- Use different representations and connections among concepts
-  Select and implement tasks that are: cognitively demanding and accessible to all students (multiple entry points and varied solution strategies)
- Enact 5 practices to support discussions: anticipating, monitoring, 

selecting, sequencing, and connecting responses. Use different talk moves and questioning strategies. 
- Use collaborative strategies of participation. Attend to patterns of participation and use of equitable strategies
- Identify and use professional mathematics education resources to inform teaching practices. Develop a professional inquiry and share with others

I have always liked math. I remember my father’s protractor set like it was yesterday. The protractor and compass were shiny and silver. The compass had a small blue pencil attached to one of its legs. It read, “DC Lottery.” Every time I attempted to touch it, I would get caught. It wasn’t until Middle School that I learned what a compass was used for. I purchased a compass for my seventh-grade geometry class. Ms. Holman was my seventh-grade geometry teacher at Roper Middle School. She weighed about 300 lbs and she always had a story to tell. Her favorite line was:

What I remember most about Ms. Holman is how she used real-life situations to help me understand math concepts. In fact, one day she told me to look out the window at the baseball field to envision a math problem. Although, I don’t remember the exact math problem. I do know, on that day, I realized that math was cool. I liked how everything could be described by using a math equation, everything from how long it takes a raindrop to land from the clouds to my body make-up. From seventh grade all the way through College, I never made below an A average in math.

At one point, in undergraduate school, I had minored in Math. In fact, I was a math tutor. I tutored math and made good money, as well.  I tutored several students with disabilities. Mostly students with cerebral palsy and autism. It was the most rewarding experience, and this is how I knew I would make a great teacher. I taught my tutees just the way I was taught, by using real-life situations and examples. Everyone that I tutored passed and graduated. Too bad, I dropped math as a minor. I dropped it because it was going to take me longer to graduate.

I graduated from Trinity College with a BA in Communications, but I have always had a passion for mathematics. I define mathematics as patterns in our everyday life that connect us to one another and other things that are stagnant or in which can be changed. Being able to make connections is one of my overall personal strengths. It is a skill that comes very easy to me.

When I was very young, my mom taught me how to connect the dots. It was through a game. The game was called, Connect the Dots. Connect the Dots required me to strategize how I could make the most boxes by connecting the dots, and then placing my initials within the box. I use to play this game with everyone I came in contact with. I loved it. Metaphorically, I have always known how to connect the dots, and it is a skill that is very important when learning and teaching mathematics. 

I remember giving my students a worksheet that required them to know how to write the numbers, 1-3. About ten minutes into the assignment, one of the students started to bust out crying. She cried out, "I don't know how to make a 3." At that very moment, I began to feel a little guilty because I did not model the lesson before asking my students to complete the worksheet. The child who was struggling was absent the day we had practiced writing the number 3. The I do, we do, you do is a great strategy, and it should be used in every lesson.

Reflection 2: Consider the eight tasks shown in figure1. How would your students go about solving these tasks? Using the four categories of cognitive demand, how would you categorize each of the tasks for your students.

Task A is High Level and it should be categorized as doing mathematics
My students would probably use number bonds to solve this equation

Task B is High Level and it should be categorized as procedures with connections
My students would convert the fractions into decimals and then they would have to figure out how to increase the percentage chance

Task C is High Level and it should be categorized as doing mathematics
My students would compare the relationships and then decide which is the better option

Task D is Low Level and it should be categorized as memorization
My students would use the formula for how to calculate the original price after a discounted item. this would require them to know how to convert percentages into decimals, multiplication and subtraction.  

Task E is High Level and it should be categorized as procedures with connections
My students would use shapes and pattern blocks to figure out the answer

Task F is Low Level and it should be categorized procedures without connections
My students would measure each block

Task G is High Level and it should be categorized as procedures with connections
My students would figure out how to make all the blocks equal the same average/height

Task H is Low Level and it should be categorized as memorization
My students would convert decimals into fractions and percentage by using the formulas they were taught

Reflection 3: Can you think of other factors that might make a task appear to be high level on the surface but that actually only require recall of memorized information or procedures?

 

After reading Goldilocks and the 3 Bears and modeling the difference between small, medium, and large, I grouped students together so that they can try categorizing items into small, medium and large bins on their own. Although many students argued about the difference between medium and large, some worked very well together because there was not any ambiguity about what I was asking them to do-- I had labeled each been. This was a low-level task because I illustrated to my students exactly how they should complete the task, so it didn't require much thinking.

Reflection 4: How might you use this tool in professional-development sessions to stimulate rich and lively discussions about mathematical tasks and the levels of thinking required to solve them?

 

To provide math examples and then have my peer teachers to categorize them into the four categories of cognitive demand would be a great interactive activity during PD. I think the discussion would be relevant when determining best practices for the implementation of a concept that is to be taught in the classroom.

Reflection 5: What do the classifications "procedures with connections" and "doing mathematics" mean to you?  How are they alike? How are they different? In what ways can these classifications be helpful in selecting and creating worthwhile mathematics tasks for use in your own classroom?

 

"Doing mathematics and "procedure with connection" are similar classifications of cognitive demand because they both require students to think deeply. With "doing mathematics, “a student would have to think about and decide which concept to use for solving a problem. This is also required for "procedures with connections," but the thinking process requires more focus on actually how to use the concepts, which develops meaning. Therefore, the difference between the two is dependent on how the problem needs to be answered--What is the problem asking students to do?

Both cognitive demands should be taken into consideration when grouping students who require differentiation. 

 

A possible standard for the lesson I watched demonstrates how students can reason abstractly and quantitatively, which is standard number 2. Students were responsible for grouping a large amount of items to show their fluency of counting within 1000. Also. I would like to add #6 Attending to precision to this post because it relates to the lesson, as well as, standard 2. With this lesson, students problem solving was done meticulously and precisely.

The teacher displayed great classroom management skills. Also, her students were very interested in the lesson. They were very engaged. It seemed as if the teacher grouped the students with their friends or someone that they worked well with. The lesson could be used at any k-5 grade level. In my kindergarten class, I would have students to group items with 30.

It is very important that students see the big picture, so this is why objectives are so important. I did not see the teacher revert back to it at all. If I were to teach fractions I would make sure that my students get a lot practice. I would provide many manipulative's. Also, I would look up games to play. In fact I would probably make a game out the cards the presenter developed in the second video. His cards helped students to see, feel and understand the difference between parts from a whole.

II.               Vertical Integration using Base 10 Blocks: Watch Video below

III.              Demonstrations of how to use Base 10 Blocks as a Vertical Integration     tool from Pre K 3 through Fifth Grade:

·         Pre-K 3: Making patterns with connecting blocks

·         Pre-K 4: Making patterns with base 10 blocks and connecting blocks

·         K-5: Adding and Subtracting. Counting with Base 10 blocks 1- 100

·         1st Grade: Adding and subtracting. Intro to Place Value. Showing students how the number system based on 10 works (ones and tens) (ten and some ones)

·         2nd Grade: Adding and subtracting two and three-digit numbers. Place Value Cont'd. regrouping and borrowing using base 10 blocks

·         3rd grade: Place Value Cont'd. Use base 10 blocks to understand how to round whole numbers

·         4th Grade: Representing decimals and fractions by using base 10 blocks (reteach regrouping/Base ten Blocks value changes)

·         5th Grade: Place value Contd' to understand how to round decimals to any place.

Works Cited
Ely, C. M. (2008). The effect of math manipulatives on first-grade student achievement (Order No. 10663076). Available from ProQuest Dissertations & Theses Global. (1984745612). Retrieved from https://search.proquest.com/docview/1984745612?accountid=28903

Fioriello, Patricia. (2017). The Return of Vertical Teaching. Retrieved from http://drpfconsults.com/return-of-vertical-teaching/

Teacher moves:
Ms. Owens reminded her students to use different strategies to count during the lesson, which stated that students should decompose numbers equal to 10, by using objects. She reminded her students how their fingers could be used as well counters, as well.

Ms. Owens, reiterated the same math problem for the first few demonstrations, but then she allowed her students to create their own number stories by using different values. So, through the use of reciprocal teaching strategies, she was able to change the mathematics in the problem to match the students’ level of understanding.

Ms. Owens was able to connect the child’s thinking to symbolic notation by asking her students to demonstrate the number story, in which another student had just recited. The students had to use their counters to symbolically illustrate it.

Objective: Decomposing numbers less than or equal to ten
Engage: Ms. Owens placed an addition (+) sign on the board and then asked the class whether or not it was a subtraction sign. The students replied, no. So, she placed the subtraction sign on the board and asked the class to tell her what the sign meant. Then the students were told that they were going to use the objects (cubes) on their desk to subtract with. But first, they had to count each cube on their desk to confirm that there were 10. Also, she reminded the students that they could use their fingers to subtract as well.
Explore: Ms. Owens selected ten students from the class to demonstrate her number story. The students performed, showing how 10 birds were flying (students were flapping their wings), then two birds flew away (Two students flapped their imaginary wings to the other side of the class), so there were eight birds left flapping their wings (the students really enjoyed that). The students were then asked to create the same number story using their counters.
Explain: Ms. Owens further explains subtraction by numerically entering it on the board 10  – 2 = 8. She asked the students, which number should be recited first and then demonstrated the problem again by using a number bond (it was a connection to their prior learning).  She explained and demonstrated, saying “we first start with a whole number and then take part of it away to end up with another part.” 
Elaboration: Students names were picked from a cup of equity sticks. They had to make up their own number stories, and then the rest of the class modeled their stories by using counters. This was a great example of reciprocal teaching.
Evaluate: Students were told to turn to Module 4 Lesson 23 in their Eureka Math workbooks. They had to complete three problems on their own.