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Why Subtraction Continues to Challenge Students

I hope you are all doing well and finding joy in the everyday things.

I think it is very easy to take things for granted especially when it comes to how we teach and how students really learn new material. Many times I have presented information to students, assumed they learned it, and was shocked later on when they hadn’t mastered the information or were clearly confused by a concept. I have realized that for most students to really gain the knowledge they need to apply their new learning in different situations they must have full understanding of each component of a concept. They must have repeated opportunities to practice new skills over time. Sometimes we have to walk away from a concept before we are ready to and circle back to it after some time has passed in order for true learning can occur.

Subtraction continues to be a challenge for many students.

Why? The skill of subtraction is very different from addition because in addition there are no “invisible” numbers, all of the values are represented, the students just have to count all of the numbers. In order to subtract accurately, students have to be aware of the “invisible” numbers inside of other numbers. Students must know for example, that when subtracting 14 – 8 that within 14 there is also an invisible ” 6.” Most students are unclear of all of the numbers embedded within another.  In order for a student to successfully subtract numbers less than 10 they must know all of the numbers that are embedded within 10 i.e., 9, 8, 7, 6 , 5, etc. and in order to successfully subtract numbers from 20 then students must know all of the numbers embedded within 20 i.e., 19, 18, 17, 16, etc.This is a skill called hierarchical inclusion and it can only be developed through counting, counting, counting. Unfortunately, most students have limited opportunities to count, especially the counting of objects which is so crucial for strong number sense. In addition, most primary grade textbooks focus very few of their pages on counting.

If you want your child to become stronger at subtraction, give them objects to count both forward and backwards. Once a student has developed the ability to count by ones forward, ask them to practice counting the same amount backwards. Once they can count by ones going forward and backwards ask them to practice counting objects forward and backwards by tens.  In virtual math classes, individual Jamboards can be created for students to touch and count, cross off and count, or circle and count forward and backwards. Providing this in class counting practice can be crucial for students with limited home support and can provide a lot of insight into their thinking.

Students must have multiple opportunities to develop an understanding of the relationships between two values.

Subtraction is not a one way street, it often involves moving back and forth between two values. However, most textbooks emphasize subtraction as taking an amount away from another rather than focusing upon the distance between two values. For example, Michelle has 27 points and Robin has 19 points. What is the difference between their points? This problem can be solved by subtracting 19 from 27 or by counting up from 19 to 27. You can move backward or forward to find the difference. Most students are not aware of this as a possibility and even when they do use it as a strategy they are often weak on explaining/understanding why it works. Providing opportunities to understand subtraction as the difference or distance between two values will help develop strategic thinking and number sense in your students.

Introducing procedures too early for students may become a roadblock as opposed to a support.

Many individuals believe that by introducing upper grade procedures to students in lower grades i.e., giving a 5th grade textbook to a 3rd grader is giving the student more advanced or sophisticated work. However, this usually introduces students to procedures and shortcuts that they cling to before they have a full understanding of how or why they work. Make every effort to delay the introduction of the standard algorithm for subtraction. The Common Core standards first introduces the standard algorithm in 4th grade although many textbooks introduce it much earlier.

The introduction of the standard algorithm was pushed up to 4th grade because it does not allow students to use reasoning, number sense, and place value to solve. Many students can arrive at the correct answer without truly understanding why the procedure works. The creators of the Common Core standards knew that much more work with place value, decomposing, and number sense had to be done before students ever practiced the standard algorithm procedure.

I have begun calling the standard algorithm the “microwave method”. If you use the microwave to cook, yes your food will be cooked quickly, but those students who learn how to use the microwave don’t like to use the oven or stove. They believe ovens and stoves take too long and create too much work. Yes, sometimes it is very helpful to have a microwave to “cook” your meals, but would you want to cook all of your meals in the microwave? There are so many benefits to having other options for subtraction than the microwave method.

On multiple occasions I have given students a subtraction problem where they have automatically applied the “microwave method”  and created extra work or mistakes because of it.
Alternative strategies for subtraction that develop number sense, place value and reasoning are:

  • decompose numbers to subtract (15 – 9) =

(10 + 5) – 9 =
(10 – 9) + 5 =

  • decomposing numbers with place values
  • compensation

(32 – 19) =
(32 – 20) = 12
(12 + 1)

Let’s stop sending the message that math and problem solving must be done quickly, it shouldn’t. I don’t think any one of us wants an engineer to quickly calculate the dimensions and suspension of a vehicle. Instead, let’s send the message that math should be done carefully, thoughtfully, with patience, and flexibility.

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You can also continue to enroll in CGI math classes through the University of San Diego.

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Stay safe, stay happy, stay healthy!