Podcast appearances and mentions of mike andy

Mike Wendt and Andy Ferg give an update on BEEFlympics, cover the Opening Ceremony of the 2024 Summer Olympic Games. Plus, comic book movies and Andy busting his knee and his boat, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Mike's Amazing Mayo & Dom's Sausage Co.! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg recap their LIVE shows from Dom's and the Cabaret Lounge. Plus, Mike's birthday, Dave Portnoy beefing, some kid trying to beef like Dave Portnoy, Andy getting FB blocked by a family member, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Mike's Amazing Mayo & Dom's Sausage Co.! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg are LIVE for a Special Bonus Show from the world famous Cabaret Lounge in Peabody, MA! The girls, the guests, the Facebook groups, and all kinds of nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Mike's Amazing Foods & Dom's Sausage Co.! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg are LIVE from Dom's Sausage Co. in Malden, MA with Dom Botticelli! We discuss the multi generational history of Dom's, steak tip culture in New England, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Mike's Amazing Foods & Dom's Sausage Co.! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

199. A bit light on this week. Doing what we can with the time we've got.

Mike Wendt and Andy Ferg recap their 4th of July Vacations. Plus, the crazy sub-culture of restaurant people that are STEAK TIPS folk... and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Mike's Amazing Mayo & Dom's Sausage Co.! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg explore restaurant dress codes, Andy riding the viral coattails of Hawk Tuah Girl, and the best hockey TV shows and movies. Plus, Andy has the funniest hot dog story ever... and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Mike's Amazing Mayo & NEW SPONSOR - Dom's Sausage Co.! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg celebrate the Celtics 18th Championship! Also, Tom Brady's Patriots Hall of Fame induction was this week, Andy is quitting Fast Food, Justin Timberlake might have an issue or two, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Priority Capital & Mike's Amazing Mayo! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg get into a lot of beef talk in the first half of the show. With the creation and popularity of the phrase "3-way" and Andy losing followers over a topless Bill & Bob's shoot and a Pride Month t-shirt. Plus, Caitlin Clark, Karen Read, Joey Chestnut and more nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Priority Capital & Mike's Amazing Mayo! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg recap the weekend that was FERG FEST 2024! The reviews are in and it was the party of the year... and next year is gonna be EVEN BIGGER! Plus, Caitlin Clark, Angel Reese and the WNBA aren't handling PR well, according to Mike and Andy. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St. & Priority Capital! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg give the final push for FERG FEST that's happening this weekend! Plus, Richard Dreyfuss upset a few people at a JAWS screening this past weekend, Mike learned that some people actually defend Jenny in the movie Forrest Gump, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St. & Priority Capital! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg discuss whether or not they'd dine on human flesh. As if that wasn't enough!! FERG FEST is a week and change away, Scott Scheffler is causing shenanigans, Boston Globe ran a ridiculous SNL bracket, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St. & Priority Capital! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg talk FERG FEST happening June 2nd. The line up, schedule, food, beer, people, good times! Plus, we're in the Golden Age of Beefing, a mayo company took advantage of Andy, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St. & Priority Capital! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music. * Some technical difficulties cut this episode short *

Mike Wendt and Andy Ferg debate one of the HOTTEST topics today, as only they can... Would a woman rather be stuck in the woods with a BEAR or a MAN? Plus, the Tom Brady Roast, Ric Flair getting tossed from a pizza joint and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Priority Capital & Pete's in Woburn! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg give updates on FERG FEST happening June 2nd! Plus, Ryan Whitney not backing down on a Bruins tweet, WNBA vs NBA pay scale, Andy's Maine house smelling like spoiled mayo, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Priority Capital & Pete's in Woburn! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike and Bob with a look at a couple of news items and Josie is back with a couple of tracks and a chat with Andy.

Mike Wendt and Andy Ferg go all in on the latest controversial review in the beef group... it's BEEFGATE 2024! Plus, a full recap of Andy's foggy week in Ireland, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Bella's Roast Beef, Priority Capital & Nick's Place - Burlington Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg recap the best BEEF MADNESS in history! Plus, Roastamania, No Diddy, Tiger Woods, Boxers vs Briefs, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Bella's Roast Beef, Priority Capital & Nick's Place - Burlington Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg cover a ton of ground this week! From the Final 4 of BEEF MADNESS, to the etiquette on sending food back at a restaurant. Plus, The Rock making wrestling cool again, interesting OnlyFans trends, April Fools jokes, April Reese, Tom Brady and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Bella's Roast Beef, Priority Capital & Nick's Place - Burlington Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg dive deep into a controversial topic amongst the North Shore Beefs community. The Elite 8 of BEEF MADNESS are upon us! Mike gives us his Ghostbusters: Frozen Empire review, as well as Mike and Andy breaking down the reboot of RoadHouse. Plus, all kinds of nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Cabaret Lounge, Nick's - Essex St., Bella's Roast Beef, Priority Capital & Nick's Place - Burlington Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg recap the shenanigans that took place on last week's LIVE show at the Cabaret. BEEF Madness '24 is down to the Sweet 16! Plus, Mike and Andy share their final thoughts on the Patriots docuseries DYNASTY. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St., Fibber McGees's, Cabaret Lounge & Bella's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg are LIVE from the world famous Cabaret Lounge in Peabody, MA! The usual format is thrown out as Mike and Andy chat with all the ladies of the Cabaret and other guests that drop in on the show. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St., Fibber McGees's, Cabaret Lounge & Bella's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg share their LIVE accounts of the March 5th Facebook Outage! Plus, Ty Law is a North Shore Beefs fan and we cover Episodes 5 & 6 of DYNASTY: The Patriots doc. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St., Fibber McGees's, Cabaret Lounge & Bella's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg announce the start of Beef Madness 2024. Mike and Andy breakdown Episodes 3 & 4 of DYNASTY the Patriots docuseries. Plus, Andy received an apology from Toucher & Hardy, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St., Fibber McGees's, Cabaret Lounge & Bella's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg discuss FERG FEST (tickets are available), North Shore Beefs making mainstream headlines, the upcoming LIVE F-Buddies Show at the Cabaret in Peabody, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St., Fibber McGees's & Cabaret Lounge for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg recap their Super Bowl weekends, including WOTW, COTW and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St. & Fibber McGees's for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg get into jaw exercises, trolls in the comments and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Nick's - Essex St. & Fibber McGees's for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

176. Bit of a different show this week. Due to work, truck shows and the insane pace of life these days it's bloody hard to pull this together. Not an excuse but a statement of reality. I'm afraid I struggle with time seeing as how I still work full time as a driver and have this, the radio shows and Truckin' Life to get sorted. have a listen to my comments at the start of the show and let me know what you think mike@ontheroadpodcast.com.au of call me 0418722488 cheers.

Mike Wendt and Andy Ferg open with a eulogy of sorts with the official FREDO of our show Adam JumpTown. McDonald's reintroduced the QUAD Mac, Shaq loves the zoo, Andy has identified as queer, Vince McMahon is a durrrty old man... and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Alcohol Armor, Nick's - Essex St. & Fibber McGees's for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg confirm Andy's upcoming Food Truck Festival happening this year. Korea JUST outlawed eating dogs, Pulling The Cork did a comedy show, public bathrooms are mostly gross,m and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Giovanni's Roast Beef & Alcohol Amor for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg announce the upcoming Food Truck Festival happening this year. A guy in Ireland thinks 81 beers in 3 days is "impressive." McD's are all over the place with their prices and are releasing the Quad Mac! Plus, snow storms, MILF life crisis, and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Giovanni's Roast Beef & Alcohol Amor for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

175. Mike and Paul talk about this weeks Australian trucking news. Andy has music from the Time Machine. a bit of Party Boys followed by Bob with Something To Talk About. we roll out of the show with the latest from Darlinghurst.

Mike Wendt and Andy Ferg get into it this week! Katt Williams, Epstein Island List, Target Stanley Cup, Golden Globes, Bill Belichick, Dry January and other NONSENSE! This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Giovanni's Roast Beef & Alcohol Amor for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg run down The Beefies and the final scores of the beef shop reviews for 2023 in North Shore Beefs (Facebook Group)... plus the being in the golden age of ejaculation and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Rizzo's Roast Beef & Giovanni's Pizza and Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg show their appreciation for all the sponsors that have supported this show. Plus, Andy's Christmas parties, strip clubs, Ricky Gervais, and more nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! HUGE THANKS to everyone that has sponsored this show over the past year! (in no particular order) The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Champions Pub, Peabody, Lazy Dog of Lynn Sports Bar, Longboards Restaurant & Bar, Foodigenous, Coastal Mass Brewing, Jamie's Roast Beef, Paddy Kellys, Sideline's Salem, Giovanni's West Peabody, Rizzo's Roast Beef & Pizza Salem, Laugh Boston & Andrew Dice Clay, Donali Food Truck, Roux Family Moving, 508 Sealcoating, Ted Londi & Londi's North Andover, Tostimo's, Nick's Place Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg celebrate the 3rd Anniversary of the podcast that would ultimately become F-Buddies... NONSENSE. Also on the show... The Mayor of Boston had a racist holiday party. Plus, Andy explains Festivus to Mike and airs his grievances. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Rizzo's Roast Beef & Giovanni's Pizza and Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg discuss Tommy Cutlets and his Jerry Maguire like agent. Plus, Oprah Whinfrey being hot, Boston sports fans attractiveness, and other NONSENSE. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Rizzo's Roast Beef & Giovanni's Pizza and Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg bombard you with topic because we are THEE greatest podcast of all time! For example... the History of North Shore Beef, algorithms, old ladies sledding, sports gambling, modern slang, and other NONSENSE. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Rizzo's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg run down Matt Rife's Netflix special, Dolly Parton, a Daddy/Daughter Bachelor/ette Party in Amsterdam. Plus, Tommy Cutlets, a Mafia Movie Draft and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Longboards & Rizzo's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Rounding Up Season 2 | Episode 6 – Multiplicative Thinking Guest: Dr. Anderson Norton   Mike Wallus: One of the most important shifts in students' thinking during their elementary years is also one of the least talked about. I'm talking about the shift from additive to multiplicative thinking. If you're not sure what I'm talking about, I suspect you're not alone. Today we talk with Dr. Anderson Norton about this important but underappreciated shift.  Mike: Welcome to the podcast, Andy. I'm excited to talk with you about additive and multiplicative thinking. Andy Norton: Oh, thank you. Thanks for inviting me. I love talking about that. Mike: So, I want to start with a basic question. When we're talking about additive and multiplicative thinking, are we just talking about strategies or operations that students would carry out to find a sum or a product of a problem? Or are we talking about something larger? Andy: Yeah, definitely something larger, and it doesn't come down to strategies. Students can solve multiplication tasks, what to us look like multiplication tasks, using additive reasoning. And they often do, I think, they get through a lot of elementary school using, for example, repeated edition. If I gave a task like what is four times five? Then they might just say that's five and five and five and five, which is fine. They're solving a multiplication problem, but their method for solving it is repeated addition, so it's basically additive reasoning. But it starts to catch up to them in later grades where that kind of additive reasoning requires them to do more and more sophisticated or complicated strategies that maybe their teachers can teach them, but it starts to add up, especially when they get to fractions or algebra. Mike: So, let's dig into this a little bit deeper. How would you describe the difference between additive and multiplicative thinking? And I'm wondering if there's an example of the differences in how a student might approach a task or a problem that could maybe highlight that distinction. Andy: The main distinction is with additive reasoning, you're working within one level of unit. So, for example, if I want to know, and going back to that four times five example, really what I'm doing is I'm working with ones. So, I say I have five ones and five ones and five ones and five ones, and that's 20 ones. But in a multiplication problem, you're really transforming across units. If I want to understand four times five as a multiplication problem, what I'm saying is, “If I measure a quantity with a unit of five, the measure is four,” just to make it a little more concrete. Suppose my unit of measure is like a stick that's 5 feet long, and then I say, “OK, I measured this length, and it was four of these sticks. So, it's four of these 5-foot sticks. But I want to know what it is in just feet.” So, I've changed my unit. I'm saying, “I measured this thing in one unit, this stick length, but I want to understand its measure in a different unit, a unit of ones.” So, you're transforming between this one kind of unit into another kind of unit, and it's a five-to-one transformation. So, I'm not just doing five plus five plus five plus five, I'm saying every one of that stick length contains 5 feet, five of these 1-foot measures. And so, it's a transformation from one unit into another, one unit for measuring into a different unit for measuring. Mike: I mean, that's a really big shift, and I'm glad that you were able to describe that with a practical example, that someone could listen to this and visualize. I think understanding that for me clarifies the importance of not thinking about this in terms of just procedural steps that kids would take to either add or multiply; that really there's a transformation in how kids are thinking about what's happening rather than just the steps that they're following. Andy: Yeah, that's right. And a lot of times as teachers or even as researchers studying children, we're frustrated like the kids are when they're solving tasks, when they're struggling. And so we try to give them those procedures. We might give them a visual model, we might give them an array model for multiplication, which can solve a lot of problems. You just sort of think about things going vertically and things going horizontally, and then you're looking at an area or a number of intersections. So, that makes it possible for them to solve these individual tasks. And there's a lot of pressure on teachers to cover curriculum. So, we feel like we have to support them by giving them these strategies. But in the end, it just becomes more and more of these complicated strategies without really necessitating the need for something we might call a “productive struggle”; that is, where students can actually start to go through developmental changes by allowing them to struggle so that they actually develop these kinds of multiplicative structures instead of just giving them a bunch of strategies for dealing with that one task at a time. Mike: I'm wondering if you might share some examples of what multiplicative thinking might look like or sound like in different scenarios. For example, with whole numbers, with fractions or decimals …  Andy: Uh-hm. Mike: … and perhaps even in a context like measurement. What might an educator who was listening or observing students' work, what might they see that would indicate to them that multiplicative reasoning or multiplicative thinking was something that was happening for the student? Andy: So, it really is that sort of transformation of units. Like imagine, I know something is nine-fifths, and nine-fifths doesn't make a whole lot of sense unless I can think about it as nine units of one-fifth. We have to think about it as a measure like it's nine of one-fifth. And then I have to somehow compare that to, OK, it's nine of this one unit, this one-fifth unit, but what is it of a whole unit? A unit of one? So, having an estimate for how big nine-fifths is, yes, it's nine units of one-fifth. But at the same time, I want to know how big that is relative to a one. So, there's this multiplicative nature kind of built into tasks like that, and it's one explanation for why students struggle so much with improper fractions. Mike: So, I'm going to put my teacher hat on for a second because what you've got me thinking is, what are the types of tasks or experiences or even questions that an educator could put in front of students that would nudge them to make this shift without potentially pushing them to a place where they're not quite ready to go yet?  Andy: Hmm. Mike: Could you talk a little bit about what types of tasks or experiences or questions might help provide a little bit of that nudge? Andy: Yeah, that's a really good question, because it goes back to this idea that students are already solving the kinds of tasks that should involve multiplicative reasoning, but they might be using additive strategies to do it. Those strategies get more and more complicated, and we as teachers facilitate students just, sort of, doing something more procedural instead of really struggling with the issue. And what the issue should be is opportunities to work with multiple levels of units and then to reflect on their activity and working with them. So, for example, one task I like to give students is, I'll cut out a piece of construction paper and I'll hand it to the student, and I'll have hidden what I'm going to label a whole, and I'll have hidden what I'm going to label to be the unit fraction that might be appropriate for measuring this thing I gave them. So, I'll give them this piece of construction paper and I'll say, “Hey, this is five-sevenths of my whole.” Now what I've given them as a rectangular strip of paper without any partitions in it, I've hidden the whole from which I created this five-sevenths. I've hidden one-seventh, and I've put them away, maybe inside of envelopes. So, it becomes like a game. Can you guess what I have in this envelope? I just gave you five-sevenths. Can you guess, what is this five of? What is the unit that this is five of and what is the whole this five-sevenths fraction is? So, it's getting them thinking about two different levels of units at once. They've been given this one measurement, but they don't know the unit in which it's measured, and they don't even have visually present for them what the whole unit would be. Andy: So, what they might do, is they might engage in partitioning activity. Sometimes they might partition what I give them into seven equal parts instead of five because I told them five sevenths and five sevenths to them, that means partition it a seventh. Well, that could lead to problems, and if they see that their unit is smaller than the one I have hidden, they might have to reason through what went wrong, “Why might have you have gotten a different answer than I did?” So, it's those kinds of activities—of partitioning or iterating a unit, measuring out with a unit, and then reflecting on that activity—that give them a basis for starting to coordinate these units at higher and higher levels and, therefore, in line with Amy Hackenberg's framing, develop multiplicative concepts. Mike: I think that example is really helpful. I was picturing it in my head, and I could see the opportunities that that affords for, kind of, pressing on some of those big ideas. One of the things that you made me think about is the idea of manipulatives, or even if we broaden it out a little bit, visual models. Because the question I was going to ask is, “What role might a visual model or a manipulative play in supporting a shift from additive to multiplicative thinking?” I'm curious about how you would respond to that initially. And then I think I have a follow-up question for you as well. Andy: OK. I can think of two important roles for visual models—or at least two for manipulatives—and at least one works with visual models as well. But before answering that, the bigger answer is, no one manipulative is going to be the silver bullet. It's how we use them. We can use manipulatives in ways where students are just following our procedures. We can use visual models where students are just doing what we tell them to do and reading off the answer on paper. That really isn't qualitatively any different than when we just teach them an algorithm. They don't know what they're doing. They get the answer, they read it off the paper. You could consider that to be a visual model, what they're doing on their paper or even a manipulative, they're just following a procedure. What manipulatives should afford is opportunities for students to manipulate. They should be able to carry out their mental actions. So, maybe when they're trying to partition something and then iterate it, or they're thinking about different units. That's too much for them to keep in mind in their visual imagination. So, a visual model or a manipulative gives them a way to carry those actions out to see how they work with each other, to notice the effects of those actions.  Andy: So, if the manipulative is used truly as a manipulative, then it's an opportunity for them to carry out their mental actions to coordinate them with a physical material and to see what happens. And visual models could be similar, gives them a way to sort of carry out their mental actions, maybe a little more abstractly because they're just using representations rather than the actual manipulative, but maybe gives them a way to keep track of what would happen if I partitioned this into three parts and then took one of those parts and partitioned into five. How would that compare to the whole? So, it's their actions that have to be afforded by the manipulative or the visual model. And to decide what is an appropriate manipulative or an appropriate task, we need to think about, “OK, what can they already do without it?” And I'm trying to push them to do the next thing where it helps them coordinate at a level they can't just do in their imagination, and then to reflect on that activity by looking at what they wrote or looking at what they did. So, it's always that: Carrying out actions in slightly more powerful ways than they could do in their mind. That's sort of the sense in which mathematics builds on itself. After they've reflected on what they've done and they've seen the results, now maybe that's something that they can take as an object, as something that's just there for them in imagination so they can do the next thing, adding complexity. Mike: OK. So, I take it back. I don't think I have a follow-up question because you answered it in that one. What I was kind of going to dig into is the thing that you said, which is, there's a larger question about the role that a manipulative plays, and I think that your description of a manipulative should be there to manipulate …  Andy: Uh-hm. Mike: … to help kids carry out the mental action and make meaning of that. I think that piece to me is one that I really needed clarified, just to think about my own teaching and the role the manipulatives are going to play when I'm using them to support student thinking. Andy: And I'll just add one thing, not to use too many fractions examples, but that is where most of my empirical research has been, was working with elementary and middle-school children with fractions. But I have to make these decisions based on the child. So, sometimes I'll use these cuisenaire rods, the old fraction rods, the colored fraction rods. Sometimes I'll use those with students because then it sort of simplifies the idea. They don't have to wonder whether a piece fits in exactly a certain number of times. The rods are made to fit exactly. And maybe I'm not as concerned about them cutting a construction paper into equal parts or whatever. So, the rods are already formed. But other times I feel like they might be relying too much on the rods, where they start to see the brown rod as a four. They're not even really comparing the red rod, which fits into it twice. They're just, “Oh, the red is a two, the brown is a four. I know it's in there twice because two and two is four.” So, you start to think about them whole numbers. And so sometimes I'll use the rods because I want them to manipulate them in certain ways, and then other times I'll switch to the construction paper to sort of productively frustrate this idea that they're just going to work with whole numbers. I actually want them to create parts and to see the measurements and actually measure things out. So, it all depends on what kind of mental action I want them to carry out that would determine what manipulative as well. Because manipulatives have certain affordances and certain constraints. So, sometimes cuisenaire rods have the affordances I want, and other times they have constraints that I want to go beyond with, say, construction paper. Mike: Absolutely. So, there's kind of a running theme that started to develop on the podcast. And one of the themes that comes to mind is this idea that it's important for us to think about what's happening with our students thinking as a progression rather than a checklist. What strikes me about this conversation is this shift from additive to multiplicative thinking has really major implications for our students beyond simple calculation. And I'm wondering if you could just afford us a view of, why does this shift in thinking matter for our students both in elementary school, and then also when they move beyond elementary school into middle and high school? Could you just talk about the ramifications of that shift and why it matters so much that we're not just building a set of procedures, we're building growth in the way that kids are thinking? Andy: Yeah. So, one big idea that comes up starting in middle school—but becomes more and more important as they move into algebra and calculus, any kind of engineering problem—is a rate of change. So, a rate of change is describing a relationship between units. It's like, take a simple example of speed. It's taking units of distance and units of time and transforming them into a third level of unit that is speed. So, it's that intensive relationship that's defining a new unit. When I talk about units coordination, I'm not usually talking about physical units like distance, time and speed. I'm just talking about different numerical units that students might have to coordinate. But to get really practical when we talk about the sciences, units coordinations have to happen all the time. So, students are able to be successful with their additive reasoning up to a point, and I would argue that point is probably around where they first see improper fractions. ( chuckles ) They're able to work with them up to a point, and then after that, things [are] going to be less and less sensible if they're just relying on these additive sort of strategies that each have a separate rule for a different task instead of being able to think more generally in terms of multiplicative relationships. Mike: Well, I will say from a former K–12 math curriculum director, thank you for making a very persuasive case for why it's important to help kids build multiplicative thinking. You certainly hit on some of the things that can be pitfalls for kids who are still thinking in an additive way when they start to move into upper elementary, middle school and beyond. Before we go, Andy, I suspect that this idea of shifting from additive to multiplicative thinking, that it's probably a new idea for our listeners. And you've hinted a bit about some of the folks who have been powerful in the field in terms of articulating some of these ideas. I'm wondering if there are any particular resources that you'd recommend for someone who wants to keep learning about this topic? Andy: Yeah. So, there are a bunch of us developing ideas and trying to even create resources that teachers can pick up and use. Selfishly, I'll mention one called “Developing Fractions Knowledge,” used by the U.S. Math Recovery Council in their professional development programs for teacher-leaders across the country. That book is probably, at least as far as fractions, that book is maybe the most comprehensive. But then beyond that, there are some research articles that people can access, even going in Google Scholar and looking up units, coordination and multiplicative reasoning, maybe put in Steffe's name for good measure, S-T-E-F-F-E. You'll find a lot of papers there. Some of them have been written in teacher journals as well, like journals published by the National Council of Teachers of Mathematics, like Teaching Children Mathematics materials that are specifically designed for teachers. Mike: Andy, thank you so much for joining us. It's really been a pleasure talking with you. Andy: OK. Yeah, thank you. This was fun. Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability. © 2023 The Math Learning Center | www.mathlearningcenter.org

Mike Wendt and Andy Ferg celebrate Harassment Day, as well as Thanksgiving. Plus, Snoop Dogg (allegedly) quit weed and more nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Longboards & Rizzo's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg discuss a particular "swinging" community in Florida. Plus, how wakes are awkward, Kim Kardashian is GQ's Man of the Year, Toucher & Rich breaking up in Boston, Andy being mad at chopsticks, and workshopping a Word of the Week. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Longboards & Rizzo's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg drag people that frivolously ask for money on social media. The HEATED debate of a boy playing girls high school sports is covered in expert form. Plus, Massachusetts Blue Laws making it the NO FUN State! . This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Longboards & Rizzo's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg crown the NEW MVB, after JumpTown's abrupt resignation of the thrown. Plus, Shoresy: Season 2, streaming exclusive sports, a possible new Podcast War, and Halloween at Universal Studios vs. Salem, MA. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef, Longboards & Rizzo's Roast Beef for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg announce the 2nd Annual Best Beef Shop in Town contest. Plus, the Horror Movie Villain Draft and a BYOB & BYOS Strip Club opened... This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Longboards for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg deal with Andy's anger issues. Plus, there's a re-draft of the Pulling the Cork's Fast Food Item Draft, a sample of the NEW Adam JumpTown hit THE BASEMENT... and other nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Champions Pub for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg recap the greatest weekend of the year, the North Shore Beefs Golf Tournament Weekend! Plus, Mike & Andy begin the story of the past 100 (plus) episodes of this AMAZING podcast! This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Champions Pub for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg introduce a couple NEW bits, such as This Week in Beefs! The North Shore Beefs Golf Tournament is this week and there's a bunch of notable celebrities appearing. Plus, Mike and Andy tease what will be happening on next week's 100th Episode of F-Buddies! This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Champions Pub for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg discuss the social media world turning into Demolition Man. Plus, the upcoming North Shore Beefs Golf Tournament, a SPORTS MOVIE draft, Andy blacking out during a comedy show, Ranch vs. Bleu Cheese and more nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Champions Pub for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.

Mike Wendt and Andy Ferg dive deep into the backlash from last week's reveal that Andy is a UberEATS Brand Partner. Plus, the OFFICIAL F-Buddies Drinking Game, the Balldo, Iglooing, and more nonsense. This ain't your average food show... this is the food show you deserve. It's F-BUDDIES! Special THANKS to The Gary Girolamo Group, Ben Franklin Print Co., Twisted Fate Brewing, Anthony's Roast Beef & Champions Pub for sponsoring this week's episode! Thanks to the very talented Mark DiChiara for the original F-Buddies theme music.