ICTM – Hands-on Division with Fractions for Student Understanding

On Saturday, October 7, 2023, I’m presenting at the ICTM Annual Conference in Naperville, IL, at 3:00 pm. I’m presenting with one of  my students, Cailey Platt. The title of our presentation is: Hands-on Division with Fractions for Student Understanding.

Links to the handouts and slides for the presentation can be found here. (or use https://bit.ly/ICTM-Olsen-2023)

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Retirement Reflections (in reverse chronological order)

Most recent dates are here at the top.

Getting Ready for Big Trip

We basically had 2 weeks from the ICTM Conference to departing for Washington, DC. I fixed the chess board in the backyard. Long overdue. Eric and I made it (for 4H) many years ago. Planted some grass. Dug out the good compost and transferred the clippings to the south side of the compost bin. Preached Oct. 22.

Started Math Olympiad at MMS

On Oct. 16, 2023 we began Math Olympiad at Macomb Middle School. It is going very well. We have 2 teachers, myself, and a physics professor on the coaching team.

Family Gathering in Des Moines/Clive

We had a fun weekend in Des Moines/Clive. Got to Rachel’s for lunch Friday. We stayed 2 nights at Jordan and Eric’s and had lots of time to play with the three grandchildren. Diane, Luke, and Nora came down from MN (and stayed at Rachel’s). Saturday morning went to the (Big) Famer’s market.

ICTM Conference Oct. 14

Attended ICTM in Naperville, IL. First ICTM Conference attended since the pandemic. Presented with Cailey Platt on Hands-on Division with Fractions for Student Understanding. Excellent conference.


I continue to enjoy preaching once a month at Point Pleasant and New Philadelphia UMCs. On Sept. 17 I was asked to preach at Hamilton and Warsaw UMCs.

To Okoboji for the closing +


Walk to Emmaus August 24–27, 2023

Only about a week before the event, Ronald Pettigrew called and asked if I wanted to go on a Walk to Emmaus. I wanted to go on a Walk for many years, had heard credit bit about it, but never been able to fit it in before. Through God’s providence, the dates worked out. I had a great 72-hour experience and I feel it helped me grow in my Christian pilgrimage. I would consider my first walk to Emmaus as a Highwater mark in my retirement. I’m fortunate it occurred in the first eight weeks.

Preach August 20

I’ve been preaching regularly (about once a month) since March 2021. It was high time that I preached a full sermon on sin. Luckily, I had made up my mind to do this many weeks in advance, and I had lots of good material.

Preach at Wesley August 13

My sermon title for my Wesley church sermon, August 13, was What can we learn from Rodney Fink? In preparation, I met with Jim and Marti Dallmeyer. We also had Bertha Fink over for dinner on August 8. It was great, Bertha opened up, and I got a lot of good ideas for my sermon. After the services on the 13th we went out for a nice dinner with Bertha, her daughter, and Pastor Melly.

To NE July 31 – Aug. 7

Visited all the Owens in NE. Played with Tate, Sophie, and Wesley. Fun time cutting down a dead tree. Dilsaver reunion as usual. On the way home we met friends, Bill and Jane Brown for lunch in Grand Island.

To Okoboji July 21-30

We had a great week with the kids and grandchildren at Lake Okoboji.

To Pittsburgh 15-18

We flew to PA to play in Table Tennis National Senior Games. I made a quick exit from the tournament, but had fun. We then enjoyed site-seeing in Pittsburgh.

July 9-14 Mission Trip

Mission trip to the Twigs Community Complex in Granite Coty, IL was great. As usual, the mission trip is a highlight of the summer and the year.

July 1-8

Retirement officially started July 1, 2023. I am very appreciative of my wife, Marilyn, so I made 5 thank you notes to give on 5 consecutive days to her.

On Friday, June 30, I went to the Singh’s house for a nice dinner, visit, and to teach them how to play Cribbage.

On Sat., July 1, I went to Bushnell for the VFW Pancake Breakfast, with Ron Pettigrew. I met some friends and former students there.

I preached July 2.

Marilyn came home on July 3. It was fun to have Hayden, Lindsay, and Kate over for the fireworks.

June 2023

I still had assistant chair duties in June and I was training my replacement, Dr. Doug Lafountain. On June 29, I had a meeting with Doug and I was moving my office. I still have an office up on third floor of Morgan so that I can keep my books. On June 29 we had the derecho. Luckily, we had no damage and our power was out only for about 7 hours. Marilyn wasn’t home and it was crazy in Macomb with lots of power outages, many of the street lights not working, and downed trees.

My last semester at Western

Spring 2023 ended up being a great final semester for me. I got a chance to teach a mask teaching methods course (middle school methods, in fact). I got a chance to have Jordan Rouse as a student for the third time. I also got to teach Geometry, Math 310, for a second time. Both classes were very good.

Furthermore, I got a chance to do an undergraduate research project with two wonderful students.

After not having our math teachers conference for three years, we re-instituted the conference in April of 2023. I got a chance to be the closing speaker. He was extra special for me since my kids came and surprised me at the presentation.

My retirement date gets set early

In the summer of 2020 (#Pandemic) our department chair, Victoria Baramidze, asked me to be the assistant chair. I agreed. It is a three-year term and I knew I would only do one term. Up until that point, I had been dragging my feet about retirement. I really like teaching and didn’t know when I would retire. my term as assistant chair would end June 30, 2023, and I determine and decided that would be my retirement date. It was a blessing because I had made up my mind and had lots of time to prepare for retirement.

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WALDO – Workflowing the Amazing Lesson Design Outline

This Google doc is comprehensive teaching methods course and education research course rolled up into one document organized into clickable links. It is compiled by Miguel Guhlin.

It include most all of the aspects of the teaching enterprise (including pre-assessment, edtech, instruction, assessment, and reflection).

Included are also the effect size of approaches and research syntheses.

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How gesture and movement help us think

From Annie Murphy Paul @anniemurphypaul

Exciting new ideas about how gesture and movement help us think

Jordan Ellenberg hates PowerPoint slides. And whiteboards. For him (and for his fellow math professors at the University of Wisconsin-Madison), it’s an old-fashioned blackboard or bust. “It’s part of the folk wisdom of mathematicians,” he told me in a recent interview. “We believe in the blackboard.” When the math department at UW recently renovated its common room, much effort was devoted to finding the right blackboard. (Some professors care about just the right kind of chalk, too, “but I don’t go that far,” says Ellenberg.)

In the classroom, the active use of the blackboard is central to Ellenberg’s teaching method. “When you’re walking around and writing things down on the blackboard, you’re there with the students in a room where math is being created. That’s what you want them to see, what you want to model for them,” he explains. “Whereas if you’re standing there showing them a series of prepped slides, the math has already been done. You don’t want students to think that math is already ‘there.’ You want them to think of it as something that you, and they, make in the moment.”

And whiteboards? “Ugh,” says Ellenberg, with feeling. “The tactile feedback is completely different.”

In his latest bookShape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, Ellenberg observes that mathematics is inherently embodied. “As abstract as we may try to make it, math is something we do with our body. Geometry most of all,” he writes. Later in the book he notes: “Mathematics is a fundamentally imaginative enterprise, which draws on every cognitive and creative ability we have. When we do geometry we use what our minds and bodies know about the size and shape of things in space.”

For Ellenberg, another way the doing of math is rooted in the body concerns the way he moves when he’s thinking. “I’m a pacer,” he says. “A lot of us mathematicians are. When I’m thinking about a difficult problem, I have to get up and walk in a circle.” This is not the same thing as going for a walk, he emphasizes. “That’s not the right rate for doing math. Walking in a tight circle is much better.”

Ellenberg is also interested in gesture as a means of thinking and communicating about math. It’s a frequent topic of conversation with his UW colleague and Madison-area neighbor, psychologist Martha Alibali, who is one of the world’s leading researchers on gesture and learning.

write about Alibali’s research in The Extended Mind, which devotes an entire chapter to “thinking with gesture.” If you’re looking for an overview of how gesture can enhance thinking and learning, that chapter is a good place to start.

The study of gesture is an astonishingly rich and ever-evolving field, and one I’ve continued to follow since completing The Extended Mind. In this issue of the “Thinking Outside the Brain” newsletter, I thought I’d share with you a handful of new ideas and insights about gesture that I’m especially excited about.

1. Gesture externalizes our thought processes.

One of the challenges of teaching and learning math is finding ways to move knowledge and skill outside the head, making it visible to others. Just as writing on a blackboard can lead students to feel that math is “happening” before their eyes, seeing and making gestures can turn internal mathematical thinking into an external resource available to others. Here are some ways this can unfold:

A teacher’s gestures externalize her thinking so that students can use it to scaffold their own thinking. Martha Alibali advises teachers to vary their use of gesture, making more gestures during parts of the lesson for which students need greater scaffolding: for example, when introducing new instructional material; when teaching material that is more complex or abstract; and when responding to students’ questions.

A student’s gestures externalize his thinking so that a teacher can use it to become more informed about the state of his understanding. Students’ gestures reveal knowledge that is not found in their speech—and often, this knowledge represents their newest and most advanced thinking on the subject. Teachers who attend to students’ gestures, and who note “mismatches” between what students are saying and what their hands are doing, are better equipped to nudge students toward explicit, verbally-articulable understanding.

A student’s gestures externalize her thinking so that she herself can use it to support and advance her thinking. The movements of our hands call our attention to salient aspects of our own thinking processes; gestures are especially effective at focusing us on spatial information. They also help us “package” ideas and impressions into coherent units of speech.

2. Gesture allows for more sophisticated thinking to emerge. 

When students are prohibited from gesturing, their speech is less fluent, their explanations are less accurate, and their problem-solving efforts are less successful. When they’re permitted to use their hands, students can use gesture as a springboard to a more advanced level of cognition.

Gesture reduces our cognitive load, thereby freeing up mental bandwidth for higher-level thinking. Especially in spatially-oriented disciplines like geometry, gesture allows “direct expression of spatial properties, lessening the need for a translation to verbal codes, and therefore alleviating working memory resources,” writes Columbia University psychologist Michael Slepian. These freed-up resources can then be applied to the more complex activities of comprehending and problem-solving.

Gesture allows thinking to progress even when the relevant words are not yet available to the speaker; indeed, gesture can help bring those words into being. Students often know more than they can yet say, especially when specialized or technical vocabulary is involved. Gesture allows them to move their thinking forward in the temporary absence of language; gesture can even help summon up the appropriate words, calling them up from memory and priming them for use.

Gesturing first, and codifying formal understanding second, is an effective way to learn. Traditional education often introduces formal definitions and concepts right off the bat—but it’s more effective to “experience first, signify later,” as UC-Berkeley education professor Dor Abrahamson puts it. And gesture is a form of physical, embodied “experience.”

3. Gestural mimicry is a powerful platform for learning.

People in conversation tend to spontaneously and unconsciously mimic one another’s gestures, allowing them to borrow some of their partner’s knowledge and understanding; such mimicry is, says psychologist David McNeil, “a tool for recovering the workings of another’s mind.” Gestural mimicry can also be employed intentionally, in these ways:

Students can mimic their teacher’s gestures, thereby appropriating their teacher’s more advanced cognitions about math. Some of what teachers know about math shows up in their gestures; by emulating their teacher’s gestures, students can begin to gain access to the teacher’s understanding. As Martha Alibali and her colleagues put it, “actions can induce cognitive states.”

Teachers can mimic students’ gesture back to them, adding more precise mathematical language as they do so. Students can see what their gesture looks like when it’s performed by another person, giving them another opportunity to learn from it; meanwhile, the teacher’s voicing of more accurate and exact language gives students another way to think and talk about what they’re seeing.

Students can mimic (or otherwise respond to) their classmates’ gestures, engaging in what psychologists call “collaborative gesture.” Students may “echo,” or reproduce, a classmate’s gesture. They may respond to a classmate’s gesture with an alternative gesture, demonstrating how their understanding differs from their classmate’s. Or multiple students may “use their hands to formulate one single mathematical object or system,” one that is “built and represented only through their bodies working in conjunction,” writes UW-Madison psychologist Mitchell Nathan and his colleagues.

I’ll leave you with one last point from Martha Alibali. In a chapter written with Mitchell Nathan, she notes that in teacher education, “one finds an overwhelming emphasis on the verbal channel. In our view, there should be a place in teacher education for the consideration of how speech and body-based resources such as gesture can work in concert to implement effective and engaging instruction that promotes deep understanding of fundamental ideas in mathematics.”

Wouldn’t it be cool if teachers were taught how to use gesture to extend their students’ minds?

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Spring 2021 Websites

Miya Christiansen: https://miyaswebpage.wordpress.com/

Brianna Gorsuch: https://mathwithmissg650681653.wordpress.com/

These are the Spring 2021 Math 304 students.

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A few notes on making web sites

Tools for making webpages (web sites).


  • (blog) Posts – more timely, often shorter, specific to one topic. Have categories and tags.
  • Pages – often more long-term. Could be more lengthy. (Pages do not have categories or tags.)
  • Can use blocks or not use blocks
  • To edit, append /wp-admin (there is more than one editing environment)

Google Sites

  • Uses Blocks (blocks of code, of information on the webpage)
  • Uses pages (no posts yet)




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Instrumental and Relational Understanding

Good article on instrumental and relational understanding.

Focus on Relational Understanding

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Putting Synthesis back in Bloom.

Benjamin Bloom published his taxonomy in 1956. In 2001, the categories were revised. Synthesis was taken out and Creating put in (among other smaller changes).

Erik Francis points out (article link below) the importance for learner to synthesize and puts it back in.  Creating is kept. Now there are 7 levels of understanding.



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An angle inscribed in a semicircle

Hereis a tweet with a nice proof of the theorem that is the namesake of this website https://twitter.com/professorbrenda/status/1323095111954288640?s=21

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3 Great WIU Graduates Teaching MS Math

These guys are teaching math and helping students in many ways.

Read the full story here (from WIU website).

Read the full story here (from McDonough County Voice Newspaper).

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