Development changes in the thinking of students and the difficulties surrounding subtraction
Children begin by thinking and learning about numbers as “an unbreakable chain” (Baroody, 1984, p. 203). That is they believe that one number comes after the previous and you cannot simply start anywhere and go forwards or backwards (Baroody, 1984). Students then develop their thinking, with practice, and are able to identify initially the number that comes after a given number and eventually the number that comes before any given number (Baroody, 1984). This is where you can see subtraction begin to emerge as students are able to work out what 1 less is from that number (Baroody, 1984).
The next stage in the development of students’ thinking about subtraction is more easily measured. Baroody (1984) has found that by the time many students enter grade one they are able to subtract one from numbers up to five and by grade two this increases to up to 10. When students are asked to subtract a number greater than one, they are able to “model directly their informal concept of subtraction as “takeaway”” (Baroody, 1984, p. 204). This is known as the “separating from” (Baroody, 1984, p. 204).
From this, students then begin to develop formal methods of mentally computing subtraction equations as often it is too difficult or impractical to visually represent the steps (Baroody, 1984). There are many different mental strategies students use. One of the more common is counting down which is based on the idea of subtraction being to take away (Baroody, 1984). This method is more difficult as not only does the student have to count backwards they also have to keep track of how far back they have to go by counting forwards (Baroody, 1984). Students are often able to track the number of steps they have taken backwards using their fingers (Baroody, 1984). Many students struggle with this method as not only do children find counting backwards challenging they may also may struggle with the fact they have to be able to complete to challenging tasks at once and thus subtraction is often considered harder than addition (Baroody, 1984).
The larger the number children are attempting to subtract, the harder they will find the task as they have to maintain the ability to monitor how far back they have gone, this is amplified when the equation involves numbers in the teens (Baroody, 1984). It is said that instead of attempting these calculations using counting backwards methods students will switch and instead begin to count up from the number being subtracted to get the initial number (Woods, Resnick & Groen, 1975 as cited in Baroody, 1984). The complication this presents is the idea that they are no longer following their previous notion that subtraction means take away (Baroody, 1984). This method is of great benefit when students are subtracting both numbers in the teens and numbers that are close together, for example 18-16 or 9-7, as counting up does not involves as many steps (Baroody, 1984). However when students are doing equations such as 13 -2 it is common for students to revert back to the counting down method as it less to keep track of (Baroody, 1984). Woods, Resnick & Groen (1975, as cited in Baroody, 1984), suggest the ability to switch between the two methods and select the best one for the problem at hand most often occurs before students reach grade three.
There is ongoing debate as to whether counting down is easier for students than counting up as the idea is presented that perhaps the curriculum and the methods students are taught i.e. that subtraction is not just take away may alter the approach they take and can lead them to instead use counting up (Baroody, 1984). It is still assumed that students are more like to use the counting down approach if it is the strategy that they are able most able to comprehend, which often it is.
There are many difficulties students may face when developing their methods of mental subtraction. One of the key issues is that if students are unable to work out what number comes before any given number they will be unable to count backwards (Baroody, 1984). This may prevent students from progressing to be able to subtract numbers greater than one and this can lead to them feeling disheartened and a lack of motivation (Baroody, 1984). Another issue students may struggle with is if they are not easily able to count backwards they will be unable to simultaneously keep track of how much they have subtracted from the beginning number (Baroody, 1984). Many students are unable to develop a strategy for keeping track of how many steps they have taken backwards when subtracting the second number from the first and thus may not know where to stop (Baroody, 1984). Baroody, 1984, suggests that even if students are able to develop a strategy they use they may still rush the process or begin counting too early thus resulting in an incorrect answer. Even those students who are able to successfully count down may find it increasingly difficult and begin to struggle when the numbers get larger (Baroody, 1984).
The next stage in the development of students’ thinking about subtraction is more easily measured. Baroody (1984) has found that by the time many students enter grade one they are able to subtract one from numbers up to five and by grade two this increases to up to 10. When students are asked to subtract a number greater than one, they are able to “model directly their informal concept of subtraction as “takeaway”” (Baroody, 1984, p. 204). This is known as the “separating from” (Baroody, 1984, p. 204).
From this, students then begin to develop formal methods of mentally computing subtraction equations as often it is too difficult or impractical to visually represent the steps (Baroody, 1984). There are many different mental strategies students use. One of the more common is counting down which is based on the idea of subtraction being to take away (Baroody, 1984). This method is more difficult as not only does the student have to count backwards they also have to keep track of how far back they have to go by counting forwards (Baroody, 1984). Students are often able to track the number of steps they have taken backwards using their fingers (Baroody, 1984). Many students struggle with this method as not only do children find counting backwards challenging they may also may struggle with the fact they have to be able to complete to challenging tasks at once and thus subtraction is often considered harder than addition (Baroody, 1984).
The larger the number children are attempting to subtract, the harder they will find the task as they have to maintain the ability to monitor how far back they have gone, this is amplified when the equation involves numbers in the teens (Baroody, 1984). It is said that instead of attempting these calculations using counting backwards methods students will switch and instead begin to count up from the number being subtracted to get the initial number (Woods, Resnick & Groen, 1975 as cited in Baroody, 1984). The complication this presents is the idea that they are no longer following their previous notion that subtraction means take away (Baroody, 1984). This method is of great benefit when students are subtracting both numbers in the teens and numbers that are close together, for example 18-16 or 9-7, as counting up does not involves as many steps (Baroody, 1984). However when students are doing equations such as 13 -2 it is common for students to revert back to the counting down method as it less to keep track of (Baroody, 1984). Woods, Resnick & Groen (1975, as cited in Baroody, 1984), suggest the ability to switch between the two methods and select the best one for the problem at hand most often occurs before students reach grade three.
There is ongoing debate as to whether counting down is easier for students than counting up as the idea is presented that perhaps the curriculum and the methods students are taught i.e. that subtraction is not just take away may alter the approach they take and can lead them to instead use counting up (Baroody, 1984). It is still assumed that students are more like to use the counting down approach if it is the strategy that they are able most able to comprehend, which often it is.
There are many difficulties students may face when developing their methods of mental subtraction. One of the key issues is that if students are unable to work out what number comes before any given number they will be unable to count backwards (Baroody, 1984). This may prevent students from progressing to be able to subtract numbers greater than one and this can lead to them feeling disheartened and a lack of motivation (Baroody, 1984). Another issue students may struggle with is if they are not easily able to count backwards they will be unable to simultaneously keep track of how much they have subtracted from the beginning number (Baroody, 1984). Many students are unable to develop a strategy for keeping track of how many steps they have taken backwards when subtracting the second number from the first and thus may not know where to stop (Baroody, 1984). Baroody, 1984, suggests that even if students are able to develop a strategy they use they may still rush the process or begin counting too early thus resulting in an incorrect answer. Even those students who are able to successfully count down may find it increasingly difficult and begin to struggle when the numbers get larger (Baroody, 1984).