Wednesday, November 23, 2016

Math That Keeps Me Up At Night

I have started many blog posts that I have not been able to finish.  Many of them center around the same questions.  I can't seem to pull my thoughts together.  Sometimes the thoughts are sparked by something I read on Twitter.  Other time a colleague says something that seems to resonate with me or perhaps is strikes a nerve.  Yesterday was no exception.  Whether I am able to complete any of these thoughts or simply babble, I am committing right now to publishing this post.  So here goes:

This year I have focused on building conceptual understanding of math.  I want students to make sense of the math and see the number 7 as more than just a 7.  Think about it.  It does mean a certain number (in this case 7) items, but what are those 7 items?  7, 700, 879, 7(4), 3 to the 7th power.  Students do not understand that these 7s while looking the same represent different things.  This is not an easy undertaking as a teacher.  I believe that if I can deepen their understanding all of the pieces of confusing math will start to come together for them.  Is this a pie in the sky thought?  Am I misguided in this thinking?  Or is it developmental?  At what age are they truly ready to understand this?  Is my expectation too high in 7th grade to think that students would be able to slow down and make sense of this?

One statement that continues to stick with me is a statement about meeting students where they are at.  I don't remember the exact quote but it was something to the effect that not all students are going to be able to understand math, so we need to give them the rules to memorize so they can do the grade level math that is being asked of them.  What is my goal as a teacher?  Depth of knowledge takes a lot of time to build.  In 7th grade students know the algorithm and (while I like to think it was built on number sense) no longer remember any of its meaning.  These students can more or less get As and Bs in math by using what they have memorized.  But what does that mean for higher level math?  Am I really doing my job as a math teacher if they don't really know math, but have simply memorized it?

Enter the models and visual representations.  These are great tools for understanding algorithms.  If a student already knows the algorithm do I care that they can show it on a model?  I do if they are able to do it on their own.  If I have shown a model and now these students are just memorizing how to show it on a model what is the point?  But after spending great amounts of time trying build deeper knowledge of decimal and fraction operations I feel my students would be scoring better on assessments if I had just given them rules to memorize and follow.  They aren't ready for the assessments because they need more time to work with numbers and make sense of it.  How long is too long?  I need to move on to the other skills.  These are actually review skills in 7th grade and I feel like I am going back to the beginning to get them to understand place value and multiplication.  Perhaps my approach is wrong.  I am trying to build concepts and they have the wrong background knowledge.  Maybe I need to look at it a different way.  I don't want to just explain their memorized rules.  I want them to understand it.  Is that a different approach than if they don't know anything about decimals (does this kid even exist?  Probably not).

Part of me has thought that I should be working in an elementary school to get a better understanding of this.  But my heart breaks for my middle school students who are so far below grade level.  Who is going to help them if I am not there?  Someone with a passion for numeracy and building number sense must be in the middle schools to help these kids see math as more than just memorized rules.

Yesterday I was talking with our computer literacy teacher.  The students were working in Excel and unable to write a formula to find a percent.  They knew to take one number divided by the other but then wanted the computer to move the decimal (or drop it) but did not know what math this was.  AAAAAAAHHHHHHH!  Our 7th graders do not know that moving a decimal is dividing by 100?  This is a travesty.  We have failed our students.

Part of the problem is our assessments and grading system.  A colleague brought up at a math team meeting yesterday that perhaps we are not asking the right types of questions on our assessment.  If students are able to memorize rules with no understanding and pass our tests we are sending the wrong message.  I love all my coworkers, but this one made my heart happy.  She suggested adding even just one question on the test that asks a little bit more than computation.  We used to have these on our tests.  Students weren't able to get them correct or found them confusing and so slowly they were removed.  One teacher made a comment that we would need to change how we teach if we put questions like that on the test.  I agree completely and think it is time for that.  We do need to change our teaching.  We seem to have simplified everything and are now teaching to the lowest common denominator (did I use that phrase right?)  We are setting the bar low so everyone can be successful, instead of helping students be successful and truly understand math.

Standards based grading versus letter grades plays into this too.  Our job as teachers is to help students be successful.  In a letter based grading system that means getting As.  In a standards based grading system that means helping students reach the standard.  Our assessments would ask deeper questions so we could figure out where they are on the spectrum of learning that skill.  Students and parents would be asking for more than rules to follow to help their kids grow in their learning.  In our letter grade system, we just simplified the test and asked, what is the most basic thing students need to do to with this skill.

There is still so much more swimming in my brain, but it did feel good to get that out.  Now I'm off to the store to buy Thanksgiving foods and try to relax my math brain so I can spend time with my family.
I truly am grateful for my wonderful coworkers, Twitter friends, and PLN who put up with rants like the one above as I continue to learn how to teach math.
Happy Thanksgiving.


  1. I'm a 4th grade teacher, and I could've written this exact post! I struggle with the same questions all year long. If I continue teaching for understanding, my students don't perform as well on standardized tests. However, they understand what they are doing, and why they are doing it, more than most. I wish we were in the same district. It would be nice to know if it pays off as the students move on from me. Do they remember all the things we worked so hard to understand? Does that save later teachers time because the understanding is already there? I feel like if we could all focus on math understanding for the first 7-8 years of school, and forget standardized test performance, the students would end up performing better when it really matters in high school and at exit level. Right? Because they would have a good, solid understanding of math . . . am I crazy?

    1. Kelli,
      You are right the part of the problem is we only have the students for one year and don't really get to see the fruits of our labor. While the research supports teaching for understanding will get the students farther in math, does it work if it is just for one year? This is why I believe the role of district math coaches, providing professional development for math teachers is so important. Unfortunately my district does not have one. Like you point out though, it would be interesting to have this conversation K-12 and focus on what math truly is instead of rules that will help with standardized tests. I think students would enjoy math so much more.