Many of my second grade classrooms are in the middle of their addition units. The classes often teach place value and addition strategies during the months of September and October. When introducing addition strategies, teachers rarely start using the standard addition algorithm (see # 4). Manipulatives and visual representations are heavily used during the first month of school. The process below differs per school, but I’m finding that this is often the case in many of the second grade classes that I’ve observed. Keep in mind that I’m missing other approaches, so perceive the following as a few highlighted strategies that are used during the first few months of school.
Students are introduced to some from of unifix cubes or counters. Students are asked to compile the groups of counters to find the sum. For the most part students find this task quickly and are ready to move onto the next portion. This is also a first grade skill that’s reviewed at the beginning of second grade.
The number line usually follows the counters. Sometimes the number line makes an appearance before the counters, but it’s usually afterwards. The number line is used extensively. Students are asked to find numbers on the number line. This builds number sense and an understanding of the reasonableness of an answer. Eventually students are asked to add numbers with hops showing the addition involved.
Base-ten blocks are then introduced to emphasize place value. Students are asked to combine base-ten blocks to find the sum. They are asked to find the sum of the base ten blocks and place them on the number line.
The four processes above aren’t necessarily mandates, but it’s found in the sequence of the textbook. I should mention that the same process is used with subtraction later in the year. What I’m finding is that there’s rarely a mention of using the standard addition algorithm to find the sum. I don’t necessarily think that’s a problem, but it raises the question of when should the algorithm be introduced? In what cases should the algorithm be introduced and is it only used in certain circumstances.
Think of 200 + 198. Would your students use the standard algorithm for this? Some might, other might prefer to use another method. Regardless of when the standard algorithm is introduced, there will still be students that would prefer to use the algorithm. Is that the most efficient method?
The topic of this post was tackled during last Thursday’s #ElemMathChat. Most of the questions revolved around when the standard algorithm should be introduced and mistakes that occur when students focus on the steps. There were many useful answers, but I’m not positive if one right answer climbed its way to the top. There are many factors at play here. Some teachers feel pressured to move through the curriculum at a high pace because of testing. They might teach the algorithm sooner, while others might not mention it and scroll through the prescribed lesson sequence. Most teachers would like students to have a conceptual understanding of numbers and systems before moving towards a standard algorithm. How much time is spent developing that truly depends on the teacher and student. I’m not judging any teacher is these situations as we are all in this together, but I believe having this discussion is important. I think this also plays a role in how other algorithms are introduced.
Educational Aspirations 
Love that you’ve created this forum to talk about something so important Matt! Here’s my swing at your pitch!
Algorithms are such a beast but it’s because they’re so widely misunderstood. I think the question that needs answering is “Are the traditional algorithm and standard algorithm the same thing?” The math progressions would suggest “no”, however many of us use both terms interchangeably and that’s mathematically incorrect.
Taken from http://commoncoretools.me/wp-content/uploads/2011/04/ccss_progression_nbt_2011_04_073_corrected2.pdf “In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.” (page 13)
If minor variations are acceptable, who are we to tell which variation is correct? So my answer to your original question….”never”.
Sorry to throw you back a Knuckleball!
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Hi Graham,
After researching this a bit further I decided to delve into my district’s adopted math series. I found that the standard algorithm isn’t actually formally introduced until third grade. I even found a performance task that asks students to use the standard algorithm to solve problems. In those cases I’m assuming that minor variations in the method are acceptable.
Similar to number rules/formulas, I’d like to have students construct their own understanding and then uncover the algorithm in the process. This isn’t always the case. I’m always looking for strategies to connect the algorithms with conceptual understanding.
Thanks for the comment and pushback. I definitely think this topic is important to discuss and I enjoy hearing other perspectives.
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