Thursday, June 22, 2017

Multiplication Charts in the Middle School Classroom

     Every year when I put up my posters, I put a multiplication chart near the front of my room.  Until a few years ago, it kind of bugged me.  You know how it goes...kids coming in to middle school should know their multiplication facts, why should I need this, etc....   But then I finally realized I needed to start looking at the good old multiplication chart not for what it might have been in elementary school (although, yes, some kids still use it like that), but for what I could use it to show in middle school.  Because now what I see when I look at that multiplication chart in the front of my room is patterns, patterns, patterns!  That's what math is all about.  Here are some of my favorite ways to use a multipication chart in my middle school classroom.

1.  Equivalent fractions--The multiplication table is filled with row upon row of equivalent fractions.

Here you can see a simple multiplication table that I created on Google Sheets.  The thing I love about my Google Sheets multiplication table is that I can customize it in whatever way is useful.  So if I want to talk about equivalent fractions for 3/8, I can highlight those rows.  But then I can easily change to something else.    What a great way for my students that may struggle with equivalent fractions to have a quick reference to find them, but also a visual way to see that the reason by 12/32 is equivalent to 3/8 is because both 3 and 8 were multiplied by 4!  These realizations that may at times seem to be no big deal for teachers can absolutely blow the minds of our students.

2.  Equivalent ratios/ratio table--Just as we can use the multiplication table for equivalent fractions, it can also be used for scaling ratios up and down to find equivalent ratios.  So now our students are trying to answer some proportional reasoning question:  "It takes James 4 minutes to solve 7 problems.  How many problems can James solve in 12 minutes?"  Again, this idea of using the multiplication chart flexibly, even attaching a label or meaning to some quantity, can be a stretch for kids at first.
This would also be a great way to teach students to think about if the answer to a proportion question like this is even reasonable.  For example, what if the question had been, "It takes James 4 minutes to solve 7 problems.  How many problem can James solve in 11 minutes?"  Now the answer is not actually on the multiplication table...but a sense of what is reasonable sure is.  If 8 minutes is 14 problems and 12 minutes if 21 problems, then the answer must be between 14 and 21 (but closer to 21!).  That sort of amazing proportional reasoning can be supported by a great visual tool like the multiplication chart.

3.  Proportional relationships versus non-proportional (but linear) relationships--Let's say you were focusing on the proportional relationship of 6x = y.  What about showing the multiplication chart as a place to see this?
The multiplication chart can show the table of values for a proportional relationship by simply looking at the column with the correct constant of proportionality.   This gives another way to think of a proportional relationship....it is a relationship that if you had an infinitely large multiplication table, it would have a row on there.  You could also build the connection between 6x = y and 6x + 1 = y by having students add one to all of the values in the 6 column.

       Ok, and honestly, some students will use the multiplication chart because they don't know their multiplication facts.  As much as I wish this weren't true, it just it.  So rather than fighting against it, I've decided to help all of my students see that the multiplication table can be a great tool to help us learn about a lot of middle school concepts far beyond simply multiplying.  After writing all of this, I think this year, we just may create a digital multiplication table in the first few weeks of school to establish right away what a great tool it can be.

How do you think multiplication tables can be helpful in middle school?  What other "elementary" tools do you rely on to make your classroom a better place?

Wednesday, June 14, 2017

Beginning of Class Routine Revamp

        At NCTM, I got several ideas that I wanted to incorporate into my beginning of class routine, and I've been finding others as well.  Here is my beginning of class routine for next year.



Wonder Monday:  This idea is the culmination of a lot of reading and listening that I have been doing.  Jo Boaler's Mathematical Mindset, as well as her growth mindset course have really opened up my eyes to the need for math to be an open and creative field.  I've also been reading "Becoming the Math Teacher You Wish You'd Had", which talks about the importance of a "notice" and "wonder"...what do kids notice about a problem?  What do they wonder about?
           So this is my thought for how to get kids thinking creatively, as well as how math is woven in to so much that we do.  My plan here is to find a crazy or interesting picture each week, and just letting the kids start to wonder about it.  I think it will get their creative juices flowing, and hopefully start to see math as an open subject, with a place for interesting questions.  I think this will be a fun way to start each week!


Two Way Tuesday:  This one came directly from a wonderful session I went to at NCTM.  The idea of the two-way puzzle is that you add going horizontally and vertically.  I think the puzzle aspect of this will keep kids engaged, and I can see it being useful for all kinds of review content....fractions, decimals, whole number, integers, and combining like terms are the first few that come to mind.

In this example, the missing box in the top row would be 22, since -8 + 22 = 14.  The bottom left square would be -5, since -8 + 3 would be -5.  From there, you can fill in the rest of the squares.
What's the Question Wednesday:  I got this idea from another blog I was reading.  Basically, you give the answer, and the kids brainstorm what the question might have been.   Again, I think this could encourage creativity and help kids see that there are all kinds of ways to get to any given answer.


Number Talk Thursday:  This is something else that I've been reading about, and something that I heard about at NCTM.  The idea is basically that you give kids a problem to solve mentally, and then you let kids share their strategies for how they solved the problem.  I tried this out a couple of times toward the end of last year, and I was amazed at what a great use of class time it was.  The kids were highly engaged, and had tons of great strategies.  It also allowed for great discussion as we compared strategies.


Quick Draw Friday:  This is also something that I got at NCTM.  The idea behind it is that you give kids a short look at a geometric drawing, and they try to reproduce it.  Then you give them one more look, and a chance to revise.  Then let kids share their vision for how they saw the picture, and how they re-drew it.  I think this one can really lead to some great vocabulary, and my artistic kids will love it!  The idea comes from this e-book.


       So these are the ideas that I plan to use next year. If you would like a copy of the Google Slides shown above for this beginning of class routine, click here.

One other idea that I would also love to incorporate (but ran out of days!) would be to have a day each week dedicated to looking at a graph and focusing on what story it tells.  I think this is really important as we live in a world surrounded by data, with graphs everywhere trying to convince us of one point or another.  I may try to work this in somehow to my routine, but I can't decide what to give up!  Why is there always more to do than there is time?????

What routines do you use at the beginning of class that you love?


Sunday, June 11, 2017

A Good Math Class Discussion: Part 2

    In my last post, I talked about my presentation norms that I use in my class.  Today, I'm going to address another important part of a class discussion:  listening.  For most kids, listening is a passive activity.  It is our job to teach them to be active listeners.  These are the strategies I use to teach my students to be active listeners in class.

1.  Listen carefully.  The first one is pretty obvious and speaks for itself.  If you're not paying attention, it's hard to hear what someone else has to say!

2.  Write down questions, comments or notes.  I think we all fall into the trap of thinking that we will remember what we want to say, what question we wanted to ask, etc.. when it is our turn to contribute.  The reality is that if we jot down notes to ourselves, we are far more likely to remember things.  Making sure that students always start with a piece of paper in front of them, even if it's just a scrap of paper or a post-it, is very important in making sure that students are active listeners.


3.  Be ready to summarize what the speaker said...    This requires a focused kind of listening.  This requires that students be more ACTIVE in their listening. As students try to do this, I think it also requires them to really think about whether or not they understand the explanation that is being give.  This leads to the second half of this expectation.

4. ......or ask the speaker a question.  It was really important to me that my classroom listening norms leave room for students to NOT understand.  I always want to send the message that it is OK to struggle and not understand, as long as you're still trying and working.  At the same time, I want students to know that not understanding doesn't mean that you don't participate.  This expectations gives students a way to stay active and involved even when they don't understand.
 
5.  Think about how your strategy compares.   I want a classroom that is open to many strategies.  By comparing strategies, students can see more clearly how strategies compare.  The more students get used to comparing strategies, the more likely they can start to pick the best strategy for the given problem.

A Good Math Class Discussion: Part 1

      Good discussion is so important in class, and it supports the standards for mathematical practice.  Yet, we all know that good discussions don't just happen by accident.  Over the years, I have learned that I need to spend time teaching my class how to have a good discussion so they can really get the most out of it.  In this post, I'm going to focus on the presentation norms that I use in my classroom.

1.  Speak loudly enough for everyone to hear.   This one is pretty obvious, and yet we all have students that seem to speak at a whisper.

2.  Speak at a reasonable pace.  Again, seems obvious, but I know that students really seem to struggle with this for a variety of reasons.  For one thing, when kids get excited, they often rush when they are talking!  Unfortunately, that can really get in the way of other people getting understanding what you're so excited to share with them.

math-practice-smp6-critique-reasoning

3.  Pause after each step and make eye contact.  This one goes hand in hand with speaking at a reasonable pace.  I can't tell you how many times I have had students completely lose everyone in the room (even me!) trying to explain their method.  I find that there are two common reasons why kids get lost during another student presentation.  One reason is that presenters give all of their steps at one time, and this puts everyone's brain on overload if they're still trying to process the second step, and the presenter is talking about the fifth step!  The other common reason that happens is that a student doesn't understand something early on, so they either stop understanding or stop listening.
         For these reasons, I teach kids that they need to pause after each step and make eye contact.  This way, the listeners have a chance to process what you're saying as you pause.  Hopefully, when you make eye contact it will be obvious if the people that you're talking to are lost!
       I also find that it is very important to tell my class that this helps everyone....including me.  I like having my students see that I also have to ask people to slow down, repeat a step, or answer a question to clarify their method.  I think it is so important to normalize the process of understanding, and that needing someone else to slow down does not make you "dumb".

4.  Ask for questions from the class.   This one closely follows the last one.  If you are pausing after each step, it is a natural time to let people ask questions.  Hopefully when you continue, there is a better chance for your audience to understand what you're saying now.  Also, if you have more chances for questions, there is a better chance more people will understand by the end.

5.  Show visuals.  This can help for different kinds of learners.  It is also helpful to have it as a reference throughout the presentation.

At the beginning of the year, we spend time talking about and practicing these norms.  In my next post, I'll look at the other side of the discussion:  listening norms.



Tuesday, April 25, 2017

My NCTM Experience Part 3: Number Talks

  When I saw sessions on number talks in the program,  I knew that I wanted to go to one of them.  We are planning a statewide book study for that will launch at the KATM conference next year, and the 4-10 book topic is on number talks.  I've looked over a copy of the book that we're planning to use, so I know the basic idea of a number talk, but really wanted more information about putting it into action.

       The idea of a number talk is fairly simple: you give students a problem, and give them time to work the problem mentally....no pencil, no paper, no calculator.  Then have a discussion about different ways that students solved the problem.


        I was eager to try this idea in my classroom, but somewhat reluctant to give up the time (isn't it always about time!).  After attending a session on number talks in middle school, I was convinced that I wanted to make this part of my classroom.  It seemed like a fairly easy idea to implement and one that could really be the center of lots of good discussion.

       The session that I went to for math talks was a good introduction.  We watched some video clips of the instructor doing number talks in a classroom and analyzed them.  One of the most helpful things that we did was practice recording the thinking of our partners.  Some of the ideas were easy to record, but others were a bit challenging.  It was definitely helpful to spend some time thinking ahead about some of the best ways to record strategies to help students understand abstract representations.

         So this week, I actually tried out a number talk for my warm up the last two days, and it was awesome!  I will definitely be incorporating number talks into my warm ups a couple of days a week from now on.  The conversations we had around different ideas was phenomenal.   My first piece of excitement came from the wide variety of hands that I had in the air of students eager to share their strategies....and some of them were kids that definitely do NOT make a habit of raising their hand.   I have one kid that has been completely disengaged since spring break....like this kid's goal for state assessment was "To try and stay awake".....and he has had his hand in the air the last two days, sharing his ideas.  Is that not amazing???!!  :)

       The other thing that was so exciting was the huge variety of strategies.  The first problem I picked was 18 x 5, which I think was a suggestion I got from the session.  It was a great problem and it led to lots of different strategies.  Our discussion has included some of the following strategies:

  • 10*5 + 8*5 = 50  + 40 = 90
  • 20*5 - 2*5 = 100 - 10 = 90
  • (2*9)(5) = (2)(9*5) = 2(45) = 90
  • (9*2)(5) = (9)(2*5) = 9(10) = 90
  • 18 + 18 + 18 + 18 + 18 = 90
  • 18 + 18 = 36, 36 + 36 = 72, 72 + 18 = 90
  • counting up by multiples of 5
  • counting up by multiples of 5, starting at 60 since they knew that 5 x 12 -= 60
I was very pleased with this many strategies coming to the surface on our very first attempt!  And this one number talk brough up important ideas and vocabulary such as distributive property, associative property and commutative property.  

        So on day 2, I chose the problem 15 x 8.  I intentionally chose a problem that had an even number and a multiple of 5, hoping to encourage rearrangement of factors  to get to a multiple of 10.  Again, I had tons of hands in the air, and a wide variety of strategies.  As with the first problem, I had a variety of strategies used.  The most common ones were probably these:
  • 10*8 + 5*8 = 80 + 40 = 120
  • 15 * 2 = 30, 30 x 2 = 60, 60 x 2 = 120
  • 15 + 15 = 30, and there are four groups of 2 15s, so you would have 30 x 4 = 120
My favorite one, however, was the very last one of the day.  It came from a student that had already shared one strategy, and as he looked as the wall, he said, "Or you could use a clock.  The 15 is like 15 minutes, and there is 4 of those in an hour.  So it would take 2 hours to have 8 sets of 15 minutes, and I know that 2 hours is 120 minutes."  I mean seriously.....could I have asked for anything more!  What awesome, creative reasoning!

          So, after 2 short days, I am quickly a believer in number talks in the middle school classroom.  I can definitely see a ton of advantages to making these a part of my classroom from day 1 next year.



Monday, April 24, 2017

My NCTM Experience Part 2: Jo Boaler

       The very first day of the NCTM conference, I was so excited about all of the sessions that I forgot to leave myself time to eat lunch.  That's not even true--I knew that I hadn't left a lunch break but I just couldn't help myself!  Jo Boaler was speaking at 12:30 and I was NOT going to miss that.  It as well worth it (and I did manage to find time to eat a quick sandwich after Jo spoke).

     Earlier this year, I read Mathematical Mindsets and it was a truly amazing read.  Listening to her talk was equally amazing.   Jo talked about many of the points from the book but I also had a few different take-aways.  Here are some of the most important points I took away from this hour.

1.  "If you're not struggling, you're not learning."--I talk a lot about making mistakes with my kids at school, and I think I have done a decent job of helping them realize that mistakes are a good way to learn.  But this phrasing helps me realize I need to take that message a step further...I need to normalize the struggle, and not just the mistakes (or right answers).




2.  "Math is not about speed, it is about depth and multiplicity of ideas."--Again, this is not a new message for me, but hearing it at this session just helped reinforce how important this is.  According to Jo, much of math anxiety onset begins  with timed tests. Interestingly, she said that math anxiety most affects the high achievers.  This matches with my beliefs....I've always been one to give kids as much time as needed.  Looking at this made me realize that although I have never really associated  speed with being good at math, this is not something that I talk a lot to the kids about.  I need to do a better job of verbalizing this message to my kids....math should be about deep thinking and understanding over speed.


3.  Teach kids to be skeptics--I love this idea.  What a great way to encourage great discussion and listening skills.  Jo gave three levels of being a critic.....convince yourself, convince a friend, and convince a skeptic.  I'm trying to figure out exactly how to incorporate this into my classroom norms for next year, but I definitely love this idea.

4.  Math freedom--This was one of my biggest take-aways.  Jo showed several clips of kids from her summer math camp, and so many of them talked about freedom being the reason that they liked the camp when they didn't like math in school.  Jo expanded on this idea into two types of math freedom:  organizational freedom and math freedom.



  • Organizational freedom included several things such as how you handle talking, sharing, recording, spending your time and movement in your room.  I'll be honest....this one gives me pause as a classroom teacher.  I understand that kids like freedom in moving around and how they spend their time....but I also know that in my classroom, structure and procedures have always been a bedrock that help my room run effectively.  I don't want to discount this idea, but I do think it is easier to do some of these things in a summer camp setting versus a regular classroom setting.  This is one I will take some time to reflect on this summer and think of ways that I can use this.
  • Math freedom included things like interpretation of problems, how kids see problems, learning new ideas, how we think about mistakes, and ideas about inquiry and creation.  I really loved this idea of math freedom....that kids begin to see math as a subject that is not just a set of rules, but there is freedom about where to start and how to proceed.  I want kids to see that math can be creative in how you think about a problem and that it needs to make sense.
I was so inspired by all of these ideas that since I've gotten back, I've taken Jo's free online course for students call "How to Learn Math for Students".  It had such great messages for students that I'm trying to figure out how to incorporate this awesome material into my classroom next year.  I also enrolled in "How To Learn Math for Teachers and Parents".  This one was not free, but I'm so excited to see what else I can learn.  I've just started the course, and I look forward to all that I will learn.

I'm already starting to have some ideas about how I want to change up some things in my classroom next year.  One definite thing is that I will be starting next year with some form of the free online course.  The other big thing I have been considering is changing up how I do homework.  I really want to make it more self-directed...I think I'll blog more about this idea later as it is still forming in my head.  I just know that I'm wanting to move towards something that is differentiated and puts it to students to examine where there are at and push themselves.

Sunday, April 16, 2017

My NCTM Experience Part 1: Dan Meyer

I feel so lucky to have been sent to the NCTM Annual Conference in San Antonio last week by the KATM Board.  I am going to do a series of blog posts about my favorite sessions and biggest conference take-aways.  I'm starting with one of the last sessions that I went to.  

 Last week I sent an email to the generic Desmos email.  Imagine my excitement when I not only got the answer I wanted, but the email came from Dan Meyer!  Yes, I in my own geeky kind of way was soooo excited.  Fast forward to the NCTM conference when I was talking to another math teacher who starts to tell me about someone (can't remember who!?!) and said, "She's my math crush.  Who's yours?"  And while I may not have thought of it in those exact words, I had to admit that it was Dan Meyer.  Now fast forward to the 8 am session on Saturday morning of the NCTM conference....what a way to start my day!

Dan with Kira and me.  He has perfect long arms to take a selfie!
The title of the session was "Math is Power, not Punishment".  The big idea of the session was based around the idea that we need to create intellectual need in our students for what we are teaching them.  As Dan said, "Math is the aspirin, but what is the headache?"  He had some really great, quick activities to illustrate this point.  The most powerful one involved the coordinate plane.  Dan started with a slide of a bunch of dots on the screen and told us each to choose one of them.
         


Then the screen changed....all those dots were there, but there was a bunch of others.  Dan got a couple of volunteers from the audience.  Volunteer #1 had the job of trying to describe which dot was hers to volunteer #2...and let's just say that was a tough job!
Then it was Volunteer #2's turn, and here is what happened.  I thought this was a great illustration of the idea of creating intellectual need.



The examples that Dan used in the presentation were very meaningful for me as a middle school teacher....the need for the coordinate grid, and another activity that looked at the value of combining like terms before solving equations.

This idea is not just powerful for secondary teachers however.  As I left this session with my K-2 Math Enrichment colleague, it got us talking about a lesson she had been telling me about earlier in the day.  She had done a lesson using non-standard measurement units, such as hands, feet and so on. As she and I talked, we realized that this lesson on non-standard measurement units would be a great way to create intellectual need in her students.  See what headaches can be cured by using standard measurement units.

As teachers, we want to help our kids find the easiest way to do things.  But perhaps we are taking some of the value of process away by not letting them experience some of the headaches first.
This is truly a powerful idea....that if kids see the value of what a method saves them it will be more meaningful to them.  Think of all those things you teach.....why did mathematicians invent those things?  What headaches did they help cure?