##### Reasoning for inclusion in MIPI+

An understanding of number involves integrating several key concepts, such as unit, place value, and one-to-one correspondence…Ideas such as unit and decomposition that children encounter…set the stage for their development of fluent use of place value, leading to number flexibility and computational proficiency. (**Dougherty, Flores, Louis,**** Sophian, 2010**)

If students do not have a true understanding of place value, which builds upon their understanding of conservation of number, operations becomes challenging.

##### Understanding the different types of questions

There are different types of questions that teachers ask around place value. In order to help you build a deeper understanding, complete the following activity.

**Materials:**

- Print the following
**document**. You only need half of it as the second half is a duplicate. Cut apart the statements

**Instructions:**

- Sort the statements into groups so that each grouping is based on a common characteristic. The words that will go on the _____’s will be provided afterwards.
- Sort the statements using a different common characteristic.
- Open the
**Place Value Sorting activity**google slideshow. - Slide 1:
- The first slide has the statements and are draggable (as long as you are not in presentation mode)

- Slide 2:
- Did you sort according to 2, 5, 8 and 20, 50, 80 and 12, 45, 78 and 123, 456, 789?
- Did you sort according to 123, 2, 20, 12 and 456, 5, 50, 45 and 789, 8, 80, 78?
- This slide shows the sorting according to both ways.
- Important! If you are asking a question that is looking for 2, 5 or 8 as the answer, that is different than the question that is asking for 20, 50 or 80 as the answer which is different than asking a question that is asking for 12, 45 or 78 as the answer.
- There are three terms at the top in grey boxes. One of these terms was created by CESD in order to help distinguish this type of question from the other two. Can you figure out which one is made up? Can you figure out which term goes with which category?

- Slide 3: Each of these words will show up individually, followed by the definition if you are in presentation mode.

Students aren’t expected to know the three terms as provided in the activity.

##### What is the misconception?

**The MIPI+ Question 2 Part 1:** Students misunderstand the difference between place value and “base value”.

**The grade 3/4 question:** How many tens are there in 125? ___________ Convince me.

The best correct answer is 12.5. However, this is not expected nor required as an answer in the MIPI+. This demonstrates a deep understanding of place value and decimals that is not expected at the beginning of grade 3 or 4.

The acceptable answer is 12. Think about 125 as money. If you only had $10 bills, how many would you have? If you skip counted, how many skips would you make?

Students, who state that there are 2 tens in the number 125, are only looking at the quantity of the number in the tens place.

Some students stated that the answer was 2 for other reasons. “I skip counted.” “Tens is the second digit.” If this happens, give them a different number, such as 346 and ask the question again.

Students who state that there are 20 tens are thinking about the value of the number in the tens place.

Students who respond 1 or 5 have not learned the place value positioning.

Students who respond 3 have added the 2 from the tens position and the 1 from the hundreds. They do not understand that the 1 is worth 100 or 10 tens.

Students who respond 26 have added the 25 and the 1.

**The grade 5 version** asks: How many tenths are there in 1.25?

**The grade 6/7 version** asks: How many hundredths are there in 0.125?

These versions still have answers as listed above: 12.5 / 12 / etc. Just shift your thinking about the reasoning behind it to reference the new placement of the “2”.

**MIPI+ Question 2 Part 2:** This question focuses on conservation of number.

**The grade 3/4 question:** Think about the number 125. You trade one TEN for ten ONES. After the trade, is your number now larger, smaller or the same? Convince me.

The correct answer is “the same because one ten is the same as ten ones.”

Students who state that the answer is larger often look at it as an exchange of quantity rather than focusing on the value of the blocks. “I got rid of one block and received ten more blocks. I have more blocks now.”

Students who state that the answer is smaller often say “I got rid of a ten so my value is less” without taking into account that the value of the ones increased.

**The grade 5 question: **Think about the number 1.25. You trade one TENTH for ten HUNDREDTHS. After the trade, is your number now larger, smaller or the same? Convince me.

The correct answer is “the same because one tenth is the same as ten hundredths.”

Students who state that the answer is larger often look at it as an exchange of quantity rather than focusing on the value of the blocks. “I got rid of one block and received ten more blocks. I have more blocks now.”

Students who state that the answer is smaller may have one of 2 main reasons.

- They say “I got rid of a tenth so my value is less” without taking into account that the value of the hundredths increased.
- They see tenths as valued less than hundredths because tens are smaller than hundreds.

Reason #2 usually made it impossible for students to provide an answer. They could not visualize exchanging what they viewed as a ten for ten hundreds. They knew that if you did that, you were changing the over all value. So, despite the fact that they didn’t understand the value of the decimal placements, they are demonstrating some understanding. If, during the assessment, students indicate this issue, give them a new number to work with. Cross out the 1.25 and give them 125. Rewrite the question so that it says you are trading one ten for ten ones. See what happens. This will help you determine students’ depth of understanding using whole numbers rather than decimals.

**The grade 6/7 question: **Think about the number 0.125. You trade one HUNDREDTH for ten THOUSANDTHS. After the trade, is your number now larger, smaller or the same? Convince me.

The correct answer is “the same because one hundredth is the same as ten thousandths.”

Students who state that the answer is smaller may have one of 2 main reasons.

- They say “I got rid of a hundredth so my value is less” without taking into account that the value of the thousandths increased.
- They see hundredths as valued less than thousandths because hundreds are smaller than thousands.

Reason #2 usually made it impossible for students to provide an answer. They could not visualize exchanging what they viewed as a hundred for ten thousands. They knew that if you did that, you were changing the over all value. So, despite the fact that they didn’t understand the value of the decimal placements, they are demonstrating some understanding. If, during the assessment, students indicate this issue, give them a new number to work with. Cross out the 0.125 and give them 125. Rewrite the question so that it says you are trading one ten for ten ones. See what happens. This will help you determine students’ depth of understanding using whole numbers rather than decimals.

##### Students need to understand:

In the number 125, there are

- 125 ones
- 12 (or 12.5) tens
- 1 (or 1.25) hundred

but there are also (not expected at beginning of grade 3/4)

- 1250 tenths
- 12 500 hundredths
- 125 000 thousandths

Each “spot” is made up of the value to the left of the number (and to the right for students with very deep understanding).

As digits move to the left in a number, their value increases by 10 times and as they move to the right, their value decreases by 10 times.

##### How is the misconception developed?

Think about the questions that resources ask students around place value. Are they asking students to “state the number in the tens place”, “state the value of the number in the tens place”, or “how many tens are there?” These are asking 3 very different questions. We should be asking all three and helping students understand the difference between each of them. The teacher needs to be clear which question he or she wants the answer to.

##### High Leverage Strategies you can use in class beginning in Kindergarten:

Click here to check out the high leverage strategies.

##### Further Learning:

Dorward, Jim. Research, Reflection, Practice – Place Value: Problem Solving and Written Assessment

Dougherty, Barbara, et al. *Developing Essential Understanding of Number and Numeration for Teaching Mathematics in Pre-K-2*. NCTM, 2010.

Hopkins, Theresa M. and Cady, Jo Ann, What is the Value of @*#? NCTM, Teaching Children Mathematics April, 2007

Ross, Sharon Parts, Wholes and Place Value: A Developmental View

##### References:

Coming soon.