Visuals to Start Interesting Conversations & Problem-Solving

I realize most teachers and students in the U.S. are just beginning their summer vacations, so thinking about math and problem-solving is most likely the last thing on their minds. But – if any of you are like me, the summer was always a time to regroup, rejuvenate, and come up with new and brilliant ideas to utilize in math class starting in the fall.  I often spent my summers taking a class or finding projects to use/create, so always looking for ways to enliven my math instruction.

This morning, with all the news about UK voting to leave the EU, shocking news to be sure, I couldn’t help notice the many different visuals being bandied about to visually show how the votes were laid out.  It’s fascinating to look at these different representations, and then to just consider all the possible questions that arise.  Here are some examples of the visuals I have seen:

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The interesting thing with these visuals is they are all showing the same “results”, but from different perspectives or different ‘groupings’.  I love the map one – it clearly shows how the countries played out in the vote.  Now – this is NOT a post about the referendum – you will have to go to your news sources for information there.  But – from the math teacher side, all these visuals about the same results just got me thinking about how really great questions and problem solving could arise from the simple act of putting up a graph of some results and asking students “what do you think or wonder?” and letting them then investigate. For example, if we look at just the map, and don’t give them any numbers, they might wonder is it half blue/half yellow? How could they then determine the actual area of each colored portion of the graph?

Here’s a couple more pulled from the Prizm Resource Page:

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If you were to just throw these up on the screen at the beginning of class and ask the students to come up with some things they wanted to know about these visuals, it would lead to some student-generated questions that then would require the use of mathematics and possibly some background/related research, to find the answers.  If we are thinking about the mathematical practices, or habits of mind we are trying to instill in our students – such as analyzing, communicating, persevering, applying, arguing, critically-thinking, problem-solving, rather than giving them all the information and then asking them to ‘calculate’ the solution, why not let them find answers to questions that interest them? They would be applying mathematics in several ways, perhaps incorporating skills they have not yet learned but need – and in the process realizing that mathematics is useful and interesting.

Try it – find an interesting visual – graph, picture, etc. that spark in you some interesting questions that need math to solve. Put them in your “things to add to my class for the fall” and then get back to summer!

 

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“I Hate Math” – We Need To Stop This Mantra!

If you are a math teacher, you have heard “I hate math” from students, parents, friends….it is often the first thing someone says when you tell them you teach math for a living. Our traditional way of teaching mathematics, through memorization of steps and skills without context or connection, is partly to blame for this.  And, unfortunately, still the prevalent way of teaching today, despite research and standards that encourage and promote thinking, questioning, and multiple approaches. It’s discouraging, it’s depressing, and it’s a disservice to students. No one should “hate math”, and when you hear it from a child as young as 6 (see the video below), it’s even more depressing, because this is someone who hasn’t even really begun to know anything about math and yet they already hate it. Probably because they are being forced to do timed drills, or worksheets (as a friend recently shared with me about their child’s math class).

I just watched this great TEDx talk I found by Dan Finkel, where he talks about bringing joy to mathematics learning.  He begins with discussing how the fear and hatred of math permeates life, and can contribute to poor decisions and immediate trust in deceiving statistics; “When we are not comfortable with math, we don’t question the authority of numbers” (Dan Finkel, TEDxRainier, The Joy of Math).  He points out that the ordinary math class begins with answers – with little opportunity for questioning or creativity. We give students the steps to skills (i.e. steps to multiply, divide, find x, etc.) and our “questions” have set answers, and once skills are grasped, we move on.  “There is no room to doubt, or imagine, or refuse…so there’s no real thinking here”. Sound familiar?  Sound like a topics needed to master for a standardized tests?

Instead, we need to give students a question and make it authentic.  (His example with the numbers 1-20 and the colors is great, so be sure to watch that). I’ve written about this previously – making math relevant, authentic, and focus on questioning (Real-world Math Applications vs. “Naked” Math; Math Questioning to Support Mathematical Practices). Finkel’s point is that math and the beauty of math can be found by asking questions. “Thinking happens only when we have time to struggle”. Time is so important – it’s the only way to teach students to be ‘tenacious’ and to persevere.  So those of you out there still stuck in this obsession with ‘timed’ skills and rote memorization, pay attention to this video and what Finkel is saying. There are many others saying the same thing (i.e. Jo Boaler from Stanford), but Dan’s message is well worth listening to.  He has so many great quotes, I could go on and on writing them in this post, but probably better if you listen yourself and take away from it that which speaks to you. For me, I am just more committed to the message that math should be about thinking, connections, questioning and providing students the opportunity and time to really explore, question and pursue authentic problems to spark their creativity. Let’s please stop the “I hate math” mantra and instead try to create joy and wonder about math so that instead we hear “Wow, look what I learned about math today!”

New Year’s Resolutions for the Classroom

I hope everyone has had a nice holiday season and are planning to do something fun for New Yearsstock-illustration-80478039-happy-new-year-2016-background-for-your-christmas Eve. I myself am planning to have a potluck dinner with many of my neighbors and spend time talking, laughing and ringing in the new year. We have built a “New Years Eve Ball” out of chicken wire, lights, and 2×4, and are planning to do our own small-town ball drop from our friends apartment, which happens to be in the center of town. Hopefully we won’t get in trouble – I will be sure to post a picture!

Anyway, as the year draws to an end, and as all of you who are teachers see the end of your winter break draw to an end, I thought it would be good to share something I would do while I was teaching in the classroom. It’s so easy to get ‘tired’ this time of year – the school year is not quite half-way through, you might be going back to face midterm exams, and after a lovely vacation, the thought of going back and facing your 30 *(or your 150 students or more) seems exhausting. But, this is a time to think of ways to rejuvenate not only yourself, but your students and your classroom. Look upon the new year as a way to make some small changes in how you teach or structure your classroom, and you will find that it keeps the energy and relaxed feeling you had during vacation going.

I always made myself some New Year’s Resolutions for my math class.  Usually only about 3-4 things I wanted to start doing differently or more often in my classroom starting that first week back after winter break. It was a challenge to myself and I found it made me more excited about facing the next semester.

What do I mean by math classroom resolutions? It can be something very simple – like adding a different question into your teaching, or incorporating technology into class twice a week if you haven’t before.  The key here is to choose some things that you don’t do currently, or know you don’t do well, and focus on doing these things on a daily/weekly basis.  Little things that can make big changes.

Here is a list of some things I use to do:

  1. Use different, thought provoking questions in each class at least two times each day (questioning is a skill I still work on, so deliberately focusing on it helps make it become a habit) (just a few example below):
    • Ask “why do you think?”
    • Ask “what if”
    • Ask “what do you wonder?”
    • Ask “why?”
    • Ask “can anyone show a different way?
  2. Incorporate technology into a lesson at least twice a week as an EXPLORATION tool (not an answer tool)(if you already do it that often, then every day…something different).  This can mean calculators, software, apps, smartphones, videos – something that provides students with a chance to explore and ask questions that expand their learning/understanding and leads to more discoveries.
  3. Have students work in pairs 2-3 times a week (more if you are already doing this).
  4. Use exit passes every day.
  5. Start each class with a real-world application.

Obviously, you need to gear your resolutions to you – what is it you don’t do now or don’t do often enough that would increase student engagement and/or student understanding. And focus on 2-3 things that you are going to do regularly. Changing a little bit consistently makes it become natural, and once you have those in your repertoire, then you can add on some more. The key here – change something.

Hope you all have a wonderful New Year’s Eve.  Be safe. Be happy. And be motivated to make small changes that will help your students and you achieve great things in the new year.

Happy New Years!

Questioning In Math – #NCTMRegionals Minneapolis Observations

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John Diehl #NCTMRegionals

Had a nice time in Minneapolis these past two days at the #NCTMRegional. It must get ridiculously cold here in the winter since they built an entire interconnected-Skywalk throughout the city. I think I only went outside twice the entire time I was here – getting in and out of the taxi! (Which was pretty terrific as the first two days were rainy).

I went to a few sessions this time that really got me thinking about the importance of questioning in mathematics. Even when utilizing technology or hands-on manipulatives/resources, the questions we ask the students are vital in order to deepen their understanding and encourage discourse and exploration. Questioning to me is the most important skill a teacher can develop to help their students – more important than any resources or technology that might be available, because it is only through questioning that we  foster rigor and develop deeper thinking to help students understand and make connections in mathematics.

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Teachers talking statistics!

What I loved about two sessions in particular that I attended, a 6-12 Statistics session with John Diehl, and a 3-5 Making Math Meaningful session with Jennifer N. Morris, was the focus on questions and how the same mathematical concept could be appropriate for students in any grade depending on the questions asked. In John’s session, where we were looking at bivariate data, we used data athletes, made a scatterplot on the Prizm, and then had great discussions about lines of fit, variables, causation, association and a multitude of other ideas around helping students understand what the data represents, in context, and how, depending on the question asked, you could address algebra content or calculus content. Students in sixth grade can be do linear regression simply by asking the right questions and allowing them to explore their conjectures with technology, such as the Prizm. The questions lead to discussion and exploration and more importantly, to more questions that the students themselves begin to ask. Something as simple as “should your graph go through (0,0)? What does that actual mean in relation to the data and does that make sense?” helps students apply math content to the real-world context and make sense of the data and the graphical representation (sounds very Common Core to me!)

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Jennifer N. Morris & Origami

In Jennifer’s session, the first part was creating an Origami pinwheel, which seemed relatively simple but the questioning throughout about area of folds, and fractions of the whole, and how do you know, what shape do you have now and what’s the ratio of this shape to the shape before – really demonstrated how a seemingly simple hands-on activity can be full of rigorous mathematics and mathematical connections. And – the same activity would be appropriate for multiple grades – up through geometry even, simply by changing the questions you ask. When we began working with the Casio fx-55Plus, Jennifer did several quick activities using the Random# generator on the calculator, but the questions asked really had the teachers (i.e. students) thinking about numbers, fractions, comparing numbers, estimation, reasonableness, probability. For example, if these random numbers (all fractions, but with varying numerators and ugly numbers like 651/1000) represented the part of a cookie your mom was going to share with you, when would you consider it big enough and why? This led to really interesting discussions on how to determine

Teachers Ordering by Random #

Teachers Ordering by Random #

what fraction you had of a whole and is just being more than 1/2 enough. She had participants stand up front with their calculators & their randomly generated fraction and rearrange themselves in numerical order – not so easy when the fractions are 516/896 and 37/52 for example. Seemingly simple activity, using technology to quickly generate numbers and have really rich discussions that help make mathematical connections. And it came down to the questions asked – engaging, in-context, and appropriate for many grade levels.

Both sessions confirmed for me something I have always believed – the questioning is the easiest way to get your students thinking, talking, applying and connecting math. Learn to ask questions and you can create an engaging learning environment that differentiates the learning and provides students with multiple pathways to make connections.