Book Study: Good Questions for Math Teaching- Algebraic Thinking, Data Analysis and Measurement

jamie and jenn

I’m excited to be back this week to share with you more about Good Questions for Math Teaching: Why Ask Them and What to Ask in Grades 5-8 by Lainie Schuster and Nancy Canavan Anderson (link below on picture of book).

If you missed my last post covering What Are Good Questions and How Do We Create Them, check it out first! Also, if you haven’t ready Jamie’s post from last week on Fractions, Decimals and Percents mixed with Geometry check it out as well!

Chapter 9 is all about Algebraic Thinking and starts with a pretty profound statement, “When we ask students to think algebraically, we are asking them to formalize patterns, analyze change, understand functions and move fluently between multiple representations of data sets.”  Doesn’t that relate to all areas of math? We are always formalizing patterns and representations of data whether we formally realize we are doing it or not.

Having students make predictions in all areas of math is a form of algebraic thinking that doesn’t force them into a formal algebra lesson or activity. When we as teachers ask the “good questions” we spark their want to dive further. Computational practice is important but it is even more important to have our students understand the meaning behind what they are doing.

Mathematical processing standards guide our students to justify their thinking and connect their thinking to prior knowledge. THis can be done at any and all grade levels if done systematically.

Chapter 10 related Data Analysis and Probability with appropriate questioning. Why are good questions important when students are discovering the concepts of Data and Probability? Students must reflect on the actions that are occurring to be able to understand the probability (or chance) in their daily lives.

Data Analysis and Probability is EXTREMELY visual! Not only should students be drawing their thoughts via charts, graphs or number lines but also writing out their thinking. This is where open-ended questions that truly make students think are important.

Incorporating manipulatives is important to help build the visuals in students minds as well… dice, dominoes, marbles, number tiles, spinners, decks of cards and so much more can be used.

And finally we come to Chapter 11 which covers one of the most important areas (in my opinion) that we as teachers need to cover with students. Measurement is a struggle for many and it continues to be a weakness for students throughout the years no matter what grade I have taught.

Questions for measurement need to not only require students to measure but also make connections to what they are measuring and prior knowledge. Measurement can easily be connected to geometry, number systems and analyzing data and therefore we can continue to enforce other skills.

So as to not give too much away, I would suggest that you all grab this book if you are looking for some solid foundations on building on concepts in your 5th-8th grade classroom. The last 5 chapters cover the different strands of mathematics and give strategies on how to stimulate these concepts to be understood with students in your classroom.

Book Study Questions

Don’t miss out on this book. It has definitely changed some of my thinking on how to guide my students to achieve their own success through asking the right questions.

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Filling Students Gaps in Math while At School

Filling Gaps at School

Last week was all about Changing the Thinking of Students in Math as well as Finding the Gaps to Fill In to strengthen their learning in Math. As we continue to proceed through the series I want to take some time to talk about filling those gaps at school (today) as well as at home (next blog post).

As a teacher you are the one that completely controls the time in your classroom. No matter whether you have set curriculum, set class periods, etc. you still control what happens from the point your students walk in the classroom to when they walk out of your classroom. Fill this time to the brim and let it spill over if needed!

I’ve talked before about how I filled my time in my class periods that were 42-48 minutes in length and I know that many of you deal with the same issues of time. You have to first set a structure in your classroom and let it be known. You can’t expect students to be able to follow your schedule if they don’t know it and you don’t remind them of it regularly.

So, how do we, as teachers, fill those gaps at school?

Every year in my state the students take our state STAAR test. After the tests are scored we eventually receive an item analysis that is aligned with the reporting categories and we can see where the weaknesses for students were. The districts that I worked in also used DMAC where we could get much more detailed information for each student which proved invaluable. Where am I going with this?

When you set up any testing in your classroom, doing an item analysis is so helpful! I have taken a day and done them with my students before as a review for those who then were going to take a retest. Simply go through the different problems provided and align them with your standards. My rule of thumb was 65% or higher achievement (giving a 5% margin of error for those who had computational issues or read the problem wrong) on a standard meant I could move on with the class.

These item analysis helped build my groups of students. If 5 of my students were struggling on exponent rules, then there is a group right there! If another 5 students were struggling with scientific notation, then there we go! Using that data helped me build my groups of students.

Small Groups in Middle School

I set my students up into six groups each week and some times they would vary based on the topic while some times they would be randomly grouped. With the six groups going on I could easily have six different activities where students were engaged in deepening their understanding. During this time I spend a lot of time moving around from group to group. I typically start with a group that I know will be able to do well with just the right guidance to start them off. This frees up more time to spend with my lower groups.

Building Students Up

Once you have determined the areas of weakness for your students you need to focus on those weaknesses within small groups and even individualized practice. I can’t stress the importance of working with students individually as well. During this time you aren’t only their teacher and mentor but you are also their coach and celebrating the small successes as they build upon their foundation.

Story time…

Dylan was a 7th grade who had not been in a real classroom since second grade. Dylan had behavioral problems and IEP paperwork that was several inches thick. No one believed in Dylan. During the beginning of 7th grade it was decided by the IEP team that we would gradually bring Dylan into the regular classroom. Dylan would start by joining my Resource Math class of 10 students. If Dylan did well in class then he could go onto Texas History with Coach, if not then he was back in the behavior unit for the day. Needless to say there was a LOT on my shoulders the first day that Dylan arrived.

Dylan came in and it was Work Station Day where my students knew that we would be in groups and working on different concepts based on their weaknesses. Where was I to put Dylan when he was on a 2nd grade level and the rest were on a 4th/5th grade level? I chose to have Dylan join in with the rest of the student and work on converting rational numbers (fractions, decimals and percents) because that is what I was expecting of the others. They were using a calculator and I had taught them how to convert using their tool so they were moving and grooving. This gave me time to teach Dylan what they had already learned.

Dylan and I worked through the process a few times and he exclaimed, “I’ve got this. You can go.” For fear of him acting out I listened but kept a close eye on what was going on while I was at the next table. When it was time to rotate to the next station Dylan refused to rotate. All of my other students were okay with not finishing something as long as they had been working but Dylan had started to learn something that was beyond his expectations and he wanted to finish. I relented and let him finish despite needing that for another student.

About ten minutes into our next station Dylan called me over. He saw that I had been taking pictures of other students working in centers and he wanted to know what I was doing. I explained that it was my way of showing to parents what students were up to in my class and allowed me to share successes with others. Dylan asked if I would take a picture of him. I noticed that he was very close to finishing his puzzle and we made a deal that when he did and could show me how to do any conversion I picked that I would take a picture.

Dylan's Finished Puzzle

Dylan called me back over in about 2 minutes telling me he was done. I was amazed to not only see all the pieces in the correct spot but when I chose one for him to show me the conversion of he easily completed it without any reminder of the steps to work through. Dylan had not only filled gaps that I didn’t even know he had but also had quickly picked up on something once I took time to teach him the process and let him show me.

To wrap it all up (because I could ramble forever):

1. Use an item analysis to determine the weaknesses of your students. These can be grade level weaknesses or even prior grade levels.

2. Set up regular group work based on those skills that students are weak in. Allow students to not only work on areas they are weak in but also those that they are successful in so that they will stay motivated.

3. Work with students individually to celebrate the small successes. You will not only build the relationship with the student but you will become their coach to motivate their learning.

4. Find resources that work for your students from trusted resources. Using resources from resources you trust will make your life easier and free up your time for planning.

5. Celebrate all successes no matter how small! Call home, share with other teachers, and brag all that you can to build student’s confidence in their learning.

Question to you…

What is an invaluable resource that you have found that has helped you fill in the gaps in your classroom? Share a link if you have it as well as the concept and grade covered as I would love to compile the list and add them to my Pinterest Boards.

Finding Gaps to Fill in Math

Finding Gaps in Students Thinking in Math

When I wrote the original post about Changing Students Thinking About Math, I never realized the feedback that I would get. Not only did I receive comments but also had emails and people asking me questions on my Facebook Fan Page. Wow, I hit a topic that people wanted more info about!

I decided to not respond right away because I wanted to plan things out and really make this a series of blog posts (currently a total of 5 are planned) and make it full of information for you to use to make a difference in your classroom.

So…. How do you go about finding gaps in your students thinking so that you can build a solid foundation of skills?

1. Assess! Yep, it’s the root of all evil to assess students over and over again but it must be done to determine if what you are teaching, or what has been taught, has stuck with them. Now when I talk about assessment I don’t always mean paper and pencil assessment. Heck, I don’t even always mean computer based assessment. You as the teacher are allowed to determine what assessment best fits the skill that you are teaching in your classroom. Yep, I give you permission. (Now, I know we all have to give school/district assessments as well but that’s another day and another story.)

Informal assessment can be anything from having a student do some practice problems, to orally explaining a process to you, writing in response to an open-ended response question (gotta love Post-It, Prove Its), CGI word problems that they can journal about and more. It doesn’t end there!

How do you know when to assess? Well, I found this amazing flow chart that I think is just the bees knees (yep, I said that!) over on C.Jayne Teach. Jump on over to her TpT store for a FREE printable copy of this and KEEP IT handy!

assessmentFLOWCHART_cjayneteach

2. Talk! Yes, sometimes it is a simple as building that relationship with your students on a one-on-one basis and just talking to them. When they have developed the level of trust with you because of building that relationship they will share with you their struggles.

Myles was one of my most stubborn students. He hated coming to my class because I was hard on him. I made him come to class with his notebook, his pencil and without his backpack (school rule). I made him get his papers as he walked in the door that he would need for the day. I made him not talk to his friends as he was preparing for class and getting his Interactive Notebook ready for the day. I made him keep his head up off his desk during the class period when he got frustrated. I made him answer questions randomly.

I didn’t make him come to tutoring, his reluctance to do homework made him come to tutoring because he wanted to play football. Myles hated tutoring because it meant he had to work or I kicked him out and I would follow through by letting his coach know. Myles knew that telling coach meant running laps and possibly not playing in that week’s game.

After serious conversations with Myles about the important of learning so that he could not only play in the game each week, Myles started coming to tutoring outside on Friday mornings (I had car duty) and sitting with a group of about 5-10 students who were getting last minute homework help because it was due that day. You know what, Myles started to talk to me. He started to tell me where his weaknesses were just by the questions he asked and because of this I was able to ask him to start coming to my after school tutorials where we played games. Myles didn’t know that those games were going to help him be a better math student. Myles came with reluctance but after the first session, he was hooked. Myles now keeps in contact with me regularly and enjoys math.

Changing Students Thinking while Filling Gaps

This is Myles in my classroom last year playing a game that not only taught him about ratios but helped fill his gaps on equivalent fractions, factors, multiples and basic multiplication facts.

3. Track and Analyze! This is probably the most boring of the three things I will share with you today but also probably one of the most important. Tracking data for each student (not just the class or your entire group of students) will help you as a teacher determine weaknesses and strengths not only in your teaching but also in your students learning.

In my classroom checklists, like these created by MissMathDork for Common Core, and my handy-dandy clipboard were my best friend. I could go through and mark the skills that students were mastering and keep dates of when it was tracked all in one location for each student.

There were also times where I would print out a class roster and use it for documentation purposes for a certain skill. Many times as I was looking at their Post It, Prove Its or Exit Tickets at the end of a day I would easily be able to mark whether a student has mastered a concept, needed a bit more practice, needed more instruction and practice, or needed a complete reteach of the skill. It was that simple to help differentiate my next days lessons in my classroom.

(More on tracking and analyzing is coming soon, I promise! This blog post has spurred some great ideas.)

I won’t say that this came to me in my first years of teaching but rather took until about my 7th year of teaching to even start doing this. Now after two years of starting to put it in here and there I saw the value of these methods in my classroom. I started to fill in those gaps that my students had from prior grades. Did it matter what caused those gaps? Nope. Was I going to dwell on it and make excuses? Nope. I knew I was at a crucial point in a students math education where if they didn’t master some of the critical skills they were going to be SUNK in high school.

If you have made it this far, thank you because as you can see I had a lot to say on this topic and I only gave you three tips. Now comes my call to action

What can you do in your classroom tomorrow to help change how you fill in the gaps for one student, a small group of students, or even all of your students?

Exit Tickets in the Classroom

Exit Tickets have come to be very popular over the past few years of teaching. I remember back when I was in high school we had a lot of “end of period quizzes” that could be considered roughly the same thing as they were allowing teachers to assess what we had mastered over the course of a period of time.

What is an Exit Ticket?

In my classroom an exit ticket can take on many different forms but it serves as a valuable tool to help develop a quick understanding of where my students are in their understanding of a topic. As a math teacher, Exit Tickets can be computational, a word problem that students must decipher and compute, OR an open-ended response where students must demonstrate their own thinking and express their process to get from the beginning to the end.

Incorporating Exit Tickets

Editable Exit Ticket with Stoplight Comprehension

 From my experience of Exit Tickets, I have done them two different ways. The most common way is that the students will take part in the lesson/activity for the day and then will complete an Exit Ticket before leaving based on the skills and objectives that we covered in class. This allows teachers to take a quick glance at the level of understanding each student is at.

Entrance Slips by Teacher Studio

Another way that I have done Exit Tickets with students is to literally give them to them as they are exiting my class and for them to use to further their learning at home by looking up information or applying prior knowledge to build upon a concept. Many times these are called Entrance Tickets as they are used to allow a student to enter the classroom for the day but also can take the place of a bellringer.

Developing Exit Tickets

When you are creating your Exit Tickets, it is best to use the same format repeatedly so that your students can get started immediately without much assistance from the teacher. I have always tended to use two different methods in my classroom that seem to work well.

First is the Stoplight Exit Ticket (seen above). This method is very simple and allows my students to also give input on where they think their level of understand is based on the red, yellow and green of a stoplight. With an exit ticket in this format, I can ask computational problems, open-ended questions and just about anything because they are being turned into our Exit Ticket tub rather than place anywhere that others can see.

Post It, Prove It Exit Ticket Strategy

The other method that I started to use in more recent years is what I call Post It, Prove It. With a Post It, Prove It I will ask the class a question based on the lesson that really allows them to show me what they know. During Exit Tickets they are completely allowed to use their Interactive Notebooks as it only furthers the amount of times they are exposed to the materials.

I’ve learned a few things over my time using the Post It, Prove It strategy in my classroom:

  1. Giving immediate feedback to students is key to keep motivated
  2. Students should write their names on the back of the Post It underneath the sticky part so that their name is not seen by all students. Once I made the Post It semi-anonymous using this method it seemed to give my students the freedom to start writing more and sometimes even using a second Post It.
  3. The only materials you need for implementing the Post It, Prove It strategy in your classroom are Post Its (or any other sticky note), chart paper, and a marker to write your question. Students will also need their writing utensil to complete their sticky note.

Post It Prove It- Similar Figures

Evaluating Exit Tickets

When it comes to evaluating Exit Tickets, a grade is very subjective. Most of the time I did not count Exit Tickets as a grade in any form or fashion where there other times I wanted to hold students accountable for their responsibilities in middle school.

I also developed a sense of grading based on a scale of a pre-composed basic rubric. I determined what I was looking for based on the question that was asked and took off points based on what wasn’t there. Remember in the end only you know your students and what will work in your classroom so you must decide what is best.

If you are looking for resources on how to implement Exit Tickets and Entrance Tickets in your classroom, feel free to checkout my Editable Entrance Tickets as well as my friend Meg (from Fourth Grade Studio) who has created a resource on Using Entrance Slips.

What is Area? Concept Review with Task Cards

As we are nearing the culmination of time before our 7th Grade STAAR testing for Math, I spent some time reviewing some key concepts that were tricking my students this past week.

area of irregular figures preview

I was so happy to come across these awesome task cards by Miss Math Dork where my students were required to find the area of irregular figures. Visual characteristics of a shape are definitely a weak point for my students but I think with the key steps I taught them, they should have it down.

What Is Area Concept Map

 Just a reminder, I teach in with Texas Virtual Academy. My students come to class in a virtual online classroom and we do a lot of our work on a virtual whiteboard. I started off asking them what area was. I wanted to see what level they were at with their use of the word as well as the previous knowledge they were bringing in to class. As you can see, my students came in with a basic knowledge and some of them even corrected each other as they were writing which is awesome!

So, next up we talked about the steps that we could go through to break down larger figures. This sort of guided them to where I provided each of them with the following handout. {Hint, hint… click on the picture to download it FREE for yourself}

Finding Area of Irregular Figures by 4mulaFun

Okay, now we have all of our tools in hand to start working on some amazing task cards. Miss Math Dork made it so easy for me to show a task card to my students and then break them down into individual shapes and talk about congruent lines all at the same time. Lots of mini-lessons all in one and it ROCKED! {I will warn you, the next paragraph shows the different conversations with my students in bold purple.}

areataskcards2

Of course my group wanted to go straight to adding up all the sides. ARGH! Let’s refer back to our stepsWhat do we do first? This is when the whining started… Oh, we need to break it down first. So what shapes do you see in this figure? Rectangle was obvious to them at first so I highlighted the border of the rectangle. 

So how do we find the area of a rectangle. That’s easy Mrs. S, it’s length x width. Show me! 5 cm x 4 cm is 20 cm. Now what other shapes do we see. This is when the student’s argued over breaking down the final area into one triangle or two. We finally decided that based on the measurements we had it would be easier to do just two triangles since we could multiply the area of one by 2 to find the total area. 

As you can see, we worked this out on the task card and my students loved showing their division skills and eventually their adding with decimals.

areataskcards

 Now, this one totally threw them for a loop at first. It was easy to pull off the rectangle and then a given triangle on the left. Which is funny because after I had done this with my students, I talked with Miss Math Dork and she originally had this planned as a rectangle and a parallelogram. Goes to show you the versatility in math. Now, I had to help my kiddos on this one because we don’t teach Pythagorean Theorem in 7th grade and therefore I had to give them the 3rd measure of the triangle. In the end the hardest part was getting them to break down the top measurement of 15km into two different measures for the two different shapes.

By the end of the class my students were ready to work on some of the task cards on their own. I gave them each a separate task card that they had to print out and submit back to me showing their steps kind of like a task card. They did awesome and I loved seeing all of the highlighted areas of various shapes to make up larger shapes.

Wow, if you have stayed with me through all of this, THANK YOU! Now, if your students struggle with Irregular Figures as much as mine do, you might want to pick up these awesome Area of Irregular Figures Task Cards and add them into your classroom for some awesome, out of the box, thinking.

missmathdork

Looking for another great math idea? Check out what Miss Math Dork has in store for you over on her blog today!

P.S. Like the FREEBIE? Leave a comment maybe…

Another Giveaway You Say?

BirthdayBashGiveaway

Are you up for another giveaway? My blogger friend, Confessions of a Teaching Junkie, is celebrating her birthday in a big way! And the surprises are for YOU!

6-8Grades_Bundle_Giveaway

I joined in with SEVERAL other bloggers to donate to a giveaway for Grades 6-8 and I urge you to join in on the fun. You can’t win if you don’t try!

DataAndGraphsBundle_Giveaway

Up for grabs from me is my Data and Graphs Bundle available for $8 on Teachers Pay Teachers! This bundle is designed to use in Interactive Notebooks and engages the students as they learn more about Data and Graphing.

Ready to join in on the giveaway? Click HERE to enter to win!