The importance of formative assessment in mathematical problem solving

"When the cook tastes the soup, that's formative; when the guests taste the soup, that's summative."

-Robert Stake

From the presentation by Lynda Watters and Jill Sanderson, 2016

Our grade 1/2 teacher (Jill Sanderson) and our grade 4 teacher (Lynda Watters) were gracious enough to put together a power point presentation of their learning from the CAFLN (Canadian Assessment for Learning Network) conference they attended. In their presentation they mentioned that Formative Assessment doesn't have to involve a lot of paper work but rather just taking the time to observe and communicate (ask questions rather than giving answers), with students about their work.  

During the presentation, Jill (grade 1/2 teacher) passed out a copy of a math problem question that she decided to try with her class having seen it at the conference. She asked us to look at the child's representation and ponder what this child knew? What questions would we ask her? How could we support her learning further?

Sometimes we feel the need to steer children towards correct answers rather than taking the time to observe their work and answer questions with further questions to support learning and understanding. 

According to the Natural Curiosity resource, Assessment for Learning is outlined as the following:

• the process of gathering and analyzing evidence for use by students and their teacher to determine where students are in their learning, where they need to go, and how best to get there.
• helps teachers differentiate and focus how to approach the learning of individual students 

(Natural Curiosity: Building Children's Understanding of the World through Environmental Inquiry/A Resource for Teachers, page 30).

I decided to take this question and ask it to my kindergarten students. I was interested in learning if they understood the question, the different strategies they may choose to use to solve it, and how they would demonstrate their answers.

Take a look at the pictures below. What do you deduce from each of their representations? What do each of them tell you about their knowledge and understanding of the question? How can these representations aid in supporting student learning further? 

"Feedback can be the vital link between the teachers' assessment of a child and the action following that assessment, which then has a formative effect on the child's learning." -Hargreaves et al. (2000)

From the presentation by Lynda Watters and Jill Sanderson, 2016

The students enjoyed the challenge question. After sharing their work with their peers I decided to try a few more questions with them.

A cube representation of twenty bees each with four wings. 

D attempted to use cubes to depict twenty bees each with four wings. This was a good start but he asked Z if he would help him with his counting as he kept getting stuck. Notice the way they used the cubes to represent the details in the question. 

This time I shared a question that required paying attention to two characteristics, number of legs and number of trunks. I again asked the students how we could figure this out?

When I viewed E's representation, I was proud that he understood to draw ten elephants. He also added one trunk and four legs for each elephant. He became tired and did not finish solving the problem. But this drawing alone is full of information about his understanding of the question and his number sense. 

  

At first she made ten pieces of wire with beads on them. She told me she was done. I then asked her to explain to me what the different coloured beads represented? How many elephants do we need? She told me that the four beads represented the legs and the green bead represented the trunk for each elephant. In telling me this she started to count the elephants and realized she made too many.

With this question I decided not to discuss as a group as I wanted to see if any of the students took some of their learning strategies from the previous questions and related them to solving this question. 

There are 15 flowers in my vase. Each flower has 5 petals. How many petals altogether?

M chose to use cubes to represent the flowers and petals. M told me he placed these in rows of five though he still chose to count the flowers and petals by one's. 

In her drawing representation, M knew to draw fifteen flowers. She told me she had to scratch some out as she realized she made a mistake and drew too many. Her telling me this information was very important as it told me a lot about her counting ability and her ability to identify when things did not add up correctly. You can see that she knew to draw five petals per flower but again she told me she drew too many on some and she had to go back and scratch them out. Checking over one's work is an important skill to have and I noticed that M was able to do this independently. She became a bit tired so she did not finish figuring out the answer. But this is not always the most important aspect as there is so much that is gained from even attempting. 

A created a lovely representation using poker chips. He made fifteen flowers each with five petals. He used jewels for the centre of each flower which he did not count as a petal. I observed him counting the petals and noticed he was counting by one's. Though we have done some work on counting by 2's, 5's, and 10's. It seems this method of counting is the the most reassuring for him.

As a fun math challenge I showed the video below to the children. I received the link to this math problem from Lynda Watters (grade 4 teacher) who received it from Lynn Strangway (TDSB Principal) and Dianne Riehl (TDSB Vice Principal). She informed me the math problem was explored by a grade 1/2 class. I wanted to know what some of the kindergarten students thought after viewing it a few times and having access to the use of materials.

Thoughts after first viewing:

"He sorted by half and split them." K. C.

"He said one for him and one for his best friend to make it equal." L. B.

"All of his friends and him go the same amount." D. F.

"But not his sister, she only got five." B. P.

"He split in half for this friends." C. T.

Thought after second viewing:

"I noticed that how many he got looks just like ten." J. K.

"When he divided with his friends they got the same amount but not his baby sister or him." C. C.

"I noticed he shared." E. B.

"Every time he split he had some left and the leftovers he kept splitting so his friends get the same amount." Z. G.

"He did a pattern, one for you and one for me." C. D.

"He split with four people." F. D.

After viewing the video a few times, I asked the students if anyone wanted to use the lids to explore this math problem further? Below is C. C. and Z. G.'s experience.

They decided to re-enact the video. They labeled three bags with the names of a few friends in the class that like gumballs. Then they proceeded to dump out all the lids on the table and split them the way they saw Charlie do in the video. They thought everyone would get the same amount of gumballs but as they kept splitting Z. G. said that there wouldn't be enough for him to split with the other bag. 

"We used to have so much gumballs but now we have less because we're sharing them with all of our friends." C. C.

"Do they all have the same amount of gumballs?" Mrs. Ralph

"No because we had a lot but when we splitted them but then they became less and less amount because I had a lot at first and I split them with A. F. but then it got less and less to share with everyone." C. C.

I asked them if there was a way that they can make it fair?

"So we gave one to everyone and keep splitting them. I think everyone will have the same amount." Z. G.

"Charlie kept splitting the gumballs but every time he did his pile got smaller and smaller so not everyone go the same amount." C. C.

I kept the bags and thought why not place them at a table with the lids and see if it would entice anyone to go and explore in their own way.  L. S. and A. R. decided to give it a try. 

"We made Charlie's gumballs. We counted by ten. We figured how many Charlie had. We put one by one. Each bag has forty gumballs. Four gumballs for Charlie. Five gumballs for his little sister." L. S. and A. R.

The video below was from the morning exploration. The photos above are from the same day but during the afternoon. In the afternoon the girls found more lids and decided to split the "gumballs" again.

What math language knowledge are they using? What do the two students understand about this math problem? What would be a possible next step to extend their learning?

A timely question as it was pizza lunch day! We were curious as to how many pieces of pizza everyone ate. I asked the kids to remember how many slices they ate. We recorded the information and got a whole bunch of numbers! I then asked them, "What are some different strategies you could use to try and figure out how many pieces of pizza we ate altogether?".

C. C. and E. B. decided to use a ten frame to help them add up all the numbers. They check off each time they added chips onto the ten frame that coincided with the information on the chart paper.

Both C. C. and E. B. did their own counting using ten frames. What do you notice in the way they represented their data? 

F. D. said that she wanted to count by two's then count the one's and add the two numbers together. I offered her the number chart to support her in adding the two numbers she got together. 


C. D. and S. T. wanted to draw pizza slices to represent the data on the chart stand. After they drew all the pizza slices they counted the slices to get the total slices.

What information can you take away from the three different strategies used? What further supports can be offered to the students? How can their thinking and learning be extended? I find myself asking these questions often. They are a good guide for me to reflect on the thinking and learning of each student.

It's important that we place just as much value on making student thinking and learning visible rather than focusing on the final product so we can better differentiate the support and guidance needed by every child.     

"What if we started from a place where learning was viewed as an interaction in which two or more individuals work together in an intellectual way to solve a problem, clarify a concept, reflect, extend a narrative, etc. Both parties must contribute to the thinking and it must develop and extend, in other words, negotiated learning." 

(Sylvia, K., Melhuish, E.C., Sammons, P., Siraj-Blatchford, I. & Taggart, B. (eds), 2010)