Curriculum Examination

Textbooks are a hot debate topic lately, as many states move away from funding textbooks for a variety of reasons. Some of the most vocal opponents of textbooks point to a lack of differentiation and focused instruction to back their resistance. Textbooks are unique in that different subjects will have better results with the text. A history textbook may be bland and unchanging in a world where history is constantly evolving, but a math textbook with problems to solve doesn't alter much over the years. The methods used to solve them may vary as new methods are discovered but the problem still remains useful. While the defenses against textbooks certainly have their merit, I believe a textbook can only be as innovative as the teacher who is using it. 

To highlight this point, I would like to evaluate a textbook I have in my classroom. This textbook is outdated with the current standards and does not include many of the advertised resources that it once came with. In my classroom, I currently have possession of approximately 25 enVision MATH textbooks, Copyright 2010. (ISBN-13: 978-0-328-42028-5)

Upon first inspection of this set, I disregarded them as useless. Without a teacher's manual and the additional resources that came with this text, what I had in my possession was a book of problems. Not only were the resources no longer accessible, but the context of the problems had either been moved up or down in grade levels with the adoption of Common Core Standards. How could that benefit me as a teacher? From previous experience, I know that a teacher does not simply assign problems from a text in order to improve student knowledge in math. There are teachers who would use a textbook in this format without any additional resources and firmly believe they are teaching their students. They might work through the demonstration problem in the beginning of a lesson, allow students to work together on the Guided Practice portion of the lesson, and expect students to work on the Independent Practice portion on their own. Theoretically, this follows the steps many teachers learned in college for creating a perfect lesson. 

In researching the tools available for use with this textbook, I have unearthed a comprehensive curriculum that can be utilized to create differentiated lessons. The textbook series comes with access to pearsonsuccessnet.com, a website teachers can use to build a complete academic plan. There are games, songs, student success trackers, exam builders, and intervention activities. All of these are excellent resources that can be used in conjunction with the textbook to create interactive lessons. 

In the current "Math Wars", this text would fit with traditional textbooks. The text does not explicitly show students how to use controversial strategies, however if that is the method the teacher is using than the problems could certainly be used in that manner. It may be important to note that this text was used prior to the adoption of Common Core Standards and therefore does not present the various solving strategies that are common to CC. This does not mean that they cannot be used in such a way; In fact, many of the example tasks are solved in what some consider to be 'new' mathematical processes. 

In my professional experience, I have not yet had the opportunity to use a math curriculum in its entirety. Every school I have been at has only the old textbooks leftover, without any additional resources. Additionally, teachers are encouraged not to use these textbooks because they 'do not align' with the current standards. To this, I beg to question, how has math changed so drastically in just a few years that a book of math problems could be considered out of date? Is it not the method in which it is to be taught that is changing? If that is the case, why can be not be provided with a digital/ print coupling that meets all of the needs of the teacher and students? What if a textbook company printed a book of problem sets. A simple compilation of problems based in all operations, to include a vast variety of word problems would be excellent. Then, in the digital tools the methods to teaching the processes would be included. This could be evolving and could include assessment measures, interactive tasks that are based on current societal ideas (i.e. problems about the paper boy's collection route would be currently outdated), and a 'teacher education' center to help teachers understand the method that is expected of them. This way, the actual textbook could be used in pieces to give students practice with the concepts they learn via lessons the teacher has prepared with the help of the digital components. In our ever changing world of education, the money spent to reprint textbooks every time someone has a new idea would no longer be wasted. 

The Skeleton Tower Task

The Skeleton Tower (A rather Halloweeny name, don't you think?) is a tower built of cubes in a pyramid fashion, with 4 stair steps extending from the center of the tower. In a tower that is 6 cubes high, there are 66 cubes altogether.

A tower 3 cubes high contains 15 cubes.

A tower 4 cubes high contains 28 cubes.

A tower 5 cubes high contains 45 cubes. 

The following table helped me in determining the number of cubes for each level. The highlighted portion was completed using mathematical equations only, without actually counting the cubes in a pyramid. 

Height	Cubes in all	Looking for Patterns
10	190	9 x 4 = 36 +1=37             *10x 19 = 190 153+37=190
9	153	8 x 4= 32 +1 = 33           *9 x 17= 153 33+ 120= 153
8	120	7 x 4=28 +1=29              *8 x 15= 120 91+29= 120
7	91	6 x 4=24 +1=25               *7 x 13= 91 66+25= 91
6	66	66-6=60          66-45=21 (the height x 4 +1) 60/4=15
5	45	45-5=40          45-28=17 (the height x 4 +1) 40/4=10
4	28	28-4=24          28-15=13 (the height x 4 +1) 24/4=6
3	15	15-3=12         15-6= 9 (the height x 4 +1) 12/4=3
2	6	6-2=4 4/4=1

The question now is whether or not there can be an equation to find the number of cubes in the pyramid at N cubes high. 

I noticed the pattern with an asterisk next to it in the table above. Obviously, 6 x 11 is 66. Initially, I thought the multiplication was just counting up odd numbers. Then I realized that 6 x 2 = 12, 7 x 2= 14, 9 x 2= 18, etc. So I wondered if I took N, multiplied it by 2, subtracted 1, then multiplied it by itself....would I get a correct answer? 

That would be [(N x 2)-1] x N.

Let's see if it is true if used for N = 4: 

[(4 x 2) - 1] x 4= [(8)-1] x 4= 7 x 4 = 28 (It works here, yay!) 

Now N= 10

[(10 x 2)-1] x 10= [(20)-1] x 10 = 19 x 10= 190 (Yay again!)

Using the table, I was able to identify patterns that would help formulate an equation to solve the problem. Oddly enough, the pattern that helped the most was just a side note when I noticed that all of the numbers in the not highlighted section were multiples of the height in cubes. 

Chicken Nuggets Task...Otherwise known as 'Now I am Hungry'

McDonald's sells chicken nuggets in boxes of 6, 9, or 20. Obviously one could purchase exactly 15 nuggets by buying a box of 6 and a box of 9. Using only combinations of boxes of 6, 9, and/or 20 nuggets:

Could you purchase exactly 17 nuggets?
     No, because two boxes of 6 would result in 12. Two boxes of 9 would result in 18, which is too much. A box of 6 and a box of 9 is already noted at a total of 15 nuggets.

How would you purchase exactly 53 nuggets?
   

What is the largest number for which it is impossible to purchase exactly that number of nuggets? 

     This question is difficult to explain, as numbers are infinite. I think the most important point here is that the number obviously cannot be prime, as that would eliminate the divisibility of it by 6, 9 and/ or 20. I did notice that continuing from 53 all the way to 100 resulted in some form of divisibility by the three numbers combined. Once I realized this, I went backwards to see if maybe that largest number was below 53. 

52: 2 boxes of 20, 2 boxes of 6

51: 1 box of 9, 7 boxes of 6

50: 1 box of 20, 5 boxes of 6

49: 2 boxes of 20, 1 box of 9

48: 2 boxes of 9, 5 boxes of 6

47: 1 box of 20, 3 boxes of 9

46: 2 boxes of 20, 1 box of 6

45: 5 boxes of 9

44: 1 box of 20, 4 boxes of 6

43: Hm. I am stumped here!  Perhaps the answer to this question is 43?

Let's say you could only buy the nuggets in boxes of 7, 11, and 17. What is the largest number for which it is impossible to purchase exactly that number of nuggets? 

   To figure this out, I looked to see if there was a pattern for the above numbers. It is possible that I am not thinking about this 'mathematically' enough, however I can't seem to determine a formula or pattern to finding 43 as the number above. The best I could come up with is using a hundreds chart, I marked out all numbers divisible by 7 (in black). I kept going with the chart, marking out every 11th, 7th, or 17th number from one that was already eliminated. The highest number I have not marked out is 37. 

In closing, there must be a more mathematically sound method of solving this problem that I have not yet discovered. As it stands, I feel fairly confident that I figured out the solution....albeit in a rather elementary fashion. 

Mathematics Curriculum Notes

   After viewing the following video, I felt that I connected to the thoughts presented by M.J. McDermott. As a fifth grade teacher, it is frustrating to see that students cannot multiply or divide fluently. We have recently worked in multiplication and division in my classroom with very scattered results. Many of my students did not know how to set up a traditional multiplication algorithm. Some knew how to use the ‘cluster math’ depicted in the video, others were familiar with the lattice method (although a precious few actually used it successfully on more than one or two problems at a time), and several were familiar with partial products. By not using the traditional algorithm, my students are not able to master any particular form of multiplying. In addition to this, when student are asked if their answer is reasonable, they cannot seem to grasp the concept of determining reasonability because they are more focused on which method they can use to determine reasonability. When we moved into long division, I saw that students looked for an alternative way to do it. One even attempted to create his own ‘division lattice’. It seems all of these ‘tricks’ have made students less willing to persist in traditional mathematics for mastery of skills.

https://www.youtube.com/watch?v=Tr1qee-bTZI#t=437

   Based on Chapter 9 from the Handbook of Research in Mathematics Education, there is a web of curricula that leads to student understanding. A few of the points in this web are the Teachers’ orientations toward curriculum, Teacher beliefs and knowledge, and Teachers’ professional identity. After reading this, I wondered how a state could have a set list of standards; yet have teachers teaching a completely different form of mathematics. I have a recent personal connection to this. My daughter is in 3^rd grade in NC. Her math homework recently included subtraction problems called “Expanded Subtraction”. Below is an illustration of the method she was instructed to use. Where you see a (c) is the place she was instructed to use a calculator.

   My question for this, is exactly what part of NC or Common Core Math standards does this fall under? Additionally, if my daughter was encouraged to use a calculator for two separate parts of this problem, why would she not logically use a calculator to simply solve 284-95 to begin with? Finally, the interpretation of this subtraction skill was different between two teachers in the same building. While my daughter’s teacher did it this way, another teacher taught it the way you see below. This exemplifies the sections we see in Explanations for Transformations from the Handbook of Research in Mathematics.

   As a home-educated student, I used Saxon Mathematics during the course of my education. Saxon Math was very straightforward, teaching the use and mechanics of all mathematics skills I use today. I cannot say that I particularly enjoyed math as a student, due in part to the literary nature of my childhood, but I did not hate it or feel inadequate using the skills acquired through traditional problem solving.  As a teacher for four years, only one of those years have I had an actual math textbook to utilize in teaching. The experience with it was strenuous at best due to a heavy focus on games, investigating math concepts and unrelated content mixed in throughout the book. Students utilized a very large workbook version of the textbook that included 1-2 problems per page. The problems always seemed either very easy or rather convoluted, neither of which was beneficial to my students.

Stein, Mary Kay, Janine Remillard, and Margaret S. Smith. "Chapter 8: How Curriculum Influences Student Learning." Handbook of Research in Mathematics Education. New York, NY: MacMillan Co, 1992. 319-23. Print.

TED 663 Project Post 1

Over the course of the next few months, I will be posting on various Math topics in relation to a graduate class I am taking.

Project Overview
1. Who's classroom? and 2. Describe your relationship with the teacher.

During the Mathematical Leadership project, I will be working with the fifth grade teammate at my school. To maintain her privacy, here I will refer to her as Nichole. This year I have moved to fifth grade in a much smaller school than I am familiar with. The fifth grade team consists of myself and a first year teacher. Due to the smaller team size and the first year status of my teammate, I am the grade level chair this year. This is a new experience for me, but I look forward to using the opportunity to assist Nichole in implementing meaningful mathematics lessons for her students.

3. What instructional problems (math concepts) do their students have?

As it is the beginning of the year, we have not yet ascertained any particular problems or mathematical concept deficits. Nichole has noticed that her students are fluent in utilizing the lattice method for multiplication. She also noted they have a decent grasp on the traditional algorithm and know most of their math facts. There are a few students who will need remediation in this concept, however. Math concepts to be covered in 5th grade include multiplication (lattice and traditional, as well as using powers of 10), order of operations in equations, patterns in numbers (with a coordinate plane), adding and subtracting unlike fractions, and several more.

4. What types of approaches or instructional modifications might help?

Instructional approaches vary based on the topic of the lesson. In the case of the multiplication standards, teacher directed instruction is utilized for at-level instruction. Students who are not performing at grade level receive independent instruction during I.E. time as well as guided math groups. During IE, the focus is on skills the students are lacking in that may be unrelated to current standards. During guided math, the focus is on the strand or standards being currently taught while instruction is broken down more and/ or increased in cognitive expectations if necessary.

5. How will you measure student growth?

For each unit, students take a pre-assessment to determine their knowledge of the standards. The information is recorded and used in grouping students for guided math. During instruction, formative assessments in the form of exit tickets assist in knowing which students are progressing and who needs additional assistance. At the end of the unit, students take a post assessment to determine final growth and to allow teachers to plan for IE time in any standard that is not yet mastered.

My whirlwind life!

This past year has been crazy busy! Have you ever had one of those days where you completely lost track of time? That was this year for me. I began grad school in January, then got married in March to the sweetest old man.

The newest change is in employment...I have moved to a closer school system and will be teaching 5th grade next year. Although it is a bit nerve wracking to change grades, I am very much ready for the new adventure. In preparation, I have been reading up on word study in the upper grades. I hope to post more on that soon!

Conquering the Parent/ Teacher Divide

   My first year of teaching I had a parent who was very involved. One parent. I only had 14 students so I was fortunate to not have a large class, but it still came as a surprise to only have one involved parent out of all the students in the classroom. I spent the year wondering why parents never responded to emails, phone calls, notes...why didn't they seem to want to contact me or attend conferences? I was extremely grateful to the parent who was there...the year would have been so much more difficult without her!

   The summer between my first and second years, I thought many times over how to get parents more involved. It occurred to me that perhaps they weren't as involved because they just didn't know where to start or how to help. The best cure for this was a deeper level of communication. So, I created a classroom-only blog and updated the school website that linked to my name...I even made a checklist of 'positive notes' to be sure I sent one home with each child regularly to keep positive contact. One of my most exciting ideas, though, turned out to be the most useful. Using Vistaprint, I created business cards that listed my name, email, school phone number and the web address of our classroom blog. The cards were free, so I purchased some cute notepads for my classroom to balance out my cart. (I am not posting a picture due to all the personal info on them.) I then put business card magnets on the back of 25 cards. In my welcome packet, each student received a magnet card and two without magnets. Ideally, this created a way for parents to easily contact me or find information on our class.

   Communication cleared up...which led to more parents volunteering. I now had a new problem...how do I keep volunteers organized?! I made a master list of who wanted to volunteer and how. One parent preferred to make copies, others didn't mind cutting out laminated materials, a couple of parents were always on hand for classroom parties. For parents who didn't mind cutting or doing other tasks at home, I created an envelope to go home with students. I have created a couple more...they aren't laminated yet, but here is an example:

In these envelopes, I included a sticky note with directions and any tools parents needed to complete the task they were helping with. Possibilities don't end with cutting laminated task cards...one parent put together research index cards by hole-punching five at a time and joining them with brads. This little envelope saved so much time! 

   In the classroom, I created a bin for volunteers and one for my assistant whom I shared with two other teachers. My assistant came to me for an hour a day unless otherwise detained, so it helped to have anything she needed perfectly lined up for efficiency. The two bins were close to the door and anything that needed to be done was sticky noted with directions. I found this was really helpful for parents who wanted to help but didn't want to interrupt class. 

   I hope some of these ideas are helpful! Here's to a great school year with fabulous parent/ teacher relationships! :)