Wednesday, May 4, 2011

Driving and deriving in math class

Here is another example of how a student can do math without the use of a textbook or a worksheet.  Many times I have, and will continue to, argue that math is not about repetition but actually deeper understanding.  Also, in math class instead of learning about or for problem solving, we should be teaching students to learn through problem solving.

Problem solving is NOT completing a word problem at the end of a class, or using an algorithm to solve a problem.  In the words of Andrew Wiles "The definition of a good mathematical problem is the mathematics it generates rather than the problem itself".  Below, I will illustrate the 6 steps of implementing a well thought out problem solving activity in a class:

Step 1:             Present an idea which students would want to solve.
Step 2:             Have students ask the question
Step 3:             Create a range of answers (students will feel safe to provide an answer)
Step 4:             Provide any other information that a student may need to solve the problem....HOWEVER....have the students ask for the information.
Step 5:             Allow for true autonomy to solve the problem.
Step 6:             When students finish, or THINK they are finished, extend the thinking by giving a question that is not more of the same work, but entirely different solving.

Recently, in my calculus class, I showed the following video and gave a laptop to each student, with the video uploaded on it, to analyze it.  Here is my lesson plan.
1)  Watch the video.
2)  I then asked the class, “Any questions about what we just watched?”  (REMEMBER: We should get the students to ask the questions; there will then be no extrinsic rewards but only true intrinsic motivation to solve the problem.)

One student asked “How fast is the other car going?”
For which I responded, “Good question

I then asked for the lowest possible speed of the car, and we determined that to be 109 km/h, while the upper bound for the speed would be 180 km/h.

As students worked they started to ask me for the length of my vehicle. Answering honestly, I informed them that I did not know this.  They then followed my response with a question about the year and make of the vehicle I was driving, which was a 2011 GMC Terrain.

Students started to pull the actual length from the GMC website.

After several conversions, and 8 minutes of work, I took a picture of a student’s work and showed it on my projector.  We then discussed alternate ideas, and where did this student have to estimate and where was he/she exact in calculations.

3) I then asked, “Is there anything else we could solve?”  Again, another student asked, “What is the change of the distance between each driver as the car drives by?” 

**Since I could not take an aerial photo of the situation, I did provide the students with a distance of 12 feet horizontally between the drivers**

Again, I asked the class to solve the problem.  Within minutes, they realized they would need a specific time, and answered with “Solve for the instantaneous change of distance of the drivers after 3 seconds.”

After some solving, I took another picture and discussed the results with the class.

4) I asked the class if there still anything else we could solve.  One student asked about the change in the angle of the arm that swung the camera.  I responded with "Let's figure out the exact change in the angle at the time above".

This task took a little longer for some, while minutes for others.  Those who did finish, or thought they finished early, I asked to calculate again but use a different method.  By using a different method, they are applying a different strategy and also checking their answer at the same time. 

We then did discuss various methods in the class. 

Overall, I believe this lesson was a great success!!

Here are some pointers I have realized about true problem solving. 

Tell the students they are correct or incorrect, ask if there is a way they could prove their answer another way.

Tell the students HOW to complete the problem, but instead ask them guiding questions.

Tell the students which way to solve the problem, but let them chose a method.

Scaffold for struggling students.

Answer questions, even if the answer is irrelevant to the problem.


  1. THIS is education. Well done. What I really like is that it's done in a way that can be applied to just about any level. It's socratic in nature and involves the kids from the word "go".

    I wish I could tatoo your pointers at the end on my eyeballs.

  2. I love it David. I'm looking forward to the next time I teach calculus, so I can apply (steal) this idea to my own teaching of rates of change as well.

  3. Rhonda PhillipsMay 4, 2011 at 8:57 AM

    Well stated. In this testing-frenzy environment, it's refreshing to see 'real' teaching and learning. @ContentLiteracy

  4. Problem solving activities have their place, but once you get to the later stages of algebra and into calculus, the body of mathematical knowledge and achievement is far too large to address in this manner. A lot of what I see here seems to be only teaching what a problem looks like, not the broad range of tools a person needs to solve problems when you actually encounter them. you need both or you have failed to teach a subject, unless your version of teaching the subject has nothing to do with the skillful use of the subject in modern society.

  5. Robert, are you implying that you can't teach a subject through problem solving?

  6. I am implying that you can't teach subjects to the level we use them in actual practice today, or in other subjects, via problem solving and discovery alone. You must teach technique as well. Civilization has advanced these subjects quite far over many centuries and whether you like it or not education in mathematics (all subjects in fact) is based on that broad advancement. Of course you want to develop fundamental problem solving abilities but after you pass that hurdle you must develop in your students the ability to study and own other people's mathematical theories and solutions to a multitude of problems because civilization doesn't reinvent the wheel over and over and over again. It is what it is. Mastery and expertise is leveraged off of all the people that came before you so you have no choice but to study their solutions. You wouldn't even touch a fraction of the possibilities if you relied on personal discovery alone.

  7. I have to ask Robert the following question. Is it really necessary to touch on ALL of the knowledge that has come before us? If we help students to become good problem solvers then we are providing them the skills to be successful in life. Part of being a good problem solver is realizing that you do not have all of the knowledge or skills to solve a problem and then seeking out the knowledge you need and learning the skills required to solve the problem.

    As I see it, we try to "teach"kids way too much of what has come before them. Instead we should be delving deeply into fewer topics and spending more time on problem solving.

  8. I agree with these comments 100%... but have difficulty implementing them in the constraints of today's classroom. I teach at a suburban school where pretty much all teachers stand at the front of the room and talk to (or at) the kids. My first year, I tried to implement group-work where students would have to work together, help each other when they didn't understand, negotiate meaning when they were all stumped, etc. They told me in no uncertain terms that they didn't like it and I wasn't going to do it. Having a standardized test at the end, I caved, realizing that I didn't have enough time to teach them HOW to learn this way, and then ALLOW them to learn this way. Am I just being a pessimist? I feel like this method works in a setting where everyone is using similar methodologies (so that students already know how to learn this way) and perhaps there aren't standardized tests covering massive amount of material (I'm not talking about the AP... I'm talking about an algebra test for students who struggle a lot or a little)

    Please tell me if i'm being the only cranky nay-sayer here. It's not that I don't like these ideas of yours... in fact I LOVE THEM. I have just become unsure of how to make them work under these types of constraints.

  9. Now THAT'S Calculus at work. Nicely done!

  10. Terry, I didn't say ALL the knowledge, that would be impossible. But the student should emerge from their education with a connection to all the knowledge and with a mastery of the processes involved in owning that knowledge. Mind you, I am talking of a student that is aspiring to a career or even a personal interest in a technical field that involves mastery of a subject like math. An analogy would be a student with musical talent. There will be a personal component to their development (like this) as well as a very large directed component that involves the musical theory and technique developed by countless individuals before them. Subjects like math are no different and this full treatment is most certainly needed by these aspiring students and these aspirations begin in early adulthood (high school), naively at first, but they do begin.

  11. Robert, I agree that we have to prepare the students that are aspiring mathematicians. However, in high school, the majority of the students in out math classes are NOT aspiring mathematicians. We have to take this into consideration. Our current HS Math curriculum serves very few of the students in our classrooms. I believe that if we focus on problem solving it will benefit ALL students in our classrooms. No matter what field of study they choose to pursue.

  12. Well Terry, you say that you agree with me that we should prepare our aspiring mathematicians and engineers, yet in the same instant you seem to be telling me that since most students don't aspire to be mathematicians and engineers we should prepare them instead. Prepare them for what? Prepare them in a subject that they are not interested in? Shouldn't we be preparing them in the subjects they are interested in, the things they aspire to? And don't get me wrong, there is certainly nothing wrong with having a course set up for those not interested because maybe they might find that they are interested, but if you take that as far as you seem to be taking it, there won't be anywhere for them to go if they do find out they are interested. You say that schools have a responsibility to teach everyone some math, and quite probably so, but I don't think that should exclude teaching aspiring students a lot of math, like we enjoyed.

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