Interview Task Context Problems-Multiplication Student #1 Bob has 4 boxes of crayons. Each box has 5 crayons in it. How many crayons does he have altogether?	Student Response 15, 25. No, 20.	Thoughts/Comments How did you get 20?
	I counted by 5 four times.	Okay.
Anna has 3 boxes of cookies. Each box has 11 cookies in it. How many cookies does she have altogether?	33	Explain to me how you figured that out?
	I said 11+11=22 and 22+11=33

Interview Task John has 18 boxes of marbles. Each box has 12 marbles in it. How many marbles does he have altogether?	Student Response (He thinks momentarily.) That’s hard.	Thoughts/Comments How do you think you could figure out the answer?
	I don’t know. Add 18+22. It’s hard.	Okay, let’s try another one.
John has 14 boxes of marbles. Each box has 3 marbles in it. How many marbles does he have altogether?	9, no 42.	How do you know?
	It’s 42 because you add 14+14+14.	Okay, let’s try a different type of problem.

Interview Task Context Problems-Division Shameka has 16 bracelets that she wants to share with her sisters. If she gives 4 to each sister, how many can she give to?	Student Response 4	Thoughts/Comments How do you know?
	I counted down from 16 four times.	Okay, break it down for me. What did you exactly do?
	16-4=12 ,12-4=8, 8-4=4	Okay.
Roshawn has 54 toy trucks that he wants to share with his friends. If he gives 5 toy trucks to each friend, how many can he give to?	I’m confused.	About what?

Interview Task	Student Response I think I should count down from 54.	Thoughts/Comments What do you mean count down?
	Subtract 5 from 54 each time.	Okay, try doing it and see what you get?
	(He uses the frog manipulatives to count out 54 –counts by 5. Once he has 54, he takes away 5 each time and loses count at 20. He then starts over and counts out 54 by 10 until he gets to 40 and then begins to count by 1. After this, he subtracts 5 each time recording his results with tally marks.) The answer is 13.	Okay. (I notice that he is looking at the two that he has left over.) What are those, leftovers?
	Yes, 2 are left over.

Interview Task	Student Response	Thoughts/Comments I know that the previous problem gave him trouble, but I wanted to see what he really knows because he appears to have some knowledge of how to solve this type of problem.
Vanessa has 65 ribbons that she wants to share with her friends. If she gives 13 to each friend, how many friends can she give ribbons to?	He draws out the ribbons. Each rectangle equals 10. After he reaches 60, each rectangle equals 1. Is the answer 6?	How did you get 6?
	I know that 7+3=10 and 10+3=13. So, I take 3 from 1 ten and add it to another 10 to make 13.	Okay. (I am somewhat understanding, but not I do not fully grasp the concept that he is using.) So, what do you think the answer is?
	(He counts the tally marks that he has made.) 6	Why 6?

Interview Task	Student Response Because 7+6=13	Thoughts/Comments I am confused, but I continue to ask him similar problems.
Shawn has 48 stickers and wants to give them to his friends. If he gives each one 8 stickers, how many can he give stickers to?	(He draws circular symbols. Each one equals 10. He writes 2 over one symbol and 8 over another. This is repeated.) I think there is a pattern.	What pattern?
	It goes 2, 8, 2.	What does this mean?
	I don’t know. Let me start over. (He begins to add 8+8+8+8+8+8.) 6	How did you get 6?
	Because 8+8+8+8+8+8=48	He has solved the problem, but I am still not clear on his rationale, so I ask one more question.
Paul has 36 fair tickets and wants to give some to his friends. If he gives 6 to each of his friends, how many of his friends can he give to?	(He begins to subtract.) (He keeps track of how many times he subtracts 6 with tally marks.) 6	Why is the answer 6?
	Because 36-6=30, 30-6=24, 24-6=18, 18-6=12, 12-6=6	I decide to move on.
Fractions What fraction of one cookie will each person get if they share equally? 2 cookies among 4 people.	(He divides the cookies that I have printed out into halves.) ½	How do you know the answer is ½?
	Because there are 2 cookies and 4 people, so you divide the cookies into 4 pieces.
4 cookies among 16 people	(He divides the cookies into fourths and counts all the pieces to make sure there are 16.) ½	Why?

Interview Task	Student Response You need 16 pieces, so the cookies should be divided into 4 pieces.	Thoughts/Comments I am thinking that he understands how to get the answer, but does not know how to put it in terms of fractions.
7 cookies among 56 people	(He divides the cookies into eighths and counts the pieces by 1 until he gets 24.) 8	(Okay, now I am sure that he is confused about how to say the answer.) You say 8. So what fraction or portion of the cookie does each person get?
	A piece.	You’re right. They do get a piece of the cookie, but how would you say it in terms of a fraction.
	8 (He is starting to get frustrated.)	Now, I realize that he may have trouble saying the answer in terms of fractions. I do not think this is major and decide to end for today.

Interview Task Flashing Student 2 ·· ·· ·· ·· ·	Student Response 9	Thoughts/Comments How did you see 9?
	I saw 4 and 5 together.
·· ·· ·· ·· ·· ·· ·· ·· ·· ·	19	In what order did you see this?
	10 and then 9	Did you notice anything?

	1 was missing, making 9	Okay.
·· ·· ·· ·· ·· ·· ·· ·	14	How did you get 15?
	I saw that 2 were missing, making 8 and I counted the other 6.	I realized that he made a mistake, but I feel that he has and firm grasp on this idea. I move on to more difficult problems.
Interview Task Fractions-Addition Bobby ate ½ of his candy bar on Monday and ½ of his candy bar on Tuesday. How much of the candy bar has he eaten?	Student Response The whole thing.	Thoughts/Comments Why is it the whole thing?
	Because if he ate ½ of it and ate the other ½ of it, he ate the whole thing.	This really doesn’t tell me what he knows, so I ask another question.
Johnnie at ¼ of his cookie on Friday and 2/4 of his candy on Saturday. How much of the cookie did he eat?	3/8	Why is the answer 3/8?
	I don’t know.	Show me how you figured it out.

Interview Task	Student Response I said ¼ + 2/4 = 3/8	Thoughts/Comments Now, I see that he is adding the fractions together. (numerator + denominator)
Becky gave 1/6 of her bubble gum to her little sister and gave 4/6 away to her little brother. How much did she give away?	5/12	Why?
	Like the others. 1/6 + 4/6 = 5/12	Okay.
Fractions Mike has a can of Mt. Dew. If he drinks ½ of it, how much does he have left?	½	And you know this because?
Interview Task	Student Response 2 halves equal a whole	Thoughts/Comments
Melissa raked 2/3 of the yard today. How much does she have left to rake tomorrow?	3/3	Explain.
	I don’t know. I haven’t really learned this.	Okay, let’s try another.
Stephanie wrote ¼ of her English paper on Wednesday. How much does she have left to write?	1/3, no 4/4.	Show me how you figured this problem out.

Interview Task	Student Response (He draws a number line like this ¼, 1/3, ½.)	Thoughts/Comments What does this mean?
	There are 4 spaces left so, the answer is 4/4.	I see that he does not understand how to subtract fractions. I ask more problems to see how he thinks.
Fractions-Subtraction ½ minus ½ equals	0	Why?
	The whole thing is gone.

Interview Task ¾ minus ¼	Student Response 4/8. Oh, minus, then 2/0.	Thoughts/Comments He is now subtracting the numerator and denominator.
15/18 - 3/18	12/0 (almost confidently)	What method did you use?
	Subtraction	I conclude that he does not do well with fractions and decide to ask him some other questions.
Context Problems-Multiplication Anna has 3 boxes of cookies. Each box has 11 cookies in it. How many cookies does she have altogether?	33	How do you know?

	I counted 10, 20, 30 +1 left over.	At first, I was confused, but I found out that he knew that there were 3 sets of 11. Instead of working with 11, he changed everything to 10 and added the leftovers.
John has 18 boxes of marbles. Each box has 12 marbles in it. How many marbles does he have altogether?	(He starts to draw lines to represent 96, then stops.) This is basically 12 x 18.	So what do you think the answer is?
	(He sets up the multiplication problem.) 96

I chose to interview an Emotionally Disabled student from Memminger Elementary School because I was interested to see what math strategies he or she would posses. I knew that the student might have a difficult time focusing on tasks for a large amount of time, so I decided to break the activity up into two days. However, the student I was working with did not attend school on the second day and I had to work with another student.

The first student that I interviewed was a 9-year-old fourth grader. He has been at Memminger for 2 years. He lives in a single parent home with his mother, 2 sisters, and brother. His favorite subjects are math and spelling.

The second student was a 13-year-old sixth grader. He has attended the school for 1 and ½ years. He lives with his older sister and 2 younger sisters. His favorite subjects are science and math. He hopes to attend college and become an actor or rap artist.

By interviewing the first student, I conclude that he uses addition to solve multiplication word problems. (I am not sure if he understands the connection between addition and multiplication.) When solving the first word problem, he counted by five. Although he did not get the correct answer on the right try, I figured that he knew what to do and may have been just a little nervous. He was able to recognize on his own that the answers that he was calling out were incorrect. He also knew that he needed to count by 5 four times to get the answer. However, I am not sure if he realized that he could have also answered the problem by multiplying 4 and 5. In the second problem, he seemed to use simple addition. He knew that he should add 11 together 3 times in order to get the answer. He made the problem somewhat complex by adding 11+11 =22 and 22+11=33 instead of adding 11+11+11. The third problem was difficult for the student. I believe that this was a result of the larger numbers that were posed. When I realized this, I began to ask him questions with smaller numbers and he was able to successfully solve them.

Division word problems confused the student a little; however, he attempted to solve some of them using subtraction. He answered the first problem correctly, but his explanation was incorrect. He explained that he “counted down from 16 four times, when in actuality it should have been three. I believe he would have noticed the mistake if he understood that the problem was division, not subtraction. The student tried to solve the second problem using frog manipulatives because it included a large number. The answer that he gave to the problem was incorrect, but the method that he was using was right. He knew that he must take away 5 from 54 as many times as he possibly could. But, since he has trouble working with large numbers, he lost track of his count causing him to give a wrong answer. I was totally confused by the student’s method for solving the third problem. He tried to use pictures and addition to solve the problem. I feel like he was trying to solve this problem like the previous one, but somehow got sidetracked.

The first student’s concept of fractions is a little shaky. He understands how to divide something up so that there are equal pieces, but he does not understand how to put the answer in fractional terms. For example, in the second fraction problem, he divides the cookies up correctly into 4 pieces, but says the answer is ½. Also in the third problem, he divides the cookies into eighths, but is unable to say that the answer is 1/8. I truly feel that his concept of fractions would be better if someone worked with him on what the divisions that he makes actually represent.

I was told that the second student that I interviewed may have been on a lower level than the first student, so I decided to start off with something less difficult such as flashing. After the flashing, I concluded that he determined some of the answers by identifying which sections were missing dots. If he was able to count the missing areas, then he was able to figure out the answer. He used this method mostly when it appeared to be an even number of missing dots. (Ex: 8). However, when there seemed to be an uneven number of missing dots, he simply counted all of the dots that were present.

Adding and subtracting fractions seemed to be foreign to the student, although I was informed that the student had a substantial amount of knowledge of them. He solved addition problems by adding the numerators and denominators. (Ex. ¼+2/4 = 3/8) He solved subtraction problems by subtracting the numerators and denominators.

(Ex: ¾-2/3= 2/0) There was no knowledge of finding a common denominator. The student did have an understanding of ½ and 1 whole. He knew that ½ + ½ =1, ½ - ½ =0, and that 1 – ½ = ½ . He also knew how to order fractions because when he tried to solve one or two problems, his method included writing ¼ , 1/3, and ½ in order on a number line.

When I asked the second student multiplication word problems, I first thought that he had little knowledge about multiplication. I thought this because he answered the first problem by using addition. However, I soon realized that the student understood multiplication well. In the second problem, he began to solve the problem using addition, but realized that he could find the answer by multiplying.

From this interview, I learned that Emotionally Disabled students have some of the same troubles with mathematics as “regular” students. They sometimes combine various math techniques and concepts that they have learned when solving problems. They also rely on manipulatives or pictures to help them solve problems that appear to be somewhat difficult or confusing. The most important thing that I learned is that Emotionally Disabled student are hardworking and focused. They take pride in being able to solve a problem correctly and seem to put forth their best effort.