Differentiating for Productive Struggle

In my current role as the math specialist and curriculum coordinator in a preK through 10th-grade school, one of the questions I am frequently asked by teachers is how to appropriately challenge strong math students.  We want all our learners to grow in their curiosity, understanding, and love for math, so how do we do this without simply accelerating through content for students who seem to "get it" more quickly than others?

“Student success occurs when you create an instructional environment that sets high expectations for each student and provides scaffolding without offering excessive help. 

The key is to incorporate productive struggle.” 

—Barbara R. Blackburn

As Blackburn states in her article “Productive Struggle Is a Learner's Sweet Spot”, student success is optimized when we set high expectations for our students by incorporating opportunities for all learners to experience productive struggle.  At our recent school-wide professional development day, we focused on how using parallel tasks, open-middle tasks, and open-ended tasks can provide all students with the opportunity to be engaged in similar tasks while providing productive struggle at differing levels based on students' readiness.

With parallel tasks, two (or more) different questions are posed that are related to the same mathematical big idea and context.  This allows various entry points to meet the needs of students while providing a common experience so everyone can participate in a rich mathematical discussion about the tasks.  For example, students may be asked to create a real-world situation that requires counting to 100 or a real-world situation that requires counting to 1000.  After having some time to consider scenarios and decide on one they would like to share, the teacher can facilitate a discussion focused on comparing the scenarios students came up with for 100 versus those for 1000 by asking questions like "What do you notice about the difference in these scenarios?" and "Could this scenario for 100 ever require counting to 1000?  If so, when?  If not, how do you know?"  Mark Chubb also wrote a blog post on parallel tasks where he gives an example related to algebraic reasoning involving pattern blocks.  The most critical component of making parallel tasks valuable learning experiences is that everyone comes together to share and discuss their problem-solving ideas so we can learn from each other, even if we have different entry points to access the math concepts.

Open-middle tasks are those which have one correct answer but have multiple entry points and possible strategies that can be used to solve.  Many problems can be open-middle problems, as long as you prioritize the time for students to share and compare the different strategies they used to find their solution.  When students share their ideas and explain why they have a viable strategy, it is then their peers that should be providing feedback and critiquing their reasoning (CCSS.MP3) rather than the teacher serving as the judge.  For some concrete examples of open-middle tasks, check out Robert Kaplinsky's Open Middle website or his problem-based lessons that allow for rich mathematical conversations about problem-solving strategies.  Andrew Stadel's Estimation 180 and 3-Act Tasks like the ones from Andrew Stadel, Graham Fletcher, Dan Meyer, Kyle Pearce, and Dane Ehlert can also be perfect resources for open-middle problems.

Image made with the
Public Math Sticker app

Open-ended tasks, on the other hand, have many possible solutions and are further opportunities brimming with potential for students to articulate and justify their mathematical thinking.  Number talks, or math talks, allow for "low entry, high ceiling" so all students can access the mathematics, but it is possible to extend the concepts to accelerated levels as well.  Which One Doesn't Belong and Fraction Talks have helpful images that can be used for math talks, and you can always use your own images as well.  The Public Math Sticker App is a fun way to add questions like "What do you notice?"  "What do you wonder?"  "How many?" and "What repeats?" to your images like I did with the gummy bear example on the right.  Brian Bushart explains in this post how numberless word problems can be another way to promote mathematical thinking through open-ended tasks.  When the numbers are removed from a word problem, the context remains.  Asking students questions like "What do you wish the numbers were (what numbers will make it easy)?  Why?" and "What numbers will make it challenging?  Why?" build students' number sense.  Asking in follow-up, "what is another question you could ask about this scenario?" allows students to challenge themselves by thinking about other ways numbers in this context can be manipulated to provide further information.  Marian Small's book Good Questions: Great Ways to Differentiate Mathematics Instruction in the Standards-Based Classroom is also a wonderful resource to see examples of both parallel tasks and open-ended tasks in different grade bands (K - 2, 3 - 5, 6 - 8) in different math strands (Counting & Cardinality and Operations in Base Tens, Number & Operations with Fractions, Ratios & Proportional Relationships, the Number System, Operations & Algebraic Thinking, Expressions & Equations and Functions, Measurement & Data, Geometry, Statistics & Probability).

In closing, I want to share one quote from a recent professional development experience with Dr. Yeap Ban Har that I continue to keep in mind.  He said, "when enrichment is done well, it leads to natural acceleration."  Our goal is to open our students' minds to multiple ways of thinking through incorporating opportunities for all students to experience productive struggle, thus allowing them to make connections among mathematical concepts and think flexibly about math.  In what other ways do you do this for your students?  I would love to hear from you in the comments!

Oh What Fun...Math Games!

With winter break approaching, I cannot help but reflect on the, "How do I keep my students engaged?" question that would run through my mind the first couple of years when I was teaching my own class.  I wanted to make the time the last day before break meaningful (and also ensure my classroom did not turn into a madhouse).  I came to realize over time that the days before winter break, spring break, and summer break were the perfect times for focused, yet fun, math games.  As Mark Chubb so eloquently lays out in this blog post, the purposeful use of math games can be a great way for students to practice computational skills, increase conceptual understanding, and build fluency in an enjoyable way.

Teachers can also support families by passing along any math-related games they have played in class.  While there are many math-related games families can enjoy together, I have compiled a list below of some of my favorite math-related games as a starting point which can be used both in the classroom and at home.  Zeno Math also has a great list of games for early math learning, and check out Kent Haine's website Games for Young Minds with tons of math-related games (he even has a newsletter you can sign up for which delivers a new math game idea to your inbox each week).

Paper/Pencil Games

Four Strikes and You're Out - This Marilyn Burns game is a mathematical twist on hangman and can easily be adapted to use lesser or greater values.  One person generates an addition/subtraction/multiplication/division equation and writes out blanks for each of the digits. For example, __ __ + __ __ = __ __.  The other players get to guess digits, one at a time, and similar to hangman the blanks are filled in if the digit is a part of the equation.  As the name of the game implies, the other players must use their understanding of computation and place value to strategically avoid getting four strikes.

The Product Game - This game from NCTM is a combination of tic-tac-toe and multiplication.  Players take turns selecting a factor from the digits 1 - 9, and then players color in the corresponding product for those factors on the game board.  The player to get four products colored in a row/column/diagonally first is the winner.  It can be played with paper and pencil by printing out this game sheet.

Strike It Out - This game from Nrich, a University of Cambridge resource, requires players to use their understanding of addition and subtraction to strategically stop their opponent from being able to take a turn.  The first player selects two numbers to cross out and then circles the sum or difference of those two numbers.  The next player must use the number just circled as a part of the next addition or subtraction problem and select one additional number to cross off before circling the sum or difference of these new numbers.  Gameplay continues until one player stops the other from making a move.

Dice Games

Sum Dice - One of the most straightforward dice games, this is accessible to our early math learners who know how to add numbers through 12 and understand the concept of even and odd numbers.  Two dice are rolled. If the sum is even player 1 gets the point, and if the sum is odd, player 2 gets the point.  To adapt this game and increase the difficulty, roll more than two dice or use dice with 10, 12, or 20 sides.  You can also find the difference or product instead (though this would change the name of the game...).  Take it to the next level after gameplay by figuring out if Sum Dice is a fair game (see Kent Haine's post about this here).

Race to 100 - This is a game that can be adapted depending on math readiness.  Players roll two dice and then find the sum/difference/product/quotient of the two numbers rolled.  The game piece is moved along the game board based on the answer to the rolled dice, and the first person to reach exactly 100 is the winner.

How Close to 100? - I love this YouCubed game because it couples computational fluency with conceptual understanding.  Players roll two dice and shade in the corresponding array on the grid (i.e. a 6 and 4 are rolled, so a 6 by 4 rectangle is shaded in).  The player also writes down the corresponding multiplication equation (6x4=24).  Gameplay alternates between players, and the goal is to fill up as much of the grid as possible.  The game ends when both players have rolled the dice and cannot fit any more arrays onto their grid.  This can easily be adapted to use larger grids or dice with 10, 12, or 20 sides to increase the complexity.

Farkle - Roll six 6-sided dice, and use the scoring guide to determine how many points each player gets during each turn.  The winner is the person who reaches 10,000 points first.  The game can also be purchased here.

Card Games

War - This can be adapted for many different levels of math readiness.  The typical War card game is perfect for younger learners who are comparing numbers, and the numeric cards even have the pictures that allow students to use 1-1 correspondence to count.  As students grow older, this can become Addition/Subtraction/Multiplication War where each player lays down two cards, and the person with the greatest sum/difference/product of the cards gets to keep both pairs.  This can be played by assigning additional numeric values to face cards (Jack=11, Queen=12, King=13, Ace=1), or the face cards can be removed.  When students are ready to work with integers, Integer War can be played with red cards representing negative numbers and black cards representing positive numbers.

5x5 - This game shared by Sara Van Der Werf requires multiple players (ideally 4 or more).  Each player has a 5x5 grid, which can easily be made by drawing it on a sheet of paper.  One person draws cards from a deck (face cards removed), and the players have to put the number drawn in one of the spaces on the grid (no erasing to move numbers around or "saving" numbers to place later).  After the 25 numbers have been drawn, each player calculates his/her score by adding any adjacent numbers that are the same in each row and column (i.e. if a 10 and a 10 are next to each other in the same row, the score for that row is 10+10=20).  The total score is calculated from adding all the sums from each row and column, and the highest score wins.

Krypto - This game from the National Council of Teachers of Mathematics can be played online or by dealing out five normal playing cards.  Players must use all 5 numbers and any combination of addition/subtraction/multiplication/division to reach the target number as the final answer.

Other Math-Related Games

24 - Use each of the four numbers show on the card once with any operations to come to an answer of 24.  This classic math game can also be played online at 4nums.com.

Battleship -Battleship is a great introduction to the coordinate plane.  By understanding how to name a location based on its horizontal and vertical coordinates, students will be ready to transfer this to the x, y coordinate plane.  For students in grades 5+, you can try playing Coordinate Graphing Battleship from MIT by printing out this template.

Mancala - This is a game of counting and strategy.  A game of counting strategy, the goal is to capture more stones than your opponent into your mancala (area of your game board).  This game also has a fascinating history if you care to read about it!

Quirkle - This game can be enjoyed by young and old math learners!  Players build lines by matching either color or shape and score points based on these matches.

Shut the Box - Players roll two dice and put down the tiles that have the same value as the sum.  For example, if a 5 and a 3 are rolled, the player can put down the 8 or the 5 and 3 or the 6 and 2 or the 7 and 1.  This is a great game for practicing addition coupled with strategy.  This game can also be played using dice and this paper game board.

Sumoku - This is a crossword-style game with numbers where players use repeated addition or multiples to determine the tiles laid down and then the corresponding score. 

Yahtzee - This classic game has players striving to get the most points possible by rolling five dice each turn.  Points are scored for rolling a straight, full house, and three/four/five of a kind.  Probability and addition are ripe opportunities for math conversations with this game.  (As long as you know how to score, you can also play Yahtzee without the formal game set as long as you have five dice.)

Scorekeeping

Any time there is any game that involves keeping score, a young mathematician in the group should serve as scorekeeper.  At the most basic level, tally marks can be used to keep score for some games, and addition/subtraction of increasingly complex numbers will be relevant for other games.  Scorekeeping is a great way to build fluency, and it can also be a valuable opportunity to talk about computational strategies.  For example, if I am playing a game with a child, and a player has 18 points with 8 more to be added, I would ask the child how she would find the score.  After the child shares her strategy (like using knowledge of doubles like 8 + 8 to do 18 + 8), I could share that I have another strategy (like adding 10 and then taking away 2).

What are some of your favorite math-related gams?  Please comment below so we can add to our list!

Helping Your Child Understand Our World Through Math

One of the questions classroom teachers and I are asked on a regular basis by parents is, "How can I support my child's math learning at home?"  It is so wonderful when parents want to support their child's growth and understanding of how mathematics is relevant outside of school!  Plus, there is a significant body of research that shows how beneficial it is when parents support the learning efforts started at school.

While some people may think of activities like practicing math facts with flashcards or purchasing math workbooks, I would like to offer some alternatives that are arguably more meaningful since these ideas allow students to make sense of our world through the lens of math.  Through counting, comparing, and composing or decomposing numbers and/or shapes, children learn how to understand mathematical relationships in authentic contexts.  Here are some of the ideas I recently provided for parents of kindergarteners, though these ideas can certainly be adapted for a variety of age levels to include more complex math concepts like fractions, decimals, percentages, ratios, multiplication, and division.  

• Have your child make math stories from book illustrations when reading together by counting, comparing, or adding/subtracting.  

• Ask your child to count the number of coins or dollar bills in your wallet.  See if your child can skip count by 5s, 10s, or 20s if you have multiple of the same bills!

• Involve your child in cooking.  Measuring the number of Tbs, tsp, cups, etc. provides both measurement and counting practice.

• Ask your child questions related to comparisons of number and size using words like most, least, bigger, smaller, more, fewer.  For example, “Who has more broccoli on their dinner plate, you or your brother?”  Then see if your child can tell you how many more/less pieces each person has when compared to the other or how many pieces will be left if your child eats a certain number.

• Count the number of pages you read after you finish a book.  To take this to the next level, see if your child can find the total number of pages read if you read multiple books.

• When checking out at a store, ask your child to tell you how many people are in each line, and then determine the best line to stand in (and remember, it may not always be the line with the fewest people based on how many items each person purchases).

• When you are out shopping, use whole number costs to have your child tell you which item is the better deal/costs less.

• Build something together!  Using blocks or Legos are great opportunities for counting, measurement, and conversations about shapes and spatial awareness.

• If you are taking a family trip, use a map to show your child where you are going compared to where you live.  Compare the distances to other locations you have visited.  For example, “We live in Chicago, and here is New York City where we will go next week.  We visited your grandmother in Indianapolis last month, which is here.  Which city is farther away?”

• Try exercising together at home!  Have your child count the number of jumping jacks/sit-ups/squats, and you can make it more competitive by comparing the number each of you can do.

• Ask your child to help you put items into equal group items.  For example, if you are plating dinner, ask your child to put an equal number of carrot sticks on each plate.  Then have your child tell you how many total carrot sticks were used.

• Look for opportunities when you can ask your child how many are missing.  For example, how many eggs are missing from the carton or how many ice cubes have been taken from the tray.

• Patterns can be found in so many places, from clothing and jewelry to floor tiles and artwork.  Ask your child to look for patterns wherever you are, and then have your child explain why it is an example of a pattern and what the sequence of the pattern is.

• Do number or shape scavenger hunts around the house or out and about, and then use the opportunities to make mathematical comparisons between numbers or shapes.
• Whenever you play a game with moving a piece on a game board, have your child count the number of spaces to be moved for each player.  If scoring is involved, ask your child to use tally marks or addition/subtraction to be the game's official scorekeeper.

The websites Math Before Bed and Bedtime Math also provide prompts for families to use to facilitate math conversations at home, and Bedtime Math has developed two free apps as well, MiniMath (ages 3 - 5) and Bedtime Math (ages 3 - 9).  Games are another great way to engage the whole family in math-related activities and conversation, so stay tuned for a future blog post on that!  What are additional ideas you give to parents to help support their child's math development at home?