All this talk over #MTBoS and #iteachmath has sparked a personal motivation to start blogging again. I believe all the conversations happening on twitter in the last week indicate a difficult yet positive shift in the Math Teacher Twitter community; this is a time we will not forget. There also appears to be an increased amount of engagement within the community, so what better time to post again?

Julie Reulbach posted a weekly blogging challenge that I want to be part of as the school year begins to rev up. This week’s focus is on goals.


During my summer session of grad school, the professor drew the following diagram:


She explained most teachers believe a student’s mathematical understanding progresses the following way; teacher introduces concept A, students show proficiency in concept A, teacher moves to concept B, students show proficiency in concept B, teacher moves to concept C and so on for the remainder of the school year. If a student happens to struggling on concept C, teacher backs up and reinforces the link between the two.

Each concept is dependent upon the previous, a single broken link causes problems with all other connections. This view forces teachers to not push too far on a concept or provide challenging material due to the fear of breaking a link.

She then erased the chain of concepts and drew the following:



The image above changed me.

The professor explained this as a Web of Understanding (If this credits back to other folks, please let me know!); the missing link between concepts B and E is not a problem because through transitivity the concepts ARE linked.

Now, if a student is struggling with concept B, there are multiple links a teacher can reinforce, since multiple concepts are linked to one another. Or maybe, there is no need to backtrack and fix a link because the others may be just fine. A broken link no longer causes issues with all other concepts.

My Goal for this year is to create a classroom where students see math not as a linear topic from one concept to another, but a web of interlinked concepts connected to one another. I hope to promote equity within my classroom in providing all students with challenging content, not just the students who have solid links from one concept to another.


Adding Polynomials

In an ongoing process to better teach the boring bits I have worked out a new way to introduce adding polynomials.  I feel like I should justify what I have been doing over the last… 7 months… but lets just say I have been busy in the classroom. Anyways; here are goods.

Start off by asking students to evaluate the following:


Walk through it to make sure they totally understand, this should be easy peasy for them.

Then throw this up:


Tell ’em you will race them and give ’em a 30 second head start, keep in mind that the previous problem has set them up to plug 2 into EVERY. SINGLE. X they see, so it is going to take them forevvvvvvvvvveeeeeeeeeeeerrrrrrrrrrrr to evaluate this expression.

After the 30 seconds is up announce that you are starting the problem (do this on a scratch sheet of paper by your desk, so they can’t see what you’re doing). Combine like terms, get this thing down to a trinomial (or possibly binomial) and then plug in 2 for x.

Announce to the class that you have finished the problem (in all my classes this was before they had even finished writing it down) then let student sit, perplexed, as to how you solved the problem so dang fast.

This gives at least some reasoning as to why mathematicians would ever want to add like terms.


This year was great. I had awesome students, great collaboration with colleagues and daily reflection. I took some time after school wrapped up to go back and read through all my posts. I noticed that first 70 or so were pretty rough. I was in a weird place; trying to establish credibility and write like all the other bloggers out there I look up to, yet at the same time make my posts unique.

At times blogging every day was stressful. There were a couple moments out there where I almost threw in the towel… I spent 5+ hours each Sunday putting together fresh and new lessons for algebra and geometry (I didn’t reuse a single set of slides I created last year… mostly because of a transition from PowerPoint to SmartNotebook). But long story short, Sundays, 1 to 2 additional hours of prep each day and blogging wore me down after 38 weeks. Thanks to all of you, I made it.

This blog held me accountable in providing top notch lessons for my students each week (I dreaded the days where I had to write about using a Pizzazz or Kuta worksheet). I was also able to reflect each day on what went well or not so well, which, in my eyes is extremely important.

Basically, I am trying to say I am happy with all the improvements I made throughout the year.

But…I still want to be better. a lot better.

Here is what the future holds Quadrant Dan and myself…

  • I am really happy with the number of lessons I have stock piled on this blog. Next year, I will not be continuing a daily blog. I do however plan to post frequently. I want to dig deeper though and increase the quality of my posts, rather than quantity.
  • The main reason for the discontinuation of daily posts is next year I gained another class. I will be teaching algebra, geometry and IB math studies. The IB class is going to add an an enormous amount of work to my plate. Plus, I still have a lot to do for algebra and geometry; three preps will be intense.

I am excited for August to roll around and start my third year. Feel free to tweet at me @danburf, comment on an old post (I love questions), or drop me an email at anytime. Let’s keep these great conversations rolling!

Day 179: A look back

Jeopardy Review today, if you need something last minute, this is a fun way to spend a period (heads up, there are a few errors, it is fun getting students fired up about them though. I am keeping ’em for next year)

Anyways, Back in August I posted about this 180 day blog thing. I want to take a little time to re-visit the goals I put in place.

My Goals:

  1. Get plugged into the math blogosphere
  2. Challenge myself to always be better
  3. Accountability
  4. Provide resources for others


1.  I have met quite a few people over the last year or so, Twitter + this blog + other blogs have served as a corner for meaningful discussions. I have neglected Twitter though, I pretty much exclusively use it as a way to get this blog out there. I wanna work on that next year.

2/3.  Looking back, 2 and 3 pretty much go hand in hand. In my eyes, a successful teacher holds themselves accountable to always bettering themselves. Long story short, I spent hours prepping each weekend, were interesting enough to write about each day. This was really hard and became exponentially harder as the year went on.

4.  The amazing amount of support the #MTBoS community has provided is unimaginable. It is excited to see the community grow and great to be a part of it.

One more day…

Day 129: Really Tall Things

A week ago I came across Kate Nowak’s awesome post on Trig. I was already into the heat of things, but pulled her activity on finding the height of some really tall things. Geometry had a lot of fun messing around with inclinometer apps. In turn we had some great conversations on real error. Students were getting heights of over 500 meters then 2 meters for the same object.

Next week we are diving into Law of Sines and Cosines, should be fun!

Day 125: Off

I was out sick Thursday and Friday of last week and came back to crazy today.

Algebra is working on properties of exponents. They had back to back worksheets, which I graded over the weekend.They were… let’s not talk about them actually. It feels like over the past few weeks I have been so caught up in focusing on the concept that I have put the students on the back-burner; trying to plow through content.

I don’t like feeling this way.

**30 minutes after that sentence**

A student came in for help while I was writing and we retested over a concept. This student raised a concept score from a 2 to a 4 (which is near mastery). This student has made a tremendous amount of progress throughout the year.

After the student left, I sent a email to the parents telling them what I wrote above. I have already had a few posts about this but January/February/March are tough. A lot of teachers talk about it and it is hard not to join in on the negativity.

Something I always forget is for every 1 aspect of teaching that goes wrong, there are 10 positives to out weigh it. I have some students see problems in amazing ways, I am even jealous sometimes of it! There are also students like the one above who have come so far this year. Recognizing and focusing on those students and positive interactions makes the January/February/March crunch no big deal.