Monday, February 25, 2013

When does "by hand" graphing or processes matter?

I am finishing up teaching polynomials to my Algebra 2 classes. We are discussing the Rational Root Theorem (i.e. p's and q's) right now. When I presented the material on Friday, I went through the whole process - finding the factors of p, finding the factors of q, finding the p/q values and testing using synthetic division. As much as I like doing synthetic division, I had forgotten how frustrating this process is for students - it is tedious and they know by now that they can find the zeros by finding the x-intercepts of the graph.

So, as I reflected over the weekend and into this morning, I decided when I assess them on this learning target, I am only going to ask them to identity the factors of p and q and the p/q values. I am not going to ask them to fully find the zeros of the polynomial from that list. The more I thought about it, the more it became clearer to me that if they were going to have to find zeros from a polynomial students would have access to technology (such as a graphing calculator or Desmos). As I was working with students today and reflecting on this learning target, I kept coming back to this question: How do we determine when it is important to have students do the processes by hand versus using technology instead? Is it really important to have students find all the zeros by hand? I had to (granted, the first easily available graphing calculators came out when I was a senior in high school) - so why shouldn't my students have to? (note - I know that's not a good reason why, just throwing my thoughts out there.)

This question is not new to me. I have wrestled with it on and off throughout my teaching career. When students have graphing calculators that can do things like graph and find zeros and maxima and minima, etc., this question returns often. However, this year, I have a mixed bag as far as who has a graphing calculator and who does not. With having (4) computers in the classroom, Desmos has been a nice addition and is far less clunky in identifying intercepts and extrema. From the bits and pieces I have caught from tweets, Desmos continues to improve and everything I have seen from them shows that they are incredibly responsive to its users (and teachers!) Bob Lochel addressed the TI vs. Desmos issue in his blog post this weekend "An Open Letter to My TI Friends." I have to say that even though I have not received quite the training and benefits that Bob has from TI, I found myself really agreeing with every point that he made in his Dear TI letter. (Go read it if you haven't already - it's worth it!) But once again, not everyone has access to the technology in my classes. We are not a 1-to-1 school, I don't have a class set of iPads or tablets or even graphing calculators. My classes range from about 30% to 50% of my students having a TI graphing calculator. I have students who do have smartphones, but they are not allowed to use them in school. I'm already starting to think about next year and how I'm going to deal with the whole technology issue. Do I have my students all get TI graphing calculators, full well knowing that many of them will not use them after my class? Do I skip the graphing calculators and find a way to work with Desmos, knowing that I have 4 computers in my class and that's it? Do I try to find funding for a class set of graphing calculators? Tablets?

But I digress from my original query. I have taught for 21 years and this is the same question that I had when I first started teaching with TI graphing calculators then. At what point do you push aside the by hand processes and let the technology take over? Are you shorting students mathematical learning by doing this or is it enhancing it? How do you structure lessons so that the technology enhances the mathematics rather than glosses over it? I'm curious to see what you all think. Please share your thoughts in the comments. Thanks!

Sunday, February 24, 2013

My Weekly Diigo Links (weekly)

Posted from Diigo. The rest of my favorite links are here.

Sunday, February 17, 2013

My Weekly Diigo Links (weekly)

Posted from Diigo. The rest of my favorite links are here.

Saturday, February 16, 2013

My Presentation on the Math Twitterblogosphere

In January, I was asked to present as part of a panel discussion about Twitter Math Camp and the Math Twitterblogosphere. Ann Drobnis, an Einstein Fellow working at the NSF, contacted me asking if I would share what makes our Community of Practice successful. I presented to mostly university professors and NSF personnel who are involved in putting together and facilitating a community of practice for Computer Science teachers who are part of an initiative called CS10K, which is trying to add 10,000 Computer Science teachers by 2015 (I believe it started in 2010). Both Steve Weimar (of the Math Forum) and myself represented parts of the Math Education community, the remaining speakers (Mark Guzdial, Neil Brown, and Shay Pokress) were from various aspects of the Computer Science community. Since Sam asked, here's my presentation:

The link to these slides is found here.

Good morning. My name is Lisa Henry and I teach high school math at Brookfield High School in northeast Ohio. I am also the lead organizer for Twitter Math Camp. Our first Twitter Math Camp was held in St. Louis, Missouri in July, 2012. Twitter Math Camp was a 3 ½ day conference that we put together ourselves. We wanted to get together in person to work through the Exeter Academy math curriculum problems and share what we are doing in our classrooms. Along the way, one of the most powerful professional development experiences for the participants happened and friendships deepened. What happened at Twitter Math Camp didn’t happen overnight. To understand what happened, I’d like to share the journey we have shared.

Around 2008-2009, there were a few math education blogs that existed. The math teachers who were online at the time would blog and comment on each other’s blogs. Conversations were taking place online, but not in real time. The main bloggers at the point included Dan Meyer, Kate Nowak, and Sam Shah. Once Twitter gained prominence, conversations moved from the comment sections in blogs to Twitter. Twitter allowed real-time conversations to happen (and some not-so real time conversations too). Friendships began to develop. For example, on Halloween, 2009, Sean Sweeney changed his twitter avatar to look like Sam Shah’s avatar, and Twittereen was born (see also here and here). After it happened, Sam shared on his blog that blog buddies had become friends. In late 2009, while grading student exams, one of the math teachers tweeted there should be a red stamp for a certain common mistake on her exams. The math teaching community responded enthusiastically, venting about common student errors using the hashtag “need a red stamp.” Eventually, Sam created a t-shirt that several of us purchased with “I only Twitter with Math Teachers” on the front and “#needaredstamp” on the back.

In the Spring and Summer of 2010, the math teacher community became very active on Twitter. Several teachers wanted to learn about Standards Based Grading and began a book chat via Twitter over the summer. It was really during that summer that the math teaching community became very active. There were many new bloggers, including myself, and lots of sharing was happening. Teachers who were using standards based grading in their classrooms were blogging about it and conversations about it happened on Twitter. We started sharing what we were doing in our classes, both via Twitter and blogs. At some point, we even crossed the line to become “Facebook Friends.” I remember it being a big deal – we had been talking about what the difference was between being “tweeps” versus being “Facebook friends” and how “Facebook friends” were “real” friends. Several people “facebook friended” each other and we continued to share more about ourselves.

At some point, it had to be that we meet. Over the years, there had been several “tweetups” – face-to-face meeting of Twitter friends. Mostly these occurred around NCTM or other math conferences or workshops. I met several of my Twitter friends at NCTM in Indianapolis in 2011. Some had attended PCMI together. Others would be traveling and arranging to meet. I think it was late in 2010 that we started talking about planning a “Twitter Cruise” – we’d all go on the same cruise and do math and talk math and visit. We did look into it a little bit and found it would be kind of expensive and figured it would be a bit difficult to get our districts to pay for this kind of professional development. Over Christmas Break 2011, we were talking about what we wanted to do over the summer. I had shared with a fellow math teacher that I wanted to work through the Exeter problem sets over the summer. She said that she wanted to also. A few others chimed in that they would be interested and someone said, “Wouldn’t it be cool to do that in person?” We all agreed and Kristen Fouss organized a Google Form to find out who was interested and where they were. We had about 10-12 who were initially interested and the central location for those who were interested was St. Louis. Once we figured out where we thought would be good to have our gathering, Shelli Temple and I started looking into possible locations. We formed a Facebook group to organize our thoughts on where and when and what we would do. Kristen had suggested a particular weekend in July since the St. Louis Cardinals would be in town and maybe we could all go to a baseball game together. Shelli realized she had a friend who taught at a private school in St. Louis and by early March, 2012, we had a confirmed location for what we dubbed “Twitter Math Camp.”

Others in the math teaching community stepped up to help. Sam Shah offered to put together a website so we would look “official” for anyone who was trying to get professional development money from their schools. His website has now morphed into our current website at, which is hosted by one of our attendees. Elizabeth Statmore offered to put together t-shirts. Registrations were taken via Google Form and Speaker Proposals were submitted via Google Form. When we first started putting this together, I think most of us expected to get about 15 people to show up. We had 37 teachers attend Twitter Math Camp – including teachers who teach in Canada, Jordan, and Argentina, and 19 different states. There were some teachers from the St. Louis area who heard of us who came, but the core group of teachers who had been on Twitter was about 30.

We spent 3 ½ days together. We worked Exeter problems. We shared what we did in our classrooms – both formally and informally. We had an amazing experience – hands down the best professional development I have ever attended or taken part in. I have never attended a professional development session where every person was engaged in every session. We socialized together. 30 math teachers toured the Budweiser Brewery. 20 of us attended a St. Louis Cardinals game, while another group of about 10 went to the City Museum. We ate dinner at Pi Pizzeria. We went to a German restaurant together. We went to the movies together. We spent 3 ½ days talking about teaching, math, our lives and growing as a community. It was hard to leave. After spending time with other people who “get” who you are, heading back to reality was difficult.

We continue to grow. Around the same time Twitter Math Camp happened, Shelli Temple started “Made for Math” – a blogging initiative where math teachers share on their blogs something they have created for their math classroom. It could be an arts-crafty type thing or a worksheet or activity. Pretty much every Monday since July, math teachers have blogged something they use in their classroom life. We started My Favorite Fridays also, where we share something that we use that’s a “favorite” – an outgrowth of one of our Twitter Math Camp sessions. We have a website to welcome math teachers to the “Twitterblogosphere” that was an outgrowth from a session that Sam Shah did at Twitter Math Camp. Megan Hayes-Golding started up the Global Math Department – a weekly meeting on Tuesday nights at 9 pm Eastern where someone or a small group of people share via video and chat about something we are doing in our classrooms or something applicable to math education. These meetings are recorded and archived online at We continue to blog and tweet and share with each other, although not as often as we would like sometimes. But we remain connected. We are looking forward to Twitter Math Camp 2013, which will be at Drexel University in Philadelphia this July. We opened registration December 26th and as of this morning 31 are registered to attend.

Why does this work? Quite simply, we want to be part of this. We have chosen to be on Twitter. We chose to attend Twitter Math Camp. We want to be better teachers. Why do people stay part of this community? The relationships we have developed over the years have kept us together. We have shared with and learned from each other. The best things that I do in my classroom are mostly a result of my interactions on Twitter and blogs. When you have worthwhile interactions, it keeps you coming back. If you don’t get anything out of it, what is the point of coming back? In today’s teacher’s world, that is wasting valuable time. We are taxed with many responsibilities related to our jobs, so if I am going to spend time somewhere, I need to get something out of it. Provide meaningful content and interactions for the participants. Encourage discussion. Make it worth their time. You can’t force people to take part – but make it engaging so that they want to. Many of our teachers teach in situations where they are looking for other input from others who are not in their districts. They may be the only (whatever) teacher or they haven’t gotten anything useful from others in their districts, so they go to the internet. Somehow they ended up at Twitter. Those who stick around are the people who engage others in conversation and get responses that they find useful. They may drop off for a while because life gets busy, but they come back because of the relationships they have developed over time. 

Thursday, February 14, 2013

Noticing and Wondering

Tuesday evening, Max spoke at the Global Math Department meeting about Noticing and Wondering. He has spoken about this before at Twitter Math Camp and I was a bit intrigued about it then. When I saw he was speaking Tuesday night, I knew I had to be there.

Now, I'll be honest, this is NOT something I have done with my students. I have had the tendency to instruct without letting them do a whole lot of exploring. Part of it for me is that there is so much material to teach in Algebra 2 (and I am having to play catch up from Algebra 1) and part of it is my own comfort level. However, as I was thinking about my lesson on Friday about the Remainder and Factor Theorem, inspiration struck me on Wednesday. My Algebra 2 students were having an assessment on Thursday and there would be enough time afterwards for them to do a little noticing and wondering. After their assessment, I asked them to complete a paper with the following questions:

Do (x^3 + 6x ^2 - 3x + 7) / (x + 3) using synthetic division.
Find f(-3) if f(x) = x^3 + 6x ^2 - 3x + 7.
What do you notice?
What do you wonder?

Do (2x^4 + 6x^3 - 15x^2 + 15x - 50) / (x - 2) using synthetic division.
If f(x) = 2x^4 + 6x^3 - 15x^2 + 15x - 50, what is f(2)?
What do you notice?
What do you wonder?

Here are some of the responses I received:


  • I don't remember how to do (the f(__)) problem.
  • same problem and you get the remainder
  • the remainder is the same as the answer of the function and they have the same number of terms
  • The remainder of the synthetic division is the same answer as the 2nd problem I worked out (the f(__) problem).
  • I noticed that the numbers are the same in the problems and f(x) is the same number as it is in the box of in synthetic division.
(I had several answers that were similar to the 2nd, 3rd, and 4th notice bullets.)

  • How to do it (the f(__) problem)
  • Could you use synthetic division to find f(x)?
  • (Written under the 1st wonder) I got the same remainder for the 2nd set of problems, but not the 1st set of problems. What did I do wrong on the top question?
  • Can you use functions to solve synthetic division?
  • What do they have in common? Why are they both remainders?
  • How does that happen? Why are they the same?
  • Are we going to do the same thing as before or is it different?
  • I wonder if the two problems are related. I wonder if you can use the 2nd problem to figure out the synthetic division problem.
  • Will this be the case every time? (Then the student added on the 2nd one when his answer did not match) Will the answer match the remainder if the number replacing x is negative?
  • Why is the answer the same as the remainder?
  • Are they connected? Is this another way?
My observations:
  • Many (TOO MANY!) of my students did not recognize function notation or how to work with it. This is something we did review earlier in the year and it dismays me how many had no clue. Even several of my top kids came up to ask how to deal with f(-3) and once I told them, they remembered. However, that they even had to ask worries me.
  • The students who didn't know how to do synthetic division (all the way) correctly obviously did not make the connection at all. Many of these students left the notice/wonder part blank.
  • I half expected some smart-aleck responses from some of my students (especially some of my struggling ones). I didn't get any. However, I had a lot of blanks in the spaces asking for their noticings and wonderings.
I had chosen to do it this way, rather than orally, for two reasons:
1) I knew I was going to have extra time after their assessment and this would help keep them focused and quietly working on something.
2) I felt that this would give my students who do not catch on as quickly as my top students the opportunity to think about it and possibly make the connection on their own. In my Algebra 2 classes, I have a range of students from very bright students (who mostly took Algebra 1 in 8th grade) to students who struggle with math. Not all of my high-ability students caught it, and some of my middle ability students did put together the connection rather nicely. I hope that will help them tomorrow when we discuss the Remainder and Factor Theorems.

Originally, I didn't think I would give them their papers back. However, after reviewing them and reflecting some, I think I will give them back to them. Hopefully we'll have some nice discussion about it tomorrow.

On Friday, my principal came in to observe me during the 1st of my 4 Algebra 2 classes. (We are a Race to the Top school and we are doing the new Ohio Teacher Evaluation System this year.) Although I'm guessing I'll get dinged for this not being as much of a class discussion since they had completed the noticings and wonderings on paper the day before, I feel like it went pretty well. I would have liked some more discussion out of them. Part of it for me is that I haven't done this before and finding the right questions to elicit discussion out of them was a little bit of a challenge for me. I feel like we answered most of their wonderings, which is a good thing. :-)

Sunday, February 10, 2013

My Weekly Diigo Links (weekly)

Posted from Diigo. The rest of my favorite links are here.

Sunday, February 03, 2013

My Weekly Diigo Links (weekly)

Posted from Diigo. The rest of my favorite links are here.