The Importance Of Why.

 

As a teacher, I am always working on finding better ways of doing things and finding out "Why" I'm having students do certain things. So why do we look at the "Why" in math. 

Why is it important to discuss the why in math class as we teach algorithms, rules, and everything else.  Most students can tell you that to divide two fractions, you have to flip one and turn it into a multiplication but they can't tell you why?  

Years ago, I watched an Algebra teacher show students they needed to divide by a fraction and then multiply both sides by the reciprocal to make the denominator equal to one.  She used the proper mathematical language rather than flip and multiply.

There are at least three reasons for explaining the why to students and the best way to do it.  The first reason is that it helps deepen student understanding.  The why helps students better understand the concept. As long as students just learn the algorithms, or process, they don't have the opportunity to understand the concept.  One example is trying to explain why a negative times a negative equals a positive.  Personally, that is something I could never picture on number lines until I finally saw a wonderful explanation I can now use with my students.

Think of the negative times a negative in this context.  You pay rent or a mortgage of $1200 per month, every month.  So each month $1200 is taken out of your account thus it is -1200.  Over a year you've paid out $14,400or -$14,400.  The next year, your boss offers to pay 12 months of your rent instead so every month  is considered a negative because you aren't paying or -12 months to you and the boss is paying out -$1200 per month or -12 times -1200 or you now have $14,400 more in your budget this year.  That is the first explanation I've seen that I can use with the kids.  The why can be done with an example rather than a mathematical proof.  

Second, knowing the why can help students retain what they are learning.  When teachers rely on teaching algorithms only, students are more likely to forget the process or misremember the steps because they don't know why they are doing things. The why helps students remember.  Third, knowing why, helps increase student confidence in their ability to do math.  Many students recognize that knowing why helps connect all the dots that the algorithm doesn't. Furthermore, knowing the why explains why students do certain things in the process.

Often times, when we try to explain the why, we resort to the long mathematical explanations.  When we do that, we loose student attention.  It is better to allow students to try problems before providing the explanation of why it works.  Research indicates that even the best students can feel overwhelmed when they've gotten too much information before they've had enough time to practice a specific topic.  In other words, don't try to give them too much information, too soon, so delay until after students have had a chance to practice or they ask for the why.

So the next time, give students a chance to practice something before you provide either a visual representation or something short and precise such as the example I gave on multiplying a negative times a negative.  Let me know what you think.  Another day, I plan to discuss teaching students shortcuts and whether it is good or bad.  Let me know what you think, I'd love to hear.

Warm-up

If the 57 links when connected make a chain that is 90 feet long.  How long is each link?

Warm-up

If a ship has an anchor attached to a chain made up of 57 links, each weighing 350 pounds, how many pounds does the whole chain weigh?

Quick Math Ideas

Yesterday, I ran a review for the upcoming test in my Algebra I class.  I had 5 minutes left, so I divided students up into groups of two, scribbled equations on pieces of paper, crumbled them up and threw them at the students. Their job was to work the problems on the paper until they got an answer.  When they brought the paper to me, I collected it and passed it out to a different student to check.  All the students were involved and active.

I vaguely remember reading about this somewhere, sometime in the past and it worked.  The students had a lot of fun.  Another way I could have done this was to have each student find an equation from the book and write it on a sheet of paper.  Then they could make a paper airplane from that paper and once everyone was done, they could fly the airplane to another person.  When the person got the airplane, they could open it up, solve the problem, write a new problem on it, fold it back into an airplane and send it off to someone else.  This could go on for several times so each student works a problem and adds to it.

I love keeping a set of index cards in my drawer for a couple of quick games.  For the first one, I've written a problem on the front with an answer on the back.  The answer does not go with the problem and I have just enough pairs for the students.  The idea is that each student is given one index card and they have to solve the problem on the front.  Once they have the answer, they have to find the person who has the answer.  I usually set it up so the problems and answers are in pairs.  For instance, on one I might have 2x + 1 = 7 one the front of one card with 5 on the back.  On another card, the front has the equation 3x + 1 = 16 with 3 on the. back and they would pair up.

Another quick game is a math version of Tic-Tac-Toe.  It doesn't take much to have a few grids set up and ready to go.  The idea is that two students are given a Tic-Tac-Toe grid filled with problems.  One student is designated as the starter and they choose a square to start such as the upper left corner.  This student solves the equation and checks their answer to prove it is correct.  Once they've proven it correct, they mark off the square with a circle or x.  The second student takes a turn, choosing a square and solving the problem on the square.  If they prove the answer is correct, they put down their mark and the students continue until they win or it's a draw.

There is also the QR code game which has a bunch of cards.  Each card has an equation with two QR codes on it.  One of the codes provides the answer and the other tells the students how many points they get if they got the correct answer.  So students work in small groups and one student selects a card.  They work the problem and once they have an answer, they check the answer.  If their answer is correct, they check the points and add it to their total but if they are wrong, they have to put the card back in the stack and a different student selects a card and works the problem.  They continue until they have finished all the cards or the teacher calls time.  The winner is the one with the most points.

The last one provides a physical result that students can see.  Create sheets of strips of paper with equations and cut them up.  The first strip has a problem on it and the student works it to find the answer.  They look for the answer on another strip, place it under the first strip and work the problem on the strip to find the answer. They again look for the matching answer, place it under the previous strip, complete the problem, and look for the answer on a different strip.  The student continues working through all the strips until they reach the strip that says finish.  When they have all the strips in order, they have the teacher check the work and if it ok'd as fully correct, they take the strips and make paper chains out of them.  The chains can be put together with other student's chains, until it reaches around the room.

It is always nice to have a few quick games available to take care of those odd bits of time that pop up or you need something to get students engaged or liven class up.  Let me know what you think, I'd love to hear.  Have a great day.

Help Build Mathematical Identify

The last webinar I took mentioned having students write a mathography to help the teacher see how they see themselves.  It gives us a bit of insight into their perception of their mathematical identity. 

Mathematical identity is defined as how a student sees themselves in regard to being able to do math.  Many will tell you  they are not "good at math" or "they can't do math" because too many believe that you have the ability or you don't. 

Instead, we need to help them see they can do math by helping them develop a mathematical identity that acknowledges they can do math.  There are many factors which contribute to students getting the idea they are not good at math.  They might have heard their parents saying they couldn't do it, or they struggled with certain new topics and concluded they couldn't do it so gave up.

It has been found that if a student has a good mathematical identity, they are more likely to succeed in their math classes.  Unfortunately, most students see mathematicians as older male geeks who wear glasses and are good at math.  They don't see themselves as mathematicians and that is an attitude that needs to be changed.  

Mathematical identity is connected to mindset.  If they have see themselves as good in mathematics, they often have a good mindset, one that is considered a growth mindset where they know they will get it whereas those who believe they are not good at math tend to have a fixed mindset because they don't think they'll ever be good at it.  Part of the process involves helping students adopt a growth mindset to help them develop their mathematical identity.

There are things that teachers can do to help students see themselves as mathematically inclined.  There are four things a teacher can do to help students.  First teachers should invest in maintaining a solid classroom community through out the year by having students work together doing worthy tasks and get all students to work equally together rather than letting one or two dominate.  Second, students should keep a math journal so they can write about warmups, classwork, open-ended reflections, and homework.

Teachers should facilitate classroom activities and act as co-learners with each other.  Finally, it is important to set high expectations for all students so they can learn math and help other learn it.  Students who develop a mathematical identity have teachers who see them as people, who take time to see how their lives effect them, and believe every student can do math.

It also means one has to redefine mathematical success because students need to believe they are capable problem solvers and thinkers and are able to contribute positively to the class.  This means going beyond thinking that success is only mastering algorithms or being able to calculate the right answers quickly.  It means recognizing when students do something right so they feel successful. It might be telling a student they did a good job by showing the idea in a new way, asking a good question, were able to restate a student's words after listening carefully, or talking about a new concept before they write it down, or they broke down a complex task into smaller steps, or they connected several ideas.  

It encourages low performing students when the teacher recognizes their contribution and it helps build their mathematical identity. We need to recognize that not all students learn math in exactly the same way and acknowledge not all of them use the same method.  In addition, we need to celebrate their mistakes because mistakes help us grow which helps a person grow as a mathematical person.

If we can help our students see themselves as "math" people who can do it with a good mind growths, we have done well. Let me know what you think, I'd love to hear.  Have a great day.

 

Google Meet and Jamboard Updates.

I just heard the other day that Google meet has released it's new version including breakout rooms but it is not available to everyone quite yet.  Google did announce the release but it will be for people who use Enterprise for Education and those who use G Suite will see the feature offered later in the year.  The breakout room feature is good news but I'd like to know why used of G Suite have to wait so long.

According to what I've read, Google Meet will allow hosts to divide attendees up into 100 smaller rooms for collaboration.  When the smaller groups are done, they can easily return to the mail room. Google is also giving moderators the ability to visit breakout rooms to check on things which is great and allows teachers to monitor the work in each room.  Moderators will also have the ability to place people in groups manually or randomly, based on need.  Although anyone with a google account can attend the meet, only the meet organizer can establish breakout rooms.  I haven't heard when the breakout room will be released for the general G-suite users but it is on the horizon.

Now for Google's Jamboard.  If you haven't seen it yet, Jamboard is a google product that allows real time collaboration of participates attending a meeting or working in the classroom.  With the distance classes in use or the required distancing in class, it allows more collaboration and interaction.  Jamboard is accessible by Chrome via an app, via an app for the iPads, or via the web based version directly at Jamboard.google.com.  

One can set up Jamboard to use in Google meet by having your admin person set things up so you can pair the two up and then use it during the class meeting.  There are several videos out there to help you get this set up.  In addition, Jamboard can be used with Google classroom since they both access material in google drive which means teachers can assign a jam to students in classroom as needed.  Furthermore, if you chrome, you can arrange for Equatio to be used with Jam by getting the extension.  In addition, Jam works with Screencastify which is another Chrome extension. 

So how do you effectively use Jam in your math classroom?  What can you do with it so it's not just the same old same old? On way which can be done either in class or by distance is to assign students to work in pairs to solve certain problems.  They could be asked to explain how to solve the problem by recording themselves and offering the product to others to see how to do the problem.

I saw Holly Clark of the Infused classroom books posted some suggestions on using Jamboard in different ways on Wakelet. There are a few for math to use on specific topics.  There were suggestions from early elementary to high school which could be used easily.  I especially liked the middle school ones.  Since I ended up on Wakelet and checked various collections on Jamboard.  I found some really nice ones with some great information.

Joanna Garza has a collection of instructional items including ones on explaining how to play live partner games on Jamboard, or running a live collaborative whiteboard session with audio using google classroom, meet, and jam board.  The one that caught my attention was the one on turning worksheets into interactive digital pages.  It provides a nice start.

This link takes you to the page filled with collections of ideas for Jamboard on wavelet. It provides a great list of collections to help you become the master of Jamboard.  Let me know what you think, I'd love to hear.  I can tell you , I'll be exploring these in more details to make the class more interactive.  Have a great day.

Warm-up

If it takes 3.5 pounds of grapes to make one quart of jelly, how many pounds of grapes are needed to make 2 gallons of jelly?