Conceptual vs Skills (The Pendulum Ever Swings)

When I first started teaching, I really didnt know what I was doing.  I do what most teachers do, I taught how I was taught math.  My second year of teacher the school adopted the Connected Math Project curriculum (CMP).  We were trained to teach the curriculum and it was fully implemented the next year.  I believe I learned more about middle school math teaching with this curriculum than I did when I was actually in middle school.  I continued to use CMP, and then CMP2 almost exclusively for almost 10 years.

I found that although the students lacked a couple skills here and there, they could really think through problems.  They were great at math reasoning and understanding the big concepts.

With more and more emphasis on standards, and with my master's research being about standards based grading, I have slowly drifted away from CMP and more and more towards a traditional looking math classroom.  I currently do not use any textbook.  I gather materials from various textbooks and sources.  While the overall plan was to stick with my favorite parts of CMP, its use has slowly diminished over the past couple years.

I didn't reallly realize how far I had drifted away from my CMP roots until about 3 weeks ago when I attended a local math conference.  There was a keynote address about Conceptual Understanding vs Skills Proficiency.  One of the tools used in this presentation was the "How Old is Your Shepard Problem."  (view video here)  It was pretty funny and convincing of the need for conceptual understanding.  I thought to myself that there is no way my students would do this terrible at the problem.  National average is that 25% of students see the problem for what it is, unsolvable.  When I polled my own students, only about one-third of them successfully recognized the problem as unsolvable.

Some of my student's responses to How Old is the Shepard:
"62, because in the Bible, shepdards look old and 62 is old."
"37 because he must be living by himself and you probably have to be so old to own sheep."
"There is no shepard."
"60, it seems like the shepard would have to be older if they have so many animals.  Unless there were a lot of shepards then there a bunch of middle aged guys."
"70 because shepards have grey beards"
"There isn't even such a thing as a flock of dogs."

The Conceptual vs Skills debate has been the eternal argument since I have started my math teaching career.   I got really sick of having this argument with parents and other teachers.   Things got really ugly form both sides for awhile.  It was the main cause of some "parent's nights" in math we had at our school where our program and myself were attacked in public.  It was not a fun time for anyone.  I always vowed to avoid that type of fighting in the future.  I now find I am having those same exact arguments but inside my own head.   I have come a realization that my own pendulum has swung too far to the skills side.  I need to try to find my center again.

The timing is kind of perfect, as in the 8th grade we are just starting our Pythagorean Theorem unit.  This was always my favorite when using CMP.  So this week we tried our first couple days using more materials based from CMP.  It got off to a rocky start.  Partly because I was a little rusty teaching in the "inquiry" style and partly because the students have not had a lot of practice at it.  However, when I saw the looks on the students' faces that discovered the Pythagorean Theorem all on their own, and could not wait to share with the class their marvelous discovery, it reminded me how powerful this can be.

I am sure at some point my pendulum will swing too far back the other way and I will need another course correction, but that worry can hold off for another time...

Something funny and kind of scary happened....

So on Monday this week, I woke up really sick at about 3 am.  Now it so happened that I had planned on getting most of my lessons ready to go during my period 2 prep that day.  So I knew that I had not much ready for a sub to work with if I just stayed home.  Throw in another fact that I was supposed to be running a junior high math league meet 45 minutes before school started.  So I ended up pulling myself together enough to get through the math league meet.  Then I would use 30 minutes to get lesson plans together for the day.  I would then head home and let the sub run the rest of the day for periods 3-9.

So after no sub signed up for my gig, I told the secretary that we only needed someone for periods 3-9 because I would be there for the first two periods because of math league.  She said okay, I ran my math league meet, created some activities and headed home.

My usual routine when I am going to be gone for the day is to email lesson plans not only to our secretary, but all the students as well.  I find it usually helps the sub out and helps the students know what is going ot happen that day even though I am not in the room.  On this Monday I did send an email to all my students giving a quick outline to the plan for class and I attached the necessary materials.

So when I get to school on Tuesday, and interesting thing happened.  I immediately had 3 students run up to me and say "We had no teacher Mr. Sieling!"  I said "What?"  They told me "all from the same class, that no teacher ever showed up to run class on Monday.  It happened in just 1 class period, but still I was a little worried about what had happened.

So I believe they came into class on Monday, realized there was no teacher.  At some point looked for the normal Monday game, realized there was no game, no teacher and then started working on their assignment.  Now, these are 7th grade students.  I doubt it was quiet.  In fact, a group of girls moved to work in the hallway because "the boys were loud."  However, they all got their work done without the teacher even being there.  I was pretty proud of them for that.

Quizzing to Promote Mastery

Its been a long time since the last post.  I have been pretty busy lately, even more than usual.  From coaching junior high and high school math league to coaching the inaugural year of our junior high robotics league to presenting at various conferences, it has been a crazy winter so far.  I thought I would try to get a quick post out while I have a moment.

In 8th grade we have been studying solving equaitons and inequalities that past 2-3 weeks.  The students took a paper/pencil quiz on equations in December.  So I tried a different way to quiz last week.  I got the foundation of this idea from a book, don't remember which one, I will try to find that title later.

The basic gist is that students answer 1 question at a time on their own.  We switch, correct that problem in class and go over how to do it.  Then we try another question (or set of questions) and repeat.  We do this over and over again so students can be remediated during their quiz.  I like the idea of this "quiz to mastery."

I took this to the next level by adding my differentiated rubric to the idea.  It went like this, the first round of the quiz everyone tried to solve 1 Level 2 question.  (leveled grading blog) When people were done we switched and corrected that question.  If the student got it right they earned at least a score of a 2 on that quiz.  If they got it wrong they got some help about their mistakes.  When we were ready (about 3-5 minutes later) round 2 began.

Round 2 now had two questions on the board.  The slide had two questions on it:   level 2 question that was similar in difficulty to the first question, as well as a level 3 question for those who got the first one question correct.  After about 3-5 minutes we switched and corrected those two questions.  We again spent some time helping each other out and figuring out mistakes.

Round 3 had three questions on the board: Level 2, Level 3 and Level 4.  We continued this way until the end of class.  We got in about 5 rounds of this style quiz.  The score the student earned on the quiz was the highest level of question they got correct.

Overall the students seemed to like this style quiz.  There were some who did not like it.  The biggest reasons seemed to be that it was different or they did not ge the score they wanted.  This type of quiz works particularly well for skills like solving equations.  I may use it again for solving inequalities, I will probably leave that choice up to the students.

Here is my google slide presentation for giving this quiz to mastery.

Parallel & Perpendicular Slope

I have always struggled getting students to really understand how to use slope to create parallel and perpendicular lines.  I have tried many things and many different activities over the years.

This year I did the typical investigation about parallel and perpendicular slopes.  It works pretty well for setting up 3 sets of parallel lines and 3 sets of perpendicular lines and asking students to analyze the equations for patterns.  This took 2 days and went pretty well.

Now on the third day I wanted them to put this new found knowledge to use.  So I created the following DESMOS graphing challenge for them.  (feel free to use and change for your use)

The challenge started pretty basic to make sure they correctly recalled the patterns for parallel and perpendicular slopes.

The next couple asked the students to start creating shapes using their patterns.

There were a couple more like this and students overall handled it pretty well.  There were a couple questions about what a parallelogram was, which I totally expected. 

A few students got off to a slow start because they did not recall anything about parallel and perpendicular slopes.  This did allow me to find them quickly and re-teach the concept.  Once they could visualize the graph on DESMOS with the slopes they seemed to do much better.  

A few student got off to a slow start because they did not understand what they were being asked to do.  This was partially because my projector lamp blew up and I had no way to project anything.  It was also because I threw this together quickly and I don't think it was as clear as it should have been.  I will have to tackle that next year.

Overall it worked pretty well.  We will see if this is the year where I can get students to remember perpendicular slopes!

Chris

Class Debates in Math Continued...

So yesterday we had a debate about math totally go off the rails.  Students kept arguing without really basing anything on logic or facts or even good common sense.  It sometimes sounded like a 24 hour news network where people yell "I''m right because I feel I am right!" and then someone else yells "No, I'm Right, because I'M LOUDER!"  Only 1 section had this, but still I couldn't stop thinking about how badly this went.

So today we started number talks.  Number talks are just a format for discussing interesting math questions that stress reasoning and estimation skills.  Fawn Nguyen is the amazing teacher behind this website.  Her blog is awesome of course.

We started with the first question on the site.

"Are there more seconds in a day, or inches in a mile?"

Before we started, I wasn't sure what to expect.  I wasn't sure the students would be engaged, participate in quiet thinking time or be able to explain their thoughts.  I was worried that students wouldn't even know where to start and give up right away.

So I put together this smartnotebook slide to help keep the class focused.

The timer is clearly displayed as well as the steps.  Also the blue box is a random name generator.  I like having the first name displayed because it gives fair warning to that student that they are going to be expected to speak about this question.  (you can also see I type to fast and have a typo in the question!)

So every section today went extremely well!  Students respected the 2 minutes of initial thinking time.  I could see some students racking their brain for an answer, a way to approach the problem, or just an idea.  Some students looked around like "He is crazy if he thinks I can do this in my head!"

The discussion time was the best.  All students were engaged.  I mean 100% engagement.  Students either could not wait to share their ideas or they could not wait to hear someone give them a clue about the answer.  I heard great discussion all day long about this problem and different and fantastic ways to approach it.  

The sharing part was a little rough in most classes, but all reasons for picking a side were grounded in math and common sense.  There were no 24-hour news channel type arguments, just reasonable debate.  

I told them I plan to do a problem like this 1-2 times a week.  Most were excited.  Some groaned.  However when I asked them why they groaned, they almost all responded about "having to think to hard"  That is just what a math teacher loves to hear!

Classroom Debates about Math...

Today the 7th graders were studying integers.  We are getting ready to add and subtract them.  Yesterday we represented adding and simple subtracting on a number line.  Today we went with a chip board, with black chips representing positive numbers, and red chips representing negative numbers.

I told them that we are going over multiple representations of adding and subtracting to get to the best understanding we could about the operations.  I emphasized we are not going to depend on memorized rules, we must strive to understand adding and subtracting.  This was of course met with some blank stares and confusion.  During their previous year in 6th grade the teachers quickly skimmed the service of arithmetic with negative numbers.  Of course they talked about rules like adding two negatives is always negative, multiplying two negatives is always positive and so forth.

Everything was going quite well with the chip board lesson for the first 30 minutes.  Then with 14 minutes left I asked them to solve -4 - -3 and represent the solution on a chip board.  This was the first negative subtracted by a negative problem of the day.  The previous section and last section of the day handled this fine.  They used the methods we had talked about and it resulted in a short discussion.

However in my middle section of 7th graders, most of the students had ended up with -7 as an answer.  The seating arrangement of the class has 6 groups of 4-5 students in each group.  Four of the groups had come to a -7 conclusion.  One of the groups came to a -1 conclusion.  The last group knew the answer was either -7 or -1 but they couldn't decide which way to go.

So we set up a little debate on the topic.  A very loud and confident 7th grader proudly proclaimed a misstated rule from 6th grade and proclaimed -7 to be the answer.  I held back any judgement and most of the students nodded in agreement.  I then called on a student to present their argument for -1.  They went to the smartboard and explained in a perfectly correct way that -4 - -3 was -1.  No mention of rules they remembered they just used the chip board.  When they finished 2 students loudly and confidently recited different rules incorrectly proclaiming -7 to be the answer.

At this point I reminded them how using rules can be confusing and we have to focus on what adding and subtracting means.  This fell on completely deaf ears.  At this point the 4 groups held fast to their wrong -7 answer, but there were now 2 groups solidly in the -1 camp.  We were also down to about 4 minutes left of class.

Now I love student debate, I love discussions in math, I love to hear the thought processes of students, and I whole heartedly believe that mistakes push learning forward.  However, I just couldn't bring myself to let the class leave thinking that -4 - -3 was -7.  So I asked for anybody to argue for the -1 answer, because there were several students still shouting out arguments for -7.  When nobody stepped up I felt I had to make an argument for -1.

So I asked them the questions that I thought would end the debate:
"If you have 4 red things, and you take away 3 red things, how many red things do you have left?"
They all correctly answered 1 red thing.  The 2 groups beamed with delight that -1 was correct.  Most of the other students looked a little confused, and 3 students starting misquoting rules to me about negative numbers to continue arguing for -7.  At this point I told them that -1 was the answer.  I showed them in a similar way to the first student using the chip board that -4 - -3 was -1.  Some students still did not believe me.

At this point there is about 1 minute left of class and I am desperate to make sure students know that -4 - -3 is -1.  So I made a last ditch effort and pulled up my smartboard calculator.  I typed in the problem, asked them if it was entered correctly and hit enter.  When -1 popped up as the answer, I actually think there were still students who thought -7 was the answer.

There is no great ending to this story yet.  I have to tackle all the misconceptions tomorrow.  I am pretty sure I completely messed up this lesson and students thinking about integers.  I am not sure what I should have done differently after the debate started.  I made sure in my next 7th grade section to take the subtraction slower and emphasize that subttracting is 'taking away.'  Now I just have to figure out how to right the ship for a group of stubborn 7th graders.

PLCs, sharing ideas, creating lessons and Statistics

I just want to share how much I love my PLC group.  In our school we do PLCs by content area.  We are a small school so we have 2 math teachers, 2 science teachers, 1 STEM teacher, and a special education teacher.  We meet about every other week for 1.5 hours.  It is always an amazing experience.

2 meetings ago, I went in to develop a lesson on mean and median.  I mentioned how I always struggle with students realizing how an outlier can effect mean much more than median.  Through discussion and brainstorming we came up with a great idea using blocks, a meter stick, and a balance point.  This lesson would have never come together the way it did without my PLC group.

The lesson involved creating sets of data with a mean of 50.  A meter stick was then balanced and blocks were placed at the data points to create a balance.  Students were asked to put on an "outlier" block and recalculate the mean and median.

This activity was also used to visualize missing data point problems with a given mean.  Students placed blocks on the meter stick and saw how it didn't balance.  Then they were asked to place a block so the meter stick would balance at 50 (have a mean of 50).

A quick bare-bones handout can be found here.