Maybe You Can Teach an Old Dog New Tricks

First of all, I need to apologize to any students that I had in my classes from 1992 to about 2010.  You are probably wondering why I feel the need to apologize.  Because when I first began teaching, I somehow used to think that it was about me.  Since my transition into a Math Teacher Educator, I have come to realize that it is about the students and their experience with the content.

Here is an article in the Chronicle documenting that there has been a true shift in instruction at the college-level.  Faculty are moving away from lecture and toward a learning-centered approach. 

http://chronicle.com/article/Professors-Place-in-the/149975/

I can vouch for the fact that it works.  Here is an example this semester from my Geometry for Teachers classroom.

We were discussing similar figures and finding the scale factor for a pair of similar figures.  I then asked the students if the ratio of the area of one of the figures over the area of the other figure would be the same as the scale factor.  One student immediately said, “I don’t think so.”

I told him to go to the board and give an example to illustrate why he doesn’t think the ratio of the areas is the same as the scale factor.

He drew the following picture:

Area of large square = 4 square units

Area of small square = 1 square unit

The ratio of the areas is 4 to 1,  but the scale factor is 2 to 1 .

Then I asked:  “What happens if your figures are triangles?” 

He drew the following picture and said that he was going to draw right isosceles triangle to ease the explanation.

Area of large triangle = 2 square units

Area of small triangle =  1/2 square unit

The ratio of the areas is 4 to 1,  but the scale factor is 2 to 1 .

Then I asked, “Can you see a pattern for the ratio of the areas of two similar figures?”
Then we discussed that since area is a two-dimensional measure that it would make sense that the ratio of the areas is the scale factor squared.

I informed the student that he had just taught a section of the textbook!  No lecture, no PowerPoint—just one good question led the student to explain an important topic.

FYI—I don’t have this great of a success story every day, but when I do, it is very rewarding!

Why Talking Math with Your Kids is so Important!

My husband and I have made a point of doing mathematics verbally with our kids since they were young.  Now that they are in third and sixth grade we are seeing some of the positive effects of this practice.  I hope that some of you will seriously consider Talking Math with Your Kids.

I don’t want you to think that we sit around at home and constantly ask our kids to solve math problems.  We do it in a more subtle way.  The best part for us is when our kids say, “Do you want to hear how I know this is true?”  Of course, our answer is always an emphatic “Yes!”

The other day when my third grader was doing her math homework, one of the problems required her to add 7 and 8.  She wrote down, 7 + 8 = 15, and then said, “I know that 7 + 8 = 15 because 7 + 7 = 14 and 8 + 8 = 16, so it has to be in the middle.”  My response, “You just found the average!”  Then, as any proud parent would do, I tweeted about it.  My Twitter is connected to Facebook so a few friends commented on it there, and then yesterday the tweet got lots of attention on Twitter thanks to Tweeps from the mathtwitterblogosphere.  Christopher Danielson, author of the blog Talking Math With Kids, said, “Follow up question:  what is 7 ½ + 7 ½?”  I still need to ask her that question and I will!

Why am I making such a big deal about what my child said?  As a parent am I bragging about her?  Well, yes, to some extent.  However, as a math educator, I am more intrigued by the fact that she thought about the problem, used knowledge of ‘double sums’, and from her knowledge of ‘double sums’, found the solution.  And the best part for me was that she freely explained her thinking—without my asking!

This is not the first time nor the last time that this will happen with her or with my sixth grader.  Why?  Because we make a regular habit of Talking Math with Our Kids.  I strongly believe that this has led our children to become better problem solvers and critical thinkers in various situations beyond mathematics.  These are the skills that our children need to be successful in life! 

Now for some bragging, here are some other ‘math stories’ from our kids:

• When my sixth grader was in first grade, he came home one day and said: “I asked _____ what 7 minus 10 was and he said it was 0.”  My response, “What is 7 – 10?”  He said, “-3.”  Then he proceeded to say that the other child said that there was no such thing as negative numbers, but he knows that there is such a thing.
• Last year as we are walking to a WNIT basketball game hosted on our campus, my children say, “Why don’t we ever get to sit in the chairs?”  I respond, “Those tickets cost $20 each.”  My second grader says, “That would be $80!”  Not only did she know the answer, she knew that her parents are too cheap to pay that much!
• This year during a MathCounts practice our 6^th grader was given the following problem.
  

Instead of figuring out how far the front wheel travels in 100 revolutions, our child took 30 divided by 4 and then multiplied that answer times 100 to get 750 revolutions. His answer was correct, and he was able to do it quickly, which is important in MathCounts.  My husband, who was helping with MathCounts that afternoon, was impressed and did explain to the students how to solve the problem two ways:  using the distance traveled by the front tire and finding how many revolutions the back tire would need to make to travel the same distance and one using our child’s method.

Some of you are going to say, “Your kids get this because their parents both have PhDs in Math.”  While that is true, you don’t need a PhD in math to talk math with your kids.  We likely do it more because we love mathematics and problem solving and we see its importance.  More parents need to talk math with their kids so that the kids build their math confidence and become better problem solvers.  We have seen first-hand that it really works!

Here are some resources that will help get you started:

• http://talkingmathwithkids.com/
• Kindle version of Danielson’s book
• http://www.jefferson.k12.ky.us/Pubs/MathTips.pdf
• http://bedtimemath.org/ --there is also a Bedtime Math app

Please share ideas on how you Talk Math with Your Kids!

It’s All About Perspective

Both pictures below represent my daughter's Flat Stanley hanging out in the tulips.  However, when you look at them, the perspective is slightly different.

When you look at a math problem, do you see something different than your friend?

One of the beauties of mathematics is that there are multiple ways to solve problems.  I really encourage our pre-service teachers to make their students aware of multiple methods as all students are unique and think differently.   Not only should teachers embrace multiple ways, they should demonstrate multiple ways.

A couple of weeks ago in my Geometry for Teachers course, we were doing a proof in class during discussion.  When I assign ‘discussion’ problems for my students, this means that they need to solve them on their own and be prepared to present their solution to the class.  One student volunteered his solution and as I was observing I noticed that his proof was different than mine.  His proof was completely correct and had all the essential elements.  After he finished, I said that I had done it a different way and we discussed my proof briefly.  Then another student chimed in, “Can’t you do it this way?” She proceeded to explain her proof.  Each of our proofs were slightly different, but they were all correct.  This is a great example of the richness of mathematics and discussing multiple solution methods—everyone learns from each other and sees a different perspective.

This idea of viewing things from another person’s perspective goes beyond mathematics—it needs to be done in culture, politics, religion, economics . . .  

Rather than condemning someone for their method or their viewpoint--WATCH and LISTEN.  You might learn something!

Pre-service Teachers' Assessment Activity

When I give my Geometry for Teachers students an exam, I always give a take-home portion that contains what I refer to as ‘teacher-type’ problems.  These are problems that they need to be able to do when they are a teacher.  Typically, they involve writing directions, grading a test question, creating a rubric, writing a test question, or evaluating curriculum.  For their first exam, I gave them the following problem.

Two of your student’s solutions to finding the radius of the inscribed circle are below.  One of the solutions is correct and one is incorrect.
     a) Write up a detailed solution and rubric to finding the radius of the inscribed circle.
     b) Grade both students' solution according to your rubric.
     c) Which student has the correct solution?

I am going to focus this post on the rubric and how they graded the students' solutions. I was very impressed with their rubrics and wanted to share them.

Natalie's solution

 

Logan's solution

Some items that we discussed about their rubrics when I returned the exam:

• Point values on the question ranged from 5 points to 15 points.  What is the best way to determine the point value of a test question?
• Was notation specifically listed in the rubric?  Some students listed it and some didn’t.  One of the students that didn’t list notation felt that she could take off a point for notation under the justification portion of her rubric.
• What is the best way to make a rubric?  We talked about how you need to make a key for a question before you could really determine the number of points that you should allot to that question?

Below are some of the rubrics and how they graded the students’ work.  I was very pleased with their work and feel that they are well on their way to becoming great teachers!

Student A

Student B

Student C