Math Misconceptions 4.NBT.4-6
Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.
By the end of the 4th grade school year, students are expected to be able to fluently add and subtract multi-digit numbers using the standard algorithm. 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 the methods of recording the standard algorithm are acceptable. Errors in multi-digit addition can range from minor mistakes in computation, to larger misconceptions where students do not fully understand the value of each digit. Students may only be looking at digits in isolation by column, rather than looking at each number in its entirety. Have students discuss, represent, and model numbers to verify that they are seeing the whole picture. MISCONCEPTION:
WHAT TO DO:
When looking at the standard algorithm for subtraction, especially when composing is needed, it is very important to work with place value, decomposing numbers, and the relationship between addition and subtraction (inverse operations). Having students “cross out the zeros and put nines” leads to a rule or procedure without proper understanding and can be forgotten or used incorrectly. Working with students on the understanding and having the students explain and show the reasoning of decomposing. Be aware that sometime hints or rules we may use can cause misconceptions as content build. “The bigger number always goes on top?” Subtraction is the difference between two numbers – their difference in value or space on a number line. Work with students to understand that when they decompose a number - such as the 4 in the hundreds place – they are taking one bundle of 100 and are combining it together with the 3 tens. The “13” that you see in the tens place now represents 13 tensor 130. Language and understanding of place value and naming numbers in more than one way (such as 13 tens and 130), is part of the thinking in NBT in the CCSS. Use of the relationship between operations will build understanding of the operations. Students need to have true understanding of the operations in order to use whatever operation they choose to solve for a situation. Students should practice composing and decomposing values to show the relationships between the numbers and operations. MISCONCEPTION: The student subtracted incorrectly, and then copied their answer to “check their work”.
WHAT TO DO:
Whole number multiplication in this grade level includes up to four-digits by a one-digit, and multiplying two two-digit whole numbers using inventive strategies embedded with place value and the properties of operations. As students calculate their products, it’s important for them to demonstrate all their thinking with models, pictures, explanations, and equations. Sometimes when students are decomposing factors and beginning to work with the distributive property, they miss some of the partial products. For example, a misconception might be that 16X13 can be interpreted as 10 groups of 10 and 6 groups of 3. Having students put problems into context and representing those groups with manipulatives or draw out solutions with area models to see their misconceptions, allows them to develop ways of solving a problem. Number lines, equal groupings, partial products, rectangular arrays, properties of operations, and area models are all great strategies for finding products of factors. As students explain their work and thinking, look to see if they are using a strategy that is both efficient and effective. Encourage students to use more than one strategy. Another common misconception is that multiplication always makes things bigger. This “rule” or “statement” is not always true, as students should conclude as they make connections to what they know about multiplication with whole numbers to multiplication with fractions. MISCONCEPTION:
WHAT TO DO:
Division standards should be worked with in connection with the multiplication standards, as they are inverse operations of each other. Students can work with up to four-digit dividends and one-digit divisors by creating equal groups, using partial quotients, and with all of the same models as multiplication. Students may conclude that dividing numbers always creates smaller quotients, but again, this “rule” or “statement” is not always true, when students begin making connections to what they now about division with whole number to division of fractions. Students mostly forget that when numbers are being divided, we can consider the point that we are creating equal groups or sharing. The left over pieces may represent a remainder, which cannot be made into an equal group. This remainder should be described in context of the problem, so students can use context to decide what to do with the remainder, which could be represented in multiple ways (even as a remaining fraction of the whole group). MISCONCEPTION:
WHAT TO DO: