But it's baking in my classroom facing the hot, hot sun, and I'm tired! So, for today's blog post, I will let the pictures do the talking...
If I were a good teacher, I'd have co-constructed criteria with them, and had them play their games and then measure the success of their games against said criteria. Or, I could have tied in my persuasive literacy unit and had them write advertisements trying to convince others to play their games. Or I could have assessed their rules for the game and used it as a mark for procedural writing.
But it's baking in my classroom facing the hot, hot sun, and I'm tired! So, for today's blog post, I will let the pictures do the talking...
Probability comprises concepts that many adults struggle to understand. (Not convinced? Just have a look at our booming lottery business!)
How, then, can we engage children in such a way that ensures they build a solid foundational understanding of this important mathematical area?
Since many of my students have developed strong oral language skills this year, especially when working with a partner or in a very small group, I developed a series of lessons that drew heavily on the instructional strategy of "think-pair-share", and banked on Van De Walle's games-rich chapter on probability. We spun spinners, rolled dice, made and tested predictions, played "hockey" and other goal-based games that depended on predicting outcomes. But first, we explored the basic vocabulary of probability: chance, possible, certain, impossible, likely, unlikely, equally likely....
A number of scenarios were presented on the board, and students had to discuss whether each was certain, possible, or impossible. Then we moved each scenario into the appropriate part of a Venn diagram on the Interactive White Board.
Lots of wait time was provided for students to consider and discuss each scenario, and as you can see from the photo above, they were quite engaged: For once, every student was either looking at the screen, or discussing the scenario with nearby neighbours!
Next, students engaged in a "continuum of probability" mini-lesson, in which we moved a series of five different spinners into sequence from impossible to certain. It was a great opportunity for them to see each of the new words they had learned in the bigger context of probability vocabulary and concepts.
Finally, after lots and lots of thinking and rich talk, I asked students to work on their own to write down a few examples of things that were likely, impossible, certain, etc. I used this exit ticket as a diagnostic assessment to see what level of understanding most of the learners were at in order to fine tune my next lesson, which involved a series of probability math games.
Communication. It’s an essential component of mathematics, especially when dealing with a student population whose first language is NOT the language of instruction!
A few weeks ago, I wrote about the simplification of criteria in order to help focus students on writing clear and succinct explanations when communicating solutions to problems they solve. That strategy seems to have worked for many of the students in my class: Below is a sample from some multiplication/division problems they solved the other day.
As one can see, it is easy to assess their understanding of the concept, as well as note where errors were made (and by following up with a short conference, one can surmise if those errors were mathematical, or linked to a misreading of the problem itself, due possibly to ESL factors).
Speaking of conferencing, my colleague, Dale, recently reminded me of the value of this: As we finished up our transformational geometry unit a few weeks back, he elected to evaluate students by meeting with them individually to orally assess their understanding of transformation understanding.
Dale showed what he had students do, including a question or to using a grid map from the text book, and a few tasks involving the translation, reflection and rotation of an object on a grid paper.
For each task, he made note of whether they were able to do it independently and confidently, whether they benefited from some teacher guidance, or whether they seemed completely at a loss, even with teacher intervention and support. (I stole his idea and created a recording template -- see photo below.)
In some sense, the conference became a bit of a teaching opportunity as well as an assessment measure. We conference daily or weekly with students for reading and writing as a way to assess and remediate as needed, why not in Math?!
I've often harped on the importance of employing "talk before writing" strategies like "think-pair-share"... Another thing I’ve noticed, both while conferencing AND while using such instructional tactics as “think-pair-share”, is the importance of wait time. It can take a while for students to focus and get on-topic. Giving them a few extra seconds or even minutes to chat with a partner, instead of rushing them through a discussion topic, can really help students to go a little deeper with their understanding. Although I know the research on wait time, I am always surprised when I actually remember to use it, and then students' responses are richer and more focused!
Simplifying success criteria, using oral conferencing as a form of assessment and integrating wait time into the instructional repertoire... three of the many ways we can increase student success as they communicate in math.
As my students were working on a problem the other day, I wandered around to video tape them, just to get a sense of how things work in my classroom during the "action" phase of a lesson. I caught a few students' explanations of their solutions on camera, then turned the camera off.
No sooner had a turned off the camera, then an intellectual debate broke out amongst four students -- I quickly turned the video camera back on, and caught the tail end of the argument.
At issue was the number of crossings a ferry captain has to make to get from one side of the water to the other in order to ferry a total of 26 cars across from the mainland to the airport where Ms. Teschow takes her flying lessons. (The ferry boat can hold a maximum of 8 cars per crossing.) The boy I had just been filming claimed it would take 8 crossings, however, the fellow in the zip up jacket with the orange collar claimed the number to be 7. In the snippet below, he attempts to show why. (Note that although he struggles with clearly articulating his thinking, everyone is very polite, even when they animatedly disagree. Note also zip-up guy's "patterned" diagram on his whiteboard, which we later shared in stages with the rest of the class.)
I continue to recognize the importance of having students engage in meaningful dialogue about their work in math. And I continue to struggle with how to support those students who demonstrate difficulty with this.
Some time ago, I wrote a blog post about rich questions, used to guide students' thinking while they work. These questions seem quite effective for individual and small groups of students as they work on a math problem, but large group debriefs continue to be an "opportunity for professional growth", as a colleague of mine would say!
A teacher from another school recently shared a resource on accountable talk, which his Early Literacy Teacher had compiled from various sources and provided to the staff there.
Although I do some of these things already, I feel that many of my students who struggle with attention and focus issues will continue to be "unready" when I call on them and invite them to join the conversation. I also wonder how many -- if any -- of the suggestions come from resources that bear in mind the types of students (high ELL, considerable socio-economic needs) that we are dealing with in the school where I currently teach.
That being said, I do wonder if some of my students merely suffer from "participation anxiety", and if the manner in which I invite them to join the large group conversation could have a more positive effect on the nature of their subsequent input.
It is certainly true that despite the learning challenges they face, my students (with extensive training) now do very well during small group conversations about texts that we've shared.
Our bulletin board is full of evidence of the grand conversations we've had about a wide variety of picture-book-inspired BIG topics: Poverty, Sexism, Racism, Illiteracy, Equity, Community and Citizenship, Peace and more...
If students can be trained to successfully and respectfully converse about such big ideas, surely they can learn to have similarly rich dialogue about mathematical thinking and problem solving???!!
I resolve to review the resources and suggestions in the document above, and try again in my classroom!
We have found the use of sentence frames extremely useful for fostering oral and written language in the math classroom. Giving our English Language Learners this scaffold allows them to fill in the blanks with the concept, and focus on the math while practising correct sentence structure.
Thanks to our Early Literacy Teacher, Jenn Black, for sharing the resource below, a handy "quick reference" of possible sentence stems for the math classroom. The frames in the doc were compiled from Rusty Bresser, Kathy Melanese, and Christine Sphar's "Supporting English Language Learners in Math Class, K-2" (pages 164-165) and "Supporting English Language Learners in Math Class, 3-5" (167-169), 2009.
One of the best activities parents can do to support their children in math (and in general) is to take them shopping. Although it is of course far easier to just do it yourself sometimes, taking children along for a grocery run, and asking them to consider food choices, costs of items, sales, etc., builds vocabulary and schema.
On of the students in my class this year is very lucky to have a neighbour who does lots of schema building with him She shared this experience with me the other day and -- with her permission (and a few edits to protect privacy) -- I am reprinting it here:
...Nilloah finally experienced real shopping.
As I'm sure you can imagine, this neighbour has the patience of a saint!
When I greeted Nlioah in class the next morning with, "hey, so I hear you went shopping!", he almost fainted. It was like magic! "HOW did you know that, Ms. Teschow?!" he exclaimed!
A little email is a miraculous thing.
Something I learned as a student work study teacher through a project with the Ministry of Ed several years ago is that – when used properly -- co-constructed criteria can dramatically increase student success in a variety of subject areas.
I decided to apply this learning in my math program this week.
First, I had my students watch another group of similar-aged students work through and share their solutions to a math problem involving addition of multi-digit numbers. I invited them to just notice whatever they wanted to pay attention to, and to subsequently make up some comparative sentences (as my class is extremely ESL, I provided some contextual vocabulary scaffolding; see this blog post for examples).
The next day, we watched the video of the same students we had observed the day before, but this time I asked my students to focus specifically on how the students in the other class communicated their thinking. In particular, I asked them to write down (as they were watching the video) what, specifically, the students in the other class said or did while they were explaining their thinking.
As we watched, I stopped the video several times to model think-alouds (“Oh, look, they are pointing to their diagram while they explain their answer to the teacher!” and, “Oh, I hear them using signal words to organize their explanation – did you hear that? She just used the word ‘first’ to tell what she did first while solving the problem!”)
At the end of this viewing, I asked students to share 2-3 ideas from their whiteboards with a neighbour. Then we discussed all the things we noticed, as a group, and compiled a list of criteria for successful communication in math.
I posted the list prominently in the room so that the students and I could easily refer to it as needed. I then invited students to solve a math problem, independently, and communicate their solution on a piece of paper, ensuring they had done all the great things from our co-constructed criteria list before submitting their work.
Although not all of the students shared a comprehensively explained solution on their paper, I was quite amazed at how many of them got “the right answer” mathematically. Many students who had been struggling with the math unit during the preceding few weeks where able to accurately solve the problem (it was a multi-digit subtraction problem). It was as though watching other students communicate effectively, and dissecting what it was that made them effective communicators, had helped my own students to think through a problem, even if they did not actually explain the process they used to solve it in great detail on the page.
I am looking forward to upcoming lessons, where we will refer back to the criteria in order to assess how well we are communicating our mathematical thinking while problem-solving. Although our next unit is measurement, the criteria we constructed together are applicable cross-strand, and we will continue to focus on communicating.
Last week I blogged about a Video of a 3-part Math Lesson by a teacher in Ottawa.
As we have been working on “communicating” in math this year, I wanted to give my students a model of what good communication looks like when students are explaining a strategy they used to solve a problem. Wendler’s students are quite articulate, so I thought I’d share the video with my grade 3s
The eventual plan is to co construct some criteria for what effective communication in math looks like, and then use these CCCs to self, peer and teacher assess students’ communication when they explain their thinking during a math problem. But I didn’t want to just throw them into this task “cold”, so to speak. So I decided to start with a warm up activity:
This morning, I showed my students Wendler’s video, and asked them to just notice some similarities and differences between his classroom and ours.
I provided the stems below, and had students first discuss several similarities and differences with a partner, then write one or two comparative sentences down. (We shared the latter in a circle later on.)
It was interesting to see the sorts of things the students focussed on (everything from seating arrangement, to availability of a carpet to similar strategies used to solve addition problems!) and it gave me a good idea of what is important to my students.
Tomorrow we will watch the video again. Since it will not be their first viewing, I’ll now ask them to focus more specifically on how the students communicate their thinking.
I will ask them whether they think the students in Wendler’s class communicate effectively, and – if so – what it is that makes their explanations “effective”. We will craft these qualities into specific criteria, which will be posted for later reference.
Dale and I already came up with a list as we previewed the video last week:
(The three near the end of the list are meant to be more "level 4-ish" criteria.)
It will be interesting to see how similar or different our students’ interpretations are to/from our own!
Two major themes emerged for me, as I watched and reflected on Wendler's Video this weekend:
N.B. To see the original Wendler Bansho blog post, and accompanying video summarising his 3-Part-Lesson, click here!