Math Misconception: The Final Answer is The Only Part That Matters

For the longest time, I really thought that getting the right answer was the goal and really the only thing that mattered.  This misconception stuck with me all the way through elementary, middle and high school and was really not challenged until college.  Through my K-12 experience I showed my work because I could not do the math without it. However, there was no reward for it.  There was not partial credit, it was all about the correct answer.  As a struggling math student (to the point where I actually started looking into careers that did not involve math) this really frustrated me because I was often unable to find the correct answer.  (Especially in Geometry when writing proofs - ugh!)

It was not until I took College Algebra my freshman year of college that the work behind the problem became just as important, if not more so, than the correct answer.  It was also, at that point, that math "clicked".  The emphasis on showing my work helped me to the understand, make connections, and to truly appreciate the beauty of mathematics.  Moreover, it also cemented my career path - to become a high school math teacher - who knew!  Now as a teacher, here is what I explain, emphasize and illustrate to my students about why the final answer is not always the best answer to the problem.

1)  Showing your work helps you to understand the "why" the answer is correct.  My favorite question to ask my student whenever they give me an answer during a class discussion, during group work or even individual practice as I am walking around assisting them is "why".  Why did you choose that formula, why did you draw that diagram,or why did you did set-up that equation?  If my students can't answer why, I will help them out with leading questions or allow other students to do so (depending on the circumstance).  I tell them upfront on day one, if you can't tell me "why" then I am going to keep asking.  Sometimes it makes them laugh, but as I am consistent with making all students answer it, it also helps them to make it to reason two.

2)  Showing your work helps you to understand the connections between  different topics within a class and different branches of math .  We as teachers, parents and adults, all know that math is circular in its connections to each other.  When I teach parallel lines, for example, I review and apply the properties of angles that we study in our foundations chapter, setting up and solving algebraic equations that they learned in a previous class as well as the idea of determining whether or not their final answer in reasonable.  If a student is not showing their work, they are not able to see these connections in action.  They are not able to see how the math keeps "coming back".  As the year goes on, these connections get even deeper.  By the time I teacher my final unit - Three-dimensional figures - we are tying together area, linear equations, squares and square roots, triangles and at least four other topics!  So often, I hear students say - wait, didn't we do x or y before and can't we use w and z here?  The more we stress showing their work, being able to answer "why" and "where" the answer came from, the more we can move students to the most important reason I see for looking beyond just the final answer.  

3)  Showing your work helps you to retain the knowledge beyond just a quiz, test or other assessment.  As much as we don't like it, standardized testing is a reality of this world that we live in.  My students are gearing up for this in the new few weeks.  Having done multiple types of practice with them through the year, I know how much inter-connection of topics and how much prior knowledge they need to retain.  But it goes beyond that, beyond a single test (normally) during their junior year of high school.  It goes on to a better attitude about math and a better appreciation how much impact math has the world around us.  I seriously cannot count the number of times that I hear "I hate math" , "ugh, math was my worst subject" or "I never use math" when people find out that I am a math teacher.  For those that I am lucky enough to continue the conversation with, the reason often comes out that they never truly understood why they had to do it or how it all worked.   

While I definitely don't have all the answers, I do think that they more we can emphasize showing how you do the math, understanding why you need to do it and how the different ideas work together we are setting our students up for better performance on tests, in college, in their career paths and more.  Yes, the final answer matters - we need to balance our checkbooks, we need to buy the right amount of paint or carpet - but we also need to know how to consistently arrive it!  

An InLinkz Link-up

Avoiding the Pitfalls of Teaching # 1 - Talking To Your Students Instead of At Them

Although  as teachers we spend many years in college learning our subject area and in teacher education classes, nothing - not even student teaching - can fully prepare you for the day that you walk into a school and have your own classroom.  Suddenly that's it.  You are in front of a room of 25 - 35 students all of whom you need to educate, engage and manage.  Unfortunately, unless you have a fantastic support system, you are likely to commit some of the common pitfalls that can make your job that much harder.  Over the next few weeks I am going to address some of these pitfalls and how to avoid them!  Additionally, I have asked some of my friends at teacherspayteachers to contribute their knowledge as well!

Topic One:  How to Talk TO your students (i.e. facilitating a discussion) instead of AT them (i.e. lecturing)! 

When we start teaching we are nervous, especially for the first year or two until we get some experience under our belts.  We want everything to go well, we want our rooms to be well managed, we want to be that super fantastic, great, wonderful, perfect teacher whose classroom looks like one of those out of a 1950's sitcom where everything is neat, tidy, with the students all raising their hands. To achieve this lofty "perfect classroom" we walk in, put the hammer down and start talking at our students through the "stand and deliver" lecture method.  We operate under the assumption that if they are silent, they are listening, learning and perfectly "under control".  Yep, sure they are. (Just like when my kids are suddenly quiet at home it means that they are being perfect angels...)   In all reality, 40% of your students are not learning anything because they don't learn that way, 50% are afraid to ask a question (or robbed of the opportunity to do so because you never pause) and 80% are about to explode from the amount of words and energy that is building up.

   
No one can sit still, silently for 45 - 60 minutes.  I know that I sure can't.  I get antsy, I start to tune out the words around me, I doddle, I start making lists, I get BORED!  Do we honestly expect our pre-teens and teens to be any different?  At the age we are when we become teachers we've learned some of the social norms, we know how to stay in our seats and at least look tuned in.  Most of the students sitting in front of us don't.  They are in ever changing bodies in an ever changing world trying to balance school, families, sports/clubs, friends, hormones and a thousand other things.  When you are trying to compete with all of that you need to teach how they learn.

So how do you do that?  How do you teach the way they learn?  Simple - you talk with them, not at them.  You allow your students to feel that they have a stake in their learning process and the ebb and flow of the classroom.

• **  To combat their need to talk you can give them quick breaks where you say "solve this question with someone sitting next to you" or have a quick class discussion about how they see the topic (when applicable) used in the "real world".  

• **  To combat their need to move ask students to come to the board to solve a problem or have them do a quick "think, pair, share".  

• ** To keep them engage you can ask "what questions do you have" instead of "does anyone have any questions"?  Ask a student to explain what you just taught in their own words.  

In other words, you involve them in the lesson which gives them the opportunity to burn off some energy and raises engagement rates.  (I talk more about effective questioning here). You talk TO them and listen when they respond back.   If you ask them a question, listen, ask more questions to draw out more information, ask other students if they agree, disagree or have something to add.  You don't want to get them talking, engaging and interested to then just shut them out by not giving their responses any credence.

I spoke with a few other teacher-authors about how they made this shift:

Nikki from Teaching Autism found success when she formed relationships and explored other methods: "For me, it's very different, 99% of our children are non verbal, meaning it's very easy to talk at them rather than with them. It's great to offer choices so that they are independently making choices with you, using prompt cards (http://bit.ly/1M85eyw) to stir up a conversation - use iPad apps such as proloquo or symbol/communication books (http://bit.ly/1hhE93w) so that they are able to answer you, get them to point to an answer when you have offered them a choice. 

At first, it can be so hard, how do you make conversation with a non verbal child? It's something that doesn't come to you at first, so don't feel deflated when you feel like you're failing. It's something that definitely grows with you as your experience does. The more you learn about the children, the more you're able to talk with them, look for different signs of communication and promote communication that they look forward to. It's through this, that you start to build strong and professional relationships with your students. Once you have that strong relationship, then you're ready to tackle the teaching world!"

Coach Christopher from Courage To Core shared that the shift happens when you engage the students strengths: "For some years I worked at the amazing Archer School for Girls in Los Angeles, a city where social capital is the subtext for almost every text, where all Instagram’s are about to go viral, and face to face interactions can feel a little like being added to someone’s LinkedIn network. The Archer girls spoke in emoji’s or hashtags or something—codes conveyed in a blink of an eye, because they knew each other and were utterly in their domain at Archer. In class they lunged to answer questions, to hear their voices and those of their peers, and it was a shame not to leverage their social skills to educational ends. When I transitioned primarily to group work in math class, it was as if they had been holding their breaths for a month, waiting to engage full-throated. It was a viral sensation."

It's Kinder Time feels that the shift happens when you make a connection: "I believe that the many years I spent in the city recreation program for children helped me learn how to have conversations with kiddos rather then talk at them. Over time I learned better ways to make connections. Finding out your students interests really helps make connection with them and can open many doors with parents once they see how much you really care."



The pitfalls are there, just waiting, but hopefully together we can navigate them, avoid them or capitalize upon them to the benefit of you and your students.  How did you learn to talk TO your students instead of AT them?  Comment below, I'd love to hear it!

More Effective Classroom Discussion and Questioning - How I'm Working to Make it Possible! (part one)

As a teacher you can find yourself in some very uncomfortable positions when standing in front of a classroom full of students, especially pre-teens or teenagers.  While you get pretty good at deflecting the inappropriate questions and comments and redirecting students who are heading off on a tangent, if you are anything like me, you still struggle with how to get responses out of students.  One of the most uncomfortable moments come when you ask a question and you get cricket eggs.  Not even crickets, just simply their eggs waiting to hatch.  It's that Ben Stein moment in Ferris Beuller...anyone, anyone, Beuller, Beuller...?   You ask, no one answers (or the same students who answers every question shouts something out), you ask again, still no response.  Then, nine times out of ten, you end up giving hints or just straight up answering your own question. 

In an attempt to combat this, I have spent this year on a quest (in that "free" time that we all don't have) to find more effective ways to facilitate classroom discussion.  I started by examining my own question techniques during a lesson to see if changing up HOW I am asking a question has any impact.  For example, I changed from asking "Does anyone have any questions" which gives students an out from asking any questions, to "What questions do you have" which assumes that there are questions that need to be answered.  This can prod students into thinking about whether they really get it or if there might be something that they are unsure about.  I have seen a rise in the amount of questions asked just by this simple change.

Taking into account that fact that many pre-teens and teens really like to "save face" in-front of their peers and not admit to not knowing something I have also expanded on this.   I will often ask  "If your friend wasn't here today, what question would they have tomorrow" or "Tell me one thing that your friend might be confused about if they just walked in and looked at the board".  Both of these questions allow students to have make it seem like it is someone else question and not their own.  Additionally, both encourage students to think outside of the box and look at a lesson or topic in a new light.  By asking students to put themselves in the brain of another person, they start to examine connections to prior knowledge and make those connections to new knowledge. 

The other major change I have made during classroom lessons and discussions is to put forth a concentrated effort to get students talking to each other and working with each other.  Once a student has answered one of my questions I will often not tell them if they are right or wrong (mean of me I know!).  Instead I will chose one a few different paths for follow. 

1)  I will ask someone else (or multiple someones) if they are agree or disagree with what the student said which serves multiple purposes. First,  it makes students really pay attention to the lesson and discussion since I call on random people not just those with their hands up.  Secondly, it makes them listen to each other since they can't explain if they agree or disagree if they don't know what the first student said.  Lastly, it helps students to focus their own thinking and understanding to figure out what they believe to be correct or not.

2)  I will ask someone to rephrase or expand upon what the student just said.  Rephrasing helps students to align their own understanding and to express it more coherently.  When I ask students to expand on what was just said, it takes the discussion to the next level and takes a simple answer to a rich mathematical inquiry.

3)  I will ask the student to convince me that they are correct.  Odd I know, but it makes students take ownership of their answer instead of just saying something for the sake of answering.  Additionally, when I go this route, I often find students pulling up prior knowledge, using better vocabulary and often asking if they can come to the board to show me.  Other students will chime in to help with the support needed or to correct misconceptions.  They love to be right and being given the opportunity to prove it is always a winner!

I don't have all of the answers, but I am seeing positive results so far so I'll keep at it and hopefully continue to improve!  Next step, improve discussion during group work and activities! :)