 1.
Enjoy – George
In today’s class we ran into the trinomial
This can be challenging for students just learning to factor because they are not familiar with the factor set of 208. Traditionally I recommend that students follow the 10-second rule:
If you cannot find the correct factor pairing within 10 seconds, you should move on to listing all factor pairs of c.
This would result in the following pairs:
That means that students need to try 13 potential factors (1 through 13) before they find the correct pair. Here’s an alternate approach: focus on the term 3x. Since c is negative, we know that m and n will have opposite signs. If we ignore those signs, we know that the two values must be 3 apart from each other or, in other words, have a difference of 3. I told my students to start multiplying numbers that were 3 apart from each other. We started with 10 & 13, but that product was too small. We bumped them up to 11 & 14, 12 & 15, and finally 13 & 16. Bingo!
Now consider the trinomial
Focusing on the middle term (29x) might not seem to be much help here. However, we could start with two numbers that have a sum of 29 like 14 & 15 and work our way back.
Give it a try and let me know what you think.
I have two sets of final exam review videos on YouTube.
The first playlist, for Elementary Algebra, can be found here: http://www.youtube.com/playlist?list=PLCE703DB5743508D7
The second playlist, for Intermediate Algebra, can be found here: http://www.youtube.com/playlist?list=PL15AA6E8E21593D3D
Please feel free to share with students and instructors who can use these.