I want my students facing confusion

Some students listen, watch a couple examples and then may have a question or two before diving into a set of problems.  There is a good chance, if you are a math teacher reading this, that you were one of these students.

I have a few students for whom this is sufficient.  You will notice I wrote a few.

Most of my students do not act like this.  They doubt themselves, frequently because they struggle with math.  These same students, when I can work with them in small groups are more willing to make mistakes.  Surprisingly, I've also found that frequently it just takes 1-2 problems for (these) students to work through the difficulties.

Here is a bold statement... In math students need to make mistakes

Students struggling need to face their misconceptions.  They need to see what doesn't work and see why it doesn't.  They need to sometimes make the mistake enough times to remember it the next time they see such a problem.

I have a student who is not shy about when he is confused.  (I love that kid). He makes great mistakes, most of the time once.  Without trying he will never clear up his misconceptions.  He needs to make his mistakes.  Looking at more problems doesn't personalize the practice, nor is it as likely to shatter the walls in his understanding.

Next to that student is another.  He sees all the mistakes the first student makes, but he doesn't get the same experience out of them.  Each student needs to be willing to make his/her own mistakes and learn from them.  Without this student's willingness to try or face his own mistakes the chances of growth are diminisihed.

Those first students I mentioned, the ones who find math easy, they need to make mistakes too.  For those students, without mistakes they lose the experience of having to practice and study.  Every math student eventually hits a wall and I want my students to know that when it happens to work through their mistakes.

So, let start celebrating our students' mistakes!

First Activity - 8th grade

Today my 8th graders were given a task.

They were asked to pick a perimeter measurement out of a hat (on index cards cut in half).

They had to draw out 3-4 rectangles with the given perimeter (for example 18).  They then had to find the area of each of the rectangles they had drawn.  Given this information they had just created they were asked the following question:

How do you find the dimensions (side lengths) that will give you a rectangle with the largest area? Explain....

Once I've demonstrated and answered questions I am mute.  I will only point out that there are other members of their group as well as the information I got when I did the original demonstration on the board.

Overwhelmingly students who don't confuse the terms Perimeter and Area are doing well.  I purposely avoided any perimeter with 4 as a factor as I didn't want the square result to be so obvious. Of approximately 20, 3-student groups, less than a quarter of them are figuring out that result.  most of the remainder are noticing that if the numbers are consecutive that the result is the largest area.

Its a start....

I have a plea and a request...but I also have an idea I would love help with PLEASE #Math #desmos #google #getkahoot #mtbos

I love how Kahoot energizes a class.   I love that it shifts the environment and lets me see my kids in a different way.

I dream of a similar game which would be usable through Google for Education, as an app.

My image is one in which all students get a copy of the same 4 graphs.  After looking at them for a teacher determined amount of time (in a small group or individually) that the students are presented with an equation - or a description - and asked to eliminate one of the choices.  At this point a timer might be set, discussion time may be given or an immediate (private) vote could be done  [A,B,C,D]  Nice formative assessment

Groups might all vote as a block.  Or maybe well considered splits might develop even within the groups.  Now ask them to discuss either why that last one was eliminated or explain why another one could be removed.  Again after discussion voting can ensue, or the teacher may ask groups to share their discussions.  More formative or even early summative assessment

They can discuss again or go right to picking which one of the two remaining graphs is correct, or options would be available for a more discussion, voting and then the reveal.

I would likely want to be able to follow up with 1 or more similar pages.  I definitely would love kooky music, with the option of being able to put in one's own.  Finding a way to export the data collected as well as add that same data to ongoing graphs of (individual) achievement (can be the same as grades, but does not have to.  ie, if I am collecting data on whether a student is achieving or losing comprehension of a particular learning goal. (types, slope, x-intercept, y-intercept, length, angle, intersection(s),domain, range, etc) more opportunities for formative/summative evaluation)

I imagined this as a website that I could put together for use in my own classroom (with google forms collecting the information) but when I started thinking of building the graphs and copying each one it just seemed easier to give a shout out and see if someone, ahem DESMOS, Kahoot or Google. might make my dream come true.

I admit I might have set the bar quite high, but don't we always tell our students to aim for the stars?




Happy Belated birthday bro (Josh)  <=== he complained that I've had a blog and don't mention him j/k

My Favorite Math Problem

I had a number of  great math ed professors,  Dr. Brumbaugh is one.  Sadly, I do not recall the name of the one whose materials I still, over 20 years later, look through for inspiration and help. (btw...UCF  Go Golden Knights)

He made every one of his students submit 3 original problems.  They had to be unique, which at the time meant that the problem, or one like it, could not be found within the class materials.  The materials for this class were not a textbook, but a copy of 2 binders that the local Kinko's had.  You walked in and told them you were in professor's class and they sold you 2 binders full of previous student's submissions as well as whatever the professor had added to them.

I was working with 5th graders and I wrote a problem that my students solved with little assistance in about 5-10 minutes.

We had to present our favorite problem in front of the class.  My turn came up and I presented my problem, but because it was the last one of the day (and the professor had a thing about promptness) I didn't get to complete giving the explanation.  He said that we should do the problem and bring a solution to class on Tuesday.  It was Saturday (this being the lab for the Tue/Thur class).   By Tuesday I had well over a dozen students ask me for help.

I've found that 5-6th graders rarely take more than 15 minutes to figure it out, whether they get the correct solution or not.

High school'ers take 30-45 minutes and rarely get a correct solution.

College people, especially those with STRONG math backgrounds fill pages and pages with work and get to the right answer, but are uncomfortable with the answer they have.

I call it the bionic bee problem.  It hinders the student in that the more math they know, the more math they try to apply to finding a solution to it.

I love it for the same reason I love www.mathwithbaddrawings.com , its all about a bad drawing and a great story.

I wish I had his ability in my blog....  maybe I till see if I can draw something and add it later...

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There's this bee.  Not a normal, everyday bee; one which has been bionically enhanced by a secret governmental agency.  Or maybe the illuminati.  It has been enhanced to have perfect reflexes and the ability to fly at 200 mph.  

In order to keep the bee from getting splatted, it is gyroscopically synchronized to turn 180 degrees in case of an imminent collision, returning exactly in the direction it was coming from.   

For some reason the bee has been released on a straight railroad track, well ahead of an oncoming train.  The train is travelling at 77 mph, so there is no way it should be a threat to the bee.  

**At this point I ask if anyone has any questions.  Invariably the question of why it was released in front of a moving train comes up.  I tell them, "hey, its not like the (FBI, CIA, NSA, Etc) tell me why they do anything"  Why do you think they released the bee on a straight section of railroad tracks?

A signalman accidentally transfers another train onto the same straight segment of tracks, this time 300 miles up the track.  This train is travelling 73 mph right toward the bee and the other train.

I'm happy to say that just as the two trains collided together, the bee falls out of the way, exhausted but safe.  Which is good, do you know how much bionic bees cost?

My question to you is, how far will the bee have flown when the trains crash together?

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For high school students used to Algebra, that the tool they use.
College students use Calculus, and sometimes Algebra as well.

For them I add...

What will the flight of this bee look like (zig-zagging back and forth between the two trains).  

I've never typed it out and given it to students.  I may, depending on the class and usually only at the middle school or below level write everything on the board.  (the bee, tracks and trains always get drawn).

6th grade CCSS (6.G.A.1 - 6.G.A4) Perimeter Area Volume

Solve real-world and mathematical problems involving area, surface area, and volume.

CCSS.MATH.CONTENT.6.G.A.1
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

CCSS.MATH.CONTENT.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

CCSS.MATH.CONTENT.6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

CCSS.MATH.CONTENT.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

These should be a Godsend to 6th grade math teachers.  Perimeter allows you to have your students practice addition (and if you are creative subtraction as well) and area does the same for multiplication. Allows you to remediate while applying a previously learned skill.  I'll admit my students do well up until surface area and then it just isn't pretty thereafter....

I have decided that this will be my first concept to polish.  I have a good lesson, I have the students square a piece of construction paper (diagonal fold, remove bottom piece) and then they cut the square into other shapes, a square, two smaller rectangles, a large triangle and two smaller ones.  We then use these shapes to discuss area and fractions and percentages (I also use these same shapes in discussing variables with my 8th graders and I even use them to discuss systems with my Algebra students).  I have a few versions of this lesson, and will share it if asked.  
Giving the students partial nets, and then complete ones to find the areas might make more sense to my students rather than starting with the 3D and then trying to backtrack.  

I showed a tissue box and then construction paper cup outs to correspond to the differently area'ed sides of the box and then introduced nets.I think starting with partial nets, then nets will more easily allow my kids to see that a net is only individual shapes added together.  Getting them comfortable with adding areas as shapes are added might be easier than trying to see adding areas while translating a 3D shape to 2D faces. 

Remediation:
I've asked nearly every class I've had for the past 15 years and only rarely have I had students who remember being taught that multiplication can be modeled beautifully through the use of blocks. When I show students that the numbers we call squares are called this because if you build the multiplication out with blocks you get a square, they are amazed.  I have a small amount of cm cubes and a small number of wood 1in cubes and I used to use these more than I now use powerpoint, or google.  When asked this year if there was any office supply I wanted for next year, I asked for 2 things.  #1 graph stamps and #2 half-inch square paper.

 I want my students building rectangles and seeing the multiplication.  I want them adding the sides in time and eventually telling me the surface are and volume of the shapes they are making.  

I see lots of multiplication practice.  I see lots of being able to visualize what the math is showing.  

8th grade Transformations

CSS.MATH.CONTENT.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Not one of the best units for my students this year.  It was basically where we started the year (which in hindsight might need to be modified).  Translations were easy enough.  Students could easily see that when a shape is moved that the basic shape remains the same, but the location changes.  

Dilations, rotations and reflections, however, were not as easy.  The first problem I came across was that students had difficulty picturing that a 1x1 square that doubled in size (becoming a larger square) didn't have an area of 2.  Even after drawing many examples in their notebooks, I still had students who struggled with seeing this.  

Next, I noticed students confusing how a polygon would look when rotated.  A triangle with an acute angle pointing in one direction in the original image should end up with its acute angle facing another direction, but this wasn't the case with many of the students. Reflections were confusing as well.

I am thinking that I will start off next year looking at shapes (capital "T" comes to mind) and go through the transformations (without dilation) before even introducing the idea of graphing this shape.  (which explains why I keep asking everyone where the district die-cut press has gone)  I figure if the students can keep picturing what happened to the "T" it might give them a visual cue to when there is a problem with their doing these transformations on a graph.  

I also will be stressing the graphing aspect of this more next year.  The way this has been done has been with the graphing being an after-thought more than a goal.  I think seeing the changes and coming up with ways of describing these changes (either by verbal or symbolic description) adds necessary depth to the topic. 

Remediation is hard

i wish I had a magic wand,  a perfect solution....the WAY!

It would be so much easier....

Remediation is a matter of understanding the student,   It's a matter of knowing how best he/she/ze will see the concept most clearly.

I've known teenagers for whom subraction still makes no sense.   Division, fraction and number sense confusion is so common as to assume that it's confusion should be expected.

I love my kids, but I ache that I cannot possibly help all of them have experiences to clarify all the misconceptions, never mind successfully develop a full understanding of all the current grade level standards as well.

I do know that disruption and sleeping as well as not doing the work is definitely positively correlated to not participating, not doing homework and not answering questions in math class.

Motivation is a killer for students.  Often-times a student's home life can make our best efforts feel like standing still.  We are not alone in this, the students feel it as well.

It's true that math is sequential, and it builds upon itself.  It's also true that many of our students memorize steps, not understanding the why.  The good news is that continuing to build basic skills and remediating misconceptions does slot in lost concepts and I've seen students with that "ah-ha" look in their eyes.

6th through 8th graders have learned the 4 basic arithmetic operations, they need opportunities to hone these skills and practice them.  My 6th graders practice divisibility, factor trees, do kenken, review vocabulary and skip-count (to practice multiples/multiplication and learn to see linear/arithmetic progression, though I don't share the motivation with them).

My 8th graders need to better explain their thinking (yes, Mathematical Practice #3, construct viable arguments - I'm looking at you).  They need to work on being better at showing their work/thinking so that in time they can be better at those explanations.