Probability Games with Monte Carlo Simulations

4th, 5th, 6th, 7th, Algebra 1, Algebra 2, Code/Programming, Estimated Grade Level, Probability, Statistics and Probability 0 Comments

I've made something. And, I think it is pretty cool.

My colleague is doing some probability investigations with her 4th grade students. She's got a number of games from Marilyn Burns' probability book, and she is having students students investigate the experimental probability of the games.

One of those games is called Shake and Spill. Grab six 2-sided chips (or coins) and flip/spill/spin them so that they land on a random side. Record the result and repeat. My colleague is having her students pool their results to better estimate the relative probabilities. Cool!

But, I wondered if they would generate enough data (and generate it fast enough) to see the Law of Large Numbers in action. So, I built a website that would simulate this game over and over and over again with a live graph. You can check out the website here.

I'm planning on building more of these. They are an awesome way for me to practice my Javascript and I think that they may be helpful to others. What other probability simulations would you want?

How Parents Can Help Their Kids Learn Their Multiplication Facts

3rd, 4th, 5th, Content, For Parents, Math Facts 0 Comments

Whenever I'm together with my sister (shout out to Lizzie!) and her fourth grader at the holidays, the conversation inevitably turns to memorizing multiplication facts. Students learn so many cool and amazing mathematical ideas in the elementary grades, but multiplication facts seem to always be on students' and parents' minds.

Learning multiplication facts is often a stressful process for elementary students. It is often the time when many kids first become anxious about math. As students become stressed, it can lead to arguments between children and their parents, and no one wants that!

It does not have to be this way. Learning multiplication facts is a wonderful opportunity for parents to have mathematical conversations with their kids! This post will help you learn how to help your child learn their facts faster and how to be happier doing it.

Math Fact Fluency

Many different researchers have developed different models for how children's brain learn multiplication facts. While some of them are complicated, it is worth giving you a quick overview here. In general, there are three levels of understanding:

Level 0 - The student cannot generate an answer to a fact. This implies that they do not yet understand how multiplication works.

Level 1 - The student can generate an answer to a fact using a laborious procedure, usually skip-counting. They know that multiplication can be thought of as repeated addition, so they use this strategy to get the solution. For example, if asked 5 x 4, they may count by fives four times (5, 10, 15, 20). This allows them to solve facts, but is slow and challenging, especially for larger facts.

Level 2 - The student can generate an answer to a fact using more efficient strategies. At this level, facts still take more than 2 seconds to solve because students are using a strategy to figure out their answer.  Their strategies will often use already memorized facts. For example, they may solve 6 x 7 by using 5 x 7 -- "5 x 7 = 35 and then I need to add one more seven, so 42".

Level 3 - The student has the answer memorized. They can recall the fact in less than two seconds and are not doing any math in their heads. They are simply retrieving a fact that they have memorized.

Level 2 Vs. Level 3

It is easy to see that level 1 will be much too slow to be a successful level of knowledge, but many parents and children wonder why level 2 is not sufficient. Research shows that students who reach level 3 have more success in future math classes.[3]To be clear, this is only a small part of what determines success in math class. But, we know that it matters. So, we want all math students to reach level 3 for the 0 to 10 facts.

So, why is memorization better? In other words, why aren't the strategies in level 1 sufficient? It comes down to the brain. When students at level 2 use multiplication facts while studying other concepts, they need to dedicate more of their brain power to answering math facts.

Take for example, learning how to multiply multi-digit numbers (ex: 123 x 45). If you multiply these two numbers, you are recalling six facts.[4]5x3, 5x2, 5x1, 4x3, 4x2, and 4x1. Students who have their math facts memorized (level 3) can dedicate all of their thinking to understanding and learning the algorithm. Students at level 2 have to learn and understand the algorithm while ALSO solving the multiplication facts using their strategies. Having level 3 knowledge of math facts likely makes learning future ideas easier.

Wait...Don't Count Out Level 2!!!

Many people hear this and come to the conclusion that they should aim right for level 3. They pull out the flashcards and the Mad Minutes and start drilling away. But, this is not the way to help!

Level 2 is a crucial stage of development for students. It is in this stage where students develop their understanding of how multiplication works. Even though level 3 is a goal, trying to skip level 2 will weaken your child's understanding of mathematics. Because, it is at this level where students learn strategies that are useful beyond just memorizing multiplication facts.

As parents, the best thing you can do to help your child learn their multiplication facts is to have conversations about the strategies you can use to solve a multiplication fact. Through these conversations you will bolster your child's understanding of mathematics and help them move up the levels of understanding introduced above. So, here are the strategies you and your child can employ while solving multiplication facts:

Zeroes, Ones, Twos, Fives, and Tens - By the time your child starts studying multiplication, they will (hopefully) have a solid understanding of addition and have reached level 3 for most addition facts. This knowledge can serve as the foundation for the first learning multiplication. Multiplying by 0, 1, 2, 5, and 10 are the easiest facts to learn. For example, multiplying by 2 is the same as adding a number to itself. Students should already know this from their addition facts. Starting with these facts gives your child easy reference points to help with other facts. This is not technically a strategy, but these facts often serve as anchors for learning other facts.

Turn-around Facts - Multiplication is commutative. This means that you can rearrange the order of the factors and get the same result. For example, 5 x 4 = 20 and 4 x 5 = 20. This knowledge nearly halves the number of facts children need to know.

Doubling - Facts can be found by doubling known facts. For example, 4 x 7 can be solved by knowing 2 x 7 = 14 and then doubling 14. This strategy can be used anytime one of the factors is divisible by 2. That's a lot!

Skip-Counting Up - You can often skip-count up from a known fact to find an unknown one. This strategy is often used for 6s and 7s. For example, I know 5 x 8 is 40, and I can use this to figure out 6 x 8 by counting up one more eight from 40. This is an extremely important strategy because it introduces students to the Distributive Property of Multiplication Over Addition. You and your child don't need to be thinking of the Distributive Property when you use this strategy, but know that you are giving them experience with an idea that is crucial in Algebra. Here's what those steps look like more formally:

6 x 8

(5 + 1) x 8

5 x 8 + 1 x 8

40 + 8

48

Skip-Counting Down - If you can skip-count up, you can also skip-count down. This is a great strategy for 4s and 9s since you can count down from 5s and 10s respectively. For example, I can figure out 9 x 7 by calculating 10 x 7 - 7. This also gives students experience using distribution.

Halve - If you or your child is a good doubler, then you may also like using a halving strategy. This is great for 5s if you know your 10s, 3s if you know your 6s, etc. Example: I know that 6 x 8 = 48, so halving 48 helps me figure out that 3 x 6 is 24.

Combining Other Strategies - As you get experience with these strategies, you may find yourself using multiple strategies together. For example, if I wanted to figure out 23 x 7 I might use doubling to figure out 20 x 7 (7 x 10 x 2 = 2 x 70 = 140) and then skip-count up three more sevens (140 + 3 x 7 = 140 + 21 = 161).

If that feels overwhelming it is likely because you have not had much experience thinking through multiplication strategies. It is not caused by a difference in math ability. After teaching for nine years, I have no reason to believe that this level of thinking is out of reach for any child or adult.

Notice that ALL of these strategies work for facts beyond ten too. That's the power of level 2: these strategies extend beyond the zero to ten facts we expect our students to memorize. Spending a lot of time developing the level 2 strategies helps children become better mathematicians while also learning their facts.

Helping Your Child Reach Level 3 Through Discussions

Alright, now that we have the background knowledge we can figure out exactly how to help. The key idea here is that you want to move your child through level 2 and into level 3 not skip level 3. To accomplish this, you will want to have conversations with your child about facts. Here is an outline for how you might discuss 6 x 8:

"Do you remember the product of 6 x 8?" - Begin by asking them to try to recall a fact. This pushes them towards level 3. But, if they can't recall it that's okay!

"I don't remember"

"Can you figure it out using a strategy?" - Since they don't have the fact memorized, it is time for them to try to use a strategy to figure it out. You may have to wait a LONG time. That's okay. We already know they do not have the fact memorized, so give them as much time as they need to think through their strategy.

"46"

"How did you figure that out?" - Notice that I didn't say "wrong." They will figure out their mistake when they discuss their strategy.

"I know that 5 x 8 is 40 so I added one more 8. And, that is 48" - They may not even notice that their answer changed. That's okay here. I am focused on the logic of their strategy.

"That's an awesome strategy! So 6 x 8 is 48." - Notice that I praised their strategy since that is what I want them to focus on.

"Are there any other strategies you could use figure out 6 x 8?" - By asking for more strategies, I will reinforce the fact and also help them develop a wider variety of strategies. Again, I'll need to wait very patiently here.

"Well, I know that 6 x 10 is 60. So I could count backwards two more 6s. [long pause] 54 [long pause] 48. 6 x 8 = 48"

"That's another cool strategy! Are there anymore that work?"

"[Long pause] I can't think of another one"

"I thought of one! I know that 3 x 8 is 24. Since 6 is double 3, I can double 24 to figure out 6 x 8. That's how I figured out 6 x 8 = 48" - If there is an obvious strategy they missed, you should introduce your child to it. You might ask them to explain the strategy back to you if they like it. Hopefully, your child is being exposed to multiple strategies in their classroom too.

"That's a cool strategy. You are the awesomest parent ever." - You probably won't get this response but you deserve it!

"Now that we have talked about strategies, can you remember 6 x 8 without using a strategy?" - Now that I've helped solidify level 2 understanding, I want to give them practice recalling the fact to practice for level 3.

"Yeah! 6 x 8 = 48."

Conclusion

It takes TONS of these conversations to move children through level 2 to level 3 knowledge of their multiplication facts. Trust your instincts on where to begin. If your child is really struggling to learn their facts, start with facts you know they can do well and repeat those facts often.

Make sure your first few discussions are successful. And, never get frustrated. If they cannot do what you are asking them, then they cannot do it. If they can't skip-count up when multiplying by six, then talk about skip-counting up for their three facts.

Most importantly, enjoy these conversations with your child. It is amazing to watch children develop and then apply new strategies as they learn. You will be astonished at how sophisticated their thinking becomes after just a few conversations!

Responding to Matt Larson's "Curricular Coherence in the Age of Open Educational Resources" NCTM Blog Post

Curricula 6 Comments

If you are a member of NCTM (or just on their email list), you probably received an email linking to Matt Larson's new blog post: "Curricular Coherence in the Age of Open Educational Resources." I found this article fascinating; it addressed so many of the things I have been wrestling with but also left me with some lingering thoughts. Let's run through some of the main points:

"Coherence, with respect to mathematics curriculum, generally means that connections are clear and receive emphasis from one year to the next, from one concept to another, and from one representation to another. High-quality materials are coherent pedagogically, logically, and conceptually."

It is hard to argue with this opening statement. We all know that students benefit from consistency as well as well-planned and organized learning experiences. When each lesson builds on prior knowledge and experiences, students are more likely to learn and understand mathematics. Understanding is building networks of connections between concepts in the brain. Coherence definitely supports that.

"The increasing availability of online instructional materials—some of which are of high quality and some of which are not, and many of which can be downloaded at no cost—has added a new dimension to the curricular landscape for mathematics teachers and school districts."

"Online instructional materials" is such a general term with lots of different materials under its umbrella: teachers-pay-teachers, desmos, auto-generating worksheets, blog posts, Khan Academy, Illustrative Mathematics, Three-act-math, WODB, Estimation 180, etc. There is a huge variety of quality and style here. I worry that lumping them together is going to muddle the point for many teachers. If you use any of these resources and feel like you are being criticized, it is so easy to get defensive and shut out new ideas.

"Some of the most engaging conversations about mathematics teaching today are taking place within online communities where teachers share instructional resources and ideas that they have either created themselves or found on their own online."

#MTBoS shout-out! Definitely agree with this statement that "engaging conversation" is happening online. However, #MTBoS benefits from being a self-selected community. The teachers that participate in it have opted-in. This greatly increases the likelihood that new members subscribe to beliefs that are similar to the overall community's. It's easy to have productive discussion when most participants are on the same page to begin with.

In contrast, a colleague that has been assigned to teach the same course will not necessarily hold the same beliefs as you. This is a great opportunity for growth but is probably less likely to produce productive discussion unless protocols are put in place like professional learning communities.

"A recent survey by the RAND Corporation found that the vast majority of math teachers, at both the elementary and secondary levels, reported they used materials that they developed or selected themselves to implement the Common Core State Standards for mathematics. There is no question that this practice is widespread."

From my anecdotal experience, it seems as if a very small group of teachers "use" a curricula. By "use" I mean follow the day-to-day lessons and activities set forth by the curriculum (with few exceptions).[7]To be clear, I am NOT saying that when you "use" a curriculum you never integrate additional lessons or activities. Just that it is NOT the norm and takes up a small percentage of your class time.

Some of this is caused by lack of access to good curricula, but much of it is not. I have encountered stigma against using curricula. It seems that there is a belief that only beginner or lazy teachers use curricula, and that the best teachers always create their own materials. It is almost a hipster-type mentality, once a curriculum is published and goes mainstream, it cannot be good. That seems insane to me. Every curriculum has its strengths and weaknesses, but some of them are REALLY good. It does not feel right to dismiss them all off-hand.

"The dilemma is that while districts, schools, and teachers have greater access than ever to tools and resources for selecting and developing instructional materials, the skill required to develop a high-quality curriculum is both complex and often underappreciated. The widespread availability of online tasks therefore makes having and working with a coherent curriculum at the school and district level even more important because it is the curriculum that establishes the learning goals in a coherent progression and helps teachers see and understand the multiple pathways that students might take through the progression."

This is the most interesting paragraph in the whole letter. I completely agree with his statement that "the skill required to develop a high-quality curriculum is both complex and often underappreciated."

Think about how you spend your time as a teacher. Mine looks roughly like this:

  • Actively teaching or working with kids: 50%
  • Email, grading, copying: 30%
  • Lesson design and preparation: 20%

Curriculum developers are spending 80% or more of their time developing lessons and have years to develop a curriculum.[8]That 80% figure is a guess, but I feel confident that curriculum developers' percentage is significantly higher than the percentage of time I spend developing lessons. Also, do not forget that they can spend years developing a curriculum whereas a teacher has to be ready for day 1 of school. If only 20% of my work time is dedicated to developing lessons, how can I possibly produce work of comparable quality consistently. I can make a few great lessons here and there, but a day-to-day program is going to be near impossible.

The answer is that I can work collaboratively with peers (in my school or over #MTBoS). That's what's happened with things like Geoff's Curriculum Maps; but, even with how well those are thought out, coherence is still a real concern. There are so many gaps and inconsistencies in those maps. That's not a criticism of Geoff, it's just an inevitable result of compiling lessons from so many different sources that are not designed to build on one another.

But, I think Larson stops short of what he actually wants to say here. He loves the active discussion happening over #MTBoS (with good reason) but does not like the trend of going away from published curricula towards collections of lessons. Even when those lessons are individually strong. His thesis is summed up in the quoted paragraphs below: given that teachers are moving away from published curricula, it is crucial that they are extremely thoughtful in how they build their own curriculum.

Personally, I think this is impossible. We already struggle to find the 3+ million qualified teachers we need to teach our students. It is not possible to ensure that all of those teachers are trained well enough to individually (or in small groups) select tasks to build their own curricula.

"Ideally, teachers who select online instructional resources and engage in online community discussions would not be working in isolation but within well-developed professional learning communities in their schools. This sustained colleague-to-colleague communication would increase the likelihood of the selection of high-quality tasks that fit within mathematical learning trajectories and support the school and district’s curricular goals for students."

"Perhaps the greatest danger is the potential for vast inconsistencies in instruction and highly variable learning experiences for students that in turn can lead to differences in student learning outcomes."

This is the politest way of saying that when teachers use different self-generated and self-organized materials there is going to be significant differences in the quality of their teaching. With so many teachers developing their own curricula or combining materials to form something like a curricula, some students are going to get lower quality instruction.

"Without question, curricular coherence is highlighted and enhanced when teachers work collaboratively and regularly with colleagues at the school level to plan instruction, implement the task, anticipate student work, respond to student learning needs, and provide consistency in curricular aims and instruction for students—no matter what teacher students might be assigned."

Collaboration is great, but I worry that it is not enough to overcome the issues with self-curated curricula. Developing cohesive curricula takes too much time and care to be done while also teaching and grading and meeting with parents and supervising recess and doing all of the other millions of things that teachers need to do.

But, even if you agree with me, there is still the issue that many many teachers are building their own curricula. One reason that is happening (I think) is that teachers are not getting what they want from the curricula they have to work with at their school. So, how can curricula be more appealing to teachers so they get used more. I have some ideas:

  • Digital - I have yet to see a curriculum that really embraces the digital world. Many now offer digital portals, practice problems, and pdf copies of the texts, but those resources are clearly built AFTER the curriculum is designed. They are not integrated into the actual designs. A modern curricula would be designed from the ground up to be a digital medium. Videos would be integrated into the lessons. Students would be using resources like Desmos, Google Docs, and GeoGebra as part of the actual lessons. Teachers are driven to use their own materials because they see the benefits of these digital tools and do not want them to be slapped on to the textbook lessons.
  • Better Hooks/Launches/Act 1s - The hook is the question or situation that sets up a lesson. It creates the need to investigate and learn some mathematics. Much of the appeal of the teacher designed online resources is that they have great hooks. Most traditional curricula have unengaging set-ups. They say things like "Two students in Mr. Schwartz's class are arguing about whether..." or "Sometimes business owners need to calculate tax..." These are hardly inspiring.
  • Include More Open Problems - Many curricula lack open questions. Every problem or prompt has an obvious answer and strategy for solving. They especially lack open-beginning and open-ended questions. Teachers know how valuable these types of questions are -- especially for engaging with the standards of mathematical practice -- so they have to work them in on their own.
  • Access - In an ideal world, a great curriculum would free to download. Is there no model for which this can happen? Couldn't a group of designers self-publish? Could we give grants to pay for the development so that the end result could be free for every teacher in every school? I do not know enough about curriculum design to answer these questions. But I suspect it is possible and that the publishing companies would REALLY dislike it.
  • Concise - Many curricula I have used suffer from being too full. They have writing prompts, side-projects, investigations, practice problems, more homework than you could ever assign, check-in quizzes, pre-assessments, grading rubrics, additional practice, lesson plans, quick lesson overviews, and on and on. It would take most classes 3+ school years to do everything provided with a curriculum. So teachers start skipping things and that undermines the coherence of the curriculum. I realize that a lot of those materials are included because they help to sell the curriculum, but I think many teachers would be happier with something that had the fat trimmed off of it.
  • Simple and Clear Teacher Materials - Trying to go through the teacher resources for a curriculum is a nightmare. There are often thousands of pages of materials to read. And, while the details you want are often there, you do not have the time to find them. A better curriculum would have much simpler lesson plans, chapter overviews, and explanations. That way, teachers can more quickly and easily see what the goal of each lesson is and why the lesson was constructed in the way it was.

Math education needs to get better, but I do not think the answer is for millions of teachers to develop their own curriculum. There are obvious benefits to working with established curricula, but curricula also has to step into the modern age and better meet the needs of teachers and students. We can get there, but there is still a lot of work to do.

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