3D Vectors- Open Middle Problem

Directions: Using the digits 1-9, each only once, fill in the blanks to make the following vector relationship true.


What vectors maximize ab?

What vectors minimize ab?


My MCTMDuluth Presentation

It was a week before the Minnesota Council Teachers of Mathematics Spring Conference when Sara VanDerWerf @saravdwerf messages me because there have been canceled sessions.  She wants to know if I will do a pop-up session, and of course I will.  The only problem- the topic.

Last year at MCTMDuluth, and this NCTM, I spoke about using Open Middle Problems to Promote Classroom Discourse.  Around Minnesota, I’m the Open Middle guy (except that one random incident where I was the Desmos guy) and that is totally OK because I truly believe in the power of Open Middle Problems in the classroom.  Those problems have led me into thinking about two other changes to make in my classroom (and these are not new in any capacity for educators)- Problems you Pose to Students and How you Ask Questions.  This is actually what my proposal for NCTM DC is next year and it’s still a stream of consciousness for me, nothing really concrete.  This is the topic I chose to present, to get a feel for how it feels as a presentation as well as to get input from my awesome peers at the MN conference.

The Google Doc presentation is here.

Too often students don’t understand what is going on in math class.  Earlier on in education, this was deemed OK- these students were slow and needed to catch up to the others.  More often than not, I think that this clip is how students feel in a math class.


When this does happen, we lose these students, this is where the Math Identity of  these students is taken away.  This is when students need us the most.

Students who are behaving in certain ways in your middle or high school math classes are not acting that way because they WANT to, they act that way because they have LEARNED to act in that manner to hide their feelings about math.  I have yet to find a student who doesn’t want to be better in math class, but it takes a lot of trust building and showing that you care to get those kids to crack and admit it.

We are teaching in the 21st Century and still use 18th Century mediums.  Books are not new, like it or not they are slowly becoming a dying technology (much like typewriters).  Our students interact with electronics, and we need to provide instruction through mediums that they interact with.  When I was young, I was trapped on a farm outside of town with no way to get anywhere.  I did a ton of stuff outside and if I wanted to “get away” I had to read a book.  They took me on journeys and adventures that I could only imagine.  My son however can pick up his iPad instead of a book.  That is how he interacts.  Right now, I have tons of difficulty getting him to read- it is a constant struggle.  Too many of my Students are the same.  They don’t process information by seeing it in a book, written on a board, or even when they hear rationalization of the process (because once again, it is referencing text).  We need to provide students with other ways to see math.

Students have to have an experience with what you are trying to teach them.  An experience they can relate to.  An experience they can visualize, manipulate, interact with.  Notice that I’m not saying we have to use digital technology here, we need to give them experiences that they can interact with.  This can be provided in many different ways, some with technology and some not.  I do agree with Dan Meyer in his age old talk about Video Games- we need to provide students with experiences in a similar fashion as video games because that is how they have learned to learn.

Give students a video to watch, activity to participate in, a photo to view, a puzzle to solve.  Allow them to notice problems and wonder about how to solve them.  Have them verbalize patterns and make conjectures on how they are related.  Allow them to be mathematicians before you introduce “formal structure.”

I’m not saying you have to totally abandon whatever curriculum or groove you currently rock in the classroom, but I am asking you to take the time to introduce a front-end stimulus for students that will allow them to have some practical experience with the mathematics you teach before you start the “I do, we do, you do” process.

After you provide students with some experience with the math, AND allow them to notice and wonder about things, THEN you can start questioning.  Notice here that I have yet to talk about calculating answers or assigning homework problems.  There has to be a lot more that happens before that- even though we all feel the crunch of getting our curriculum in “by this time” and get kids to master skills “for the test.”  Trust me, if you can get students talking, wondering, and verbalizing their thinking through questioning you will have a TON less work in the form of student questions on “how do I do number 13?”

One of the biggest thing many teachers can do to help their students be successful in the classroom is to implement a universal lesson design.  This may seem like a special education method but it is truly best practices.  If you learn to design lessons taking all of these considerations into mind, you will have provided students the best possible learning experience and environment.  (And I’m willing to throw out there that you will enjoy your lessons more as well).

Like I said earlier, this is just a stream of thought, please drop some comments on your thoughts on this- I’m still learning and would appreciate any input on this.

My Favorites of NCTM- Global Math Department Presentation

I just finished talking at Global Math Dept and it was great!  After listening to my fellow presenters, I think I was the kid in the back of the room who doesn’t fully listen to directions and gave a half-finished assignment.  Whatever the outcome, it was what I have been thinking of since I left NCTM in San Antonio.

My presentation slides are here…




Overall, as my first experience at NCTM this was awesome, and has me hooked.  I would present there any time I was able (and I’m currently working on my proposal for next year!).

Drop my any questions you have…

My NCTM Presentation

In case you, like the rest of the mathematical community, was at Jo’s presentation Thrusday @12:30, here is my powerpoint for my presentation at NCTM.



Some big ideas from it:

Change the way you question to promote student thinking and conversations.  This is my new thinking kick, and now I am constantly looking at problems and trying to determine “how can I ask this better?”

Once we ask student for an answer, we ask them to stop thinking.  They become focused on one goal, and will no longer notice and wonder to make connections to mathematical meanings and possible solution paths.

Please try out an  Open Middle problem.  They can fit so seamlessly into your curriculum.  I use them in flexible ways: as warm ups, practice problems, exit slips and for formal assessments.  When you are assigning homework for students, examine your text’s problems and then check out our site- see if you can get them to practice in a more meaningful way that promotes understanding without burying them in paperwork.

It was a great experience presenting for the first time at the national conference, I really enjoyed NCTM and would like to thank everyone that made the conference possible.  I am definitely submitting a proposal for D.C.



The Moment…

One of my good friends once said that “we need to let go of the things we can’t control and focus on those we can” she also said that “until we want to, we truly can’t change anything” (OK, so maybe I paraphrased a bit @veganmathbeagle).  It wasn’t until NCTM in San Antonio that I had that moment, the moment where I wanted to change how I approach conferences.


I flew into San Antonio on Wednesday and picked up my program.  I first double-checked when my speaking session was , the time and where, then I started looking through the program to find sessions.  While I was looking I had my moment.  The moment I am talking about is when you realize that you are stuck in a rut, that you are doing the same things over and over even though you are wanting to change.  It’s no one’s fault that this happens, it seems to be a condition of being human- finding safety and security in doing the same things, finding a pattern to your life.  The bad thing is when you do these in a profession such as ours (that is, unless you are a super-teacher whose students are exceeding in their learning).

My moment was realizing that when I come to these conferences, even though I want to find new things to implement in my classroom I also want to talk with and hang around my friends on the #MTBoS that I never see.  As such, I revert to the high school student who takes all their friends’ classes- and while that is not necessarily bad it also doesn’t fully address MY needs, I am attending sessions that satisfy their needs (and as we know sometimes those overlap in areas).


Looking at my program, I was highlighting people I interact with through the #MTBoS (and by interacting, I also mean stalking because there are times I don’t feel like I can approach them).  I do this at my local conference a lot- I go to my friend’s sessions.  That hasn’t been bad so far because I still need those connections, my friends do push me to become a better teacher.  The problem is that, for the most part, I am already aware of their passions and know about their session.  This is not NEW material to push myself as a teacher.  So when I caught myself highlighting my “peeps”; Dan Meyer, Robert Kaplinsky, Andrew Stadel, etc, I realized that perhaps I need to find a different direction for this conference.  This feeling was further enforced as I stood on the second floor balcony and observed how many math people were present at the conference.  I thought I had a large network of math people because of my involvement in #MTBoS, but here was concrete evidence that my potential mathematical network could be much, much larger.

So I took a different approach, I sat down and really read through the program and found sessions that really touched on areas for myself as a teacher.  I didn’t hang out with all my twitter people as I usually do (and I will say I am sorry for that, I was being purposeful this conference but I always tried to say Hi when I saw you guys around).  I went to sessions of people I didn’t know (or perhaps I did but I’m getting old and didn’t remember).  I looked for sessions that would directly impact my needs as a teacher working at a juvenile center where 90% of my students have special needs.  A funny thing happened, there were a couple of sessions where I did go to my “twitter friends”, and there were sessions that friends also attended.  The big thing is that I met new friends, heard new voices, got fresh ideas.

I am hoping that I will be able to attend NCTM again next year in DC, if I do I will plan a balance of “being with my friends” and “finding something new”.  I also hope that ShadowCon will consider @delta_dc’s comment on “the Next Generation”, allowing newcomers to present and add their voice to our great community.

If we truly want to change, we can’t continue to travel in the same circles we always do.  I find it funny that we, as a community, lament the fact that teaching always seems to revert to those strategies and curriculum that we strongly feel is wrong- but that we are unable to see those same qualities within ourselves.  Perhaps this blog will inspire the moment within you where you realize that you also need to step outside of your current zone- that every one of us needs to in order to challenge our ideas, our absolute belief that we are teaching the only way possible to teach our students.  Because in reality, there is so much more we can learn from those on the sidelines of our #MTBoS community that can enrich our own learning and that of our students.


Following up on Changing Questions

I haven’t heard back from anyone (and I’m not sure I really expected to) so I’ll just relate what happened in my classroom when changed the way I asked a question.  The first thing I’ll say is that changing the way you question definitely changes the way your students think- and in most cases it is for the best.  Take for example this convo Christopher and I had yesterday…

I was writing an Open Middle problem, and as I often do I hit publish too quickly.  This is what the first draft was…


Now, I want you to know that this was up maybe 2 minutes until I realized that I hadn’t put any real constraints on the problem.  Typically with Open Middle problems you are allowed to only use a digit once, and I was modifying this for a smaller number set.  OF COURSE Christopher would be online and checking my tweets because this is what happened next…


I had also realized that it wasn’t possible if you weren’t able to repeat numbers- which would be fine for the classroom environment, but not for the Open Middle format.  This is what I was originally thinking of the problem so I published it instead (which caused Christopher’s reaction)


I was trying to reduce the numbers usable because I was wanting to target 3rd grade and was trying to put a smaller constraint on possible values for the boxes.  Unfortunately that wasn’t working, and I had to go back to the original format.  But like Christopher stated, it changed the problem and some of the thinking students needed to do.

Back to the original set of questions.  The first:

Create an addition problem where 2, 3 and 4 appear somewhere in the problem or answer.

Let me remind you that I work at a juvenile center where I have mixed classrooms.  There is a wide range of student ages and ability levels in one class.  Most of them were VERY uncomfortable with this problem.  They weren’t asked to make a direct computation, and there wasn’t any “answer” for them to work with.  After their initial shock, and Mr. A’s many many times of explaining 2,3 and 4 need to appear somewhere within the problem, they produced some nice examples.


The top left response was overwhelming, with over 92% of the class producing it.  It was the other responses that I really found interesting.  Students used combinations of grouping 2 and 3 to create a new number (like the bottom left), and others introduced additional digits with 2, 3 and 4 to create a different sum than 9 (like the top right).  I was also interested in students who created a sum equaling 234, asking students how they determined what numbers to use as addends.  Many students replied that the number they used was their favorite number, which made me a bit concerned about the student on the bottom right- students always try to sneak in inappropriate things to check and see if you are paying attention.  The bottom response was the one I found most interesting.  This is a student who HATES math class and is working at a proficiency level that is 3 grade levels lower than her enrolled grade level.  She has an IEP for Emotional/Behavior Disorders as well as Learning Disability in Mathematics.  Yet she is the one who provided me with the most elegant and interesting solution.  She loves patterns, and told me she wanted her numbers to be sequential, so she needed to use 5.  She sat and worked for over 10 minutes trying to figure out how to create a sum of only 5.  She asked me many times if she could multiply, subtract or divide- and was referred back to the wording of the problem many times.  She did not give up or throw a fit however, and I believe that is because she set the parameters she was trying to achieve.  Then, the light bulb went off, her face lit up and she asked me, Mr’ A,- can we use negative numbers?  I referred her back to the problem and asked if it indicated that she couldn’t.  She reread it and said no.  So I told her she had her answer and she was immensely pleased with herself.  She was in an incredible mood all day and even came into class the next day asking if anyone else had used negative numbers (the answer to which was no).

That second day, I once again asked them the “same” question, and highlighted how I made it different.

Create an addition problem where 2, 3 and 4 appear somewhere in the problem or  and  answer.

Even though many students found the question easy yesterday, once again the sky was falling and the math gods were against my poor innocent students.  Once again I calmly took it all in stride (while laughing to myself inside because I expected this response) and verbally explained that 2, 3 and 4 needed to appear on both sides of the equal sign.  They then went to work.


By changing one word, it created such a variety of thinking and solutions that I could have never elicited with the first question without very guided questions.  Some students used the same thinking as the first question, but I also loved how some natural properties of mathematics was produced: place value, additive identity, reflexive property.  Students introduced decimals!  When does that EVER willingly happen?  Once again, I was surprised by a student who took an extra step and made their sum by using only 2, 3 or 4 in all of their numbers (except for the 100 Mr. A, my brain started hurting).  This has left me with many a rich discussion to take with my students as we reflect on the difference of these questions, their thinking, and their solutions.

I hope you find the same in your classroom.