Archive for the ‘Uncategorized’ Category

Is It 1957 or 2017? (Part 2)

Posted: January 21, 2017 in Uncategorized

In Part 1 of “Is It 1957 or 2017?” I pose the following, “How should the mathematical content we teach be different in 2017 compared to 1957? Should we be teaching new things? Should we be teaching old things in new ways?” Let’s explore this a bit. Note that one could write for days in response to these questions…this is just a teaser to start some thinking and maybe some conversations.

What is math?

Conrad Wolfram, in a Ted Talk, poses the question, “What is math?” You might take some time to answer this for yourself before you continue reading.

Wolfram argues that mathematics is

  • Posing the right question.
  • Converting the real world situation to a mathematical representation or model
  • Computation

I would like to add a fourth bullet:

  • Posing the right question.
  • Converting the real world situation to a mathematical representation or model
  • Computation
  • Analyze, interpret, connect back to the real world situation.

Typically…traditionally…even still in 2017…students spend a lot of time…most of their time…on one of these four bullets…COMPUTATION! And computation is the one thing that a computer can do better than any human (with apologies to Scott Flansburg…the Human Calculator)! Why not spend more time grappling with interesting situations, posing questions, creating a mathematical model, computing/solving/doing something (even using technology), analyzing/interpreting results?

An Example

Suppose you are teaching a 7th-8th grade math class and want students to experience these four things (posing questions, convert to math, compute, analyze). You can see the classroom resources for this lesson here. Here is a way that it might play out:

  1. Show students the following video (stopping after just a couple of minutes)

It probably doesn’t take long before a question emerges in viewer’s mind…how long does it take to preheat this oven to 400 degrees so I can cook my frozen pizza?

2. In the classroom resources, students are provided with an increasing amount of data so that models can be created. Is the temperature increasing at a constant rate? If so, we can use a linear model. Given this model, can we predict the time needed to preheat the oven?

3. Once models and assumptions are determined and articulated, computations can be performed to make the prediction.

4. Take the result and clearly describe what it means and also recognize any limitations. Is a linear function, for example, really the best choice? Why or why not? What would that mean? What could we do to determine a more accurate estimate?

What Math Do We Teach?

When we start thinking about mathematics as a tool to model real-world phenomena and to make predictions, new math ideas might become relevant. For example, when modeling real-world data with function models, students might encounter a statistic known as the coefficient of determination. On the TI-84, it looks like this when creating a linear regression model and is designated by “r squared”:


What does this value mean? How is it computed? Why is it computed that way? If mathematical modeling becomes a focus, I will argue that ideas like understanding the coefficient of determination should become a part of a student’s mathematical experience!  I you are interested in making sense of the coefficient of determination, consider reading this post.

What other mathematical ideas might become important or necessary in 2017? How about topics in discrete mathematics and statistics? As we have recently (mid-late 2016) been focused on presidential election politics, it could be very useful and important for the general public to have a greater understanding of polling techniques. In fact, right after the election, many news outlets discussed the “failure” in the polling process since most of the polls predicted a win for Hillary Clinton. What is a poll? How are they conducted? Why do they only ask around 1000 people? What is a random sample? What is the margin of error? There are so many good discussions and real-world contexts that could be used to engage students in statistical thinking!

There are many initiatives pushing to include quantitative literacy as a critical component of a student’s educational experience  (read Lynn Steen, Deb Hughes-Hallet, AMATYC, MAA).

For a crash course, here is an example of why quantitative literacy is important…consider the following showing Average monthly temperature in New Haven, CT. Image: Yale University:

Global Warming Out of Control


Do you see what is happening here? The data/graph only show temperatures for the first half of the year! Of course temperatures are increasing!

What to Teach or Not to Teach?

There is so much to say…but…just to capture my thinking I will post the following for the purpose of discussion. This is not meant to be an exhaustive list. Rather, the items shown are representative of the idea of what we might think about starting/keeping or stopping.

start-stop* Obsolete procedures or certain procedures/computations may be studied from a historical perspective

As stated in Part 1, the intent is to have a conversation that I think is worthy of our attention so that the education our children receive isn’t simply what it is because that is the way we have always done it. Rather, it is what it is because what we have thought carefully about what expect students to learn to do and understand.






Raising Arizona Kids

Posted: June 16, 2014 in Uncategorized

This article appeared in the online magazine in June 2013:

Math teachers prepare for Common Core standards

By Daniel Friedman | June 3, 2013

Your kids need to be smarter and better educated to get decent jobs when they graduate from college. Many states, including Arizona, have adopted Common Core State Standards to better prepare children for the job market.
Implementing new standards in education isn’t like flipping a switch. At the most basic level, teachers need to know if the textbooks and teaching materials they already have will work for the new standards. On a more complex level, they need to modify their the lessons, sequence and teaching approach to meet the new standards.

The Arizona Mathematics Partnership, led by Scottsdale Community College, was awarded an $8.7 million, five-year grant by the National Science Foundation to help 300 middle school math teachers understand and teach the new mathematics Common Core Standards.
Math gets more difficult during the middle school years. Algebra rears it’s ugly head and equations move beyond basic operations. Parents often bow out of helping their kids in middle school because it gets more complex and they haven’t factored polynomials recently.
I spoke to Scott Adamson, a math teacher at Chandler-Gilbert Community College. He is working with teachers this summer at a five-day Summer Institute at Scottsdale Community College, but that is just a small part of the program. He also visits teachers at their schools, observing lessons and giving feedback on how to implement the new math standards.
The new standards include the usual content standards. like this one for sixth grade:
6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
This is a standard for third grade:
3.OA.7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8 ) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Adamson says the current math textbooks don’t really line up with the new standards but having teachers create their own entirely new lessons to match the new standards isn’t practical. It takes hours to design a lesson, create the materials and the assessments to go with it.
He works with teachers to help them implement the new standards as efficiently as possible. If they try to create all new lessons “they’ll be burned out by October,” he says.
In addition to the content standards, the Common Core standards include Mathematical Practices:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

Adamson says getting the correct answer is not enough “The heart of it is, what do we want out of our students? It is not to mindlessly perform steps and arrive at answers without understanding why and how they got their answers,” he says. One example he gave is students should understand why multiplying a negative number times a negative number yields a positive number. (You remember that, right? -3 x -7 = 21)
Really, the new standards with the mathematical practices are building an intellectual framework within which the content standards are learned and understood, says Adamson. A calculator won’t help “construct viable arguments and critique the reasoning of others.”
The new standards will demand more of your children as well as their teachers.
Teachers in the Arizona Mathematics Partnership program will get 200 hours of professional development, and a $2,500 stipend for their efforts. In addition to SCC and Chandler-Gilbert, other partners in the project are Glendale Community College, Chandler Unified School District, Deer Valley Unified School District, Florence Unified School District, Fountain Hills Unified School District, J.O. Combs Unified School District, Salt River Pima-Maricopa Indian Community Schools and Scottsdale Unified School District.
Parents will need to support their kids and send them to school willing and able to work to meet the new standards in math and English language arts which will be assessed for the first time in the spring of 2015.
Read the Arizona Common Core Math standards.
Find out everything you can about the standards.
Tags: Arizona Mathematics Partnership, Arizona schools, Common Core Standards, math, Scottsdale Community College, teachers

Daniel Friedman
Daniel Friedman is a staff writer and photographer for RAISING ARIZONA KIDS magazine.

Apparently, an English teacher in Colorado has resigned because of the Common Core Standards. She appeared on Fox News April 14, 2014 ( and reported the following:

1. First, she cited her 3rd grade son’s experience by saying, “the changes at the elementary school level are just insane…”

Insane? Let’s consider the math standards as I am not well versed on English/Language Arts Standards (and, in the TV segment, they do not differentiate between math and ELA):

  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

These are certainly rigorous, challenging, advanced…standards that we have not had in place in years past. But insane? What is insane about making sense of problems, constructing arguments, modeling, etc.? The teacher says that her son is “struggling to keep up to where they want him to be.”  It is not insane to challenge children to learn at high levels of expectations. Furthermore, we have to allow for some time of productive struggle as we transition to the new standards. I admit that the transition will be challenging…but that doesn’t make anything bad. In fact, challenges are often good once we overcome a difficulty, positive challenge.

2. The news anchor asks the teacher, “what is it about the changes…how has the Common Core changed teaching?” Her response:

  • It is more specific…
  • It has certain things they want us to teach…from kindergarten all the way up to high school…
  • It has certain skills they want the kids to have at a certain time, certain age, certain grade level…
  • Kids aren’t made like that…they are all different…

Wait a minute…standards that are specific and have things to teach, skills to have? That’s what standards are for! They give us something to aim for…high expectations to strive to meet!

3. She says that her job is to help students to think for themselves, help them to find solutions to problems, help them to become productive members of society. Again, she is right but this is an argument for the value of having rigorous standards. Again, I point to the Standards for Mathematical Practices listed above…they support the idea of thinking, reasoning, and problem solving!

4.  She also says that she has a problem with the Common Core Standards promoting a “teach to the test” focus in schools. This is a major misconception…people confound standards with assessment and curriculum. These are separate, albeit, connected entities. If she wants to argue against high-stakes testing, then have at it! But don’t confound rigorous standards with assessing these standards nor with curriculum based on standards.

5. In the interview, the news anchor pushes back on the “standardized tests are bad” issue by bringing up the fact that the US compares very poorly to other countries especially in math and science. The teacher makes a claim about standardized state testing (again confounding assessment with standards). Let’s be clear…there has been standardized testing for decades! This is nothing new because of Common Core. In fact, the standardized testing that is to accompany the Common Core Standards has only recently (this month) been field tested in the US. Most of us don’t even know what the new assessments will look like! She continues by saying that “…it has become about a right answer rather than a way to think.” I would agree that the old standards and accompanying assessments in most states did this. The Common Core addresses this very clearly and puts the focus on ways of thinking rather than memorizing and regurgitating procedures and facts. Again, I point to the Standards for Mathematical Practices. Also, read the content standards. You will see expectations like “understand…” and “compare…” and other ways of thinking that are much higher on most critical thinking taxonomies.

Please read the standards for yourself…separate the standards from assessment and from curriculum (although they do have to be connected)…then see if there are such “insane” problems with the Common Core Standards.



Cramer’s Rule

Posted: October 26, 2013 in Uncategorized

For those that attended AMATYC in Anaheim in 2013, here is the Cramer’s Rule document mentioned…

Determinants and Cramer’s Rule



My 8th grade son was doing his homework last night. I was washing dishes and he said, “Dad…this homework is easy!” I said, “Great! What are you doing?” My son, replies “We have to find out if a proportion is true or not.” I looked at the worksheet…it had things like this:

7/21 = 1/3

and he had to say “True” or “False”

I then had a struggle in my mind…do I ask him if he understands what it means for two quantities to be in proportion or do I just relax that he knows how to DO his homework? I chose to ask…I said, “Zach…what does it mean to say that the two fractions are proportional?”

Zach: It means that they are the same.
Me: Ok…would you say that 2 and 6 – 4 is proportional?
Zach: Yes…they are the same…

Zach then asked me the following question as he continued to work on his homework that is “easy” and he knows how to “do it.”

Zach: Do I cross multiply to see if the numbers are the same or do I multiply straight across the tops and the bottoms?
Zach: What’s wrong with you?
Me: I can’t answer that question…I need to leave now…

Maybe because I have been teaching for so long now…but I am becoming more and more impatient with this kind of thing with my students at school, my colleagues, and my own kids! Think about it…Zach is really saying, “I can’t remember what to do…I have learned so many meaningless procedures in my math classes that I am having trouble remembering what meaningless procedure to do and when to do it.”

Where are the big ideas behind proportional reasoning that should help to inform Zach as to what to do? I wish he as taught in such a way so that his FIRST thought is not, “what should I do…” but his first thought should be “what does this mean?” Then, based on what this means, he determines what to do because it makes sense to do it!

Thanks for letting me vent…parenting is hard…