Students have varying preferences with regards to solving multi-digit addition problems. These variations may stem from the perceived simplicity of a method or a misconception about the other methods. In some instances, the students may obtain incorrect solutions even by using the seemingly less complicated or their preferred method. As a teacher, it is important to critically comprehend and evaluate the key methods applicable to multi-digit problems to effectively address the misconceptions and concerns that lead to erroneous solutions. Three notable multi-digit addition methods will be highlighted in this paper: new groups above, new groups below, and the show all totals method (Fuson et al. 2011).
New Groups Above Multi-Digit Addition Method
The numbers to be added are arranged in a vertical strategy. The numbers are lined up appropriately using the place values as the determinants, with ones over ones, tens over tens and so forth. The numbers in the corresponding place values are then added, starting from the right (ones) position going left. These additions will likely result in values beyond the place value being added, prompting a ‘spill over’ of some values to the next place value. These ‘spill over’ values are called the new groups.
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The new groups are placed above the numbers in their corresponding place values. They are then added together with the numbers in that place value. The sequence is repeated if the addition results in a value that is beyond the place value being added (Beckmann, 2018).
In the problem 89 + 238, the figures are arranged vertically. Starting with the numbers in the ones place value, 9 + 8 yields 17. However, the result has 17 ones, which translates to 1 ten and 7 ones. Since the addition was being done on the ones column, the 7 goes under the ones column, and the 1 becomes the new group which is moved above the tens column. The same applies to all the other place values until the final answer is arrived at.
1 1
2 3 8
+8 9
327
New Groups Below Method
Like in the new groups above method, the numbers to be added are vertically stacked one above the other, taking into consideration the place values of the numbers. However, the new groups in this method are placed under the vertical columns in their corresponding place values (Beckmann, 2018).
2 3 8
+ 8 9
1 1
3 2 7
A major misconception in the two methods is the idea that the base values in the sets of numbers are of equal value. The students may therefore incorrectly vertically align the numbers with wrong overlaps in the base values. Also, the students may disregard the new groups as having no significant value and fail to add them to the corresponding place value numbers or fail to indicate them as is required (Fuson et al. 2018).
Show All Totals Method
In this method, the numbers are arranged in a vertical strategy. It is important to take into consideration the place values of the numbers. The base values are added starting from either side of the columns. If the addition is to begin from the right side (ones place value), the figures in that column are added, and the resulting number is indicated below the vertical stack of numbers being added. The next column (tens place value) is then added similarly, and the resulting figure indicated below the previous answer from the ones place value. These steps are replicated to the n th place value. In so doing, the resulting number of rows below the problem at hand should correspond to the extent of the base values in the numbers constituting the problem. Say, for example, if one of the problem parameters has values to the thousandth base value, there will be four rows below the problem, base value translating to a single row.
2 3 8
+ 8 9
1 7
1 1 0
+2 0 0
3 2 7
The rows, in this case, represent the total value of the corresponding base value. These total values are then added to obtain the final solution to the problem. For the problem 89 + 238, the sets of numbers are stacked vertically in their corresponding base values as previously explained. One set of the numbers has figures to the hundredth value. This means that there will be three rows for the total values, as demonstrated in the solution below.
This method of multi-digit addition breaks down large and intricate problems into simple, workable numbers, making it easier for the students to solve the problems. Suffice to say; it is an accurate and faster method of multiple-digit addition (Common Core Standards Writing Team, 2011).
Similar to the new groups above and new groups below methods, the numbers are arranged in a vertical strategy when solving the problem. The correct stacking of numbers with their corresponding base values is therefore imperative. The biggest potential undoing of this method is the incorrect vertical arrangement of the numbers. Students may perceive that all the values in the problem have the same base values, and they will consequently stack the hundredth value over the tens value, or in any other incorrect way. Their final solution will be incorrect.
Other misconceptions emanate from psychological factors. Negative attitudes and the perception that the method and mathematics as a whole are difficult may cloud the students’ mind, impeding their ability to correctly analyze and solve the problem (Fuson et al., 2011).
Classroom Case Study
When solving the problem 89 + 238, two students obtained different answers and were discussing their answers. Student 1 got 1, 128 while student 2 got 327 but was not sure whether his/her answer was right. Neither of the students had a mathematical drawing to illustrate how they arrived at their answer.
The best approach of intervening is asking the students what they understood from the topic. They both have a shred of the concept, but their application into solving the problem was not right. Student one most likely did not align the figures correctly in their corresponding place values. He/she must have considered the 8 in 89 as a hundredth value. Hence the incorrect solution arrived at.
Guiding the students to obtain the correct solution to the problem requires explaining to them in depth the significance of the place values in each set of numbers. To effectively solve this, the set with the lower total value should be stack above the set with the higher total value. The students should also ensure that they use mathematical drawings when solving the problem. That way, it can be easier to backtrace where they went wrong (Fuson et al. 2011).
References
Beckmann, S. (2018). Mathematics for elementary teachers with activities (5th ed.). Boston, MA: Pearson.
Fuson, K. C., Clements, D. H., & Beckmann, S. (2011). The focus in grade 2: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Common Core Standards Writing Team. (2011a). Progressions for the Common Core State Standards in Mathematics: K–5, number and operations in base ten, pp. 6–10.
Progressions for the Common Core State Standards in Mathematics: K–5, Number and Operations in Base Ten by the Common Core Standards Writing Team, 2012.
