Technology provides me with wonderful opportunities when it comes to assessment. I had a supply teacher in this morning for my math block. We’re just starting our new unit on fractions, and while she did an introductory activity with the students using fraction strips, I got a chance to do the reflect and connect stage when I got back this afternoon. While I started this part of the math lesson with an anchor chart, assessment helped me quickly make changes to this lesson.

As we created this anchor chart together, I realized that the students understood how the fraction size changes based on the denominator, but I wanted to see if they could apply their knowledge in a real life situation. That’s when I posed this question to them: **For a special year end party, I’d like to order in pizza for the class. I called up the pizza place, and the man said that he could make me large pizzas (all the same size), but I could choose to get them cut into 8, 10, or 12 pieces. Which option should I choose? Why?**

I thought that the students were going to say 8 pieces (because the slices were bigger), but right away, I got a 12 piece answer. It was then that I knew that I needed to record the conversation so that I could reflect back on the lesson later. I quickly grabbed my iPad, used a recording app, and backtracked with the problem and the first answer. Things took off from there!

Here’s where assessment really matters! I may have asked similar questions if I was recording or not recording the conversation, but when I press record, I know that I plan to share the podcast. This makes me even more aware of the questions that I ask (*something that I continue to work on*) and the responses that the students give. *I listen more.* With listening, I can assess during the discussion and make changes to my questions based on what I hear.

As our discussion evolved, I saw an opportunity for proportional reasoning, and my questions changed based on this opportunity. I also noticed that students were focused on the money amount (*I guess that this is where a real world problem becomes very “real”*), but I wanted them to make the connection between the denominator (the number of slices in the pizza) and the size of each slice (I want a big slice if we’re ordering pizza 🙂 ). To see if students understood this, I needed to get them to think in this way, so that’s when I changed things. Instead of continuing to ask what option they would choose and why, I gave them the option that I would choose, and I had them explain why. Now they got to see the problem from a different perspective, and I got to see if they understood certain components of fractions.

In the podcast, you can also hear the question about the medium pizza. This again had ties to proportional reasoning, so I wanted to see if students could figure out why I wouldn’t choose this option. Listening to the student questions and answers in the moment, allowed me to assess what they knew and tailor my questions to address next steps in thinking. But with a recording, I can also be brave, and listen back afterwards (*and yes, I need to be brave because the more I listen to myself, the more problems I notice* 🙂 ). Tonight, I listened back to this recording, looked back at the tweeted anchor chart, and realized some questions that we could explore next:

**1) What might be the best option if I wanted two slices per person? Why? Then I could get the students to look at later why the results change between one and two slices. (This has links to proportional reasoning.)**

**2) To reduce extra slices of pizza, I’m thinking about ordering some 8 slice, some 10 slice, and some 12 slice pieces of pizza. What do the students think of this option? Why? This again has links to proportional reasoning and helps students see that the size of the slice varies depending on the total number of slices in each pizza.**

**3) I could extend the pizza problem — looking at dividing pizzas cut into different numbers of slices between groups of children of various sizes — to help students see the purpose for equivalent fractions. We explored equivalent fractions today, and students can tell me what they notice, but the big question is, why does this matter?**

Today I realized just how much I rely on documentation of student learning. Having this documentation tells me what the students know, tells me what questions/concepts I may have missed, and helps me see where we need to go next (as a class and with small groups of students).

**How do you document student learning and use it to inform your teaching practices? How do you assess during and after a lesson? What possible next steps would you suggest based on what you see and hear (for documenting student learning and sharing it publicly, also lets us ask others for help)?**** **I’d love to hear your thoughts!

Aviva

Hi aviva, I am going to go off topic here from your questions and ask you, how does your problem discuss equivalent fractions? I am having a hard time seeing this with them deciding how many slices to divide a pizza into. To me this is a understanding that fractions represent division with a quotient less than one. It also discusses how as a denominator gets larger the piece gets smaller.

In addition to this I find that pizza or more importantly circles are not the best way to represent fractions (at least in this stage in their development). Many of the reasons is that it is very easy to do this when it’s an even denominator but ask them to do this for an odd denominator and they will struggle. This is because they don’t have the understanding of how to divide a circle properly. Also students have a hard time comparing circles because the whole is never the same and they can’t accurately see the representation. This is often seen with 4/5 and 3/4 or any really close fractions.

If it’s equivalent you want them to see then have them play uncover. Or build a number line and use benchmarks (how much more to a whole, half). If it was understanding proportional reasoning of fractions I would go back to the fraction strips and ask them if the noticed anything as the denominator gets larger or how it is growing. Students will notice that as the denominator doubles then you are halving the fraction. You can also use the strips to ask what relationships do they notice between the equivalent fractions? Even the ones that you did in the anchor. What do they noticed? Why are they equivalent? (Besides being equal) can they make any other ones?

Sorry again for going off topic but just a few questions and observations.

Thanks for the comment, Jonathan! I’m so sorry! When you asked the first question, I thought to myself, this problem wasn’t about equivalent fractions at all. I wanted them to see the link between the denominator and the size of the pieces. That’s when I re-read my blog post and noticed my error. Apparently writing a blog post when you’re sick is not a good idea. 🙂

I do love your follow-up questions. I agree with you about the circles and students actually used the fraction strips today. We also drew the fractions in the strip format and not in circles for our initial problem in class today and for our anchor chart. I asked the pizza question today, but students didn’t actually draw the circles. They used the anchor chart (with the strips) to see the size differences. What’s a good real world connection that has rectangular pieces instead of circular pieces? I have the wonderful document that you shared with me via email, and I’m going to use that problem next week. I just wondered if you have any other suggestions. Thanks for the additional question suggestions as well! We’re really focusing on proportional reasoning as a Board, so your proportional reasoning questions will definitely be useful ones.

Aviva

Hi aviva,

Just listened to the podcast and your right it was about the smaller denominator, which makes perfect sense now.

As for problems,

1) have they built the kit? If not I use the sub context to build the kit as I think even to middle school they need this. I would also have them build 3rds, 5ths, 12ths 9ths, 6ths 10ths. Ask them question about the relationships they see with those. The more students understand that fractions are ratios/division/multiplication the easier it will become for them.

As for more problems. The sub problem is amazing for this. I also have them do training for races, where they have to construct a numberline and use benchmarks of a whole, 1/2 and 0.

I have more articles on fractions if you ever need it but read Fosnot’s book or vandewalle’s chapter on fractions.

Thanks so much, Jonathan! They have built the kit, and I plan on doing the sub problem next week. I can’t wait to see where this goes!

Thanks for the resources and the support!

Aviva

Hi Aviva,

I am teaching a grade 5 class fractions right now and would love to see the “sub” question that you were discussing above (if neither of you mind sharing it, that is).

By the way, I am really enjoying all of the sharing you do on Twitter. I am getting some great ideas for my own class. I only wish I had started following you earlier! 😉

Shannon

Thanks Shannon! Glad you find my Twitter sharing helpful. I learn so much from others, and I’m glad that I can share with others as well.

Let me just check with Jonathan, and then I’ll email you the fraction information that he shared with me.

Have a great weekend!

Aviva

Always happy to share. All of my resources can be found at bit.ly/Soresources. The problem insent mine, I just adapted it from Fosnot’s sub problem in her fundraiser book. She has some great fraction problems there. Always happy to share and I too love Aviva’s reflections and sharing on twitter. I have learned so much from reading hers and am always in awe.

Thanks Jonathan! I appreciate you sharing your wonderful resources. I also love what you share on Twitter and through your blog (and through you comments on this blog). It’s great that we can learn so much from each other. That’s how we all get better!

Aviva

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This is an assessment story as much as a math story…the documenting you do allows you to be incredibly precise with your questioning and lesson sequence.

What’s always interesting in math is: what consistent mistakes or misunderstandings are you uncovering? Are these mathematical problems they’re having, and if so, how do you course correct.

I often find with fractions that connections to other representations like proportions (as you said), decimals, and percents can help!

Also agree- move on from pizza next to other wholes or sets, and see if the learning transfers!

Thanks Matthew! I always appreciate your feedback. I’m going with a sub problem next, as suggested by Jonathan. Then we can further explore proportions, but also start looking at equivalent fractions. For now, we’ll move on from the pizzas … 🙂

Aviva

As always, great reflections Aviva. In terms of documenting and assessment – it’s great to see assessment and documentation happening throughout a lesson or learning activity. All too often, although we claim to understand the difference between formative and summative, our documentation of learning comes at the end – end of a lesson, end of a period, end of an activity. Stopping mid-stride to make sure we are on the correct path – or just on a path at all – is not an area of comfort for people….and consequently not for students as well.

If you’re looking for alternate methods to document learning at various “checkpoints” perhaps you could involve the students more. Get them to be more cognisant of their learning by documenting it themselves. Something I’ve never tried but have wanted to is keeping a “continuum of learning” or a “learning matrix” posted somewhere in the classroom where students can go to record or document their discovery at any point in an activity or task. Making it public would serve the collaborative culture we are promoting and stopping to honour the things students contribute may help others to contribute their findings as well. Of course some gradual release of responsibility would most likely be necessary but I think there’s potential for student voice, links to inquiry, meta cognition,and encouraging risk taking.

Just a thought….not sure if it would work…..just a thought.

Thanks for the comment, AJ! I agree with you about documenting during the learning process. I often record podcasts and videos during the activity as I conference with students. Sometimes I take photographs of their work and document their learning as well. I love your idea too! This is definitely something to try. It also gets the students involved in the documentation process, which I love!

Aviva