Math Instruction at HTeM

Mocha w/ Monique 11/19/2019


Cognitively Guided Instruction (CGI) is a problem-solving approach to teaching mathematics for understanding- based on an integrated program of research of (Carpenter et al , 2000)

  • the development of students’ mathematical thinking.
  • instruction that influences that development
  • teachers’ knowledge and beliefs that influence their instructional practice,
  • the way that teachers. knowledge. beliefs, and practices are influenced by their understanding of students’ mathematical thinking

In CGI instruction, a story problem is given, teachers make sure all students comprehend the question. Afterwards, students are able to share their strategies with others and answer questions from their peers. See sample work from K-2 below.

What K-2 Math Instruction looks like at HTeM. Click to enlarge an image.

At HTeM we work on building a flexible mind set. Students tackle a problem in different ways as shown in the sample work above. There is no unique solution to a problem, they are able to critique their own work and learn from others. This enables them to learn with an open and flexible mind. kids are encouraged to think deeply about a problem and understand the concept in depth instead of finding an easy solution and answer. Struggling and making mistakes is an essential part in the process of gaining deeper understanding.

Jo Boaler on Creative and Flexible Mathematics

Sample CGI Story Problems:


Beginning of the year:

  • Pedro had 4 marbles. He found 6 more marbles under his bed. How many marbles does Pedro have now?
  • I had 11 animal crackers. I ate 4 of them How many animal crackers do I have left?
  • Morn bought 3 cartons of eggs_ Each carton had 6 eggs in it How many eggs did Mom buy?
  • I had 8 markers I put the markers into 2 boxes with the same number of markers in each box. How many markers are in each box?

Middle of the year:

  • 13 children were sitting on the rug. 17 more children joined them on the rug. How many children are sitting on the rug now?
  • The store had 23 bottles of juice on the shelf. 17 of the bottles fell off the shelf and broke. How many bottles of juice are left on the shelf’?
  • Ms Lang ordered 4 trays of sandwiches. Each tray has 8 sandwiches How many sandwiches did she order?
  • Kristian had 26 jelly beans. She put them into 5 bags with the same number of jelly beans in each bag How many jelly beans did she put in each bag?

End of the year:

  • Makayla put 16 pennies in her piggy bank. Then she put 24 more pennies in her piggy bank. How many pennies did Makayla put in her piggy bank altogether?
  • Our class has 32 bouncy balls We gave another class 13 of our bouncy balls. How many bouncy balls does our class have left?
  • There are 4 tables set up in the lunch room At each table there are 10 children. How many children are in the lunch room?
  • Elijah had 24 milk cartons He put the milk cartons in 4 boxes with the same number of milk cartons in each box. How many milk cartons were in each box?

First Grade:

Beginning of the year:

  • I had 32 pennies in my bank During the week I earned 46 more pennies How many pennies do I have now?
  • Mekhars mom brought 38 cupcakes to school for Mekhai’s birthday and gave 21 of them to the children. How many cupcakes does Mekhai’s mom have left’?
  • I have 4 bags of cookies There are 4 cookies in each bag How many cookies do I have?
  • There are 24 children in Ms Scott’s class She puts them into four groups with the same number of children in each group How many children are in each group?

Middle of the year:

  • On Friday, Jamari did 27 jumping jacks On Saturday Jaman did 21 jumping jacks. and on Sunday he did 32 jumping jacks How many jumping jacks did Jaman do altogether’?
  • Naomi had 87 crayons She broke 59 of them How many unbroken crayons does Naomi have?
  • My aunt bought 9 packs of gum Each pack has 4 pieces of gum in it How many pieces of gum does my aunt have?
  • I had 32 lunches I put the lunches into 8 bags with the same number of lunches in each bag. How many lunches are in each bag’?

End of the Year:

  • A school is going on a field trip First grade needs 34 lunches for the trip, second grade needs 26 lunches, and third grade needs 37 lunches How many school lunches does the school need altogether?
  • Jonathan had 113 chocolate chips in a bag He used 74 of them for a batch of muffins How many chocolate chips does he have left?
  • Jonathan had 113 chocolate chips in a bag He used 74 of them for a batch of muffins How many chocolate chips does he have left?
  • Mckenzie had 64 crayons She put the crayons in 7 boxes with the same number of crayons in each box How many crayons were in each box?

Second Grade:

Beginning of the year

  • Grade 2 collected 58 cans for the food drive on Monday On Tuesday they collected 67 more How many cans did they collect for the food drive?
  • Third Grade has 112 bouncy balls They gave 54 of them to Second Grade How many bouncy balls does Third Grade have left?
  • I have 5 bags of animal crackers There are 6 crackers in each bag How many animal crackers do I have?

The cafeteria had 41 lunches They put the lunches onto 5 trays with the same number of lunches on each tray. How many lunches are on each tray?

Middle of the year

  • The candy shop sold 142 chocolates on Thursday. 149 on Friday. and 168 on Saturday. How many chocolates did the shop sell?
  • Ms. Sullivan has 71 prizes in her prize box. Ms. Richardson has 45 prizes in her prize box. How many more prizes does Ms. Sullivan have than Ms. Richardson?
  • There are 77 children going to the museum. If 9 children can travel in each minibus, how many minibuses will be needed?
  • There are 63 children at field day. Ms Haynes put them into 3 groups with the same number of children in each group How many children are in each group? End of the Year
  • Giovanni had some marbles. His mother bought him 79 more marbles Now he has 111 marbles. How many marbles did Giovanni start with?
  • Macy had a bag of mini pretzels. She and her friend ate 59 of them. Now there are 165 mini pretzels in the bag. How many mini pretzels were in Macy’s bag at first?
  • 2 children share 1 brownie so that everyone gets exactly the same amount HOW much brownie will each child get? 4 children share 1 brownie so that everyone gets exactly the same amount How much brownie will each child get?
  • The T-ball league has 6 teams with 13 players on each team. How many players are in the T-ball league altogether?

What are our Math routines at HTeM?

Counting Jar Routine

Students are independently counting, the teacher is conferring with each student and then there is whole group share at the end. During this time students share their counting strategies with one another. 

Goals: Encourage increasing proficiency and understanding in counting a quantity to answer the question “How many…”, The list below includes all skills addressed during this routine:

  • Standard sequence of number names.
  • One-to-one correspondence.
  • Pairing each object counted with one and only one number name.
  • Deliberate and careful keeping track of the objects counted.
  • Accurate number reported.
  • Understanding cardinality (last number name said answers “How many?”)
  • Understanding conservation of number and counting order irrelevance.
  • Consistent use of groups larger than 1 (e.g., 2s, 5s, 10s).
  • Understand that a quantity is expressed as a number and a unit.
  • Creating another set with the same number using different objects.

Expectations by Grade Level:


  1. Minimum per the Common Core: Rote counting to 100 by 1s and by 10s; rational counting to tell how many up to 20.
  2. Recommended additional expectations: Rational counting up to at least 43 to fully establish the pattern in counting; rote counting to at least 121 by 1s to establish the repeating pattern beyond 100; rote counting to 121 by groups of 10s, 5s, and 2s; counting on by 1s following group counting (e.g. 10, 20, 30, 40, 41, 42, 43); half of all K children should be rational counting to 43 at the exemplary level, and the rest should be rational counting at the proficient level.

First Grade:

  1. Minimum per the Common Core: Rational counting up to 120; rote counting to 120 starting at any number less than 120; and in this range, read and write numerals and represent a number of objects with a written numeral.
  2. Recommended additional expectations: Rational counting at an exemplary level to at least 43 and rational counting at a proficient level to 200.

Second Grade:

  1. Minimum per the Common Core: Count within 1000, including skip counting by 5s, 10s, and 100s.
  2. Recommended additional expectations: Demonstrate the Common expectations in exemplary rational counting up to 1000 (by 5s, 10, and 100s).

Money Jar Routine

Money Jar Objectives for Second Grade: Encourage increasing proficiency in counting

  • Correct sequence of number names and one-to-one correspondence
  • Carefully keeping track and accuracy in the number of coins counted
  • Purposeful grouping of coins for combining equivalent money values
  • Accurate amount of money reported in cents or dollars and cents
  • Appropriate use of words and symbols for cents and dollars ($)

Common Core Goals: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, symbols appropriately

Example: If you have 2 dimes and 3 pennies, how many cents do you have?The Money Jar Routine supports development of thoughtful strategies for solving other word problems involving money in the context of the problem “How much money is in your jar?”

Counting Around the Class: The Activity

Students count around the class by a particular number (depending on the grade level this may be by 1, 2. 4, 5, 10. 15, 25, 150 etc.). That is, if counting by 2, the first person says “2,” the next person say “4,” the next “6,” and so forth. Before they count students discuss the relationships among the factors and their multiples. Counting Around the Class is designed to give students practice with counting by many different numbers and to foster numerical reasoning about the relationships among factors and their multiples. Students focus on.

  • becoming familiar with multiplication patterns
  • relating factors to their multiples
  • developing number sense about multiplication and division relationships

Counting Around the Class: The Mathematics – There are many opportunities in Counting Around the Room to focus on different aspects of mathematics that are embedded in the activity. For example in Kindergarten and Grade 1, students regularly count around the circle. This is a way to count and double check the number of students in a group In this activity students are working on the sequence of numbers, understanding that the number you say represents the place you hold in the sequence (many students will say “I’m not 11, I’m 5”), and knowing what the total should be (the total number of students in the class). There are many opportunities in this activity to discuss questions such as• Is everyone here today? How many students are not here? Is this number lower or higher than our count yesterday? What do you think will happen tomorrow? In Grades 2 and 3 students begin to count around by the multiples they are beginning to explore at other times in math

class In some cases the purpose is to practice finding the next multiple of a number and gain familiarity with the patterns in the multiplication tables At first students may be adding on by l’s to get to the next number but over time may be utilizing a pattern or remember the sequence because of repeatedly constructing and hearing it. Also, students begin to consider the relationships between numbers. That is, they count by 5’s and then 10’s and consider how the final number is related (that is, if you are counting by 5 with a group of 23 students, the total will be half as much as when you count around by 10’s) Similarly, counting by 2 and then 4

Counting Collections:

In Counting Collections, children are given a collection of objects to count The activity often begins with a mini-lesson in which the teacher highlights a particular idea like how to share the counting task with a partner, an efficient counting strategy they have seen students using, or a way to record their count

Questions and Answers during Mocha w/ Monique

How do you provide support for teachers who have to spend time redirecting? There are moves that teachers can make that can help alleviate distractions i.e., pacing to make sure your students are timely, being clear and consist with students. Every class is different and the question is what it looks like for the teacher and what it looks like to parents. There are different ideas to show what students look like. There is also a network of support that teachers can utilize if they feel they need more help in the moment, as well as consistent support for teachers such as ACs.

Next Steps:
Create a pool of parents and blocks of time and have teachers ask for that support from parents.
Create a network of parents at all who is needed.
Put it in the newsletter.
Teachers can get together and discuss what volunteer options they have for parents-ask them on Wednesdays.
Create a survey for teachers-best times and blocks, what work they need help with, across grade wide needs would help.
Create a group on Konstella for Volunteers.

Teachers: Plan ahead and have groups/stations for parents can come in (maybe while teachers are pulling small groups) parents can help confer and monitor. Especially during exhibition times (peak times for teachers). Look at curriculum and say this is when students need a little extra support. (In school versus at home) In classroom versus and take home assignments. Determine best times for parents.

Parent Math Night: Have an extended learning for parents to be able to practice these strategies at home.
Make a night and video strategies for parents who can’t be there.
What are kids are learning (with link to video).