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One major takeaway from Chapter 7 was the strategy for adding two numbers with renaming . While it would not be helpful in my classroom since students are at a higher level, I wonder how it would look with an actual group of elementary school kids. I am interested to see how this would actually play out in a classroom setting because my initial reaction is that is seems like a much more complicated approach then simply teaching kids to carry the one. On the other hand, I wonder if this creates a deeper more conceptual  understanding of what it means to carry numbers in an addition problem which would possibly be more beneficial for kids.

While I have some questions about the effectiveness of the strategies presented, I do appreciate the way in which the chapters are organized to provide a scaffolded instructional strategy for teaching seemingly basic mathematical concepts such as addition and subtraction. Maybe if we used more complexed approached to allow students to gain a deep conceptual understanding of whats happening when performing these operations it can better allow students to make the necessary connections to these foundational skills when moving on to more complex material.

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Two major take-aways from these chapters were the need for students to master the concept of addition and subtraction before engaging in sustained mental math and the possibility of using tally marks to help with this.

As I read this, I tried to remember if I learned the concept of addition first or started memorizing facts first. For addition, I do not remember but for multiplication, I definitely started by memorizing facts.

I agree that student need the concept to understand what they should do with the facts. As the book points out, this ensures that they are able to complete more complex problems.

I also like that they encourage the use of tally marks. This makes a vague concept much more concrete!

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One make take-away from these chapters is that even for a skill as basic as addition, the teaching and practice of this skill provides an opportunity to discuss equivalency across an equation.  While students initially may seem to just be recounting a total from two separate amounts,  experimenting with different pieces as the missing pieces is a great way to establish the idea that one side must equal the other, no matter what.  Equivalence plays a big role in solving for quadratics in Algebra 1, and I see how a better foundation in this can benefit students on the high school level.
Another take-away I got from the latter part of the reading is the way the skills move from more intuitive to more memorizable. In subtraction, for example, the text begins the sequence with examples where the "subtrahend is smaller than the minuend in each column." This allows students to use what they can already understand/picture about taking away something from another value. As the level progresses and "renaming" because necessary, students must draw on their addition skills to find the missing pieces, but must also remember certain renaming steps to account for right-most digits that are larger than whatever they're being subtracted from. In short, the book acknowledges that the "why" behind renaming gets put on hold until students can complete the skills well. While I'm not sure if I agree that the "why" should come afterwards, I do appreciate that the book doesn't speak about these strategies as singular ways of doing things; they are merely the beginning of the fluency that is used for more complicated understanding.

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From a quick glance at this text, it would be easy to assume that the person writing the text is only arguing for directly instructing the students in how to use the addition/subtraction algorithms, and how to regroup (or "carry" numbers). However, a major takeaway I had from really delving into the chapters here, is that a conceptual understanding of both subtraction and addition are critical "preskills"  to being able to add and subtract "quickly."  Both chapters focused on modeling using tally marks, both counting the total number of tally marks, as well as learning to "cross out" the lines to find the remaining ones, in order to help scholars understand how subtract and addition work conceptually.  However, as the work becomes slightly more complex, I am unsure where that conceptual understanding comes in. As a clear example in Chapter 7, it states "when the missing addend is zero, as in 8 + ? = 8. the teacher can use this wording to replace steps....'this is a special kind of problem. The sides are already equal. Eight on both sides'" (p 99). Instead of asking a student to explain their own understanding of addition to determine what they think the answer is. Based on our previous readings, the student should already have a firm understanding of zero at this point in their mathematical instruction, so they should be able to recognize the type of problem without being explicitly told. This leads to my question - as a math teacher in the era of the Common Core, and in the interest of making students better mathematicians - should we be focusing on teaching students STRATEGIES on on understanding how math works. This chapter seems contradictory to what "rules" the 13 Rules that Expire article was trying to explain.

A second takeaway that I had from reading these chapters is how complex the act of remediation in math is. The text offers multiple pages and examples on how to remediate based on skill gaps, strategy gaps, and even more basic understandings - such as math facts - for both addition and subtraction (pages 100-101, 105-107, Figure 7.2 on 108, 125, 129 - 130, Figure 8.4 on 131) . Remediation is not simply an act of more practice - but of diagnosing a student's true needs by studying their work, and understanding where the deficit lies.

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Addition and Subtraction go hand in hand. Therefore instruction should be built similarly. There are an array of ways to introduce addition and subtraction. The basic understanding of these concepts must be established to reach the second stage, which requires students be able compute mentally, which may include memorization of the basic facts.
While teaching Special Education, I have encountered students who have trouble at both stages. What I have realized is that most of the time, students struggle with the mental math because they did not understand the conceptual reason behind it. However, I have found some students who understand addition and subtraction and still struggle with mental computation. This plays a large role as they progress in their academic career.
Math builds on itself, and as it gets more difficult, the computation mental math of basic addition and subtraction is very vital in the ability for students to gain further knowledge and to be able to solve through the higher level math.

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Response From Alex Bauer:      One major takeaway from this reading is using the strategy, addition the fast way. This strategy teaches students to draw lines to represent the math problem. I like this strategy and believe that drawing and diagrams in math are very helpful. This allows the students to visual the addition problem and can making solving the problem easier. Additionally, this strategy provides a good foundation for learning additional math concepts. According to Stein, Kinder, Silbert and Carnine (2006), "After students work addition problems the fast way accurately, subtraction instruction can begin." (p.100)Another takeaway from this reading is using the beginning subtraction strategy. This strategy instructs the students to draw lines to represent the mathematical problem. The student first draws the lines for the minuend and then crosses off the number that needs to be subtracted. I like this strategy because it also allows the students to visual the mathematics problem. They can visualize the total number and then cross off the amount to be subtracted, leaving the lines left to be the answer. These two strategies are important when teaching beginner math, but the use of visuals and diagrams continue to be very helpful to students as they learn more difficult math and can apply visuals to solving more complex math problems. 

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Response From Alex Bauer:      One major takeaway from this reading is using the strategy, addition the fast way. This strategy teaches students to draw lines to represent the math problem. I like this strategy and believe that drawing and diagrams in math are very helpful. This allows the students to visual the addition problem and can making solving the problem easier. Additionally, this strategy provides a good foundation for learning additional math concepts. According to Stein, Kinder, Silbert and Carnine (2006), "After students work addition problems the fast way accurately, subtraction instruction can begin." (p.100)Another takeaway from this reading is using the beginning subtraction strategy. This strategy instructs the students to draw lines to represent the mathematical problem. The student first draws the lines for the minuend and then crosses off the number that needs to be subtracted. I like this strategy because it also allows the students to visual the mathematics problem. They can visualize the total number and then cross off the amount to be subtracted, leaving the lines left to be the answer. These two strategies are important when teaching beginner math, but the use of visuals and diagrams continue to be very helpful to students as they learn more difficult math and can apply visuals to solving more complex math problems. 

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Response to Designing Effective Mathematics Instruction - Chapters 7 & 8

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Response to Designing Effective Mathematics Instruction - Chapters 7 & 8

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What were 2-3 major takeaways from the math reading done to date in Chapters 7-8 from Stein, M , Kinder, D. Silbert, J. Carnine, D. W. (2006) Designing Effective Mathematics Instruction: A Direct Instruction Approach?

What are the implications for your teaching of math?  What questions do you have about the content read so far? Be sure to reference specific points and chapters!

Respond to at least 2 of your peers!