|The Problem in Question|
A few weeks ago, much attention has been given to a story about a father who shared his frustration on his Facebook page about an assignment sent home with his child for homework that's been labeled as a "Common Core Math" that required analyzing how a problem was solved, evaluating what they did right and wrong, and expressing your conclusions in a letter.
The Facebook post went viral, and many e-news outlets, detractors of Common Core, and conservatives such as Glenn Beck who object to any kind of presumed "big government" influence on society and culture have turned this social media posting that was probably meant to be shared only with friends of the father into a political campaign.
I had a similar situation a couple of weeks ago when my 10 year old daughter came home needing to complete a worksheet full of math problems that looked like this.
I Googled "Lattice Multiplication" and clicked through a few of the web pages to learn what exactly this mathematical method is and how it helps kids understand multiplication, including this link to Khan Academy that offered a video I could watch to learn and understand deeply. Once I felt I had the conceptual and procedural understanding I felt I needed to help my daughter complete her homework by explaining the procedures to apply to solve the problem. What started as a very frustrating experience full of tears and yelling ended as a very enjoyable experience for my daughter and a rewarding one for me as we worked together to complete the worksheet and learned something new.
Now I'm not saying what the Frustrated Facebook Father did was wrong by sharing his feelings and frustration (though his appearance on Glenn Beck may have been a little much - not to mention attention-seeking and self-serving). In fact, if you truly think about it, the father did address the prompt.
He told "Jack", the person to whom the letter should be address, what he did wrong, which was use the mathematical featured in the question. He argued that using the more traditional method for subtracting three digit numbers - using an mathematical algorithm rather than an mathematical model -would have been simpler. He responded to the prompt in the appropriate format - a letter - and his response not only demonstrated the highest level of thinking according to Bloom's Revised Taxonomy - create - but he also communicated the highest depth of knowledge according to Webb - extended thinking - by defending his argument with his own expert opinion and his personal experience of how such a problem would be handled in the workforce.
I will also agree is NOT an exemplar assignment. However, it's the kind of stuff offered by the textbook companies who profess their curriculum packages are "Common Core-aligned" even though there's no scientifically-based research to prove that claim yet because the practice tests from the PARCC and the Smarter Balanced Assessment Consortium are currently being taken in some K-12 schools across the nation. It's also the kind of stuff we teachers are given by the publishers and our schools in our desperate attempt to address the cognitive rigor of the Common Core State Standards, which I believe are challenging and engaging.
My problem is not with the standards or even the politics behind them but rather how poorly the roll out and implementation has been, the lack of support in regards to funding and high quality training provided to us educators to implement these new standards, and how the partisan politics and misconceptions about the involvement of "big government" in education has created confusion and caused misconceptions about the intent and quality of the standards.
|Mathematical Process Standards |
Teaching mathematics reasoning and proofing truly determines whether the student deeply understands mathematics by having the student explain their thought process behind how they solved the problem. Proofing challenges and engages them to determine the accuracy and validity of their answers. Traditional math instruction called this "checking your answers", which used to be a suggestion but has now become an essential part of the deeper teaching and learning experience.
Reasoning and proofing also requires challenging and engaging students to analyze and evaluate the mathematical thinking of others. This means students will delve deeper into the concepts by learning about different mathematical concepts, methods, models, practices, and procedures as well as analyze and evaluate how math problems are solved using these different processes.
Not only will teaching for conceptual knowledge and metacognition involve reasoning and proofing but also communication, which challenges and engages students to express their deeper knowledge, understanding, thinking, and awareness.
|Use place value understanding to round |
multi-digit numbers to any place
(By the way, this problem is an original problem I created based on a sample problem for the PARCC. I changed the stadiums from baseball to football, Googled the seating capacity of U.S. football stadiums, and found these three to be the closest in size. The name of the people in the problem were also changed to my brother and sister's names. We will address how to create original questions, problems, and tasks aligned to the CCSS in future blogs.)
Students still need to solve and work with algorithms, equations, numbers and formulas to demonstrate procedural knowledge. However, they should also be provided opportunities to demonstrate and communicate their deeper mathematical knowledge, understanding, thinking, and awareness by explaining their reasoning and defending their reasons through proofing. These questions should be abstract, complex,and even "messy", encouraging students to think deeply and communicate clearly using oral, written, creative, and technical expression.
Problem solving, which is another NCTM process standard students should be challenges and engaged to demonstrate and communicate, should be deepened by providing a combination of algorithmic and story problems student should and open-ended, text dependent questions that challenge and engage students to delve deeper into the reasoning and thinking behind their answers. However, these problems should establish connections between mathematics and the real world to show students how math is used to address, handle, settle, or solve real world issues, problems, and situations.
So what exactly did the father do wrong?
First, he made a presumption and generalization that this problem is an example of "Common Core Math", which truly is not a curriculum. The Common Core State Standards are performance objectives that define what students should be able to know, understand, and be able to do and how deeply they need to know, understand, think, and be aware of a concept, idea, subject, or topic in order to answer the question posed, address the problem presented, and accomplish the task provided by the teacher. It's not a concept or idea. It's performance objectives.
Second, he took the time to express his frustration by writing a sarcastic response to the prompt that was given to his child to do for homework, shared his feelings on Facebook, and then made the rounds in media to share his story and dismissing "Common Core Math", which, again, is a misnomer.
He could have used that time more wisely and even productively if he Googled "subtracting three digit numbers with a number line". He would have found an instructional video on LearnZillion.com that's less than that minutes long that explains how to subtract numbers on a number line. If he scrolled down a little further on Google, he would have found a link to a webpage on K-5 Math Teaching Resources that explains what is an empty number line and how this mathematical model for addition an subtraction was developed by researchers from the Netherlands, and even more examples of how to use this model to perform addition and subtraction.
This is what we parents are going to need to do in order to understand the mathematical concepts, methods, and models that our children are learning in the class and bringing home for homework. We're going to have to devote the time to familiarize ourselves with these methods, models, and strategies, which could be a struggle for us because we were taught math to simply listen, learn, and do. Our children's math has them research, investigate, examine, experiment, explore, argue, defend, justify, refute, and come up with their own original ideas and processes for using math and how it can be used to address, handle, settle, or sole real world issues, problems, and situations - which, incidentally, is what mathematicians do.
Now the father - and Glenn Beck, for that matter - can retort back that he should not have to engage in such in-depth investigation to learn how to help his child use some mathematical method, model, or practice to do their homework - and he's right! Those investigations should happen in the classroom with the teacher, who has the student either read and respond to the informational text, "What Is An Empty Number Line?" (or, since these are 2nd Graders, read it to them and ask questions for checks for understanding), and perhaps have students work on mastering such complex problems under the guidance of the teacher before sending them home to work independently.
Of course, the Frustrated Facebook Father is entitled to his opinion and can express it as freely as he wants. I even don't disagree with how he feels about the work that was sent home with his child. I can't criticize him for what he did because I felt the same way when my daughter brought him that Lattice Multiplication worksheet and thought, "This is just stupid! She knows her times tables and how to multiply numbers! Why can't she just do it the old-fashioned way to get the correct answer?!!"
Then I realized that's not what the assignment wanted her to do.
It challenged and engaged her to look for and make use of structure (CCSS.Math.Practice.MP7) by determining the pattern of the mathematical model and how it can be used to multiply two and three-digit numbers .
It challenged and engaged her to reason abstractly (CCSS.Math.Practice.MP2) by considering the units involved in the algorithm, attending to the meaning of the quantities using a lattice, and demonstrating flexibility by using this new method.
It challenged and engaged her make sense of problems and persevere in solving them (CCSS.Math.Practice.MP1). Yes, she was frustrated - and so did I - but we both took the time to learn what exactly Lattice Multiplication involved and how it could be used to multiply two and three-digit numbers. To verify whether our responses were correct, we used the traditional method by organizing the numbers in an algorithm and solving the equation "the old-fashioned way" her father (me) learned how to do multiplication.
That's what the Mathematics Common Core State Standards do. They challenge and engage students to demonstrate and communicate their deeper knowledge, understanding, thinking, and awareness in mathematics, which involves more than simply solving problems to find the correct answer.
Mathematical thinking demonstrating and communicating deeper knowledge, understanding, and awareness of representation by selecting, applying, and translating mathematical representations to solve problems, which is what my daughter's homework on Lattice Multiplication challenged and engaged her to do and the Frustrated Facebook Father's son's assignment did using the Empty Number Line.
Mathematical thinking involves making and investigating mathematical conjectures and developing and evaluating mathematical arguments - or reasoning - and proofs, which is what the Frustrated Facebook Father's son's assignment challenged and engaged him to do.
Mathematical thinking involves analyzing and evaluating the mathematical thinking of others, which is what the Frustrated Facebook Father's son's assignment challenged and engaged him to do by having him communicate his analysis and evaluation of what "Jack" did wrong and what he could have done differently in the form of a letter.
Mathematical thinking involves engaging in problem solving that challenges students to adapt a variety of appropriate strategies to solve problems, which is what both my daughter and the Frustrated Facebook Father's son was expected to do.
Unfortunately, what the assignments both my daughter and the Frustrated Facebook Father's son were given failed to do is make a connection for the Facebook Frustrated Father and me to understand the relevance of these methods and models and how they can be used in a context beyond mathematics.
Who's to fault for that? It's not the teacher. In fact, according to the interview on Glenn Beck, the teacher found the Frustrated Facebook Father's response on the assignment quite amusing and valid. She also shared her frustration with how she is required to provide problems such as the one she assigned to the Frustrated Facebook Father's son or the one assigned to my daughter. She's experiencing what millions of teachers around our nation are experiencing - grand expectations for teaching and learning with minimal support or training in how to teach concepts, ideas, subjects, or topics such as Lattice Mathematics or adding and subtracting using the Empty Number Line.
But that is another discussion for another day. Here's hoping The Blaze - who broke the story about the Frustrated Facebook Father's post - or Glenn Beck invites me to have that discussion with them.