**Parent**: I don’t understand why you won’t answer my son’s questions in class.

**Teacher**: This is the new common core *teaching method.* Students need to figure things out on their own.

**Parent**: I am unclear on the way you want my son to complete this homework assignment. Why can’t he just follow the traditional algorithm?

**Teacher**: We are using the new common core *curriculum*. This is how common core wants students to do that particular computation.

**Parent/Teacher/Student**: (with much frustration) &%!@# common core *&!!@($^ math *@##!$%^% stupid…

Parents, if you have had this kind of a conversation with a math teacher recently, I encourage you to challenge that teacher to explain the difference between common core * STANDARDS*, common core

*, and common core*

**CURRICULUM**

**TEACHING METHODS.**The interesting thing is that one of these things is not like the others…STANDARDS! The common core STANDARDS are not like the others because the others *don’t even exist!*

The common core standards for mathematics include content standards for grades K-12…that is, * what *content should be taught. The content standards are written in such a way that mathematical ideas and understanding can be developed in a coherent, structured way through what they call “progressions.” For example, the ways in which students are to think about, make sense of, and ultimately compute multiplication problems are developed so that the way students are thinking in 3rd grade are extended to 4th grade, etc.

What the common core standards DO NOT do is tell teachers HOW to teach in order to provide the opportunity for students to master the standards. Nor do the common core standards provide curricular material to support the learning of the content standards.

In addition to the content standards, the common core includes Standards for Mathematical Practices.

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

Will teachers need to find/create/adapt high quality curricular material to meet these standards? Will teachers have to think about their teaching practice and possibly adjust so that these practices can be developed? YES!

The question is…so what DOES good teaching and learning look like (whether held to the common core standards or not)?

I will try to capture the essence of this in the following vignette:

You open the door to the classroom and your initial reaction is shock. It seems that chaos reigns as you notice students out of their seats as they talk with one another. Furthermore, the desks are not neatly arranged in rows, but seem to be randomly clustered into small groups as the students use the desk tops for what appears to be some sort of technical equipment set-up. Where is the teacher? You are not sure. Concerned with the outward appearance of this classroom, you enter the classroom to investigate. You stop at the first group of students and try to figure out what is going on. Quickly you realize that the topic of their conversations is highly technical. They are discussing the result of an experiment where two tuning forks of different frequency are struck and the combined frequency is measured. Would the frequency be the sum of the frequencies of the two tuning forks? The difference? Or some combination that will be difficult to determine? They follow through with the experiment using a data collection device and a graphing calculator and the students begin to analyze the results. They are discussing ideas and concepts foreign to you as they search for the beat pattern, calculate the period and frequency, determine the equation of the sinusoid which fits over the beat pattern, and ultimately discover the relationship between the frequency of the individual tuning forks and the combined frequency. You ask one of the students, “why are you doing this?” Annoyed by the interruption, she responds that they are trying to solve the problem at hand where, in the situation given, an old World War II mine needs to be deactivated. This can only be accomplished by directing the proper sound frequency at the timing mechanism in the bomb in order to disable it. You find it refreshing to see students motivated to solve a problem and notice that students are simultaneously using technology, paper and pencil calculations, and teamwork to find the solution to a problem that has obviously engaged the interest of the students.

You move on to the next group of students. They are equally engaged in what appears to be a different problem. You first notice that the students are huddled around a graphing calculator screen. Suddenly, a shout of celebration is heard as students triumphantly give each other “high-fives”. You get the attention of one of the students and ask what they are doing. He explains that their team was given data related to the height of waves in the ocean during a particularly dangerous storm. Their job was to create a scatter plot of the data and determine a sinusoidal regression model for this data so that they could calculate the period, frequency, and height of the waves at the moment in time a small touring vessel was capsized. Given the manufacturers specifications of this vessel, the students would be able to determine if it was built to withstand such a storm or not. This would aid in the investigation and eventual lawsuit brought by passengers of the vessel who believed that the vessel’s skipper used poor judgment in venturing out into the storm.

Impressed by the level of engagement of the students, the high-level discussions that were taking place between students, and the interesting combination of technology and paper-and-pencil calculations that were taking place, you still have one more concern: where is the teacher?

You move on to the next group and find that they are working with tuning forks also. Peering in to see what this group is discussing, you find that one of the members of the group seems to be asking questions. “What would you know about the frequency if the period is 0.123 seconds?” “Do you see a pattern when you consider the different combinations of two tuning forks and their resulting beat pattern?” “Can you explain the difference between period and frequency?” You realize that this is the instructor asking probing questions to the students to help them construct their own understanding of the mathematical concepts being explored.

Standing back to observe, as a whole, the classroom setting that initially brought disgust (nearly), you now feel joy (nearly) as you realize the high engagement of students as they study difficult math (and physics) concepts. But the students seem to be enjoying it. Could this really be a math classroom? What curriculum development and implementation steps were employed by the instructor in this classroom to get to this point?

The National Council of Teachers of Mathematics recently released a book (Principles to Actions: Ensuring Mathematical Success for All) describing the Mathematics Teaching Practices that “provide a framework for strengthening the teaching and learning of mathematics:

- Establish mathematics goals to focus learning.
- Implement tasks that promote reasoning and problem solving.
- Use and connect mathematical representations.
- Facilitate meaningful mathematical discourse.
- Pose purposeful questions.
- Build procedural fluency from conceptual understanding.
- Support productive struggle in learning mathematics.
- Elicit and use evidence of student learning.

Much support and patience is required as mathematics education is improved in the USA. Let’s all commit to making sense of the problem of mathematics education and then persevere in solving it!

Scott

Really great blog. I invite you to read mine as well – especially the one on “Doing Math vs. Thinking Mathematically”. Perhaps we can talk some time and trade ideas? I am an ASCD author whose book has been recently commissioned to be published by ASCD.

Here is my problem with your defense of Common Core: You claim that Common Core has standards of knowledge for each grade (true) and that it also teaches students to think deeply (false). A student cannot be taught to think deeply before the student learns to think linearly. Common Core CURRICULUM often attempts to teach both linear and deep thinking at the same time, wasting class time, and, yes, confusing the student in the process. Sorry, but a third grader doesn’t want to know 4 different ways to subtract 316 from 427. He wants to know the fastest way. Deep thinking comes later. Attempts to teach deep thinking causes confusion and frustration for the majority of young students and slows the learning process to a crawl.

Thanks for your response! Please allow me to clarify and/or add to my post.

You say that I say that “…it (the common core standards) teaches students to think deeply (false).” I don’t think I said that the standards teach anything. Standards don’t teach…teachers teach students. However, I stand by my claim that the Common Core Standards for Mathematical Practices provide the opportunity for students to develop sense making, problem solving strategies, reasoning, modeling, etc. I am excited that these standards for mathematical practices are part of the expectation for teachers and students.

You say that students cannot be taught to “think deeply”. Is this a research based claim? You say that students must learn to think linearly before thinking deeply. What does it mean to think linearly? And again, do you have research to back up your claim?

I can provide research that supports my classroom experience that students can indeed reason, problem solve, make sense, etc. and learn mathematics as part of a well connected network of understanding. For example, consider the topic of fraction division. When students learn to divide fractions as part of a well connected web/network of understanding of what a fraction really is, what division means, equivalent fractions, etc. they can first create a procedure for dividing fractions in the context of a problem solving situation. They can also work to make sense of the tradition algorithm for dividing fractions (invert and multiply). It is not the case that students first have to be told the traditional algorithm (if that is what is meant by thinking linearly) and then later learn how to “think deeply”.

You claim that it is a waste of time to “teach both linear and deep thinking at the same time”. I would like to understand more of what you mean by linear thinking and to better understand your vision of deep thinking. It seems to me that deep thinking should never be a waste of time!

Finally, you claim that “a third grader doesn’t want to know 4 different ways to subtract 316 from 427.” Where do you get the idea that the common core standards create a curriculum and/or a teaching method that require this? Maybe I missed it…but can you point to a place in the standards where 4 different methods is required? I think the idea is that teachers would help students to develop an idea of what it means to subtract (e.g. take away model, comparison model – see my other posts for discussion about this). Building from the foundation of place value, students develop a way to think about subtraction. Certainly, multiple methods might be shared among students as students “make viable arguments (MP#3)” and then students work to make sense of different methods as they “critique the reasoning of others (MP#3)”. Let’s face it…there are different ways of thinking depending on the problem situation. If the nature of the problem was “I have 427 dollars and I spend 316 dollars”, I might think about “take away” and employ a traditional subtraction algorithm to determine how much money is left over. But, if the context of the problem was something like, “You have 427 dollars and I have 316 dollars, how much more money do you have?”, I might think “well…84 more dollars gets me up to 400 dollars…another 27 dollars takes me up to 427…so 84+27 is 111. You have 111 more dollars than I do.” It’s not that I want students to have to solve a problem in many ways, rather I want students to have flexible, fluent, efficient ways to solve problems, in general.

It’s not about just teaching students to subtract…FAST! It’s about teaching students to think, reason, solve problems, make sense…developing flexible, fluent, and efficient ways to think mathematically AND compute arithmetically!