Update 7/29/2025

We have two new columns! Use the link to access all ICRSME Newsletters containing our column each quarter.

My colleague, Michael Kamen and I have a regular column to support pre-service elementary teachers learning about Math and Science. All can be found here either in Current Issues or in the Archives. Learning to Teach Math and Science: Thoughts from an Emerging Elementary Teacher

They can all be found in the archived newsletters using the link above. I have really enjoyed using them to spark conversations with my pre-service students. I think they'd make good conversation pieces for practicing teachers as well! Enjoy!

1. Kamen, M., & Plowman, D. L. (Summer 2023). Issue #1 Learning to Teach Math and Science: Thoughts from an Emerging Elementary Teacher-What’s Happening? First Experiences in Math and Science Methods Classes. ICRSME Newsletter Issue 110 (pp. 6–8). Online: International Consortium for Research in Math and Science Education.

2. Plowman, D. L., & Kamen, M. (Fall 2023). Issue #2: Learning to Teach Math and Science Thoughts from an Emerging Elementary Teacher Solving Word Problems with Understanding. ICRSME Newsletter Issue 11(pp. 8–10). Online: International Consortium for Research in Math and Science Education. 

3.  Kamen, M., & Plowman, D. L. (Winter 2024). Issue #3: Learning to Teach Math and Science Thoughts from an Emerging Elementary Teacher: It’s Done with Mirrors. ICRSME Newsletter Issue 12 (pp. 8-11). Online: International Consortium for Research in Math and Science Education. 

4. Plowman, D. L., & Kamen, M. (Spring 2024). Issue #4: Learning to Teach Math and Science Thoughts from an Emerging Elementary Teacher: There Are No Real Word Problems in Life. ICRSME Newsletter Issue 13(pp. 5–8). Online: International Consortium for Research in Math and Science Education.

5. Kamen, M., & Plowman, D. L. (Summer 2024). Issue #5: Learning to Teach Math and Science Thoughts from an Emerging Elementary Teacher: Respecting Children’s Questions and Adapting K-W-L Charts. ICRSME Newsletter Issue 14 (pp. 8–11). Online: International Consortium for Research in Math and Science Education. 

6. Plowman, D. L., & Kamen, M. (Fall 2024). Issue #6: Learning to Teach Math and Science Thoughts from an Emerging Elementary Teacher: Never Too Old to Play with Blocks. ICRSME Newsletter Issue 15 (pp. 9–13). Online: International Consortium for Research in Math and Science Education.

7. Plowman, D. L., & Kamen, M. (Winter 2025). Issue #7: The 5E Instructional Model Gets an F? Is it Time to Move On? ICRSME Newsletter Issue 16(pp. 8–11). Online: International Consortium for Research in Math and Science Education.

8. Plowman, D. L., & Kamen, M. (Spring 2025). Issue #8: Math is a 4-Letter Word ! ICRSME Newsletter Issue 17 (pp. 6-8). Online: International Consortium for Research in Math and Science Education.

9. Summer 2025 is Submitted and will be available soon! "Data Charts: Making Connections?"

Stressed out? Try some math!

Stressed out? Try some math!

(Developing a case for using math to generate cognitive load distraction

from the ‘news of the day’)

Debra Plowman, PhD

COEHD

Curriculum, Instruction and Learning Sciences

On top of the demands of work and school, daily news of the COVID-19  Pandemic brings us has created a lot of stress in our everyday lives. When I am stressed my wonderful partner will give me a math problem.  

“What?” you say, “A math problem? That would make me more stressed!” 

Sadly, it is true that for many people the mere thought of solving a math problem stresses them out. But hang in with me for a few paragraphs and I’ll explain why that is not so for us and how math works as a de-stressor for me. Perhaps it can also help you, too.

How many people have used counting sheep or even counting backwards to go to sleep? In my personal experience concentrating on one thing, by counting backwards from 1000 for example, can occupy just enough brain power for a moment that I let other thoughts go. In other words, counting provides just enough “cognitive load” that I let go of thoughts that are preventing me from sleep. 

I have often wondered about why it works.  While not much research has been conducted on this particular strategy, the idea aligns with what is known about concentrative meditation as a strategy to relieve stress. Research on these types of meditation have found increased theta wave activity --  an indicator of relaxation  -- associated with attentional focus on  simple cognitive tasks (e.g. Baijal & Srinivasan 2010; Cuthbert, et  al 1981) .  Using math to relax helps me forget about other things if the problem is “just right”. The cognitive load encountered in doing these “just right” problems allows an immersion in thought that can provide a needed break from the heavier challenges everyday life bombards us with. 

As an example of a ‘just right’ mathematical tasks, I like to play with 99  + anything. 99 + 3 is 102 because 99 and one more makes 100 and just two more is 102. 99 + 56 works the same way.  Use any starting number you want to and you will find a pattern. Try 67  + anything. What patterns will arise? I encourage you to do these in your head. I also enjoy multiplication puzzles like 4 times anything or 50 times anything. For example, 50 times a number is  the same as 100 times the number divided by 2.

There are several problems that Number Theorists have yet to solve, but are easily  studied by a common person. Here are a couple of examples of famous problems that make what I call “just right” for relieving stress. A “just right” problem begins with an easy idea that you can use simple mathematics to begin to explore. The Collatz[†] Conjecture is a famous unsolved problem in mathematics that anyone can explore using a sequence rules: a) pick any whole number (1,2,3,4,5,6,7,….) b) if your number is even, then divide by two and if the number is odd then multiply by 3 and add 1, c) if the next result is an even number, divide by 2, if not, multiply by 3 and add 1. Keep doing those steps until you get an answer of 1. The conjecture is that any whole number selected will always end at 1. Another interesting question related to these sequences is the predictability of the length of sequences given any number.  

Okay, so let’s give a number a try using the rules involved in the Collatz Conjecture. Let’s try starting with  10.

Example with the Number  10

·      So, 10 is even, divide by 2 and you get 5. 

·      Five is odd so multiply by 3 and add 1 and you get 16. 

·      16 is even and now divide by 2, that’s 8 so divide by 2 again and you get 4. 

·      Four is  even so divide by 2 again you get 2 and then 2 again and you get 1. Done!

Number theorists record these sequences to look for patterns and the pattern we created here is: 10,5,16,8,4,2,1- a 7-number long sequence. Doing this in your head is fairly easy, the rules are simple and you have to  concentrate just enough to keep the numbers straight, but it is complex enough so that other thoughts cannot intrude. This meets my criteria for a math stress reliever. Another thing to notice is if I had begun with the number 16 a larger number, my sequence would have been shorter at just 5 numbers in length (16, 8,4,2,1).

Try  a couple of numbers yourself to see. Try it while you are walking somewhere, or sitting on the couch trying to not check your phone for news alerts, or even, trying to fall asleep. Share the love! Play around with the conjecture with a friend so you both are not looking at the news! 

Another ‘Big Unsolved’ is the Goldbach Conjecture, that all even numbers can be written at the sum of two primes. This conjecture has been around since the early 1730s, and the largest number ever tested is 4 x 10¹⁴ (that is 4 with 14 zeros!). And yes, it worked. An interesting thing about working with this problem is that it takes a little more exploration, rather than the straight-forward, mechanical operation of Collatz. This can start with a conversation on an afternoon walk, and end with writing some simple calculations down and looking for patterns. 

So, I have shared a few examples of what works for me, and I hope that you will find solace in these ideas. I also want to share some other math diversions that can help you get through these times. The first two are YouTube channels which fall into the category of “Math-tainment” and the third is an invaluable resource to find fun and engaging math tasks to do with the whole family. I have presented more ideas and thoughts in this vein on my blog: Math Nerd Under Construction (http://debbieplowman.blogspot.com).

Enjoy your Cognitive Load!

Numberphile (https://www.numberphile.com)

The host is video-journalist Brady Haran. He interviews mathematicians from around the world who are willing to explain in engaging ways about many topics often using plain brown paper and simple drawings and calculations.

3 Blue 1 Brown (https://www.3blue1brown.com)

Grant Sanderson, author of this channel uses visualization to share big mathematical ideas. Math “eye-candy”, if you will, that allows you to see the mathematics even if you are not ready to make any calculations.

NRich Maths (https://nrich.maths.org)

I have used this website to do some math on my own as well as curate for lessons with my students  as well as families. The description directly from the website explains it best: 

“NRICH is an innovative collaboration between the Faculties of Mathematics and Education at the University of Cambridge, part of the University’s Millennium Mathematics Project. NRICH provides thousands of free online mathematics resources for ages 3 to 18, covering all stages of early years, primary and secondary school education - completely free and available to all.”

References

Cuthbert, B., Kristeller, J., Simons, R., Hodes, R., & Lang, P. J. (1981). Strategies of arousal control: Biofeedback, meditation, and motivation. Journal of Experimental Psychology: General, 110(4), 518.

Baijal, S., & Srinivasan, N. (2010). Theta activity and meditative states: spectral changes during concentrative meditation. Cognitive processing, 11(1), 31-38.

Thanks Tony for finding my de-stress zone and for the counseling reference help!

---

[†] Named after Luther Collatz, but also explored by  other mathematicians and  other names such as the Syracuse problem, and the hailstone sequence or numbers

This a cross posting of an item I wrote for the "Improving Your AIMM" blog for the Advancing Inquiry in Middle Mathematics project I am part of in East Texas.

Teaching is Improvisational

I want to share what I learned while writing a chapter about using improvisational games in educational settings. Teaching in discussion-based classrooms is a lot like an improv performance. To make this comparison, in formal theater everyone has specific lines and has practiced until perfect. Everyone on stage knows how the performance will end. The audience’s task is to absorb. Contrasting that type of theater is improvisation. During improv there are rarely any props, no script and the acting team needs to listen and respond to each other to keep the act going. However, there are rules and guidelines that the acting team must follow, it’s not just a free-for-all. This comparison parallels the contrast between teaching as directive and teaching in a discussion-based lesson. And, like improv, the task and goals of a mathematics lesson form the rules that the team or class follows.

However, as the AIMM team well knows, improvisation during math lessons is a demanding task. As we have taught your students while you watch, you have observed how we have to be ready to respond authentically to students. You have all seen us struggle at times to be open to an unexpected idea or an idea that is incorrect or partially formed without shutting down the student contributions. By the way, these struggles become an important part of our follow-up discussion of the lesson, as well and the unexpected student responses.

The “yes-and…”/”yes-but…” improv game is one of my favorites for building understanding of what we need to be doing and not doing with our students during math conversations and discussions. When learning how to have these discussions, we might be tempted to respond to a student idea and say, “Yes but…” and go on to insert the correct term, or redirect the student to a more efficient way.  But what if, instead, we said, “Yes, and what else are you thinking?” or “Yes, and can we hear from another student?” or “Yes and, I like how you have used what we have learned about graphs to explain the pattern…” The object of thinking “yes, and…” is to keep the conversation going, and to be inclusive of the other players’ (students’) ideas. 

This game can be played between students in a fun way first to help them develop better, more supportive communication skills between each other. In the beginning, students can play the game using a non-mathematical context, like planning a vacation together. It is important to play the “yes, but…” scenes too so students can have a discussion about what that feels like in comparison to the “yes and…” scenes. 

The Bridging Project was a Mathematics PD that used an improvisational framework along with content sessions to develop middle school teachers understanding of mathematics argumentation (conjecturing, explaining, justifying and generalizing). The most compelling finding was that students of teachers who learned how to use improv in math class had higher academic achievement that students of teachers who did not have the training in improv. These teachers also held substantive discussions more often in their classrooms. Both sets of teachers received the same PD in content. Interestingly, teachers did not have to teach the students improv games directly (some did and some did not) as that aspect did not influence the results at all (Knudsen and Shectman, 2016). This indicates that it may be as or more important for teachers to practice and build improvisational skills when learning these new and complex discussion practices.

My chapter, Improv games in educational settings: Creative play and academic learning,

has been accepted and will soon be published in the Springer online publication, Encyclopedia

of Educational Innovation: Teaching and Learning Innovation Through Play.

I hope to have permission to share it directly with you when it is finalized. Until then, the rules of the game I highlighted here in this post are described via this link: http://www.yesandyourbusiness.com/portfolio/yes-but-yes-and/). 

References

Knudsen, J., & Shechtman, N. (2016). Professional development that bridges the gap between workshop and classroom through disciplined improvisation. Taking Design Thinking to School: How the Technology of Design Can Transform Teachers, Learners, and Classrooms, 163.

What I learned this summer: Probability and Statistics

This summer I taught something new. Statistics for middle school mathematics. I had a blast! The approach we used was to use investigations of data to build statistical reasoning. Many curricular tasks focus in hard on the vocabulary and definitions statistics and then quickly move on to the procedural instruction on how to calculate these statistics and how to create graphical representations from that data. Very little time is spent designing and enacting investigations, creating good meaty questions, and then on the other side choosing representations and interpreting the results.

I quickly realized that the separation of probability from statistics is a mistake. This may seem obvious to others, but interpreting probabilities is directly linked to interpreting results from investigations. Trends and likelihoods are the same thing. Fair games are like designing an investigation that collects data equitably and without bias as well as choosing a statistical tool that fits the question. I don't know if teachers have been teaching these as linked ideas or not. My middle school teacher group reported that in their curriculum, probability tasks are separated from statistics tasks by months in the scope and sequence, and that they purposely teach statistics right before the state test to make sure kids will remember the procedures. **SIGH**

I also realize without a strong understanding of probability (e.g. certainly is rarely guaranteed, and that chance has no memory), the general population does not understand the responsibility of science to be transparent about their predictions, and honest about the level of certainty. For example, if a study on climate change shows a 99 percent chance that a cause is linked to the effect, this is pretty certain! The slightest chance of something being one way or the other make people doubt the predicted outcome. This happens in gambling or lotteries as well. People think "Well there is a one-percent chance that the prediction/correlation could not be true (we have human influenced climate change OR you will not win money), so I am going to believe the less likely outcome instead.

Last, I learned that I LOVE box and whisker graphs and dot plots. Honestly, it is true, until you teach something you don't really know it. Yes, I have used these tools in research, but it felt different teaching it, and teaching them conceptually, created from real messy data. Below are some photos taken from my group. We collected data on the first day about the groups favorite math topic, years of experience and certification. Teachers, the next day, created an investigation based in the data (yes, I know the investigation could have begun with a question, design of data collection, blah-blah-blah, but this was truly spontaneous). One question arose as they studied this data was, "Does alternate certification have a relationship with years of experience?" Here is the table of data and the results. Does certification type predict years of experience? Why might it be related?

Next, we wondered about the relationship of favorite topic to the other two questions. What do you think? Is this a good question? Why? Why not? What kind of representation could support answering this question?

In a couple of weeks, I am teaching another teacher group, and then after that we will be interviewing middle grades students to investigate their statistical reasoning and then teach a lesson based on what we learn about them. Can't wait!

It's been a while!

I have been looking at my own blog during the past couple of weeks, and realized that I have not posted anything in two years. My how time flies.

Updates: I am currently working with middle grades teachers on an algebraic and proportional reasoning project, as well as teaching a Math Methods class for a local university. Both of these activities are tremendously important to me. I truly appreciate the kind of dedication I see from both practicing and "about to be" practicing teachers. I continue my work with the Texas Regional Collaboratives and see the same kind of dedication in the Mathematics Professional Development Communities as in the teachers they teach. I am lucky to work with all of them

I visited with educators from all over the world and the International Conference for Math Education in Hamburg Germany last summer and also got the opportunity to be one of 15 math educators to visit in a  3-day colloquium with Finnish Math professors and educators in Helsinki.

I am proud to say that may of the things we are doing align with the best practices of many of the top performing countries, and that we all need to learn from each other.

In closing, I hope to begin again posting stories of math education on this blog and that just maybe some people will read it! LOL

Cheers--

--Debbie

Do you like math?

Just the other day we were using some conversation starter cards at the dinner table. One of the cards asked if there were any subject in the world that you'd want to be an expert at what would it be? My middle daughter, who has struggled with math throughout high school and college said that her subject would be math. On the one hand it surprised me because for someone who goes around saying, "I hate math!" , I would not have guessed that she would choose it. On the other hand she sees that it is important enough to want to be come good at it. Thoughts?

The Growth Mindset in Mathematics

This is a MUST read for all parents, teachers and older kids. I am even thinking that high school teachers should assign this as a reading. The main idea is the false belief that you cannot change one's math ability. Just like we can become better at any sport if we work out, we can become better at math too--

See this article: "There’s one key difference between kids who excel at math and those who don’t"

http://qz.com/139453/theres-one-key-difference-between-kids-who-excel-at-math-and-those-who-dont/

ReplyDelete

Update from the Second Grade Landscape

So this year I have been working with second grade again. I am attempting to learn from what we did last year. I think that our understanding about how the students work with base ten ( and especially my host teacher's understanding) has grown. A recurring issue is the representation of base ten blocks with stick and ball drawings that almost immediately replaces the actual blocks after a few limited counting experiences with the blocks. This, apparently, has become common practice and is partly a result of teachers feeling pressured to increase the number sizes too fast, which keeps the students in the direct modeling stages longer since they have not developed a good understanding of the use of facts and related facts to solve the smaller number problems. Teachers also feel that the time spent in getting the blocks out, managing them and putting them back wastes precious time. Of course I am constantly pushing against this practice, but it is not my classroom and I am only there once a week.

Nevertheless the kids and their teacher are making progress, the conversations and discussions are pretty complex and meaningful and the kids love doing real math. No M&Ms needed.  Here is an example of the board work from one of our problem solving discussions. This year, we frequently write problems after reading a book. The photo is the board work from a problem written by a student after reading the book Porkenstein by Kathryn Laskey (the gist of the book is friendship). The problem was:

Porkenstein has 35 friends. Some don't like him anymore. Now he has 19 friends. How many friends don't like him anymore?

The strategies were presented from left to right. The sticks in the upper left were a problem for the girl who drew them, so as a class we talked about what she did instead (insert: the problem was that she did not believe that a stick had ten in it, and she was treating the problem as a separate result unknown: 35-19 = ?). When she explained her second drawing (5 by 7 circles below), she said she covered up 19 and counted how many needed to be taken away (now she was treating the problem as a separate change unknown: 35 - ? = 19). Two more students shared, one who subtracted 19 from 35 using the base ten representation and another who used the number line to count back 19, and then as a class we used the numberline to count from 19 to 35. My goal for this lesson was to help kids see that problem can be solved a number of ways, and I hoped that they would treat the problem like a join change unknown and use the anchor of 19 close to 20 ( e.g. 19 + 1 >>20 + 10 >>30+ 5 >> 35; so 1 + 10 + 5 = 16.) Instead we ended up talking more about base ten approaches and recording number sentences that matched their thinking.

                   

I am reposting this very nice explanation of the difference between teaching algorithms versus teaching conceptual understanding. The author is Sam Otten from The University of Missouri, and is responding to a viral post by a dad earlier this year (that I commented in in my May 16th post). The context is within the Common Core, but the issue is common to any math teacher no matter if you are in a Common Core state or not:

https://www.youtube.com/watch?v=dmybO35F_EI

Patty Paper

Patty Paper can be used in elementary, middle school of high school settings. It is a paper that was used to separate hamburger patties and is sort of like parchment paper, but really thin. You can do more with it than folding origami or regular paper for two reasons. It is cheap, and the lines show really well for the folds. You can use it for teaching many mathematical ideas including fractions, multiplication, algebra and geometry.

LINKS

This You-Tube video demonstrates how to find the line that bisects another line. This is fairly simple and direct teaching of the activity but there are lots of other things you can do that are more open ended. Copy and paste this link:  https://www.youtube.com/watch?v=WvgAvVKUISQ

EDUCATIONAL VALUE

Once you get to know the uses of patty paper you can actually use it for developing geometric proof and other geometric properties. We used patty paper for a study of young children's multiplicative thinking is a study that was later published as an article in Teaching Children Mathematics (Turner, E., Junk, D. & Empson, S. (2007) The Power of Paper Folding Tasks to Support Multiplicative Thinking and Rich Mathematical Discussion Teaching Children Mathematics, 13, 6, 322-329.). A side note to the study is that patty paper serves as an instant feedback mechanism for the learner. We had two basic tasks: predicting what would happen given a particular sequence of folds, and propose what sequence of folds would be needed to make a certain number of sections. (approximate examples of each type: "If you fold paper into 2 equal parts and then 3 equal parts how many equal parts will you see when you open the paper back up?" and "If you want the paper to show exactly 8 folds, what are the steps to folding it?") Students make predictions or make the folds then reflect on what happened and make appropriate adjustments. Class time can be spent talking about why the sequences of folds worked or did not work. Also records can be made of the folds that will eventually reflect the relationship of multiplication, division and fractions.

Supporting Struggling Students

I have been participating in an online course, and one of the assignments was to outline an issue in my work that needs attention, and that the attention could be in the form of an innovative technology or use of a current technology in an innovative way. Here is my post about supporting teachers to learn how to support students who are struggling:

Helping teachers to learn how to effectively responding to students who are struggling is a difficult thing to do in a teacher workshop setting. Teachers need lots of practice in doing this and also need to do it in a supportive non-stress environment. In a workshop setting there is just not enough time to help 20 to 25 teachers learn this important skill. Also just saying "they will get it with practicing with their own students" is only a partial solution, since, those particular students NEED competent responses NOW, and teachers who are actively teaching need time to reflect and perhaps some real time coaching at those very moments which are often not predictable. One way might be to develop a computer program that allows teachers to view student teacher interactions, choose how they would respond, then try it out virtually, and see the results.

Here is a description of the issue:
Teachers typically respond to struggling students in math class by providing instructions on how to get the answer. These instructions often remove the challenge of the problem in such a way that a student avoids actually learning the thing he or she needs to be able to solve future problems. Complicating the issue is teachers’ perceptions of struggle and their role as teachers. For example, when a child is attempting solve word problem, it is natural for a student to model the problem as it is stated by physically representing the objects and then counting them. Sometimes the child struggles to understand in what order to do things. For example this problem can be challenging for young children:

Jeff has some toy animals in a box. He gives 5 of them to his little brother to play with. Now he has 4 little animals in his box. How many did he have to start with?

If a teacher intervenes and tells the child the order (e.g. “first put out 5 blocks… then put out 4 blocks… now count them all”), the very thing the activity was supposed motivate the child to do has now been done by the teacher. All the child is doing now is following the teacher (which can be cognitively challenging), but not thinking about the problem’s meaning and how that might used to make the decision about what to do first. Teachers need to know several things about teaching and learning mathematics and employ them as they are interacting with students. First they need to understand that struggle isn’t necessarily a bad thing, and part of the job while intervening is to maintain the present cognitive demand of the problem, but possibly create a more accessible entry point. Often that entails a subtle suggesting of a problem solving tool (“yesterday I saw you using blocks to figure a problem do you think that might help you now?”); supporting the child to understand what the story in the problem is about (“so have you ever collected little toy animals like the boy in this story? can you tell me what is happening in this story?”); understanding why this problem might be hard—it is structurally a subtraction problem, but to solve it the two given amounts need to be added, so going back to an slightly easier problem with the same context might be helpful (“jeff has 8 toys and gives 3 to his brother, how many does he have left?”)

What teachers need to be able to do:
Teaching teachers how to respond to students without taking away the challenges in the task that are designed to help kids learn the concept is challenging.  Each teacher-student interaction can feel different and teachers feel that to be responsive they need to experience every situation or be provided with endless sets of prompts and questions that will solve all of the issues with struggling kids. OR (and this is sort of worse) teachers see all struggles as virtually the same and responses become all of one kind (usually just provide the child with the appropriate steps to solve the problem). Neither way of responding is ideal. Being responsive means that teachers can customize their responses, but also be able to see a set of general ways children behave when offered particular problems, understand the goals of the task the child is having difficulty with, and then provide the most minimum “intervention” so that the child learns the hard stuff through thinking, and doing rather than following.

Check out my new page under the Adventures in Second Grade tab!

Here is my response to yet another uninformed journalist/parent ranting about mathematics instruction:

Oh please. I am so tired of uninformed people posting stuff like this! We have all seen kids who create "another way" to solve a problem that is not as efficient as another way. Again the point of solving problems by decomposition is partly to become efficient at computing, but another reason to do it is to build a robust understanding of numbers. The only problem I have with this example is the judgement of the US Standard Algorithm as "old fashioned", as it communicates a negative attitude toward a pretty slick way of calculating. And, saying "new way" (these terms were probably introduced by a well-meaning teacher, but misguided) also seems mislabeled, since the goal is not to do something new, but something smart. Impressively (and probably not understood by the everyday person), the representation describes a strategy related to the number line, which can become more and more efficient over time.

One more thing: I have seen teachers (again, well-intentioned, but misguided) interpret "use a numberline", or "decompose numbers", or "use addition to solve subtraction", as an entry point for showing kids a specific way to solve problems rather than letting kids analyze the numbers and make computing decisions and then discuss those decisions as a mathematical activity.

Real World Photos and Counting Brownies with Second Graders

I have been fortunate to work with a bunch of really great second graders as a guest teacher this year. It has been super fun to see them grow in math. They love to be challenged with big numbers, and the number zero makes them giggle. The fact that you can start any counting sequence with zero groups of anything and get zero is a hoot for them. Last week, I decided to try out some of my photos I have posted under the Math in the World tab. I logged in on the classroom computer and pulled up the blog to show the photos through the projector.

I scrolled through all of them stopping for a moment to consider the math that was there in each one. The were intrigued by the children's classroom in Nicaragua, and overwhelmed by the crowd of people at the UT game.

We stopped on the photo on the cut brownies. I love this one because many fraction problems and array problems talk about a "tray of brownies". But let's face it nobody except a chef can cut a pan of brownies straight. My photo comes from a time my daughter made brownies and of course there is a chunk out of the middle too. She probably decided to take a taste!

When I asked how many brownies were there (they are just starting to talk about multiplication), they were excited. One student said he could count by ones and would have to double check that a few times. Then a couples of kids said they could count by twos (the 3 x 5 array was not immediately obvious to most of them--), and showed me how that would work. Then another child showed how he could group them by 5's, and make a ten, then 5 more. Next someone said, count by 5's! I kept probing for more and asking the children to show how each strategy would work.  By the end of the discussion, the children saw that 3 x 5 was the same as 5 x 3, and the whole array (pun intended) of strategies that could be used to solve the problem.

I think using an authentic photo like this kept the problem real and fun. The discussions were light, and kids felt free to offer all kinds of ideas.  I am not sure how this would work for fractions though, since the pieces are not cut evenly.

Zombie already?

Ugh sorry this blog is fast becoming a "zombie blog" ! I am working on a blog about counting games and shall post soon!

PME-NA Part II: Tale of Three Great Tasks

One reason I enjoy going to PME-NA is seeing the work of my colleagues on children’s thinking. Inevitably what arises out of these studies are incredibly rich tasks that teachers can use in their teaching. I might add that these tasks are also good for teachers and prospective teachers to explore as they are learning about the ways children think.

Higinio Dominguez at Michigan State is an excellent interviewer of children-- :)Somewhere between Herb Ginsburg and Mister Rogers. He just gets those kids to talk, and does it with very little interference so what the kids say is remarkably close to what they really think. One thing he routinely does is leave off the question that usually ends the typical work problem. So for instance, when introducing this problem: “Joey has 14 cars. He lost some, so now he has 5 cars," and then asking, "How many cars did he lose?” Higinio says he simply asks, "So what so the problem?" The kids say all sorts of things such as, “Joey should really keep better track of his cars,” or “Probably his brother took them,” or “They are probably just under the bed.” Higinio’s point is that is you want kids to bring in their own personal connections (or resources--this is a much deeper construct) and this approach is one way to give them the opportunity to do that. Eventually the child or children will ask, "How many cars did he lose?" Or the teacher/interviewer can ask, "So how many are under the bed?”

Steven Greenstein, from Montclair State University has been investigating young children’s understanding of geometry, specifically the geometric reasoning involved in topology. He has noted that young children can identify important properties of sameness between 3-d objects. His research methodology included the use of a microworld, Configure, he developed to support children’s exploration on 3-D shapes. It is free and it looks like such a great way to get children to talk about their ways of thinking in geometry. The website includes suggestions for teachers who wish to use the program with their students. Here is the link: PLAYWITH SHAPES.com

Last, this idea comes from some researchers at Virginia Tech, Andrew Norton and Steven Boyce. The task is about unitizing, and children use their thinking about multiplication, and units to solve these problems mentally. I was familiar with the candy roll problem used to develop base ten understanding,

(see an explanation here: http://www-rohan.sdsu.edu/faculty/jbowers/macpics/intro.htm)

but was unfamiliar with this packing scenario:

A chip is worth 2 cents. When they are packed there are 3 chips to a cup, and 5 cups to a box. How many cents is the box worth? OR If you have 12 cents, how many more cents do you need to make a box? With this scenario, you can change the numbers around, and also you can begin the problem with actual chips, cups, and boxes. Eventually, kids begin to re-unitize and can solve them mentally. The problems can also extend to boxes per crate. I can see a relationship with this task to factoring and even prime numbers as well as fractions and folding problems!

PMENA 2013 in Chicago Part I: Lacking Legos™

I attended the North American Chapter of the international Group for Psychology of Mathematics Education. Thank goodness we just say "PME-NA" for short!

Anyhow, I attended a session about the parents' role in math education. This investigation focused on activities children do at home that could be resources for the development of young children's learning of math. To introduce herself to the parents the researcher planned a home visit to just talk to them in a more casual way before interviewing about what they do with their kids. What she realized after her home visits was that all but one of the 8 families had lots of access to computers, video games, smart phones, and TV (of course). Electronic access was not an issue at all. 

What she did notice was a lack of blocks and other such building and counting toys. When parents have a limited budget, they do not want to spend lots of money on blocks. She said that when parents have choice between spending dollars on these constructing type of toys (which are expensive) or electronics, they invest in electronics because those will be appealing to their kids in a variety of ways and will still be used as the child grows. The presenter of this research made this comment, “Schools should be interested in providing check out materials of these resources. Also, during school time, kids don’t actually need more time on screens.” She added that since her investigation seems to indicate that blocks and other physical materials are what is lacking in these homes, schools need to protect those spaces where children can be actively engaged in working and playing with physical materials. 

I really enjoyed this insight. It makes sense. Better off kids have lots of Legos ™ and K’nex ™, and tinker toys and building stuff, for the simple fact that these materials are expensive. The kids that have these toys also have the electronic devices and may not play with them much either.  Maybe Lego needs to start a campaign to get these things into classrooms to counteract the heavy marketing of tablets and smart devices that Gates and Apple seem to be pushing in these early childhood classrooms.