### The developmental continuum for teaching addition.

 The developmental continuum for teaching addition. Bruce Williams Creating Real Mathematicians bruce@creatingrealmathematicians.com The teaching of the operations is generally still very traditional. Educators lament that students do not understand place value. These two points are inherently linked as the strategies we are teaching children is not place value-based. Here is the approach to teaching addition that is developmentally based, that uses place value as the basis for understanding, and that allows students to progress developmentally through a range of progressive place value and mentally computationally based strategies. Algorithms have been traditionally taught through rote memorisation. This is because we needed the maximum of people with minimal mathematical understanding to have a suitable strategy for solving algorithmic problems. Think about what mathematical understanding is needed to perform the traditional right to left column addition. All that is really required is to able to add two one-digit numerals. Before technology, the pencil and paper was the most efficient method for this calculation. With technology now at everyone’s fingertips, the need for traditional algorithms has diminished, while the need for a range of strategies for mental computation and approximation is as important as it ever.   There is a logical progression in teaching algorithms to students that greatly assists the developmental understandings of each process without any of the traditional rote memorisation of a series of steps without understanding. In summary, the progression for teaching addition strategies through the primary years is: 1.      Counting all with models 2.      Counting on by ones, (Number line) 3.      Counting on by ones, (Hundreds Chart) 4.      Counting on by tens and ones, (Hundreds Chart) 5.      Addition using the Jump Strategy (Hundreds Chart supported) (Partial Sums), 6.      Addition using the W method (Partial Sums) pulled apart, 7.      Addition using the W Method (Partial Sums) kept whole, 8.      Horizontal addition working Left to Right (Partial Sums). 9.      Column addition working Left to Right (Partial Sums). 10.  Traditional Column Algorithm working Right to Left.     10.                    Traditional Column Algorithm working Right to Left. If we begin at the end, that is, where we would like all students to end up understanding, is the traditional right to left column addition. The mathematics required to perform this algorithm is, as mentioned, simply to be able to add two one-digit numerals, or three, where one of those numerals is a one if there is “carrying” or “renaming” is involved. The most difficult mathematics in this problem is adding three one-digit number, one of which is 1. Students do not need place value understanding to perform this algorithm. The traditional right to left algorithm is the last strategy to teach students as it does not develop any place value understanding and is not a mental computational strategy i.e., it is very difficult to perform mentally. No place value understanding is required in this algorithm, which is why it was widely taught as a strategy. People with minimal mathematical understanding are able to perform mathematical calculations to achieve the correct answer. It is often more efficient to take out our phone and perform the calculation than write it on a paper and calculate. It is even more efficient to perform mentally, even for an approximate answer, which is why we encourage the following strategies before tackling the traditional vertical left to right algorithm. This traditional vertical left to right algorithm strategy is fine if students understand all the mathematics behind the strategy, but this strategy in itself does not develop any of the underlying understandings, especially place value understanding, and hence should be the final strategy taught to students.   9.    Column addition working Left to Right (Partial Sums) The step before understanding the right to left traditional addition algorithm is the same strategy but left to right. This makes a huge difference in the required mathematical understanding to perform this algorithm. It forces place value understanding, in that it makes the student recognise the place value of each numeral. In the example below, students must recognise that the 1 represents one hundred, the 2 two tens and the 3 three ones. Students must recognise that in the number 123, the 1 represents one hundred, the 2 two tens and the 3 three ones. Note that this working left to right algorithm is also useful for finding an approximation of the solution, even with just the hundreds added mentally. Once understood, this algorithm leads to the formal traditional left to right algorithm for addition, now with a full understanding of the process and of place value. Place value understanding is important in understanding this algorithm. We often hear that students do not understand place value. This would largely stem from the traditional strategies taught to them that do not require any place value understanding, other than maybe in reading the final answer.     8. Horizontal addition working Left to Right (Partial Sums). This same left to right vertical algorithm can also be written horizontally. In fact, there is no need to write vertically unless you are using one of these algorithms above. Otherwise, I would always encourage writing in a horizontal format, the same way we write everything else. Place value understanding is developed through this Partial Sums approach to addition. The pulling apart each number into its place value components forces place value understanding and the whole algorithm is able to be performed mentally. This strategy emphasises use of place value and mental computation. Students begin with two-digit numbers before moving onto numbers with three digits and more. 7.    Addition using the Jump Strategy (Hundreds Chart supported) (Partial Sums). Pulling one number into Place Value components and adding onto the other number whole: This is a “counting on” strategy. Students should be able to alternate between the above strategies before transitioning into the vertical algorithms. This counting on method is also excellent for adding time. E.g., 7.45am + 2 hours and 35 minutes, while the traditional algorithm does not work when the change in units is not a place value step, such as when adding non-metric units. When introducing these partial sums strategies, ensure students are supported with a hands on hundreds chart. Ideally one they can write on and reuse to support their thinking and calculations.     6.    Addition using the W Method (Partial Sums) kept whole Leave both numbers whole and pull apart visually to add each place value component separately. Combine the hundreds, then tens, then ones., through counting on while leaving the original numbers whole.         5.  Addition using the W method (Partial Sums) pulled apart   This is the same strategy as above but separating the numbers into their place value components. Eventually, students do not need to perform this step and can separate mentally. Combine the hundreds, then tens, then ones, through counting on after pulling the initial numbers into their place value components.         4.    Counting on by tens and ones, (Hundreds Chart) These Partial Sums strategies flow logically from a counting on partial sums strategy using tens and ones. If this is supported with a hundreds chart, and students are gaining confidence with the “tenness” of numbers they are able to separate two-digit numbers into their tens and ones place value components. Then it is a matter of counting down the hundreds chart to add the tens, and across to the right to add the ones. Once familiar with this concept students are able to perform this task mentally. Student understanding will allow them to realise it is generally quicker to add the smaller number to the larger, but either way you still get the same result. It is recommended you provide individual hundred charts for students to work with. The aim is for the student to have a mental image of the hundred chart they can use to count with, that is, visualise their counting. Before students are able to visualise they should work with a hard copy, and once transitioning to visual counting on, have a hard copy of the hundred chart on hand to check their calculations. Ideally, students should be able to write on their charts. Laminated charts with erasable markers allowing the teacher to see student understandings as they perform calculations are ideal. We want students to have a mental image of the hundreds chart in their mind and be able to perform two-digit mental addition before moving onto more advanced strategies. This is a counting on strategy, where students start with one number and add the tens then the ones. Remember to always use authentic contexts for the addition of numbers. It is important to give a purpose for wanting to find the solution. This applies to all problems. An easy way for addition is to provide students with catalogs that contain only whole numbers. Many toy sale catalogues and electronic good and homewares catalogs contain prices without cents, which is ideal for this purpose. Students are then able to self-differentiate the numbers they select to add. Less confident students can select easier numbers while more confident students select more difficult numbers. Once students have a satisfactory understanding of the counting on strategy with tens and ones, allow students to practice their strategy with numbers into the hundreds.     3.    Counting on by ones, (Hundreds Chart) 2.    Counting on by ones, (Number line) 1.    Counting all with models Students begin to make a number sentence verbally using manipulatives without using formal mathematical language. We want them to understand that addition is counting on, combining numbers or counting forwards. Once established, we practice writing this mathematically and matching with a worded equation. Bears in the house Big Book. Students model the problem onto their own house picture and answer the problem. Students then turn the page to reveal the answer. This is best done sitting in a circle and modeling the problem with counting bears. The teacher is then able to monitor the progress of all students. Students can then model this with a number line, beginning at one number and “jumping” to the solution. Students practice between the models, worded, number sentence and number line and transition between all four. A very useful understanding at this stage is learning their “friends of ten” (1 & 9, 2 & 8, 3 & 7,  …). While much fun can be had developing this concept at this stage, knowing these friends greatly assists later when working with numbers into the tens and hundreds. Activities such as “friends of ten” rainbows, “ten bears at the picnic, some red and some blue, how many of teach could there have been?”, and similar. When individual students are beginning to count to numbers above twenty, progress students to a hundreds chart using similar activities of adding numbers. Always giving the numbers a context and allowing the students to select their own numbers allows for self-differentiation as well as a more interesting scaffolded share at the end of the lesson. A question such as “I had two fishbowls. Each bowl had many fish. How many fish did I have altogether?” demonstrates an example of an open question that can be answered at many levels. Using models, diagrams, number lines, hundred charts, worded sentences and number sentences are all acceptable strategies depending of the level of the understanding of the individual student. It is the role of the teacher to identify where each child’s understanding is and progress to the next level as appropriate.  Preceding the hundreds chart counting by tens and ones, is by ones only. I would have my earliest school students using the hundreds chart by the end of their first year at school. This would be assisted by undertaking “Hundred Days” activities every day for their first year or two of their schooling, developing the essential understanding of the place valueness of our number system, that “ten of these makes of those”. Students counting by tens and ones, as well as hundreds, tens and ones, also need to understand the concept of “fact families”. “Fact family” houses are a fun way to develop this understanding of the commutative law of mathematics. If we know 1 + 2 = 3, we also know 2 + 1 = 3, as well as 3 – 2 = 1 and 3 – 1 = 2. That’s the “three for free” concept when using fact families, if we know one, we immediately know the other three. This strategy is used extensively when solving subtraction problems. Calculating money change mentally is generally performed through counting on rather than subtracting as it is a more efficient method of calculation. These strategies are developed in progression throughout a student’s primary schooling. The aim would be to have all students fluent with the traditional left to right algorithm by the time they complete year 6. Realistically, students will have developed a range of addition strategies they are confident in using depending on the numbers in question and the accuracy of the answer required. At the end of each lesson, best practice dictates we use the final minutes of each lesson to “share” their findings and strategies. This verbal and visual sharing of individual student’s work enhances both the understanding of the student sharing as well as the rest of the class being exposed to other strategies, as generally, not all students would be at the same level of understanding and hence would be using a range of strategies within the one classroom. Begin with the easier strategies when sharing and progress through to more difficult strategies, thus exposing all students except those at the most advanced strategy to the next strategy in their conceptual development. I have always found students are keen to try the next strategy in the continuum and will attempt at the next opportunity.