Sunday, May 18, 2014

Don't Forget to Remember

Is there no longer a need to remember?

That seems to be the impression - or rather, misconception - based upon the educational research and literature that encourages and promotes teaching and learning that challenges and engages students to think analytically, creatively, critically, and strategically about what they are learning.  We educators are constantly being encouraged, prompted, and even pushed to have our students think about what they are learning.

However, remembering is thinking, too.

There's also a perception that students do not need to remember details, facts, and information.  They can just Google what they need to know or store all data and information on a computer from where they can access and retrieve the details and facts when needed instead of keeping it all in their head.

However, that's not remembering.  That's referencing or retrieving externally, which is correlated to remembering.  Remembering involves intrinsic referencing and retrieving data, facts, and information from the computer in our head -- our mind.  Search engines and computer hard drives allow us to store vast amounts of information and keep our mind open and free to think about the knowledge we have acquired.

Remembering is also not knowing.  When we remember, we are demonstrating and communicating our ability to recall, recognize, or retrieve from our memory data, facts, and information.  When we remember, we are also actively learning by defining, identifying, locating, naming, reproducing, and stating specific details, elements, procedures, terminology, and vocabulary.  Knowing addresses demonstrating factual, conceptual, procedural, or metacognitive knowledge of concepts or content.  In fact, in their revision of Bloom's Taxonomy, Anderson and Krathwohl (2001) separated knowing and knowledge into its own dimension within the Cognitive Process Domain.

The misconception about remembering is that we associate it with memorizing, which are two completely different concepts.   When we memorize, we are just committing  data, facts, and information to memory to use at a later time.   When we remember, we are drawing upon these bits of information from our memory to engage in deeper thinking that challenges and engages us to apply, analyze, evaluate, and create.

Memorizing is a process or way we remember, which is a very difficult yet simple process to commit facts and information to memory.  Interestingly, the way we remember things actually involve higher level thinking such as the following:

  • summarizing (understand)
  • making connections (apply)
  • associating (analyze)
  • correlating (evaluate)
  • designing or developing mental models (create)
  • drawing visual representations (create)
  • making acronyms, similes, and metaphors (create)
In fact, if you really think about it, when we're asked to remember something, are we not actually creating by picturing and visualizing models or coming up with original limericks, lines, saying or to make remembering easier?  How many of us know what the function of a conjunction is because of the song?  How many of us make acronyms to help us remember chronological order or processes?

Remembering also allows us to be self-reliant and responsible for what we know, understand, and can do without having to ask for help or seek out more information constantly.  There's nothing wrong about having a good memory or being able to remember things.

Remembering is the foundation as well as the launching pad of higher order thinking.  We can't demonstrate and communicate our ability to understand, apply, analyze, evaluate, or create without remembering the details, facts, and information that has been developed into background knowledge.  We can't think strategically about concepts, ideas, subjects, or topics without first demonstrating and communicating our ability to reproduce and apply learned facts and procedures.  We can't extend thinking across the curriculum and beyond the classroom without being able to draw from foundational knowledge.

So why is teaching and learning for remembering so controversial and contested?

Remembering is considered to be a lower-level thinking skill, and the term lower has a negative connotation.  Lower is not synonymous with easy.  In fact, remembering is much more difficult than the other categories within the cognitive domain of Bloom's Revised Taxonomy.  It takes a lot of effort to remember, which makes such thinking hard - or difficult.  However, the outcome or result of remembering is simple in that the response and result is very straightforward by being correct or incorrect.

When it comes to complexity, remembering is at the lowest level not because of the level of thinking but rather the depth expected to be demonstrated and communicated.  Remembering requires only recalling and recognizing facts and information.  Questions for remembering are closed-endedhaving one specific answer that is irrefutable and easy, quick, and simple to verify.  They are also convergent, requiring students to reproduce and apply facts and procedures to achieve or attain the one acceptable, correct response or result. 

However, when it comes to remembering, we must remember a few key facts.

Remembering is an essential skill.  There's no debate or doubt about it.  The ability to retain vast amounts of information is as impressive as being able to run a marathon or lift a heavy weight.  It's an admirable feat.  Remembering is also still highly respected and valued in society.  Being able to remember is a positive quality and trait.  People feel honored, respected, and cared for when a person can remember their name, stories they shared, or key events.

However, being able to use what you can remember is just as positive, respectful, and valued.

So what exactly all the fuss about teaching and learning for remembering?

It's not about remembering.  The fuss is actually about teaching and learning for memorizing.  We don't want our students to remember facts, information, and procedures in isolation.  We want them to remember these elements so they can develop background knowledge from which they can draw and establish connections within and beyond the classroom and curriculum.

Higher order thinking and depth of knowledge are the ideal.  It's what we want students to do with what they have learned and how deeply we want teaching and learning to go.  However, we still need to remember what is the information; remember who provided the information; remember where it is stored; remember why the information is needed; and remember how it relates to the question we are answering, the problem we are addressing, or task we are accomplishing.

When it comes to remembering, just think about this classic Eric Clapton song.

- E.M.F.

Saturday, May 3, 2014

Doing Math vs.Thinking Mathematically: What's the Difference?

There's a lot of talk about the dilemma and discord surrounding "Common Core Math", which actually is a misnomer (see my blog "This Is Not the Frustrated Facebook Father's Math - Or Yours Either").  Primarily, the discussions have been pedagogical and political. However, now even performers such as Louis C.K. are engaging in the discussion (Actually, it's surprising more celebrities or comedians are not espousing their opinions about the Common Core.)

We've been warned that the college and career standards our states have adopted and the assessments that will test our students on these standards will expected them do math differently than what they and even we educators and our students parents have experienced.  In fact, this comedic monologue and tune from Tom Lehrer actually sums up how the term "new math" and how it has been presented not only to us educators and our students but also the general public.

However, is it that are students are expected to do math differently or rather think differently about mathematics?

We've also been told this is a "new math", which brings some baggage since the term has been used so loosely and incorrectly over the years every time there's a new educational initiative or reform.

However, is there new math to learn to do or are there new expectations for demonstrating and communicating mathematical thinking?

What's the difference?

Doing math is an operation. It's about arithmetic and applying mathematical procedures such as addition, subtraction, multiplication, division, estimation, and measurement to solve an algorithmic or story problem correctly and successfully.  It's all about the reproducing and applying facts and procedures to achieve or attain that correct answer because, in the end, that's all that mattered - get the correct answer!

What Is Mathematics?
Thinking mathematically is an art - specifically, as Lockhart (2002) states, "the art of explanation.  It's about actively developing deeper knowledge, understanding, and awareness of mathematical concepts, practices, and processes - more specifically, analyzing how, evaluating why, and creating new ways of thinking about and using mathematics.  It focuses on deeper understanding of procedural knowledge, deeper thinking about conceptual knowledge, and deeper awareness of how mathematics can address, handle, settle, or solve real world issues, problems, and situations.

Our primary concern should be that we do not confuse doing math with thinking mathematically.  Doing math is about using math to answer algorithmic questions and solve story or word problems.  Thinking mathematically is about how analyzing how and evaluating why mathematical concepts, practices, and processes are used to address math problems and creating new ideas, procedures, and ways of thinking about math.

What Is Mathematical Thinking?
Mathematical thinking correlates to the process standards identified by the National Council of Teachers of Mathematics (2000).  Students should be challenged and engaged to apply mathematical concepts, practices, and processes to solve problems in mathematics and in other contexts.  They should demonstrate their ability to think mathematically by developing and evaluating mathematical arguments and proof.  They should communicate their depth of knowledge of mathematics using oral, written, creative, and technical expression to explain how they achieved and attained their responses and results.  They should realize and establish connections between mathematics and the real world.  They should represent mathematical phenomena with an algorithm or formula that explains how and why and determines relationships.

Mathematical Thinking in Problem-Based Learning
The core of mathematical thinking is problem solving.  However, the problems presented to students should represent actual real world issues, problems, and situations that would require students to engage in doing math and thinking mathematically rather than a story problem that has students do math in a real world context.  With the adjacent story problem, the problem is not the difference in how many bouquets are sold but rather what is the least amount of bouquets they must sell the subsequent month in order achieve their goal or quota for the quarter.  With this problem, not only are students learning how to subtract but also how subtraction is used in business (financial literacy).  They are also learning academic vocabulary (quota), subject-specific terminology (2nd quarter, which is how time is measured in business). 

Mathematical Reasoning and Proofing
Students should also be challenged and engaged in mathematical reasoning that has them achieve responses and prove the results they have attained.  Take a look at this routine algorithmic problem involving dividing fractions. Students are expected to do the math by reproducing and applying the facts and procedures in order to find the single correct answer - in this case, 1/2.  Students would be challenged and engaged to think mathematically if they were expected to evaluate why 1/2 is the answer and analyze how the result is attained.

Communicating Mathematical Thinking
Mathematical thinking also challenges and engages students to communicate their depth of knowledge as well as demonstrate higher levels of thinking.  For example, instead of providing students with a list of algorithmic problems to solve, provide the students with an open-ended, text dependent question that challenges and engages them to defend and justify their mathematical thinking.  Not only will students learn that math is not just about finding the answer or solution but also being able to explain how responses and results are attained.

Cognitive Rigor Questions for Mathematical Thinking
Communication has become as essential in mathematics as counting, calculation, and computation.  We educators can challenge and engage students to demonstrate and communicate mathematical thinking not only by expecting them to solve mathematical problems and explain their processes by providing them open-ended, text dependent questions in which the solution is provided and they have to analyze how the answer was attained and evaluate why the outcome is correct or incorrect.

Communicating Mathematical Thinking
To encourage communication in mathematics, provide students with a format that allows them to engage in technical writing in which they first identify what the problem is and the mathematical concept, practice, or process that is being addressed in the first paragraph, list the procedures or steps they took to solve the problem, then defend and justify their responses and results using the facts and procedures they reproduced and applied.

Communicating and Connecting Mathematical Thinking
The new assessments students will be taking during the 2014-2015 academic year will expect them not only to solve mathematical equations and problems but also use verbal language to explain their reasoning and proofs.  On the PARCC and Smarter Balanced assessments, some of the questions will look like this one to the right, challenging and engaging students to express how and why a response is correct using oral, written, creative, or technical expression rather than reproduce and apply facts and procedures to attain the result.  

Connecting Mathematical Thinking to Real World Situations
The mathematical problems presented should also show students the connection between academic mathematical concepts and the real world.  Connecting mathematical thinking fosters and supports students' ability to engage in data analysis and interpretation.  It also allows students to use other forms of academic thinking and process such as the scientific method or historical analysis and research to interpret statistical information and draw their own conclusions.

Representing Mathematical Thinking
Students should also learn how to create and use representations to organize, record, and communicate mathematical ideas as well as use representations to model and interpret physical, social, and mathematical phenomena.  In the accompanying example, the algorithmic formula for speed represents how long it would take me to drive from my house in Phoenix, Arizona, to Disneyland in Anaheim, California, (357 mi / x hrs.) if I drove 65 miles per hour (65 mi / 1 hr).  The problem also allows the student delve deeper into the problem by changing the variable measures (e.g. the average speed from 65 mph to a faster or slower speed).

If there is a "new math" our students are learning, it's that thinking mathematically through problem solving, defending and justifying response and results through reasoning and proofing; analyzing how and evaluating why those responses and results were achieved or attained using oral, written creative, and technical communication; establishing connections between mathematical concepts and ideas as well as between math and the real world, and using concrete representations to model and interpret not only mathematical but also physical and social phenomena are just as essential as doing the math to find those responses or results.

So how can we get our students to think mathematically as well as do the math?  Use these problems as a guide to create mathematical experiences that encourage students to think critically, creatively, and deeply.  Connect each mathematical concept or idea to a real world issue, problem, or situation and have them examine and explore how math can be used to explain what happens in life.  Show, don't tell, how math is an essential concept they will use in some aspect of their personal and professional lives.  Make math real, and encourage them to think about as well as do the math.

The true challenge may not to be the children but rather us adults who were most likely not expected to think as deeply about math.  We were just expected to do the math just as we were taught to get the correct answer.  However, math is no longer about correct and incorrect answers.  It's also about how responses and results can be defended, explained, and justified using mathematical thinking.

How can we have the adults develop deeper knowledge, understanding, and awareness of the new expectations for math?  At your next staff development or parent meeting, present this quote and the corresponding questions to the adults in your audience.  Not only will they hopefully develop depth of knowledge about the new expectations for mathematics but also the importance and value of going beyond calculating and computing to communicating, connecting, and considering mathematics.

What Is Mathematics?

-- E.M.F.